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KOVACEVIC AND CHEBIRA: LIFE BEYOND BASES: THE ADVENT OF FRAMES 1

Life Beyond Bases: The Advent of FramesJelena Kovacevic and Amina Chebira

Redundancy is a common tool in our daily lives. We double-and triple-check that we turned off gas and lights, took ourkeys, money, etc. (at least those worrywarts among us do).When an important date is coming up, we drive our lovedones crazy by confirming “just once more” they are on topof it. Of course, the reason we are doing that is to avoid adisaster by missing or forgetting something, not to drive ourloved ones crazy.

The same idea of removing doubt is present in signalrepresentations. Given a signal, we represent it in anothersystem, typically a basis, where its characteristics are morereadily apparent in the transform coefficients (for example,wavelet-based compression). However, these representationsare typically nonredundant, and thus corruption or loss oftransform coefficients can be fatal. In comes redundancy; webuild a safety net into our representation so that we can avoidthose fatal disasters. The redundant counterpart of a basisiscalled a frame (no one seems to know why they are calledframes, perhaps because they are bounded from two sides asin (15)?).

It is generally acknowledged1 that frames were born in 1952in the paper by Duffin and Schaeffer [57]. Despite being overhalf a century old, frames gained popularity only in the lastdecade, due mostly to the work of the three wavelet pioneers—Daubechies, Grossman and Meyer [49]. Frame-like ideas, thatis, building redundancy into a signal expansion, can be seenin pyramid coding [26], quantization [47], [99], [15], [69],[43], [48], [72], [14], denoising [128], [36], [56], [84], [65],robust transmission [68], [79], [113], [16], [17], [18], [119],[31], [19], [101], CDMA systems [98], [122], [123], [115],multiantenna code design [74], [78], segmentation [116], [92],[50], classification [116], [92], [33], prediction of epilepticseizures [12], [13], restoration and enhancement [85], motionestimation [96], signal reconstruction [6], coding theory[75],[103], operator theory [2] and quantum theory and comput-ing [59], [110].

While frames are often associated with wavelet frames, itis important to remember that frames are more general thanthat. Wavelet frames possess structure; frames are redundantrepresentations that only need to represent signals in a givenspace with a certain amount of redundancy. The simplestframe, appropriately namedMercedes-Benz (MB), is given inBox I; just have a peek now, we will go into more details later.

The question now is: Why and where would one use frames?The answer is obvious: anywhere where redundancy is a must.

This work was supported by NSF through award 0515152. The authors arewith the Center for Bioimage Informatics, Dept. of Biomedical Engineering.Jelena Kovacevic is also with the Dept. of Electrical & Computer Engineeringat Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213.Email: [email protected], [email protected]

1At least in the signal processing and harmonic analysis communities.

The host of the applications mentioned above and discussedlater in the article illustrate that richly.

Now a word about what you are reading: why an introduc-tory article? The sources on frames are the beautiful bookby Daubechies (our wavelet Bible) [47], a recent book byChristensen [35] as well as a number of classic papers [29],[77], [46], [73], among others. Although wonderful material,none of the above sources offer an introduction to framesgeared primarily to engineering students and those who justwant an introduction into the area. Thus our emphasis in thisarticle; this is a tutorial, rather than a comprehensive surveyof the state of the field. Although we will try to go into anumber of applications and touch upon a number of theoreticalresults, we will do so only for the sake of teaching. We willgo slowly, whenever possible using the simplest examples.Generalizations will follow naturally. We will be selective andwill necessarily give our personal view of frames. We will berigorous when necessary; however, we will not insist upon itat all times.

As often as possible, we will be living in the finite-dimensional world; it is rich enough to give a flavor ofthe basic concepts. When we do venture into the infinite-dimensional one, we will do so only using filter banks—structured expansions used in applications. We will stay awayfrom all other infinite-dimensional settings.

I. WHAT ’ S WRONG WITH BASES?

The reason we try to represent our signals in a differentdomain is, typically, because certain signal characteristicsbecome obvious in that other domain facilitating various signalprocessing tasks. For example, we perform Fourier analysistouncover the harmonic composition of a signal. If our signalhappens to be a sum of a finite number of tones, the Fourier-domain representation will be nonzero at exactly those tonesand will be zero at all other frequencies. However, if oursignal is a sum of, say a pure frequency and a pulse of veryshort duration (think Dirac), the Fourier transform will bean inefficient representation; the signal energy will be, moreor less, spread evenly across all frequencies. Thus, the rightrepresentation is absolutely critical if we are to perform oursignal processing task effectively and efficiently.

To understand frames, it helps to go back to what we alreadyknow: bases. In this section, we review basic concepts on bases(we assume basic notions on vector spaces, inner products,norms). If you are familiar with those, you may skip thissection and go directly to the frame section which comes next.We stress that often we will forgo formal language in favor ofmaking the material as accessible as possible. An introductorytreatment is also given in [121].

When modeling a problem, one needs to identify a spaceof objects on which certain operations will be performed. For


example, in image compression, our objects are images, whilein some other tasks, our objects can be audio signals, movies,and many others. Initially, we will assume that these objectsare vectors in a vector space. In this paper, we consider almostexclusively finite-dimensional vector spacesRn and Cn aswell as the infinite-dimensional vector spaceℓ2(Z) (as theseare commonly used in discrete-time signal processing, seeSection V). By itself, a vector space will not afford much,except for the ability to add two vectors to form a new vector(still belonging to the same vector space) and to multiply bya scalar. To do anything meaningful, we must equip such aspace with an inner product and a norm, which will allow usto “measure” things. These functions turn the vector space intoan inner product space. By introducing the distance betweentwo vectors (as the norm of the difference between those twovectors), we get a precise measurement tool and turn our innerproduct space into a metric space. Finally, by considering thequestion of completeness, that is, whether a representative setof vectors can describe every other vector from our space, wereach the Hilbert space stage, which we denote byH. Thisprogression allows us to do things such as measure similaritybetween two images by finding the distance between them, astep present in compression algorithms, systems for retrievaland matching, and many others.

We need even more: tools which will allow us to look at allthe vectors in a common representation system. These toolsalready exist as bases in a Hilbert space. Bases are sets ofvectors used to uniquely represent any vector in a given Hilbertspace in terms of the basis vectors. An orthonormal basis, inparticular, will allow us not only to represent vectors but toapproximate them as well. This is useful when resources donot allow us to deal with the object directly, but rather withitsapproximation only. For example, we might not have enoughbits to representπ to the 10th digit but only to the5th oneyielding3.14159. Obviously,3.14159 is just an approximationto 3.1415926535, which in turn is an approximation toπ.Another example is compression of images. An “instant”approximation of a natural image is just its lowpassed version– we get a blurry image.

A. Bases

A subsetΦ = {ϕi}i∈I of a finite-dimensional vector spaceV (whereI is some index set) is called abasisfor V if V =span(Φ) and the vectors inΦ are linearly independent (givenS ⊂ V, the spanof S is the subspace ofV consisting of allfinite linear combinations of vectors inS). If I = {1, . . . , n},we say thatV has dimensionn.

A vector spaceU is infinite dimensionalif it contains aninfinite linearly independent set of vectors. IfU is equippedwith a norm, then a subsetΦ = {ϕj}j∈J of U is called abasis(or a Schauder basis)2 if for every u in U, there exist uniquescalarsuj such thatu =

∑∞j=1 ujϕj .

Spaces We Consider:As we already mentioned, in thispaper, we consider exclusively the finite-dimensional Hilbert

2Note that here we need a normed vector space because the definitionimplicitly uses the notion of convergence: the series converges to the vectoru in the norm ofU.

spacesH = Rn,Cn with I = {1, . . . , n}, as well asthe infinite-dimensional space of square-summable sequencesH = ℓ2(Z) with I = Z.

Rn and Cn are the most intuitive Hilbert spaces whichwe deal with on a daily basis. Their dimension isn. Forexample, the complex spaceCn is the set of alln-tuplesx = (x1, . . . , xn)T , with xi in C (similarly for Rn).

In discrete-time signal processing we deal almost exclu-sively with sequencesx having finite square sum or finite en-ergy, wherex = (. . . , x−1, x0, x1, . . .) is, in general, complex-valued. Such a sequencex is a vector in the Hilbert spaceℓ2(Z).

For the above spaces, the inner product is defined as:

〈x, y〉 =∑

i∈I

x∗i yi,

while the norm is

‖x‖ =√

〈x, x〉 =

√∑

i∈I

|xi|2.

A note of caution: In the definition of a basis, we have topay attention to the use of terms “span” and “independence”when we deal with infinite-dimensional spaces as both of thesewords implyfinite linear combinations. Many subtleties arisein infinite dimensions that are not present in finite dimensions.For instance, the infinite set{δi−k}k∈Z (here, δi = 1 fori = 0 and is0 otherwise) is a Schauder basis forℓ2(Z) butdoes not spanℓ2(Z) because we cannot write every square-summable sequence as a finite linear combination ofδi’s. Formore details, we refer the reader to [76].

B. Orthonormal Bases

A basisΦ = {ϕi}i∈I where the vectors are orthonormal:

〈ϕi, ϕj〉 = δi−j ,

is called anorthonormal basis (ONB). In other words, anorthonormal system is called an ONB forH, if for everyx inH,

x =∑

i∈I

Xiϕi, (1)

for some scalarsXi. These scalars are called thetransformorexpansion coefficientsof x with respect toΦ and it followsfrom orthonormality that they are given by

Xi = 〈ϕi, x〉, (2)

for all i ∈ I.We now discuss a few properties of orthonormal bases.1) Projections: A characteristic of orthonormal bases al-

lowing us to approximate signals is that an orthogonal pro-jection onto a subspace spanned by a subset of basis vectors,{ϕi}i∈J , whereJ is the index set of that subset is:

Px =∑

i∈J

〈ϕi, x〉ϕi, (3)

that is, it is a sum of projections onto individual one-dimensional subspaces spanned by eachϕi. Beware that thisis not true when{ϕi}i∈J do not form an orthonormal system.


2) Bessel’s Inequality:If we have an orthonormal systemof vectors{ϕi}i∈J in V, then, for everyx in V, the followinginequality, known as Bessel’s inequality, holds:

∑

i∈J

|〈ϕi, x〉|2 ≤ ‖x‖2.

3) Parseval’s Equality:If we have an orthonormal systemthat is complete inH, then we have an ONB forH, andBessel’s relation becomes an equality, often calledParseval’sequality(or Plancherel’s). This is simply thenorm-preservingproperty of ONBs. In other words

‖x‖2 =∑

i∈I

|〈ϕi, x〉|2. (4)

As an example, you might recognize this in the case of theFourier series as

‖x‖2 =∑

k∈Z

|Xk|2, (5)

whereXk are Fourier coefficients.4) Least-Squares Approximation:Suppose that we want to

approximate a vector from a Hilbert spaceH by a vectorlying in the (closed) subspaceS = {ϕi}i∈J . The orthogonalprojection ofx ∈ H onto S is given by (3). The differencevector d = x − x satisfiesd ⊥ S. This approximationis best in the least-squares sense, that is,min ‖x − y‖ fory in S is attained fory =

∑

i αiϕi with αi = 〈ϕi, x〉being the expansion coefficients. In other words, the bestapproximation is ourx = Px previously defined in (3).An immediate consequence of this result is the successiveapproximation property of orthogonal expansions. Callx(k)

the best approximation ofx on the subspace spanned by{ϕ1, ϕ2, . . . , ϕk}. Define the coefficients{X1, X2, . . . , Xk}by Xi = 〈ϕi, x〉. Then the approximationx(k+1) is given by

x(k+1) = x(k) + 〈ϕk+1, x〉ϕk+1,

that is, the previous approximation plus the projection alongthe added vectorϕk+1.

A note of caution:The successive approximation propertydoes not hold for nonorthogonal bases. When calculating theapproximationx(k+1), one cannot simply add one term tothe previous approximation, but has to recalculate the wholeapproximation.

C. General Bases

We are now going to relax the constraint of orthogonalityand see what happens. The reasons for doing that are numer-ous, the most obvious one being that we have more freedom inchoosing our basis vectors. For example, inR2, once a vectoris chosen, to get an orthonormal basis, we basically have onlyone choice (within a sign); on the other hand, for a generalbasis, it is enough not to choose the second vector colinear tothe first.

Example: As a simple example, consider the following setin R

2: Φ = {ϕ1, ϕ2} = {(1, 0)T , (√

2/2,√

2/2)T }. We haveseen how orthonormal bases expand vectors. This is not an

ONB but can we still use these two vectors to represent anyreal vectorx? The answer is yes:

x = 〈ϕ1, x〉ϕ1 + 〈ϕ2, x〉ϕ2,

with ϕ1 = (1,−1) and ϕ2 = (0,√

2). Thus, we can representany real vector with our initial pair of vectorsΦ = {ϕ1, ϕ2};however, they need helpers, an extra pair of vectorsΦ ={ϕ1, ϕ2}.

So what can we say about these two couples? It is obviousthat they work in concert to representx. Another interestingobservation is that, while not orthogonal within the couple,they are orthogonal across couples;ϕ1 is orthogonal toϕ2

while ϕ2 is orthogonal toϕ1. Moreover, the inner productsbetween corresponding vectors in a couple are of unit norm,that is,〈ϕi, ϕi〉 = 1 for i = 1, 2.

In general, thesebiorthogonalityrelations can be compactlyrepresented as

〈ϕi, ϕj〉 = δi−j .

The representation expression can then be written as

x =∑

i∈I

〈ϕi, x〉ϕi =∑

i∈I

〈ϕi, x〉ϕi,

that is, the roles ofϕi and ϕi are interchangeable. These twosets of vectors,Φ and Φ, are calledbiorthogonal basesandare said to bedual to each other. If the dual basisΦ is thesame asΦ, we get an ONB. Thus, orthonormal bases are selfdual.

While orthonormal bases are norm preserving, that is, theysatisfy Parseval’s equality, this is not true in the biorthogonalcase. This is one of the reasons successive approximationdoes not work here. In the orthonormal case, the norm of theoriginal vector is sliced up into pieces, each of which is thenorm of the corresponding expansion coefficient (and equal tothe length of the appropriate projection). Here, we know thatdoes not work.

From the above discussion, we see that biorthogonal basesoffer a larger choice, since they are less constrained thanthe orthonormal ones. However, this comes at the price oflosing the norm-preserving property as well as the successiveapproximation property. This trade-off is often tackled inpractice and depending on the problem at hand, you mightdecide to go with either orthonormal or biorthogonal basis.

D. From Representations to Matrices

While we are great fans of equations, we like matrices evenbetter, as equations can be hard to parse. We strongly believethat visualizing our representations is more intuitive andhelpsus understand the concepts better. Thus, we rephrase our basisnotions in matrix notation.

Example: Suppose we are given an ONBΦ ={(1,−1)T/

√2, (1, 1)T /

√2}. Given this basis and an arbitrary

vectorx in the plane, we might want to ask ourselves, whatis this point in this new basis (new coordinate system)? Weanswer this question by projectingx onto the new basis.


Suppose thatx = (1, 0)T . Then,xΦ1= 〈ϕ1, x〉 = 1/

√2 and

xΦ2= 〈ϕ2, x〉 = 1/

√2. Thus, in this new coordinate system,

our point(1, 0)T , becomesxΦ = (xΦ1, xΦ2

) = (1, 1)T /√

2. Itis still the same point in the plane, we only read its coordinatesdepending on which basis we are considering. We can expressthe above process of figuring out the coordinates in the newcoordinate system a bit more elegantly:

X = xΦ =

(xΦ1

xΦ2

)

=

(〈ϕ1, x〉〈ϕ2, x〉

)

=

(ϕ11x1 + ϕ12x2

ϕ21x1 + ϕ22x2

)

=

(ϕ11 ϕ12

ϕ21 ϕ22

) (x1

x2

)

=1√2

(1 −11 1

)(x1

x2

)

= Φ∗x,

where ∗ denotes Hermitian transposition. Observe that thematrix Φ describes an ONB in the real plane3. The columnsof the matrix are the basis vectors (the rows are as well),that is, the process of finding coordinates of a vector in adifferent coordinate system can be conveniently representedusing a matrixΦ whose columns are the new basis vectors,xΦ = Φ∗x.

We now summarize what we learned in this example ina more general case: Any Hilbert space basis (orthonormalor biorthogonal) can be represented as a matrix having basisvectors as its columns. If the matrix is singular, it does notrepresent a basis.

Given that we haveX = Φ∗x, we can go back tox byinverting Φ∗ (this is why we requireΦ to be nonsingular),x = (Φ∗)−1X . If the original basis is orthonormal, thenΦ isunitary andΦ−1 = Φ∗. The representation formula can thenbe written as

x =∑

i∈I

〈ϕi, x〉ϕi = ΦΦ∗x = Φ∗Φx. (6)

If, on the other hand, the original basis is biorthogonal, thereis not much more we can say aboutΦ. The representationformula is (the two basesΦ and Φ are interchangeable):

x =∑

i

〈ϕi, x〉ϕi = ΦΦ∗x = ΦΦ∗x =∑

i

〈ϕi, x〉ϕi.

E. Summary

To summarize what we have done in this section:• We represented our signal in another domain to more

easily extract its salient characteristics.• Given a pair of biorthogonal bases(Φ, Φ), the coordinates

of our signal in the new domain (or, with respect to thenew basis) are given by

X = Φ∗x, (7)

whereΦ is a linear operator describing the basis changeand it contains the dual basis vectors as its columns, whileX collects all the transform coefficients together. ForH =Rn,Cn, Φ is an n × n matrix; for H = ℓ2(Z), Φ isan infinite matrix. The above is called theanalysisordecompositionexpression.

3By abuse of language, we useΦ to denote both the set of vectors as wellas the matrix representing those vectors.

• The synthesis, or, reconstructionis given by

x = ΦX, (8)

whereΦ is a linear operator as well, and it contains thebasis vectors as its columns.

• If the expansion is into anorthonormal basis, then

Φ = Φ, and ΦΦ∗ = I,

that is,Φ is a unitary operator (matrix).• If the expansion is into a pair ofbiorthogonal bases, then

Φ∗ = Φ−1.

Example: [DFT as an Orthonormal Basis Expansion] TheDiscrete Fourier Transform is ubiquitous; however, it is rarelylooked upon as a signal expansion or written in matrix form.The easiest way to do that is to look at how the reconstructionis obtained:

xk =1

n

n−1∑

i=0

XiWikn , k = 0, . . . , n− 1, (9)

whereWn = ej2π/n is thenth root of unity. In matrix notationwe could write it as

x =1

n

1 1 · · · 11 Wn · · · Wn−1

n...

......

...1 Wn−1

n · · · Wn

︸︷︷︸

Φ=DFTn

X0

X1

...Xn−1

︸︷︷︸

X

.

Note that the DFT matrix defined as above is not normalized,that is (1/n)(DFTn)(DFTn)∗ = I. If we normalized theabove matrix by1/

√n, the DFT would exactly implement

an orthonormal basis.The decomposition formula is usually given as

Xi =n−1∑

k=0

xkW−ikn , i = 0, . . . , n− 1, (10)

and, in matrix notation:

X = DFT∗n x.

Consider now the normalized version. In basis parlance, ourbasis would beΦ = {ϕi}n−1

i=0 where the basis vectors are:

ϕi =1√n

(W 0n ,W

in, . . . ,W

i(n−1)n )T , i = 0, . . . , n− 1.

(11)Then, the expansion formula (10) can be seen as

Xi = 〈ϕi, x〉 i = 0, . . . , n− 1,

and the reconstruction formula (9) forx = (x0, . . . , xn−1)T :

x =

n−1∑

i=0

Xiϕi =

n−1∑

i=0

〈ϕi, x〉ϕi =1√n

DFTn

︸︷︷︸

Φ

1√n

DFT∗n

︸︷︷︸

Φ∗

x.

(12)


1

−1

1

ϕ1

ϕ2

ϕ3

(a)

1

−1 1 2

ϕ1

ϕ2

ϕ3

(b)

Fig. 1. A pair of general frames. (a) FrameΦ = {ϕ1, ϕ2, ϕ3}. (b) Dualframe Φ = {ϕ1, ϕ2, ϕ3}.

II. I NTRODUCTION TOFRAMES

The notion of bases in finite-dimensional spaces impliesthat the number of representative vectors is the same as thedimension of the space. When this number is larger, wecan still have a representative set of vectors, except that thevectors are no longer linearly independent and the resultingset is no longer called a basis but aframe. Frames are signalrepresentation tools which are redundant, and since they areless constrained than bases, they are used when more flexibilityin choosing a representation is needed.

In this section, we introduce frames through simple ex-amples and considerH = Rn,Cn only. In Section III, wewill define frames more formally and discuss a number oftheir properties. In Section IV, we examine finite-dimensionalframes in some detail. Then, in Section V, we look at theonly instance of infinite-dimensional frames we discuss in thisarticle, those inH = ℓ2(Z) implemented using filter banks.

A. General Frames

Example: Let us take an ONB, add a vector to it and seewhat happens. Suppose our system is as given in Fig. 1(a),with Φ = {ϕ1, ϕ2, ϕ3} = {(1, 0)T , (0, 1)T , (1,−1)T}. Thefirst two vectorsϕ1, ϕ2 are the ones forming the ONB andthe third oneϕ3 was added to the ONB. What can we sayabout such a system?

First, it is clear that by having three vectors inR2, thosevectors must necessarily be linearly dependent; indeed,ϕ3 =ϕ1 − ϕ2. It is also clear that these three vectors must be ableto represent every vector inR2 since their subset is able todo so (which also means that we could have added any othervectorϕ3 to our ONB with the same result.) In other words,we know that the following is true:

x = 〈ϕ1, x〉ϕ1 + 〈ϕ2, x〉ϕ2.

Nothing stops us from adding a zero to the above expression:

x = 〈ϕ1, x〉ϕ1 + 〈ϕ2, x〉ϕ2 + (〈ϕ1, x〉 − 〈ϕ1, x〉)(ϕ1 − ϕ2)︸︷︷︸

0

.

We now rearrange the above expression slightly to read:

x = 〈2ϕ1, x〉ϕ1 + 〈(−ϕ1 +ϕ2), x〉ϕ2 + 〈−ϕ1, x〉(ϕ1 −ϕ2).

In the above, we can recognize(−ϕ1 + ϕ2) as−ϕ3, and thevectors inside the inner products we will call

ϕ1 = 2ϕ1, ϕ2 = −ϕ1 + ϕ2, ϕ3 = −ϕ1.

With this notation, we can rewrite the expansion as

x = 〈ϕ1, x〉ϕ1 + 〈ϕ2, x〉ϕ2 + 〈ϕ3, x〉ϕ3 =

3∑

i=1

〈ϕi, x〉ϕi,

or, if we introduce matrix notation as before:

Φ =

(1 0 10 1 −1

)

, Φ =

(2 −1 −10 1 0

)

and

x =

3∑

i=1

〈ϕi, x〉ϕi, = ΦΦ∗x.

The only difference between the above expression and theone for general bases is that matricesΦ and Φ are nowrectangular. Fig. 1 shows this example pictorially.

Therefore, we have shown that starting with an ONBand adding a vector, we obtained another expansion with3vectors. This expansion is reminiscent of the one for generalbiorthogonal bases we have seen earlier, except that the vectorsinvolved in the expansion are now linearly dependent. Thisredundant set of vectorsΦ = {ϕi}i∈I is called aframewhileΦ = {ϕi}i∈I is called thedual frame. As for biorthogonalbases, these two are interchangeable, and thus,x = ΦΦ∗x =ΦΦ∗x.

B. Tight Frames

Well, adding a vector worked but we ended up with anexpansion that does not look very elegant. Is it possible tohave frames which would somehow mimic ONBs? To do that,let us think for a moment what characterizes ONBs. It is notlinear independence since that is true for biorthogonal basesas well. How about the following two facts:

• ONBs are self dual, and• ONBs preserve the norm?

Example: Consider now the system given in Box I and thefigure within, with vectorsΦPTF as in (45). We can easilycomputeΦPTFΦ∗

PTF = I and thus,ΦPTF can representany x from R2 (real plane). Since the same set of vectors isused both for expansion and reconstruction (see (46)),ΦPTFis self dual. We can think of the expansion in (46) as ageneralization of an ONB except that the vectors are notlinearly independent anymore. The frame of this type is calleda tight frame(Parseval tight)4 and this particular one is calledtheMercedes-Benz (MB) frame. We can normalize the lengthsof all the frame vectors to 1, leading to the unit-norm versionof this frame given in (42). One can compare the expansioninto an ONB with the expansion into a unit-norm version ofthe MB frame and see that the frame version has an extrascaling of 2/3 (see (43)). When the frame is tight and allthe vectors have unit norm as in this case, the inverse of thisscaling factor denotes the redundancy of the system: we have3/2 or 50% more vectors than needed to represent any vectorin R2.

4We will define all classes of frames in the next section.


This discussion took care of the first question, whether wecan have a self-dual frame. To check the question about norms,we compute the sum of the squared transform coefficients asin (47). We see that, indeed, this frame preserves the norm.To make the comparison to orthonormal bases fair, again wetake the unit-norm (UTF) version of the frame and computethe sum of the squared transform coefficients as in (44). Nowthere is extra scaling of3/2; this is fairly intuitive, as in thetransform domain, where we have more coefficients than westarted with, the energy is3/2 times higher than in the originaldomain.

Thus, the tight frame we constructed is very similar to anONB, with a linearly dependent set of vectors. Actually, tightframes are redundant sets of vectors closest to ONBs (we willmake this statement precise in Section IV-B).

One more interesting tidbit about this particular frame;note how all its vectors have the same norm. This is notnecessary for tightness but if it is true, then the frame iscalled anequal-norm tight frame (ENTF).

C. Summary

To summarize what we have done until now, assume thatwe are dealing with a finite-dimensional space of dimensionn andm > n linearly-dependent frame vectors:

• We represented our signal in another domain to moreeasily extract its salient characteristics. We did that in aredundant fashion.

• Given a pair of dual frames(Φ, Φ), the coordinates ofour signal in the new domain (that is, with respect to thenew frame) are given by

X = Φ∗x, l (13)

where Φ is a rectangularn × m matrix describing theframe change and it contains the dual frame vectors as itscolumns, whileX collects all the transform coefficientstogether. This is called theanalysis or decompositionexpression.

• The synthesis, or reconstructionis given by

x = ΦX, (14)

where Φ is again a rectangularn × m matrix, and itcontains frame vectors as its columns.

• If the expansion is into atight frame, then

Φ = Φ, and ΦΦ∗ = In×n.

Note that, unlike for bases,Φ∗Φ is not identity (why?).• If the expansion is into ageneral frame, then

ΦΦ∗ = I.

A note of caution: In frame theory, the frame change isusually denoted byΦ, not Φ∗. Given that Φ and Φ areinterchangeable, we can use one or the other without risk ofconfusion. Since

∑

i∈I Xiϕi is really the expansion in terms ofthe basis/frameΦ, it is natural to useΦ on the reconstructionside andΦ∗ on the decomposition side.

All frames

ENF TF

ENTF

UNF PTF

UNTF ENPTFONB

Fig. 2. Frames at a glance. ENF: Equal-norm frames TF: Tight frames.ENTF: Equal-norm tight frames. UNF: Unit-norm frames. PTF:Parseval tightframes. UNTF: Unit-norm tight frames. ENPTF: Equal-norm Parseval tightframes. ONB: Orthonormal bases.

III. F RAME DEFINITIONS AND PROPERTIES

In the last section, we introduced frames through examplesand developed some intuition. We now discuss frames moregenerally and examine a few of their properties. We formallydefine frames as follows: A familyΦ = {ϕi}i∈I in a HilbertspaceH is called aframeif there exist two constants0 < A ≤B <∞, such that for allx in H,

A ‖x‖2 ≤∑

i∈I

|〈ϕi, x〉|2 ≤ B ‖x‖2. (15)

A, B are calledframe bounds.Frame nomenclature is far from uniform and can result in

confusion. For example, frames with unit-norm frame vectorshave been called normalized frames (normalized as in allvectors normalized to norm 1, similarly to the meaning ofnormalized in orthonormal bases), uniform, as well as uniformframes with norm 1 (the first author of this paper is as guilty ofthis as anyone else). We now define various classes of frames.Their names, as well as alternate names under which they havebeen used in the literature, are given in Table I. Fig. 2 showsthose same classes of frames and their relationships.

Tight frames (TF)are frames with equal frame bounds, thatis, A = B. Equal-norm frames (ENF)are those frames whereall the elements have the same norm,‖ϕi‖ = ‖ϕj‖, for i, j ∈I. Unit-norm frames (UNF)are those frames where all theelements have norm1, ‖ϕi‖ = 1, for i ∈ I. A-tight frames(A-TF) are tight frames with frame boundA. Parseval tightframes (PTF)are tight frames with frame boundA = 1 andcould also be denoted as1-tight frames.

From (15), in a tight frame (that is, whenA = B), we have∑

i∈I

|〈ϕi, x〉|2 = A ‖x‖2. (16)

By pulling 1/A into the sum, this is equivalent to:∑

i∈I

|〈 1√Aϕi, x〉|2 = ‖x‖2, (17)

that is, the familyΦ = {(1/√A)ϕi}i∈I is a 1-tight frame. In

other words, any tight frame can be rescaled to be a tight frame


Name Abbr. Description Alternate Names

Equal-norm frame ENF ‖ϕi‖ = ‖ϕj‖, for all i, j Uniform frame [89]

Unit-norm frame UNF ‖ϕi‖ = 1, for all i Uniform frame with norm1 [89]Uniform frame [68]Normalized frame [11]

Tight frame TF A = B

A-tight frame A-TF A = B = A

Parseval tight frame PTF A = B = 1 Normalized frame [7]

Unit-norm tight frame UNTF A = B, ‖ϕi‖ = 1, for all i Uniform tight frame with norm1 [89]Uniform tight frame [68]Normalized tight frame [11]

TABLE I

FRAME NOMENCLATURE.

with frame bound1—a Parseval tight frame. WithA = 1, theabove looks similar to (4), Parseval’s equality, thus the nameParseval tight frame.

In an A-tight frame,x ∈ H is expanded as follows:

x =1

A

∑

i∈I

〈ϕi, x〉ϕi. (18)

While this last equation resembles the expansion formula inthe case of an ONB as in (1)-(2) (except for the factor1/A),a frame does not constitute an orthonormal basis in general.In particular, vectors may be linearly dependent and thus notform a basis. If all the vectors in a tight frame have unit norm,then the constantA gives the redundancy ratio. For example,A = 2 means there are twice as many vectors than neededto cover the space. For the MB frame we discussed earlier,redundancy is3/2, that is, we have3/2 times more vectorsthan needed to represent vectors in a two-dimensional space.Note that ifA = B = 1 (PTF), and‖ϕi‖ = 1 for all i (UTF),thenΦ = {ϕi}i∈I is an ONB (see Fig. 2).

Because of the linear dependence which exists among thevectors used in the expansion, the expansion is not uniqueanymore. ConsiderΦ = {ϕi}i∈I where

∑

i∈I αiϕi = 0(where not allαi’s are zero because of linear dependence). Ifx can be written asx =

∑

i∈I Xiϕi, then one can addαi toeachXi without changing the decomposition. The expansion(18) is unique in the sense that it minimizes the norm of theexpansion among all valid expansions. Similarly, for generalframes, there exists a unique canonical dual frame, which isdiscussed later in this section (in the tight frame case, theframe and its dual are equal).

Before we proceed, we settle on notation (see Table II).Note that some of the concepts in the table have not beendefined yet.

A. Frame Operators

The analysis operatorΦ∗ maps the Hilbert spaceH intoℓ2(I):

Xi = (Φ∗x)i = 〈ϕi, x〉, i ∈ I.

Symbol Explanation

H = Rn Real Hilbert spaceCn Complex Hilbert spaceℓ2(Z) Space of square-summable sequences

I = {1, . . . , m} Index set forRn, Cn

Z Index set forℓ2(Z)

WhenH = Rn, Cn n Dimension of the spacem Number of frame vectors

ϕi ∈ H Frame vectorΦ = {ϕi}i∈I Frame familyΦ∗ Analysis operatorS = ΦΦ∗ Frame operatorG = Φ∗Φ Grammian

ϕi ∈ H S−1ϕi Dual frame vectorΦ = {ϕi}i∈I Dual frame familyΦ∗ = Φ∗S−1 Dual analysis operatorS = S−1 Dual frame operatorG = Φ∗S−2Φ Dual Grammian

TABLE II

FRAME NOTATION.

As a matrix, the analysis operatorΦ∗ has rows which are theHermitian-transposed frame vectorsϕ∗

i :

Φ∗ =

ϕ∗11 · · · ϕ∗

1n · · ·ϕ∗

21 · · · ϕ∗2n · · ·

.... . .

. . . · · ·ϕ∗

m1 · · · ϕ∗mn · · ·

......

.... . .

.

When H = Rn,Cn, the above is anm × n matrix. WhenH = ℓ2(Z), it is an infinite matrix.

The frame operator, defined asS = ΦΦ∗, plays an impor-tant role. The productG = Φ∗Φ is called theGrammian.

B. Useful Frame Facts

When manipulating frame expressions, the frame facts givenbelow often come in handy. It is a useful exercise for you totry to derive some of these on your own.


• For any matrixΦ∗ with rowsϕ∗i

S = ΦΦ∗ =∑

i∈I

ϕiϕ∗i .

• If S is a frame operator, then

Sx = ΦΦ∗x =∑

i∈I

〈ϕi, x〉ϕi,

〈Sx, x〉 = 〈ΦΦ∗x, x〉 = 〈Φ∗x,Φ∗x〉= ‖Φ∗x‖2 =

∑

i∈I

|〈ϕi, x〉|2,∑

i∈I

〈Sϕi, ϕi〉 =∑

i∈I

〈Φ∗ϕi,Φ∗ϕi〉 =

∑

i,j∈I

|〈ϕi, ϕj〉|2.

• From (15), we have that

AI ≤ S = ΦΦ∗ ≤ BI,

as well asB−1I ≤ S−1 ≤ A−1I.

• We say that two framesΦ and Ψ for H are equivalentif there is a bounded linear bijectionL on H

5 forwhich Lϕi = ψi for i ∈ I. Two framesΦ and Ψare unitarily equivalent if L can be chosen to be aunitary operator. AnyA-TF is equivalent to a PTF asϕPTF = (1/

√A)ϕA−TF. In other words,{S−1/2ϕi}i∈I

is a Parseval tight frame for any frameΦ.• The nonzero eigenvalues{λi}i∈I , of S = ΦΦ∗ andG =

Φ∗Φ are the same. Thus,

tr(ΦΦ∗) = tr(Φ∗Φ). (19)

• A TF has orthonormal columns. In finite dimensions, thisis equivalent to the Naimark Theorem (see Section IV-A), which says that every TF is obtained by projectingan ONB from a larger space.

C. Dual Frame Operators

The canonical dual frameof Φ is a frame defined asΦ ={ϕi}i∈I = {S−1ϕi}i∈I , where

ϕi = S−1ϕi, i ∈ I. (20)

Noting thatϕ∗i = ϕ∗

i S−1 and stackingϕ∗

1, ϕ∗2, . . ., in a matrix,

the analysis frame operator associated withΦ is

Φ∗ = Φ∗S−1,

while its frame operator isS−1, with B−1 andA−1 its framebounds. Since

ΦΦ∗ = ΦΦ∗︸︷︷︸

S

S−1 = I,

thenx =

∑

i∈I

〈ϕi, x〉ϕi = ΦΦ∗x = ΦΦ∗x.

5This a mathematically simple (albeit possibly scary sounding) way totranslate the notion of “invertibility” to an infinite-dimensional Hilbert space.

IV. F INITE-DIMENSIONAL FRAMES

The most studied class of frames is the finite-dimensionalone, that is, whenH = Rn,Cn. We now examine few of theirproperties.

For example, for an ENTF with norm-a vectors, sinceS =ΦΦ∗ = AIn×n,

tr(S) =n∑

j=1

λj = nA, (21)

whereλj are the eigenvalues ofS = ΦΦ∗. On the other hand,because of (19)

tr(S) = tr(G) =

m∑

i=1

‖ϕi‖2 = ma2. (22)

Combining (21) and (22), we get

A =m

na2. (23)

Then, for a UNTF, that is, whena = 1, (23) yields theredundancy ratio:

A =m

n.

Recall that for the MB frame,A = 3/2.These, and other trace identities for all frames classes are

given in Table III.

A. Naimark Theorem

The following theorem tells us that every Parseval tightframe can be realized as a projection of an orthonormal basisfrom a larger space. The theorem has been rediscovered byseveral people in the past decade: The first author heard it fromDaubechies in the mid-90’s. Han and Larson rediscovered itin [73]; they came up with the idea that a frame could beobtained by compressing a basis in a larger space and thatthe process is reversible. Finally, it wasSoljanin [110] whopointed out to the first author that this is, in fact, Naimark’stheorem, which has been widely known in operator algebra andused in quantum information theory. In this paper, we consideronly the finite-dimensional instantiation of the theorem.

Theorem 1 (Naimark [2], Han & Larson [73]) A setΦ ={ϕi}i∈I in a Hilbert spaceH is a Parseval tight frame forHif and only if there is a larger Hilbert spaceK,H ⊂ K, andan orthonormal basis{ei}i∈I for K so that the orthogonalprojectionP of K onto H satisfies:Pei = ϕi, for all i ∈ I.

While the above theorem specifies how all tight frames areobtained, the same is true in general, that is, any frame canbe obtained by projecting a biorthogonal basis from a largerspace [73] (we are talking here about finite dimensions only).We will call this processseedingand will say that a frameΦis obtained by seeding from a basisΨ by deleting a suitableset of columns ofΨ [101]. We denote this as

Φ∗ = Ψ[J ],

where J ⊂ {1, . . . ,m} is the index set of the retainedcolumns.


Frame Constraints Properties

General {ϕi}i∈I A‖x‖2 ≤P

i∈I |〈ϕi, x〉|2 ≤ B‖x‖2

is a Riesz basis forH AI ≤ S ≤ BI

tr(S) =Pn

j=1λj = tr(G) =

Pmi=1

‖ϕi‖2

ENF ‖ϕi‖ = ‖ϕj‖ = a A‖x‖2 ≤P

i∈I |〈ϕi, x〉|2 ≤ B‖x‖2

for all i and j AI ≤ S ≤ BI

tr(S) =Pn

j=1λj = tr(G) =

Pmi=1

‖ϕi‖2 = ma2

TF A = BP

i∈I |〈ϕi, x〉|2 = A‖x‖2

S = AI

tr(S) =Pn

j=1λj = nA = tr(G) =

Pmi=1

‖ϕi‖2

PTF A = B = 1P

i∈I |〈ϕi, x〉|2 = ‖x‖2

S = I

tr(S) =Pn

j=1λj = n = tr(G) =

Pmi=1

‖ϕi‖2

ENTF A = BP

i∈I |〈ϕi, x〉|2 = A‖x‖2

‖ϕi‖ = ‖ϕj‖ = a S = AI

for all i and j tr(S) =Pn


Pmi=1

‖ϕi‖2 = ma2

A = (m/n)a2

UNTF A = BP

i∈I |〈ϕi, x〉|2 = A‖x‖2

‖ϕi‖ = 1 S = AI

for all i tr(S) =Pn


Pmi=1

‖ϕi‖2 = m

A = m/n

ENPTF A = B = 1P

i∈I |〈ϕi, x〉|2 = ‖x‖2

‖ϕi‖ = ‖ϕj‖ = a S = I

for all i and j tr(S) =Pn

j=1λj = n = tr(G) =

Pmi=1

‖ϕi‖2 = ma2

a =p

n/m

UNPTF A = B = 1P

i∈I |〈ϕi, x〉|2 = ‖x‖2

⇔ ‖ϕi‖ = 1 S = I

ONB for all i tr(S) =Pn

j=1λj = n = tr(G) =

Pmi=1

‖ϕi‖2 = m

n = m

TABLE III

SUMMARY OF PROPERTIES FOR VARIOUS CLASSES OF FRAMES. ALL TRACE IDENTITIES ARE GIVEN FORH = Rn, Cn .

We can now reinterpret the Parseval tight frame identityΦΦ∗ = I (see Table III): It says that the columns ofΦ∗ areorthonormal. In view of the above theorem, this makes a lotof sense as that frame was obtained by deleting columns froman ONB from a larger space.

For example, for the MB frame given in Box I, the three-dimensional ONB from which it is obtained is given in (48)and the projection operator in (49). The MB frame obtainedis in its PTF version given in (45).

B. What Can Coulomb Teach Us?

As the ONBs have specific characteristics highly prizedamong bases, the same distinction belongs to tight framesamong all frames. As such, they have been studied extensively,but only recently have Benedetto and Fickus [11] formallyshown why tight frames and ONBs indeed belong together. Intheir work, they characterized all UNTFs, while in [30], theauthors did the same for nonequal norm TFs.

To characterize UNTFs, as a starting point, the authorslooked atharmonic tight frames(we will introduce those inSection VI-A), obtained by takingmth roots of unity inRn.These lead to regular arrangement of points on a circle. Anexample is obviously the MB frame from Box I. The PTFversion given in (45) was obtained by deleting the last columnof a three-dimensional ONB given in (48). Trying to generalizethe notion of geometric regularity to three dimensions, theylooked at vertices of regular polyhedra but came short asthere are only five such Platonic solids. Considering othersets of high symmetry such as the “soccer ball” (a truncatedicosahedron), they found that all these proved to be UNTFs.

As the geometric intuition could lead them only so far, theauthors in [11] refocused their attention on the equidistributionproperties of these highly symmetric objects and thought ofthenotion of equilibrium. To formalize that notion, they turned toclassical physics and considered the example ofm electronson a conductive spherical shell. In the absence of external


forces, electrons move according to the Coulomb force lawuntil they reach the state of minimum potential energy (thoughthat minimum might only be a local minimum leading to anunstable equilibrium). The intuition developed through thisexample lead them to the final result.

They tried to replicate the physical world for the simplestUNTFs—orthonormal bases, and thought of what kind ofequilibrium they possessed. Clearly, whichever “force” actson the vectors in an ONB, it tries to promote orthogonality.For example, the Coulomb force would not keep the ONBin a state of equilibrium. (Thinkn = 2, the Coulomb forcewould position the two vectors to be colinear of oppositesign.) Thus, the authors set to find another such force—theorthogonality–promoting one. This force should be repulsiveif vectors form an acute angle, while it should be attractiveifthey form an obtuse angle. Since points are restricted to moveonly on the circle (unit-norm constraint), one can consideronly the tangential component of the force. Even if vectors donot all have equal norm,‖ϕi‖ = ai, for i ∈ I, one can definethe frame force FFon the whole space:

FF (ϕi, ϕj) = 2〈ϕi, ϕj〉(ϕi − ϕj)

= (a2i + a2

j − ‖ϕi − ϕj‖2)(ϕi − ϕj).

Following the physical trail, one can now define the potentialbetween two points as:

P (ϕi, ϕj) = p(‖ϕi − ϕj‖).This is found by usingp′(x) = −xf(x), where f(x) isthe scalar part of the frame force andp(x) is obtained byintegrating the above and evaluating at‖ϕi−ϕj‖2. After somemanipulations, the result is:

P (ϕi, ϕj) = 〈ϕi, ϕj〉2 −1

4(a2

i + a2j)

2.

Then, the total potential contained within a sequence is:

TP (Φ = {ϕi}i∈I) =∑

i,j∈I,i6=j

|〈ϕi, ϕj〉|2 − 1

4

∑

i,j

(a2i +a

2j )

2.

Physically, we can interpret the total potential as follows:Given two sequences of points, the difference in potentialsbetween these two sequences is the energy needed to movethe points from one configuration to the other. As potentialenergy is defined in terms of differences, it is unique up toadditive constants and thus we can neglect the constants andadd the diagonal terms to obtain the final expression for theframe potential:

FP (Φ = {ϕi}i∈I) =∑

i,j∈I

|〈ϕi, ϕj〉|2. (24)

Thus, what we are looking for are those sequences in equilib-rium under the frame force, and these will be minimizers ofthe frame potential.

For UNTFs, Benedetto and Fickus discovered the following:

Theorem 2 ([11]) Given Φ = {ϕi}mi=1, with ϕi ∈ H

n,consider the frame potential given in(24). Then:

1) Every local minimizer of the frame potential is also aglobal minimizer.

2) If m ≤ n, the minimum value of the frame potential is

FP = n,

and the minimizers are precisely the orthonormal se-quences inRn.

3) If m ≥ n, the minimum value of the frame potential is

FP =m2

n,

and the minimizers are precisely the UNTFs forRn.

The above result tells us a few things:

1) Minimizing the frame potential amounts to finding se-quences whose elements are “as orthogonal” to eachother as possible.

2) UNTFs are a natural extension of ONBs, that is, thetheorem formalizes the intuitive notion that UNTFs area generalization of ONBs.

3) Both ONBs and UNTFs are products of the minimiza-tion of the frame potential, with different parameters(number of elements equal/larger then the dimension ofthe space).

What happens if points live on different spheres,ϕi = ai

(vectors are not of equal norm)? Again, we can try to minimizethe frame potential. Since now points live on spheres ofdifferent radii, it is intuitive that stronger points (withalarger norm) will be able to be “more orthogonal” than theweaker ones. If the strongest point is strong enough, it grabsa dimension to itself and leaves the others to squabble overwhat is left. We start all over with the second one and continueuntil those points left have to share. This is governed by theFundamental Inequality given in (26), which says that if nopoint is stronger than the rest they immediately have to share,leading to tight frames. In other words, whenm points in ann-dimensional space are in equilibrium, we can divide thosepoints into two sets. (a) Those “stronger” than the rest. These(i0 − 1) points get a dimension each and are thus orthogonalto each other. (b) Those “weaker” than the rest. These pointsget the rest of the(n − i0 + 1) dimensions and form a tightframe for their span. If no point is the “strongest”, they allhave to share the space leading to a tight frame, as per theFundamental Inequality. This discussion is summarized in thetheorem below:

Theorem 3 ([30]) Given a sequence{ai = ‖ϕi‖}mi=1 in R,

such thata1 ≥ . . . ≥ am ≥ 0, and anyn ≤ m, let i0 denotethe smallest indexi for which

(n− i)a2i ≤

m∑

j=i+1

a2j , (25)

holds. Then, any local minimizer of the frame potential is ofthe form

Φ = {ϕi}mi=1 = {ϕi}i0−1

i=1 ∪ {ϕi}mi=i0 ,

whereΦo = {ϕi}i0−1i=1 is an orthogonal set andΦf = {ϕi}m

i=i0forms a tight frame for the orthogonal complement of the spanof Φo.


The immediate corollary is theFundamental Inequalitywetalked about, which says that if no point is stronger than therest, the vectors have to share the space, leading to a tightframe:

maxi∈I

a2i ≤ 1

n

∑

i∈I

a2i . (26)

The frame potential defined in (24) is a concept introducedby Benedetto and Fickus, and it proved immediately useful.For example, it was used in [31] to show how to packetizecoefficients in transmission with erasures to minimize theerror of reconstruction. A decade before [11], Massey andMittelholzer [98] used the frame potential (albeit not callingit the frame potential) as the total user interference in code-division multiple access systems (CDMA). Minimizing thatinterferences lead to the spreading sequences (of lengthn)being a tight frame (minimum of the Welch’s Bound). This isdiscussed in more detail in Section VII-D.

C. Design Constraints: What Might We Ask of a Frame?

When designing a frame, particularly if we have a specificapplication in mind, it is useful to list potential requirementswe might impose on our frame.

• Tightness (T): This is a very common requirement.Typically, tightness is imposed when we do need toreconstruct and stability of reconstruction is an issue.Since tight frames do not require inversion of matrices,they seem a natural choice.

• Equal norm (EN): In the real world, the squared normof a vector is usually associated with power. Thus, insituations where equal-power signals are desirable, equalnorm is a must.

• Maximum robustness (MR): We call a framemaximallyrobust to erasures (MR), if everyn× n submatrix ofΦ∗

is invertible. This requirement arose in using frames forrobust transmission [68] and will be discussed in moredetail in Section VII-C.

• Equiangularity (EA): This is a geometrically intuitiverequirement. We ask for angles between any two vectorsto be the same. There are many more (tight) framesthan those which are equiangular, so this leads to a veryparticular class of frames. These are discussed in moredetail in Section VI-B.

• Symmetry (S): Symmetries in a frame are typicallyconnected to its geometric configuration. Harmonic andequiangular frames are good examples. See the work ofVale and Waldron [118] for details.

Invariance of Frame Properties:When designing frames,it is useful to know which transformations will not destroyproperties our frame already possesses. For that reason, welist below a number of frame invariance properties [101].

Let Φ be a frame. In all matrix products below, we assumethe sizes to be compatible.

• AΦB is a frame for any invertible matricesA,B.• If Φ is TF/UNTF, thenaUΦV is TF/UNTF for any

unitary matricesU, V anda 6= 0.• If Φ is EN, thenaDΦU is EN for any diagonal unitary

matrixD, unitary matrixU , anda 6= 0.

• If Φ is MR, thenDΦA is MR for any invertible diagonalmatrixD and any invertible matrixA.

• If Φ is UNTF MR, thenDΦU is UNTF MR for anyunitary diagonal matrixD and any unitary matrixU .

V. I NFINITE-DIMENSIONAL FRAMES VIA FILTER BANKS

We now consider the only infinite-dimensional class offrames discussed in this paper, those implemented by filterbanks, the reason being that these are frames used in applica-tions and our only link to the real world. The vectors (signals)live in the infinite-dimensional Hilbert spaceH = ℓ2(Z). Anin-depth treatment of filter banks is given in [117], while amore expansion-oriented approach is followed in [120], [121].

A. Filter Bank View of Bases

As we have done earlier in the paper, we will first examinehow filter banks implement bases, and then move onto frames.

We have seen that we want to find representations ormatricesΦ and Φ such thatΦΦ∗ = I. As of now, we havepresented a generic matrixΦ, but how do we choose it? Ofcourse, we want it to have some structure and lead to efficientrepresentations of signals. Since now we are dealing withinfinite-dimensional matrices, this might be easier said thandone.

Example: Suppose we have the following two vectors:ϕ0 =(· · · , 0, 1, 1, 0, · · · )T /

√2, ϕ1 = (· · · , 0, 1,−1, 0, · · · )T /

√2.

These vectors form a basis for their span, that is, theycan represent any two-dimensional vector, but not any vec-tor in ℓ2(Z). Now, define τi as a shift by i, that is,if x = (. . . , x−1, x0, x1, . . .)

T ∈ ℓ2(Z) , then τix =(. . . , x−i−1, x−i, x−i+1, . . . )

T is its shifted version byi. Letus form the following matrix:

Φ∗ =

...(τ−2ϕ0)

∗

(τ−2ϕ1)∗

(ϕ0)∗

(ϕ1)∗

(τ2ϕ0)∗

(τ2ϕ1)∗

...

,

that is, the columns ofΦ are the two vectorsϕ0, ϕ1 and alltheir even shifts:

Φ∗ =1√2

. . ....

......

......

...· · · 1 1 0 0 0 0 · · ·· · · 1 −1 0 0 0 0 · · ·· · · 0 0 1 1 0 0 · · ·· · · 0 0 1 −1 0 0 · · ·· · · 0 0 0 0 1 1 · · ·· · · 0 0 0 0 1 −1 · · ·

......

......

......

. . .

.

From this expression we can see thatΦ is block diagonal. Ifwe denote by

Φ∗0 =

1√2

(1 11 −1

)

, (27)


x

Analysis

h

g

2

2

2

2

Synthesis

h

g

+x

Fig. 3. Two-channel filter bank with downsampling by2.

then,Φ can be written as:

Φ∗ =

. . .Φ∗

0

Φ∗0

Φ∗0

. . .

.

This is known as theHaar transform and is an exampleof a block transform. The matrix Φ∗ above is unitary andcorresponds to an orthonormal expansion. The basisΦ is givenby Φ = {ϕ2i, ϕ2i+1}i∈Z = {τ2iϕ0, τ2iϕ1}i∈Z. Therefore, anyx ∈ ℓ2(Z) can be represented using the Haar ONB as:

x = ΦΦ∗x =∑

i∈Z

〈ϕi, x〉ϕi,

and can be implemented using the two-channel filter bankshown in Fig. 3. The decomposition is implemented using theanalysisfilter bank, while the reconstruction is implementedusing thesynthesisfilter bank (we will make this more preciseshortly).

In general, in such a filter bank, one branch is a lowpasschannel that captures the coarse representation of the inputsignal and the other branch is a highpass channel that capturesa complementary, detailed representation. The input into thefilter bank is a square-summable infinite sequencex ∈ ℓ2(Z).Assuming that the filter lengthl = 2, the two analysis filtersact on2 samples at a time and then, due to downsampling by2, the same filters act on the following2 samples. In otherwords, there is no overlap. On the synthesis side, the reverseis true. This is an example of a block transform. Iterating thisblock (the two-channel FB) on either channel or both leadsto various signal transforms, each of which is adapted to aclass of signals with different energy concentrations in timeand in frequency (this is usually referred to as “tiling of thetime-frequency plane”).

So how exactly is the filter bank related to the matrixΦ?In our discussion above and the Haar example, we assumedthat the filter length is equal to the shift. This is not true ingeneral, and now, we lift that restriction and allow filters tobe of arbitrary lengthl (without loss of generality, we willassume that filters are causal, that is, they are nonzero onlyfor positive indices). However, we do leave the restrictionthat

the filters are finitely supported, that is, they are FIR filters6.Consider an inner product between two sequencesx andy (onthe left), and filtering a sequencex by a filter f and havingthe output at timek (on the right):

〈x, y〉 =∑

i∈Z

x∗i yi x ∗ f =∑

i∈Z

x∗i fk−i.

By comparing the above two expressions, we see that we couldexpress filtering a sequencex by a filter f and having theoutput at timek as:

∑

i∈Z

x∗i fk−i = 〈fk−i, xi〉.

Thus, to express the analysis part of the filter bank, we cando the following:

X =

...X0

X1

X2

X3

...

=

...〈ϕ0, x〉〈ϕ1, x〉〈ϕ2, x〉〈ϕ3, x〉

...

=

...〈g−i, xi〉〈h−i, xi〉〈g2−i, xi〉〈h2−i, xi〉

...

=

. . ....

......

.... . .

· · · g3 g2 g1 g0 · · ·· · · h3 h2 h1 h0 · · ·· · · g5 g4 g3 g2 · · ·· · · h5 h4 h3 h2 · · ·. . .

......

......

. . .

︸︷︷︸

Φ∗

...x0

x1

x2

x3

...

︸︷︷︸

x

= Φ∗x.

Similarly, the reconstruction part can be expressed as

x =

. . ....

......

......

.... . .

· · · g2 h2 g0 h0 0 0 · · ·· · · g3 h3 g1 h1 0 0 · · ·· · · g4 h4 g2 h2 g0 h0 · · ·· · · g5 h5 g3 h3 g1 h1 · · ·. . .

......

......

......

. . .

︸︷︷︸

Φ

...X0

X1

X2

X3

...

︸︷︷︸

X

=(· · · τ−2g τ−2h g h τ2g τ2h · · ·

)X

=

. . ....

......

. . .· · · Φ1 Φ0 0 · · ·· · · Φ2 Φ1 Φ0 · · ·. . .

......

.... . .

X = ΦX. (28)

whereΦi aren × m matrices withn being the shift andmthe number of channels/filters in the filter bank. The matricesare formed by taking theith block ofn coefficients from eachof them filters. Heren = 2 andm = 2.

Recall again, we have assumed our filters to be causal. Fromabove, we can conclude the following:

6IIR filters also fit in this framework, we concentrate on FIR only forsimplicity. Moreover, this restriction makes all the operators bounded and allthe series converge.


• The basis isΦ = {τ2iϕ0, τ2iϕ1}i∈Z = {τ2ig, τ2ih}i∈Z .In other words, the impulse responses of the templatefilters g and h and their even shifts form the basisΦ(they are the columns ofΦ).

• The dual basis is Φ = {τ2iϕ0, τ2iϕ1}i∈Z ={τ2ig, τ2ih}i∈Z . In other words, the impulse responsesof the template filtersg and h and their even shifts formthe basisΦ (they are the columns ofΦ).

• When Φ = Φ, the basis is orthonormal. In that case,gi = g−i, that is, the impulse responses of the analysisfilters are time-reversed impulse responses of synthesisfilters.

• The even shifts appear because of down/upsampling by2.

• When the filters are of lengthl = 2 (l = n ingeneral),Φ∗ or Φ∗ contain only one block,Φ∗

0 or Φ∗0,

along the diagonal, making it a block-diagonal matrix(as in the Haar transform). The effect of this is that theinput is processed in nonoverlapping pieces of length2.Effectively, this is equivalent to dealing with bases in thetwo-dimensional space.

• We discussed here a specific case with2 template filtersand shifts by2. In filter bank parlance, we discussedtwo-channel filter banks with sampling7 by 2. Of course,more general options are out there and one can havem-channel filter banks with sampling bym. We then havemtemplate filters (basis vectors) from which all the basisvectors are obtained by shifts by multiples ofm. TheblocksΦ∗

i then become of sizem ×m. Again, if filtersare of lengthl = m, this leads to the block-diagonalΦ∗,and effectively, finite-dimensional bases.

B. z-Domain View of Signal Processing

Historically, the above, basis-centric view of filter bankscame very recently. Initially, when the filter banks weredeveloped to deal with speech coding [40], [60], the analysiswas done inz-domain (for easier algebraic manipulation).The mapping that takes us from the original domain to thez-domain is thez-transform and is defined for a sequencex ∈ ℓ2(Z) as

X(z) =∑

i∈Z

xiz−i. (29)

You can think of the z-transform as a generalized discrete-time Fourier transform (DTFT), whereejω has been replacedby the complex numberz = rejω . Just like the DTFT, thez-transform possesses nice properties (such as the convolutionproperty) making it a useful analysis tool. More precisely,thez-transform allows us to deal with polynomial-like objectsinstead of convolutions.

In particular,z-transform comes in handy when we haveto deal with shift-varying systems such as filter banks. Shiftvariance is introduced into the system due to downsamplers(or shifts). A tool used to transform a filter bank from a single-input single-output (SISO) linear periodically shift-variant

7By sampling, we mean the two sampling operations, downsampling andupsampling.

system into a multiple-input multiple-output (MIMO) linearshift-invariant systems is called thepolyphase transform.

For i = 0, . . . ,m − 1, for the ith synthesis filter (templatebasis vector),ϕi(z) = (ϕi0(z), . . . , ϕi,m−1(z))

T is called thepolyphase representation of theith synthesis filterwhere

ϕik(z) =∑

p∈Z

ϕi,mp+kz−p, (30)

are thepolyphase componentsfor i, k = 0, . . . ,m−1. To relateϕik(z) to a time-domain object, note that it is the discrete-timeFourier transform of the template basis vectorϕi obtained byretaining only the indices congruent tok modulom. ThenΦp(z) is the correspondingm× m synthesis polyphase matrixwith elementsϕik(z). In other words, a polyphase decompo-sition is a decomposition intom subsequences modulom. Wecan do the same on the analysis side, leading to the polyphasematrix Φ∗

p(z). Then, the input/output relationship is given by

x(z) = (1 z−1 . . . z−(m−1))Φp(z)Φ∗p(z)xp(z), (31)

where xp(z) is the vector of polyphase components of thesignal (there arem of them) and∗ denotes conjugation ofcoefficients but not ofz. Note that the polyphase componentsof the analysis bank are defined in reverse order from thoseof the synthesis bank. When the filter length isl = m, then,each polyphase sequence is of length1. Each polyphase matrixthen reduces toΦp(z) = Φ0, Φ∗

p(z) = Φ∗0, that is, bothΦp(z)

and Φ∗p(z) become independent ofz. It is clear from above,

that to obtain perfect reconstruction, that is, to have a basisexpansion, the polyphase matrices must satisfy:

Φp(z)Φ∗p(z) = I. (32)

If the filter length l = m, the above implements a finite-dimensional expansion (block transform). For example, ifwe wanted to implement theDFTm using a filter bank,we would use anm-channel filter bank with sampling bym, and prototype synthesis filtersϕi given in (11). Sinceeach prototype filter is of lengthm, each of its polyphasecomponents will be of length1 and a constant, leading to aconstant polyphase matrix.

If a filter bank implements an ONB, thenΦp(z) = Φp(z−1),

and (32) reduces to

Φp(z)Φ∗p(z

−1) = I. (33)

A matrix satisfying the above is called aparaunitary matrix,that is, it is unitary on the unit circle.

Example: As a first example, go back to the Haar expansiondiscussed earlier. Sincem = 2, ϕ0(z) = (1 + z−1)/

√2,

ϕ1(z) = (1− z−1)/√

2, and the polyphase matrix isΦ∗p(z) =

Φ∗0 from (27).As a more involved example, supposem = 2 again and we

are given the following set of template filters:

G(z) = z−2 + 4z−1 + 6 + 4z + z2,

H(z) =1

4z(

1

4z−1 + 1 +

1

4z),

G(z) =1

4(−1

4z−1 + 1 − 1

4z),

H(z) = z−1(z−2 − 4z−1 + 6 − 4z + z2).


2

2

h

g

level j + b b

2

2

h

g

level 1 +x

Fig. 4. The synthesis part of the filter bank implementing theDWT with jlevels. The analysis part is analogous.

Having the polyphase decomposition for each filter beingwritten as:G(z) = G0(z

2) + z−1G1(z2), H(z) = H0(z

2) +z−1H1(z

2), G(z) = G0(z2) + zG1(z

2), H(z) = H0(z2) +

zH1(z2), the polyphase matrices are then (they have polyphase

components of the above filters as their columns):

Φp(z) =

(z−1 + 6 + z 1

16 (1 + z)4(1 + z) 1

4z

)

,

Φp(z) =

(14 −4(1 + z−1)

− 116 (1 + z−1) 1 + 6z−1 + z−2

)

.

Thus, the filter bank with filters as defined above implementsa biorthogonal expansion. The dual bases are:

Φ = {ϕ2i, ϕ2i+1}i∈Z = {τ2ig, τ2ih}i∈Z,

Φ = {ϕ2i, ϕ2i+1}i∈Z = {τ2ig, τ2ih}i∈Z,

and they are interchangeable.

Filter Bank Trees:Many of the bases (and frames later on),are built by using two- andm-channel filter banks as buildingblocks. For example, the dyadic (with scale factor2) DWT isbuilt by iterating the two-channel filter bank on the lowpasschannel (Fig. 4 depicts the synthesis part). The DWT is a basisexpansion and as such nonredundant (critically sampled). Todescribe the redundancy of various frame families later on,we introduce sampling grids in Fig. 12, each depicting timepositions of basis vectors at each level. Thus, for example,thetop plot in Fig. 12 depicts the grid for the DWT. At level 1, wehave half as many points as at level 0, at level 2, half as manyas at level 1, and so on. Because of appropriate sampling, thegrid has exactly as many points as needed to represent anyx ∈ ℓ2(Z) and is thus nonredundant.

We can also build arbitrary trees by, at each level, iteratingon any subset of the branches (typically known aswaveletpackets[37]). In order to analyze these tree-structured filterbanks, we typically collect all the filters and samplers along apath into a branch with a single filter and single sampler. Thisis possible using the so-called Noble identities [117] whichallow us to exchange filtering and sampling.

Example: Assume we have a DWT with 2 levels, that is, thelowpass branch is iterated only once (see Fig. 4). Then, theequivalent filter bank has3 channels as in Fig. 5 with sampling

2

4

4

ϕ2 = h

ϕ1 = g ∗ (↑ 2)h

ϕ0 = g ∗ (↑ 2)g

+x

Fig. 5. The synthesis part of the equivalent three-channel filter bankimplementing the DWT with2 levels. The analysis part is analogous.

x

ϕ3

ϕ2

ϕ1

2

2

2

2

2

2

ϕ3

ϕ2

ϕ1

+x

Fig. 6. 3-channel filter bank with downsampling by2.

by 2, 4 and4, respectively. The equivalent filters are then (call(↑ m) the operator upsampling a filter bym):

ϕ2 = h, ϕ1 = g ∗ (↑ 2)h, ϕ0 = g ∗ (↑ 2)g.

Assuming for simplicity that the filters have only two taps, thematrix Φ in (28) is block diagonal with:

Φ∗0 =

h0 h1 0 00 0 h0 h1

g0h0 g1h0 g0h1 g1h1

g20 g0g1 g0g1 g2

1

.

We see here that even though we have only 3 branches, thefilter bank behaves as a4-channel filter bank with samplingby 4.

C. Filter Bank View of Frames

The filter bank expansions we just discussed were basesand thus nonredundant. Now, nothing stops us from beingredundant (for reasons stated earlier) by simply adding morevectors.

Example: Let us look at the simplest case using our favoriteexample: the MB frame given in Box I. OurΦ∗ is now block-diagonal, with Φ∗

0 = Φ∗UNTF from (45) on the diagonal.

In contrast to finite-dimensional bases implemented by filterbanks (see Haar in (27)), the blockΦ∗

0 is now rectangular ofsize 3 × 2. This finite-dimensional frame is equivalent to thefilter bank shown in Fig. 6, with{ϕi} = {ϕi}, given in (45).

As we could for finite-dimensional bases, we can investigatefinite-dimensional frames within the filter bank framework (seeFig. 7). In other words, all cases we consider in this article,


both finite dimensional and infinite dimensional, we can lookat as filter banks.

Similarly to bases, if in (28)Φ is not block diagonal, weresort to the polyphase-domain analysis. Assume that the filterlength is l = kn (if not, we can always pad with zeros), andwrite the frame as (causal filters)

Φ∗ =

. . ....

......

......

. . .· · · Φ∗

0 0 · · · 0 0 · · ·· · · Φ∗

1 Φ∗0 · · · 0 0 · · ·

· · ·...

......

...... · · ·

· · · Φ∗k−1 Φ∗

k−2 · · · Φ∗0 0 · · ·

· · · 0 Φ∗k−1 · · · Φ∗

1 Φ∗0 · · ·

. . .... 0

......

.... . .

(34)

where each blockΦi is of sizen×m:

Φ0 =

ϕ00 . . . ϕ0,m−1

.... . .

...ϕn−1,0 . . . ϕn−1,m−1

.

In the above, we enumerate template frame vectors from0, . . . ,m−1. A thorough analysis of oversampled filter banksseen as frames is given in [44], [23], [45].

D. Summary

To summarize, the class of multiresolution transforms ob-tained using a filter bank depends on three parameters: thenumber of vectorsm, the shift or sampling factorn and thelength l of the nonzero support of the vectors:

1) Basesm = n: The filter bank in this case is calledcrit-ically sampledand implements a nonredundant expansion—basis. The basisΦ has a dual basis associated with it,Φ,leading tobiorthogonalfilter banks. The associated matricesΦ, Φ are invertible. In thez-domain, this is expressed asfollows: A filter bank implements a basis expansion if andonly if (32) is satisfied [120].

An important subcase is when the basisΦ is orthonormal,in which case it is self-dual, that is,Φ = Φ. The filter bankis called orthogonaland the associated matrixΦ is unitary,ΦΦ∗ = I. In the z-domain, this is expressed as follows: Afilter bank implements an orthonormal basis expansion if andonly if its polyphase matrix is paraunitary, that is, if and onlyif (33) holds [120]. Well-known subcases are the following:

• When l = m, we have ablock transform. In this case, in(28), onlyΦ0 exists, makingΦ block-diagonal. In effect,since there is no overlap between processed signal blocks,this can be analyzed as a finite-dimensional case, whereboth the input and the output arem-dimensional vectors.A famous example is the Discrete Fourier Transform(DFT), which we discussed earlier.

• Whenm = 2, we get two-channel filter banks. In (28),Φi

is of size2× 2 and by iterating on the lowpass channel,we get the Discrete Wavelet Transform (DWT) [120] (seeFig. 4).

• When l = 2m, we getLapped Orthogonal Transforms(LOT), efficient transforms developed to deal with the

x

ϕm−1

ϕ0

b

b

b

n

n

n

n

ϕm−1

ϕ0

b

b

b

+x

Fig. 7. A filter bank implementation of a frame expansion: It is anm-channelfilter bank with sampling byn.

blocking artifacts introduced by block transforms, whilekeeping the efficient computational algorithm of theDFT [120]. In this case, in (28), onlyΦ0 and Φ1 arenonzero.

2) Framesm > n: The filter bank in this case implementsa redundant expansion—frame. The frame Φ has a dualframe associated with it,Φ. The associated matricesΦ, Φ arerectangular and left/right invertible. This has been formalizedin z-domain in [44], as the following result: A filter bankimplements a frame decomposition inℓ2(Z) if and only if itspolyphase matrix is of full rank on the unit circle.

An important subcase is when the frameΦ is tight, in whichcase it is self-dual, that is,Φ = Φ, and the associated matrixΦ satisfiesΦΦ∗ = I. This has been formalized inz-domain in[44], as the following result: A filter bank implements a tightframe expansion inℓ2(Z) if and only if its polyphase matrixis paraunitary. A well-known subcase of tight frames is thefollowing:

• When l = n, we have ablock transform. Then, in (34),only Φ0 is nonzero, makingΦ block-diagonal. In effect,since there is no overlap between processed blocks, thiscan be analyzed as a finite-dimensional case, where boththe input and the output aren-dimensional vectors.

VI. A LL IN THE FAMILY

We now consider particular frame families. The first two,harmonic tight frames and equiangular frames are purely finitedimensional, while the rest are, in general, infinite dimen-sional. For some of the families, we will consider the UNTFversion and give the frame boundA yielding the redundancy ofthe frame family. We will denote byAj the redundancy/framebound at levelj when iterated filter banks are used.

A. Harmonic Tight Frames and Variations

Harmonic tight frames (HTF)are obtained by seeding fromΨ = DFTm given in (10)-(12), by deleting the last(m − n)columns:

ϕi =

√m

n(W 0

m,Wim, . . . ,W

i(n−1)m ), (35)

for i = 0, . . . ,m − 1. Since obtained as an instance of theNaimark Theorem, this is thus a PTF, that is,ΦΦ∗ = I. Thesimplest example of an HTF is the MB frame given in Box I.In [31], the authors define a more general version of the HTF,called general harmonic frames. They also show that the HTFs


are unique up to a permutation of the orthonormal basis andthat every general harmonic frame is unitarily equivalent to asimple variation of an HTF.

HTFs have a number of interesting properties: (a) Form =n+1, all ENTFs are unitarily equivalent to it; in other words,since we have HTFs for alln,m, we have all ENTFs form = n+1. (2) It is the only ENPTF such that its elements aregenerated by a group of unitary operators with one generator.(3) HTFs are maximally robust (MR) to erasures.

These frames have been generalized in an exhaustive workby Vale and Waldron [118], where the authors look at frameswith symmetries. Some of these they term HTFs (their def-inition is more general than what is given in (35)), and arethe result of the operation of a unitaryU on a finite AbeliangroupG. WhenG is cyclic, the resulting frames arecyclic.In [31], the HTFs we showed above are withU = I andgeneralized HTFs are withU = D diagonal. These are cyclicin the parlance of [118]. An example of a cyclic frame are(n+1) vertices of a regular simplex inRn. There exist HTFswhich are not cyclic.

Similar ideas have appeared in the work by Eldar andBolcskei [58] under the namegeometrically uniform frames(GU), frames defined over a finite Abelian group of unitarymatrices both with a single generator as well as multiplegenerators. The authors also consider constructions of suchframes from given frames, closest in the least-squares sense,a sort of a “Gram-Schmidt” procedure for GU frames.

B. Grassmanian Packings and Equiangular Frames

Equiangular (referring to|〈ϕi, ϕj〉| = const.), frame fami-lies have become popular recently due to their use in quantumcomputing, where rank-1 measurements are represented bypositive operator valued measures (POVMs). Each rank-1POVM is a tight frame.

The first family is symmetric informationally completePOVMs (SIC-POVMs)[102]. A SIC-POVM is a familyΦ ofm = n2 vectors inCn such that

|〈ϕi, ϕj〉|2 =1

n+ 1(36)

holds for all i, j, i 6= j. From the measurement statistics ofsuch a POVM, it is known that the density matrix can bereconstructed completely. Even without the restriction ontheinner products, it is known that if an arbitrary state can bereconstructed, thenΦ forms a frame. At this point, it is notknown whether SIC-POVMs exist for all finite dimensions.

The second family aremutually unbiased bases (MUBs). AMUB is a family Φ of m = n(n+ 1) vectors inCn such that

|〈ϕi, ϕj〉|2 =1

n, (37)

or 0 (these are(n+ 1) ONBs).Both harmonic tight frames and equiangular frames have

strong connections toGrassmanian frames. In a comprehen-sive paper [113], Strohmer and Heath discuss those frames andtheir connection to Grassmanian packings, spherical codes,graph theory, Welch Bound sequences (see also [79]). Theseframes are of unit norm (not a necessary restriction) and

ϕ2 = h

ϕ1 = g ∗ (↑ 2)h

ϕ0 = g ∗ (↑ 2)g

+x

Fig. 8. The synthesis part of the filter bank implementing thea trousalgorithm. The analysis part is analogous. This is equivalent to Fig. 5 withsampling removed.

minimize the maximum angle (correlation)|〈ϕi, ϕj〉| amongall frames. The problem arises from looking at overcompletesystems closest to orthonormal bases (which have minimumcorrelation). A simple example is an HTF inHn. Theorem 2.3in [113] states that, given a frameΦ:

minΦ

( max(ϕi,ϕj)

|〈ϕi, ϕj〉|) ≥√

m− n

n(m− 1). (38)

The equality in (38) is achieved if and only ifΦ is equiangularand tight. In particular, forH = R, equality is possible only form ≤ n(n+ 1)/2, while for H = C, equality is possible onlyfor m ≤ n2. Note that the above inequality is exactly the oneWelch proved in [125] and which later lead to what is todaycommonly referred to as theWelch’s Boundgiven in (41) byminimizing interuser interference in a CDMA system [98] (seethe discussion on the Welch’s Bound in Section VII-D). In amore recent work, Xia, Zhou and Giannakis [127] constructedsome new frames meeting the original Welch’s Bound (41).

These frames coincide with some optimal packings inGrassmanian spaces [38], spherical codes [39], equiangularlines [94], and many others. The equiangular lines are equiv-alent to the SIC-POVMs we discussed above.

C. The Algorithmea Trous

The algorithme a trous is a fast implementation of the dyadic(continuous) wavelet transform. It was first introduced byHolschneider, Kronland-Martinet, Morlet, and Tchamitchian in1989 [80]. The transform is implemented via a biorthogonal,nondownsampled filter bank. An example forj = 2 levels isgiven in Fig. 8 (this is essentially the same as the2-level DWTas in Fig. 5 with samplers removed).

Let g andh be the filters used in this filter bank. At leveli we will have equivalent upsampling by2i which means thatthe filter moved across the upsampler will be upsampled by2i, inserting(2i − 1) zeros between every two samples andthus creating holes (“trous” means “hole” in French).

The bottom plot in Fig. 12 shows the sampling grid for the atrous algorithm. It is clear from the figure, that this schemeiscompletely redundant, as all the points exist. This is in contrastto a completely nonredundant scheme such as the DWT, givenin the top plot of the figure. In fact, the redundancy (orframe bound) of this algorithm grows exponentially sinceA1 = 2, A2 = 4, . . . , Aj = 2j, . . . (note that here we use atwo-channel filter bank and thatAj is the frame bound whenwe usej levels). This growing redundancy is the price we pay


for redundancy as well as the simplicity of the algorithm. The2D version of the algorithm is obtained by extending the 1Dversion in a separable manner.

D. Gabor and Cosine-Modulated Frames

The idea behind this class of frames, consisting of manyfamilies, dates back to Gabor [67] and the insight of con-structing bases by modulation of a single prototype func-tion. Gabor originally used complex modulation, and thus,all those families with complex modulation will be termedGabor frames. Other types of modulation are possible, suchas cosine modulation, and again, all those families with cosinemodulation will be termedcosine-modulated frames8. Theconnection between these two classes is deep as there existsageneral decomposition of the frame operator correspondingtoa cosine-modulated filter bank as the sum of the frame operatorof the underlying Gabor frame (with the same generatorand twice the redundancy) and an additional operator, whichvanishes if the generator satisfies certain symmetry properties.While this decomposition has first been used by Auscher in thecontext of Wilson bases [5], it is valid more generally. Bothofthese classes can be seen as general oversampled filter bankswith m channels and sampling byn (see Fig. 7).

1) Gabor Frames:A Gabor frame isΦ = {ϕi}m−1i=0 , with

ϕi,k = W−ikm ϕ0,k, (39)

where indexi = 0, . . . ,m− 1 refers to the number of frameelements,k ∈ Z is the discrete-time index,Wm is themth rootof unity andϕ0 is the template frame function. Comparing(39) with (35), we see that for filter lengthl = n andϕ0,k =1, k = 0 and 0 otherwise, the Gabor system is equivalent toa HTF frame. Thus, it is sometimes called theoversampledDFT frame.

For the critically-sampled case it is known that one cannothave Gabor bases with good time and frequency localizationat the same time (this is similar in spirit to the Balian-Low theorem which holds forL2(R) [47]); this promptedthe development of oversampled (redundant) Gabor systems(frames). They are known under various names:oversampledDFT FBs, complex-modulated FBs, short-time Fourier FBsand Gabor FBs and have been studied in [42], [23], [21],[20], [63] (see also [112] and references within). More recentwork includes [93], where the authors study finite-dimensionalGabor systems and show a family inC

n, withm = n2 vectors,which allows forn2 − n erasures, wheren is prime. In [90],new classes of Gabor ENTFs are shown, which are also MR.

2) Cosine-Modulated Frames:The other kind of modula-tion, cosine, was used with great success within critically-sampled filter banks due to efficient implementation algo-rithms. Its oversampled version was introduced in [21], wherethe authors define the frame elements as:

ϕi,k =√

2 cos((i+ 1/2)π

m+ αi)ϕ0,k, (40)

where indexi = 0, . . . ,m− 1 refers to the number of frameelements,k ∈ Z is the discrete-time index andϕ0 is the

8Cosine-modulated bases are also often called Wilson bases.

x

Analysis DWT i

Analysis DWTr

Synthesis DWTi

Synthesis DWTr

+x

Fig. 9. The filter bank implementing theCWT. The two branches have twodifferent two-channel FBs as in Fig. 3. The analysis part is analogous.

template frame function. Equation (40) defines the so-calledodd-stacked cosine modulated FBs; even-stacked ones exist aswell.

Cosine-modulated filter banks do not suffer from time-frequency localization problems, given by a general resultstating that the generating window of an orthogonal cosinemodulated FB can be obtained by constructing a tight complexfilter bank with oversampling factor 2 while making sure thewindow function satisfies a certain symmetry property (formore details, see [21]). Since we can get well-localized Gaborframes for redundancy 2, this also shows that we can get well-localized cosine-modulated filter banks.

E. The Dual-Tree Complex Wavelet Transform

The dual-tree complex wavelet transform (DT-CWT) wasfirst introduced by Kingsbury in 1998 [85], [86], [87]. Thebasic idea is to have two DWT trees working in parallel. Onetree represents the real part of the complex transform whilethe second tree represents the imaginary part. That is, whenthe DT-CWT is applied to a real signal, the output of thefirst tree is the real part of the complex transform whereas theoutput of the second tree is its imaginary part. Shown in Fig.9is the synthesis filter bank for the DT-CWT. Each tree usesa different pair of lowpass and highpass filters. These filtersare designed so that they satisfy the perfect reconstructioncondition (32).

Let Φr andΦi be the square matrices representing each ofthe DWTs in the DT-CWT. Then,

Φ =1√2

(Φr Φi

),

is a rectangular matrix, and thus a frame, representing theDT-CWT9. The right inverse ofΦ is the analysis FB (analysisoperator) and is given byΦ∗ = 1/

√2[(Φr)

−1(Φi)−1]T . If Φr

andΦi are unitary matrices, thenΦ = Φ, ΦΦ∗ = I, andΦ isa PTF.

Because the two DWT trees used in the DT-CWT are fullydownsampled, the redundancy is only2 for the 1D case (it is2d for the d-dimensional case). We can see that in the thirdplot in Fig. 12, where the redundancy at each level is twicethat of the DWT, that isA1 = A2 = . . . Aj = 2. Unlike the atrous algorithm, however, here the redundancy is independentof the number of levels used in the transform.

9The indicesr and i stem from real and imaginary.


When the two DWTs used are orthonormal, the DT-CWT isa tight frame. The DT-CWT overcomes one of the main draw-backs of the DWT: shift variance. Since the DT-CWT containstwo fully downsampled DWTs which satisfy the half-sampledelay condition (see below), aliasing due to downsamplingcan be largely eliminated and the transform thus becomesnearly shift invariant. One advantage that the DT-CWT hasover other complex transforms is that it has a fast invertibleimplementation. Moreover, when the signal is real valued, thereal and imaginary parts of its transform coefficients can becomputed and stored separately.

As mentioned previously, the pairs of filters(hr, gr) and(hi, gi) of each DWT have to satisfy the perfect reconstructioncondition. In addition, the filters have to be FIR and satisfythe so-calledhalf sample delay condition, which implies thatall of the filters have to be designed simultaneously. From thiscondition it also follows that the two highpass filters form anapproximate Hilbert transform pair, and it thus makes sensetoregard the outputs of the two trees as the real and imaginaryparts of complex functions. Different design solutions exist,amongst them the linear phase biorthogonal one and thequarter-shift one [87], [106]. Moreover, we can use different-flavor trees to implement the DT-CWT. For example, it ispossible to use a different pair of filters at each level, oralternate filters between the trees at each stage except for thefirst one.

In 2D (or mD), the DT-CWT possesses directional selec-tivity allowing us to capture edge or curve information, aproperty clearly absent from the usual separable DWT. In thereal case, orientation selectivity is simply achieved by usingtwo real separable 2D DWTs in parallel. Two pairs of filters areused to implement each DWT. These two transforms producesix subbands, three pairs of subbands from the same space-frequency region. By taking the sums and differences of eachpair, one obtains the oriented wavelet transform.

The near shift invariance and orientation selectivity proper-ties of the DT-CWT open up a window into a wide rangeof applications, among them denoising, motion estimation,image segmentation as well as building feature, texture andobject detectors for images (see Section VII-F and referencestherein).

F. Double-Density Frames and Variations

The DT-CWT appears to be the most notable among theoversampled FB transforms. It is joined by a host of others:In particular, Selesnick in [104] introduces thedouble-densityDWT (DD-DWT), which can approximately be implementedusing a three-channel FB with sampling by2 as in Fig. 6.The filters in the analysis bank are time-reversed versions ofthose in the synthesis bank. The redundancy of this FB tendstowards2 when iterated on the channel withϕ1. Actually,we have thatA1 = 3

2 , A2 = 74 , . . . A∞ = 2 (see second plot

in Fig. 12). Like the DT-CWT, the DD-DWT is nearly shiftinvariant when compared to the a trous construction. In [105],Selesnick introduces the combination of the DD-DWT and theDT-CWT which he callsdouble-density, DT-CWT (DD-DT-CWT). This transform can be seen as the one in Fig. 9 (DT-CWT), with individual filter banks being overcomplete ones

2

2

h

g

level j + b b

h

g

level 1 +x

Fig. 10. The synthesis part of the filter bank implementing the power-shiftableDWT. The samplers are omitted at the first level but exist at all other levels.The analysis part is analogous.

given in Fig. 6 (DD-DWT). In [1], Abdelnour and Selesnickintroduce symmetric, nearly shift-invariant FBs implementingtight frames. These filter banks have 4 filters in two couples,obtained from each other by modulation. Sampling is by 2 andthus the total redundancy is 2.

Another variation on a theme is thepower-shiftable DWT(PSDWT)[109] orpartial DWT (PDWT)[111], which removessamplers at the first level but leaves them at all other levels(seeFig. 10). The sampling grid of the PSDWT/PDWT is shown inthe third plot in Fig. 12. We see that is has redundancyAj = 2at each level (similarly to theCWT). The PSDWT/PDWTachieves near shift invariance.

Bradley in [25] introducesovercomplete DWT (OC-DWT),the DWT with critical sampling for the firstk levels followedby a trous for the lastj − k levels. The OC-DWT becomesthe a trous algorithm whenk = 0 or the DWT whenk = j.

G. Multidimensional Frames

Apart from obvious, tensor-like, constructions (separateapplication of 1D methods in each dimension) of multidimen-sional (mD) frames, we are interested in true mD solutions.The oldest mD frame seems to be thesteerable pyramidintroduced by Simoncelli, Freeman, Adelson and Heeger in1992 [109], following on the previous work by Burt andAdelson on pyramid coding (see Box II and [26]). Thesteerable pyramid possesses many nice properties, such asjoint space-frequency localization, approximate shift invari-ance, approximate tightness, oriented kernels, approximaterotation invariance and a redundancy factor of4j/3 wherej is the number of orientation subbands. The transform isimplemented by a first stage of lowpass/highpass filteringfollowed by oriented bandpass filters in the lowpass branchplus another lowpass filter in the same branch followed bydownsampling. An excellent overview of the steerable pyramidand its applications is given on Simoncelli’s web page [108].

Another beautiful example is the recent work of Do andVetterli on contourlets[53], [41]. This work was motivatedby the need to construct efficient and sparse representations ofintrinsic geometric structure of information within an image.The authors combine the ideas of pyramid coding (see Box II)and pyramid filter banks [52] with directional processing, toobtain contourlets, expansions capturing contour segments.The transform is a frame composed of a pyramid FB and


m

m

ϕm−1

ϕ0

b

b

b

+

Pyramid

filter bank +x

Fig. 11. The synthesis part of the pyramid directional filterbank (PDFB).The pyramid FB is given in Box II. The scheme can be iterated and theanalysis part is analogous.

a directional FB. Thus, first a wavelet-like method is usedfor edge detection (pyramid) followed by local directionaltransform for contour segment detection. It is almost criticallysampled, with redundancy of1.33. It draws on the ideasof a pyramidal directional filter banks (PDFB)which is aParseval tight frame when all the filters used are orthogonal(see Fig. 11).

Some other examples include [95] where the authors buildboth critically-sampled and nonsampled (a trous like) 2DDWT. It is obtained by a separable 2D DWT producing4 subbands. The lowest subband is left as is, while thethree higher ones are split into two subbands each usinga quincunx FB (checkerboard sampling). The resulting FBpossesses good directionality with low redundancy. Many ”-lets” are also multidimensional frames, such ascurvelets[28],[27] and shearlets[91]. As the name implies, curvelets areused to approximate curved singularities in an efficient man-ner [28], [27]. As opposed to wavelets which use dilationand translation, shearlets use dilation, shear transformation andtranslation, and possess useful properties such as directionality,elongated shapes and many others [91].

H. Discussion and Notes

While in this section we aimed to present a comprehensiveoverview of frame families implementable by filter banks,omissions are probable. We note here some other develop-ments, which, while not necessarily yet in the realm of filterbank frames, are related to them nevertheless.

For example, Casazza and Leonhard keep a tab on all equal-norm Parseval frames in [32]. In [70], [8], [9], the authorsintroduce the notion oflocalized frames, as an important newdirection in frame theory, with possible filter bank instantia-tions in the future.

VII. A PPLICATIONS

We now present a glimpse at application domains whereframes have been used with success. As with the previousmaterial, we make no attempt to be exhaustive; we merely givea representative sample. These applications illustrate whichbasic properties of frames have found use in the real world.In some of these, frames have been used deliberately; byconsidering the requirements posed by applications, framesemerged as a natural choice. In others, only later have webecome aware that the tools used were actually frames.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

12345

leve

l

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

12345

leve

l

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

12345

leve

l

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

12345

k

leve

l

Fig. 12. Sampling grids corresponding to time-frequency tilings of(top to bottom): DWT (nonredundant), DD-DWT/Laplacian pyramid, DT-CWT/PSDWT/PDWT, a trous family (completely redundant). Black dotscorrespond to the nonredundant (DWT-like) sampling grid. Red dots denoteredundant points.

A. Quantization

An area where frames were immediately recognized as natu-ral signal expansions was that of quantization. The successwasmotivated by the fact that frames show resilience to additivenoise as well as numerical stability of reconstruction [47]. Westart by illustrating resilience to noise. Intuitively, ifa certainamount of noise is present, distributed over the transformcoefficients (inner products), then it stands to reason thatwhenthere arem coefficients (frame) as opposed ton (m > n,bases), it is easier to deal with that lower level of noise percoefficient.

Example:[Mercedes-Benz Frame] We go back to our MBframe and consider its UNTF version given in (42). Supposewe perturb our frame coefficients by adding white noisewi

to the channeli, whereE[wi] = 0, E[wiwk] = σ2δik fori, k = 1, 2, 3. We can now find the error of the reconstruction,

x− x =2

3

3∑

i=1

〈ϕi, x〉ϕi − 2

3

3∑

i=1

(〈ϕi, x〉 + wi)ϕi

= −2

3

3∑

i=1

wiϕi.

Then the averaged mean-squared error per component is

MSE =1

2E ‖x− x‖2 =

1

2E

∥∥∥∥

2

3

3∑

i=1

wiϕi

∥∥∥∥

2

=1

2σ2 4

9

3∑

i=1

‖ϕi‖2=

2

3σ2,

since all the frame vectors have norm1. Compare this to thesame MSE obtained with an ONB:σ2. In other words:

MSEONB =3

2MSEMB ,


that is, the amount of error per component has been reducedusing a frame.�

Frames are thus generally considered to be robust underadditive noise [47], [99], [15], and more generally quanti-zation [69], [43], [48], [72], [14], and have been used withsuccess in the context of sigma-delta quantization [22], [71],[48].

B. Denoising

Denoising with wavelets can be traced back to the workby Weaver et al. [124] (and even earlier to Witkin [126]),and was later on popularized by Donoho and Johnstone [55],[54]. Even then, sophisticated use of overcomplete expansionsshowed excellent results, and thus one of the first works ondenoising with frames is [128], where the authors combinedthe overcomplete expansion with a variation of the techniquefrom [97] to reconstruct the image from its wavelet maxima.

More recent works include cycle spinning introduced byCoifman and Donoho [36]. They suggest that when usinga j-stage wavelet transform, one can take advantage of thefact that there are effectively2j different wavelet bases, eachone corresponding to one of the2j shifts. Thus one candenoise in each of those2j wavelet bases and then average theresult. Even though errors of individual estimates are generallypositively correlated, one gets an advantage from averagingthe estimators. Another effect of this is that the shift-varyingbasis gives way to a shift-invariant frame (collection of bases).In [56], Dragotti et al. construct separable multidimensionaldirectional frames for image denoising. The algorithm is inspirit similar to cycle spinning.

Ishwar and Moulin take a slightly different approach todevelop a general framework for image modeling and es-timation by fusing deterministic and statistical informationfrom subband coefficients in multiple wavelet bases usingmaximum-entropy and set-theoretic ideas [84], [81], [82],[83].For instance, in [84] natural images are modeled as havingsparse representations simultaneously in multiple orthonormalwavelet bases. Closed convex confidence tubes are designedaround the wavelet coefficients of sparse initial estimatesinmultiple wavelet bases (frames). A POCS algorithm is thenused to arrive at a globally consistent sparse signal estimate.Denoising and restoration algorithms based on these imagemodels produced visually sharper estimates with about 1-2dB PSNR gains over competitive denoising algorithms suchas the spatially adaptive Wiener filter.

Some other works include that by Fletcher et al. [65], wherethe authors analyze denoising by sparse approximation withframes. The known apriori information about the signalx isthat it has known sparsityk, that is, it can be represented viaknonzero frame coefficients (with respect to a given frameΦ).Then, after having been corrupted by noise yieldingX, thesignal can be estimated by finding the best sparse approxima-tion of X. This work is essentially an attempt to understandhow large a frame should be for denoising with a frame to beeffective. In [34], the authors use the shift-invariant propertiesof the DT-CWT to provide better persistence across scalewithin the Hidden Markov tree, and hence better denoising

Fig. 13. Denoising with frames. From left to right: (1) Lena with 34.0dB white Gaussian noise. (2) Denoised Lena with 25.4 dB noise, using asoft threshold in a single basis. (3) Denoised Lena with 24.2dB noise, usingcycle spinning (frame) from [36]. (4) Denoised Lena with 23.2 dB noise, usingdifferential cycle spinning (frame). The technique used here is an extensionof the work in [64] (figure courtesy of Vivek Goyal).

performance, while in [100], the steerable pyramid is used(see Section VI-G). An example of denoising by frames isgiven in Fig. 13 (courtesy of Vivek Goyal).

C. Robust Transmission

Another application where frames found a natural homewas that of robust transmission in communications. It waspioneered by Goyal, Kovacevic and Kelner in [68], and wasfollowed by works in [79], [113], [16], [17], [18], [119], [31],[19], [101], [90]. The problem was that of creating multipledescriptions of the source so that when transmitted, and in thepresence of losses, the source could be reconstructed basedon received material. This clearly means that some amount ofredundancy needs to be present in the system, since, if not,the loss of even one description would be irreversible.

In the initial work, the Rn–valued source vectorx isrepresented through a frame expansion with frame a operatorΦ∗, yielding X = Φ∗x ∈ Rm. The scalar quantization ofthe frame expansion coefficients givesX lying in a discretesubset ofRm. One abstracts the effect of the network to bethe erasure of some components ofX. This implies that thecomponents ofX are placed in more than one packet, forotherwise all ofX could be lost at once. If they are placed inmseparate packets, then any subset of the components ofX maybe received; otherwise only certain subsets will be possible.The authors assume that linear reconstruction is used, thatis,the dual frame is used to reconstruct. The authors model thenoise as additiveη = X − X as in Section VII-A withthe assumptions that every noise component is of zero meanand varianceσ2 and that they are uncorrelated. The canonicaldual frame operator (20) is used as it minimizes the error of


reconstruction. Losses in the network are modeled as erasuresof a subset of quantized frame coefficients; to the decoder, itappears as if a quantized frame expansion were computed withthe frame missing the elements which produced the erasedones, and thus, assuming it is a frame, a dual frame can befound. As a result, the authors concentrated on questions suchas which deletions still leave a frame, which are the framesremaining frames under deletions of any subset of elements(up tom−n), etc. An example of this discussion is given forthe MB frame in Box I.

D. CDMA Systems

The use of frames in CDMA systems dates back to Masseyand Mittelholzer [98] on Welch’s Bound and sequence sets forCDMA systems.

In a CDMA system, there arem users who share theavailable spectrum. The sharing is achieved by “scrambling”m-dimensionaluser vectorsinto smaller,n-dimensional vec-tors. In terms of frame theory, this scrambling correspondsto the application of a synthesis operator corresponding tom distinct n-dimensionalsignaturevectorsϕi of length

√n.

Noise-corrupted versions of these synthesized vectors arriveat a receiver, where the signature vectors are used to helpextract the original user vectors. The variance of the interuserinterference for useri is:

σ2i =

m∑

j=1

|〈ϕi, ϕj〉|2 − n2,

leading to the total interuser interference:

σ2tot =

m∑

i,j=1

|〈ϕi, ϕj〉|2 −mn2 = FP ({ϕi}mi=1) −mn2.

In the above, we recognize the frame potential from (24). Thegoal is to minimize the interferences and make them equal.

It is obvious that no interference is possible if and only if allϕi are orthogonal, and in turn, this is possible only ifm ≤ n,or, whenϕi either form an orthogonal set or an orthonormalbasis. Whenm > n, FP−mn2 ≥ FP−m2n and the result isknown asWelch’s Bound10: The sequences all have the samenorm and

m∑

i,j=1

|〈ϕi, ϕj〉|2 ≥ m2n, (41)

with equality if and only if them×nmatrixΦ∗ whose rows areϕ∗

i has orthogonal columns of norm√n. If we normalize every

vector to be unit norm, we can immediately translate the aboveinto frame parlance (see Theorem 2): (a) Welch’s Bound isequivalent to the frame potential inequality. (b) Frame potentialis minimized at tight frames. (c)m×n matrix is the analysisoperatorΦ∗. (d) Columns of the analysis operator of a tightframe are orthogonal (consequence of the Naimark Theorem).

10The question of which expression is the actually Welch’s Bound fre-quently leads to confusion. In his original paper [125], Welch found the lowerbound on the maximum value of the cross-correlation of complex sequences,given in (38). In 1992, Massey and Mittelholzer [98] rephrased it in terms ofthe bound on the maximum user interference as given in (41).

This work was followed by many others, among those,by Viswanath and Anantharam’s [122] discovery of the Fun-damental Inequality (26) during their investigation of thecapacity region in synchronous CDMA systems. The authorsshowed that the design of the optimal signature matrixSdepends upon the powers{pi = ‖ϕi‖2}m

i=1 of the individualusers. In particular, they divided the users into two classes:those that areoversizedand those that are not. While theoversized users are assigned orthogonal channels for theirpersonal use, the remaining users have their signature vectorsdesigned so as to be Welch Bound Equality (WBE) sequences,namely, sequences which achieve the lower bound for theframe potential, and are thus tight frames (see Theorem 2).

When no user is oversized, that is, when the FundamentalInequality is satisfied, their problem reduces to finding a tightframe for H with norms {√pi}m

i=1. The authors gave onesolution to the problem using an explicit construction; char-acterization of all solutions to this problem using a physicalinterpretation of frame theory was given in Theorem 3 [30],Section IV.

The equivalence between UNTFs and Welch Bound se-quences was shown in [113]. Waldron formalized that equiva-lence for general tight frames in [123], and consequently, tightframes have been referred in some works as Welch Boundsequences [115].

E. Multiantenna Code Design

An important application of equal-norm Parseval tightframes is in multiple-antenna code design [74]. Much theoret-ical work has been done to show that communication systemswhich employ multiple antennas can have very high channelcapacities [66], [114]. These methods rely on the assumptionthat the receiver knows the complex-valued Rayleigh fadingcoefficients. To remove this assumption, in [78] new classesof unitary space-time signals are proposed. If we haventransmitting antennas and we transmit in blocks ofm timesamples (over which the fading coefficients are approximatelyconstant), then aconstellation ofK unitary space-time signalsis a (weighted by

√m) collection ofn×m complex matrices

{Φk} for which ΦkΦ∗k = I, a PTF in other words. Theith

row of anyΦk contains the signal transmitted on antennai asa function of time. The only structure required in general isthe time-orthogonality of the signals.

Originally it was believed that designing such constellationswas a too cumbersome and difficult optimization problem forpractice. However, in [78], it was shown that constellationsarising in a “systematic” fashion can be done with relativelylittle effort. Systematic here means that we need to designhigh-rate space-time constellations with low encoding anddecoding complexity. It is known that full transmitter diversity(that is, where the constellation is a set of unitary matriceswhose differences have nonzero determinant) is a desirableproperty for good performance. In a tour-de-force, in [74],theauthors used fixed-point-free groups and their representationsto design high-rate constellations with full diversity. Moreover,they classified all full-diversity constellations that form agroup, for all rates and numbers of transmitting antennas.


F. Other Applications

We now just briefly touch upon a host of other applica-tions from standard to fairly esoteric ones such as quantumteleportation.

Although unintuitive, frames were used for compressionin the 1980s (unintuitive since frames are redundant and thewhole purpose of compression is to remove redundancy). Burtand Adelson proposed pyramid coding of images [26] whichused redundant linear transforms and was quite successful fora while (see Box II).

If one considers the segmentation problem as classificationinto object and background, the work of [116], [92] then usesframes for segmentation. In a more recent work, de Rivazand Kingsbury use the the complex wavelet transform (seeSection VI-E) to formulate the energy terms for the level-setbased active contour segmentation approach [50]. They usea limited redundancy transform with a fast implementation.Both Laine and Unser used frames to decompose textures inorder to characterize them across scales [116], [92]. In [33] theauthors use frames to significantly improve the classificationaccuracy of protein subcellular location images, to close to96%.

Regularized inversion problems such as deblurring in noisecan also greatly benefit from the ability of redundant framesto provide signal models that allow Bayesian regularizationconstraints to be applied efficiently to complicated signals suchas images, as illustrated in [51].

Another application of frames has been in signal reconstruc-tion from nonuniform samples, see [3], [10], [62] and refer-ences therein. Benedetto and Pfander used redundant wavelettransforms (frames) to predict epileptic seizures [12], [13].Kingsbury used his complex wavelet transform for restorationand enhancement [85], motion estimation [96] as well asbuilding feature, texture and object detectors for images [88],[4], [61]. Balan, Casazza and Edidin used frames for signalreconstruction without noisy phase within speech recognitionproblems [6]. Many connections have been made betweenframes and coding theory [75], [103].

Recently, certain quantum measurement results have beenrecast in terms of frames [59], [110]. They have applicationsin quantum computing and have to do with positive operatorvalued measures (POVMs). The SIC-POVMs as well as mu-tually unbiased bases were discussed in Section VI-B. Whoknows, maybe Star Trek comes to life, and frames play a rolein quantum teleportation [24]!

VIII. C ONCLUSIONS

Coming to the end of this article, we hope you have adifferent picture of a frame in your mind from a “pictureframe”. While necessarily colored by our personal bias, weintended this tutorial as a basic introduction to frames, gearedprimarily toward engineering students and those without ex-tensive mathematical training. Frames are here to stay; aswavelet bases before them, they are becoming a standard toolin the signal processing toolbox, spurred by a host of recentapplications requiring some level of redundancy. We hope thisarticle will be of help when deciding whether frames are theright tool for your application.

ACKNOWLEDGMENTS

We gratefully acknowledge comments from RiccardoBernardini, Bernhard Bodmann, Helmut Bolcskei, PeteCasazza, Minh Do, Matt Fickus, Vivek Goyal, Chris Heil,Prakash Ishwar, Nick Kingsbury, Gotz Pfander, EminaSoljanin, Thomas Strohmer and Martin Vetterli; their collec-tive wisdom improved the original manuscript significantly.In particular, Chris Heil’s help was invaluable in making thepaper mathematically tight and precise. We are also indebtedto Vivek Goyal for providing us with Fig. 13.

BOX I: THE MERCEDES-BENZ FRAME

The Mercedes-Benz (MB)frame11 is arguably the mostfamous frame. It is a collectionΦ of three vectors inR2,and is an excellent representative for many classes of frames.For example, the MB frames is the simplest HTF frame (seeSection VI-A), and as such, it is also unitarily equivalent toall ENTF’s in R2 with three vectors. Again, as an HTF frame,it can be obtained by a group operation on a single element(choose one frame vector and use rotations of2π/3).

The two equivalent versions of the MB frame are:1) UNTF Version:The unit-norm version of the MB frame

is ΦUNTF = {ϕ1, ϕ2, ϕ3} (see Fig. 14):

Φ∗UNTF =

0 1

−√

3/2 −1/2√3/2 −1/2

=

ϕ∗1

ϕ∗2

ϕ∗3,

, (42)

with the corresponding expansion:

x =2

3

3∑

i=1

〈ϕi, x〉ϕi =2

3ΦUNTFΦ∗

UNTF, (43)

and the norm:

‖X‖2 =3∑

i=1

|〈ϕi, x〉|2 =3

2‖x‖2. (44)

2) PTF Version: The Parseval tight frame version of theMB frame is ΦPTF = {ϕ1, ϕ2, ϕ3}, where the frame hasbeen scaled so thatΦPTFΦ∗

PTF = I:

Φ∗PTF =

√

2

3Φ∗

UNTF =

0√

2/3

−1/√

2 −1/√

6

1/√

2 −1/√

6

, (45)

and thus the expansion expression is

x =

3∑

i=1

〈ϕi, x〉ϕi = ΦPTFΦ∗PTFx, (46)

and the norm:

‖X‖2 =3∑

i=1

|〈ϕi, x〉|2 = ‖x‖2. (47)

11MB frames are also known as Peres-Wooters states in quantum informa-tion theory [107].


1

− 1

2

−√

3

2

√3

2

ϕ1 = (0, 1)T

ϕ2 = (−√

32 ,

12 )T ϕ3 = (−

√3

2 ,12 )T

Fig. 14. Simplest unit-norm tight frame—Mercedes Benz frame (MB). Thisis also an example of a harmonic tight frame.

3) Seeding: The PTF version of the MB frame can beobtained by projecting an ONB from a three-dimensionalspace (see Naimark Theorem 1):

Ψ =

0√

2/3 1/√

3

−1/√

2 −1/√

6 1/√

3

1/√

2 −1/√

6 1/√

3

, (48)

using the following projection operatorP :

P =1√3

2/3 −1/3 −1/3−1/3 2/3 −1/3−1/3 −1/3 2/3

, (49)

that is, the MB frame seen as a collection of vectors inthe three-dimensional space isΦ3D = PΨ. The projectionoperator essentially “deletes” the last column ofΨ to createthe frame operatorΦ∗.

4) Resilience to Noise:In Section VII-A, we looked atthe properties of this frame in the presence of additive noise;we found that by using this frame, MSE per component hasbeen reduced using the MB frame. That result shows anotherparticular property of the MB frame. Namely, among all otherframes with three norm-1 frame vectors inR2, this particularone (and the others in the same class [68]) minimizes theMSE. With an orthonormal basis, theMSE = σ2, while withthe MB frame frame, theMSE = (2/3)σ2.

5) Resilience to Losses:Assume now that one of thequantized coefficients is lost, for example,X2. Does our MBframe have further nice properties when it comes to losses?Note first, that even withϕ2 not present, we can still useϕ1

andϕ3 to represent any vector inR2. The expansion formulais just not as elegant:

x =∑

i=1,3

〈ϕi, x〉ϕi, (50)

with

ϕ1 =

(1/

√3

1

)

, ϕ3 =

(2/

√3

0

)

. (51)

xg

g

2

2

2 h

−+

+X1

X0

Fig. 15. The analysis part of the pyramid filter bank [26] withorthonormalfilters g andh, corresponding to a tight frame.

Repeating the same calculations as above for theMSE, weget that

MSE{2} =1

2E ‖x− x‖2

=1

2E

∥∥∥∥

∑

i=1,3

wiϕi

∥∥∥∥

2

=1

2σ2

∑

i=1,3

‖ϕi‖2=

4

3σ2,

that is, twice theMSE without erasures. However, the abovecalculations do not tell us anything about whether there isanother frame with a lowerMSE. In fact, given that oneelement is erased, does it really matter what the original framewas? It turns out that it does. In fact, among all frames withthree norm-1 frame vectors inR2, the MSE averaged overall possible erasures of one coefficient is minimized when theoriginal frame is tight [68].

BOX II: PYRAMID CODING

Pyramid coding was introduced in 1983 by Burt and Adel-son [26]. Although redundant, the pyramid coding scheme wasdeveloped for compression of images and was recognized inthe late 1980s as one of the precursors of wavelet octave-band decompositions. The scheme works as follows: First, acoarse approximation is derived (an example of how this couldbe done is in Fig. 15). While in Fig. 15 the intensity of thecoarse approximationX0 is obtained by linear filtering anddownsampling, this need not be so; in fact, one of the powerfulfeatures of the original scheme is that any operator can beused, not necessarily linear. Then, from this coarse version, theoriginal is predicted (in the figure, this is done by upsamplingand filtering) followed by calculating the prediction errorX1.If the prediction is good (which will be the case for mostnatural images which have a lowpass characteristic), the errorwill have a small variance and can thus be well compressed.The process can be iterated on the coarse version. In theabsence of quantization ofX1, the original is obtained bysimply adding back the prediction at the synthesis side.

The pyramid coding scheme is fairly intuitive, thus itssuccess. There are several advantages to pyramid coding: Thequantization error depends only on the last quantizer in theiterated scheme. As we mentioned above, nonlinear operatorscan be used, opening the door to the whole host of possibilities(edge detectors, ...) The redundancy in 2D is only 1.33, farless then the a trous construction, for example. Thanks tothe above, pyramid coding has been recently used togetherwith directional coding to form the basis for nonseparable MDframes called contourlets (see Section VI-G).


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