07-1 - wavelets fundamentals

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Wavelets fundamentals

Spring 2008 ELEN 4304/5365 DIP 1

by Gleb V. Tcheslavski: [email protected]

http://ee.lamar.edu/gleb/dip/index.htm

Preliminaries When looking at images, we generally see connected regions of similar texture and intensity levels combined to formobjects. Small or low-contrast objects are better viewed at high resolution. If small and large objects are present it can be


are present, it can be advantageous to study them at different resolutions.From math viewpoint, images are 2D arrays of intensity values with locally varying statistics that result from different features.

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Image pyramidsAn image pyramid is a collection of decreasing resolution images arranged in the shape of a pyramid.The base of a pyramid is a high resolution imagebeing processed; the apex contains a low-resolution approximation. While moving up, both size and resolution decrease


resolution decrease.

Base level J is of size

where

2 2J J N N⋅ = ⋅

2logJ N=

Image pyramidsThe apex level 0 is of size 1x1. Most pyramids are truncated to P + 1 levels, where 1 ≤ P ≤ J. The total number of pixels in a P + 1 level pyramid is

1 1 1 4⎛ ⎞2 22

1 1 1 41 ...4 4 4 3PN N⎛ ⎞+ + + + ≤⎜ ⎟

⎝ ⎠On the diagram for constructing two image pyramids, the “level j-1 approximation output” provides the images needed to build an approximation pyramid, while the “level j prediction residual output” is used to build a complementary prediction residual pyramid. Unlike


is used to build a complementary prediction residual pyramid. Unlike approximation pyramids, prediction residual pyramids contain only one reduced-resolution approximation of the input image (top of the pyramid, level J-P). All other levels contain prediction residuals where the level j prediction residual (J-P+1 ≤ j ≤ J) is defined as the difference between level j approximation and its estimate.

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Image pyramidsAn estimate of the level j approximation is computed based on the level j-1 approximation.Approximation and prediction residual pyramids can be computed by iterations. The original image is placed in level J of the approximation pyramid. The pyramids then are generated in Piterations for the following steps for j = J, J-1,…, J-P+1:1. Compute a reduced-resolution approximation of the level j image

by filtering and downsampling the filtered image by 2.2. Estimate the level j input image from the reduced-resolution

i i b li b 2 d fil i


approximation by upsampling by 2 and filtering.3. Compute the difference between the output of step 2 and the

input of step 1. place this result in level j of the prediction residual pyramid.

Image pyramids

For a 1D sequence f(n), the upsampled by 2 sequence is

( )2f n if n is even ⎧⎪ ( )2

2( )

0f n if n is even

f notherwise↑

⎧⎪= ⎨ ⎪⎩

The downsampling by 2 is discarding every second sample and is defined as

2 ( ) (2 )f n f n↓ =


Both operations are usually followed by approximation and interpolation filters.

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Image pyramids4-level approximation pyramid: an original 512x512 image and its

i ti tapproximations at 256x256, 128x128, and 64x64. A Gaussian smoothing filter was used.

Prediction residual pyramid A bilinear


pyramid. A bilinear interpolation filter was used.

Subband coding

A 2-band subband coding and decodingcoding and decoding system (analysis and synthesis filter banks)

Magnitude frequency


g q yresponses of the analysis filter bank FIR filters (half-band filters).

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Subband codingFilter h0(n) is a low-pass (half-band) filter, whose output flp(n) is an approximation of the input f(n); h1(n) is a high-pass (half-band) filter, whose output fhp(n) is a high-frequency or detail part of the input f(n). Synthesis filters glp(n) and ghp(n) combine two subband signals to produce ˆ ( )f n

The goal of subband coding is to select filters such that Which is called perfect reconstruction conditions that require

ˆ ( ) ( )f n f n=

0 1( ) ( 1) ( )ng n h n= −


11 0( ) ( 1) ( )ng n h n+= −

10 1

1 0

( ) ( 1) ( )

( ) ( 1) ( )

n

n

g n h n

g n h n

+= −

= −

or

Subband codingFilters must satisfy biorthogonality condition

{ }(2 ), ( ) ( ) ( ); , 0,1i jh n k g k i j n i jδ δ− = − =

inner product

Of special interest are filters satisfying orthonormality condition

{ }( ), ( 2 ) ( ) ( ); , 0,1i jg n g n m i j m i jδ δ+ = − =

which satisfy the conditions

Spring 2008 ELEN 4304/5365 DIP 10

{ }1 0( ) ( 1) ( 1 )( ) ( 1 ), 0,1

neven

i i even

g n g K nh n g K n i

= − − −

= − − =

An orthonormal filter bank can be designed from a single prototypefilter; all other filters are computed from the prototype.

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Subband coding

A 2D 4-band subband coding (analysis) filt b kfilter bank.

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A 2D 4-band subband decoding (synthesis) filter bank will have a reversed structure.

Subband coding

A 4-band approximation

t

A 4-band vertical d t ilcomponent

A 4-band

detail component

A 4-band

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horizontal detail component

diagonal detail component

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Haar transformThe Haar transform can be expressed in the matrix form:

TT = HFHT HFHNxN image

NxN Haar transformation matrixNxN transformed image

The Haar basis functions are

[ ]0 001( ) ( ) , 0,1h z h z zN

= = ∈

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[ ]

2

2

2 ( 1) 2 ( 0.5) 21( ) ( ) 2 ( 0.5) 2 2

0 , 0,1

p p p

p p pk pq

q z qh z h z q z q

Notherwise z

⎧ − ≤ < −⎪

= = − − ≤ <⎨⎪ ∈⎩

Haar transformA 2x2 Haar transformation matrix is:

2

1 11 ⎡ ⎤= ⎢ ⎥H2 1 12 ⎢ ⎥−⎣ ⎦

H

And a 4x4 Haar transformation matrix is:

1 1 1 11 1 1 11

⎡ ⎤⎢ ⎥− −⎢ ⎥H

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4 2 2 0 040 0 2 2

⎢ ⎥= ⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

H

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Haar transformDiscrete wavelet transform (DWT) using Haar functions: average (almost uniform histogram) and detail images having very similar histograms.Note: these figures are not exactly a Haar transform!

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64x64, 128x128, and 256x256 approximations derived from the DWT.

DWT in 1Ddetails

A two-stage analysis filter bank:

approximation

analysis filter-bank: Direct DWT

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Bandwidths

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DWT in 1D

A two-stage synthesis filter-bank: Inverse DWT

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DWT in 2DDWT is a separable transform…One stage 2D analysis bank LPF

LPF

HPF

HPF

HPF

One-stage 2D analysis bank

Two-stage decomposition

LPF

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One-stage 2D synthesis bank

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DWT in 2D

Original One-scaleOriginal image

One scale 2D DWT

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Two-scale 2D DWT

Three-scale 2D DWT

DWT in 2DZeroing the lowest scale approximation component of the DWT and computing IDWT, it is possible to enhance edges of the image.

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DWT in 2DA CT image uniformly corrupted with white noise;

It is possible to attenuate (orIt is possible to attenuate (or completely remove) noise by the thresholding detail coefficients at a selected level.

Hard (zeroing details) or soft thresholding (zeroing details and

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thresholding (zeroing details and scaling the non-zero coefficients toward zero) can be implemented

Information removed is shown on the right.

Wavelet packetsDetail channels can be further split…

Structure and bandwidths of a three-scale full wavelet

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packet analysis tree.

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Wavelet packets

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A fingerprint and its three-scale full wavelet packet decomposition.We see that some channels contains nearly no data!

Wavelet packets

An optimal wavelet packet decomposition

Spring 2008 ELEN 4304/5365 DIP 24

07-1 - wavelets fundamentals

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