Digital Image Procesing

The Karhunen-Loeve Transform (KLT) in Image Processing

DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON

The concepts of eigenvalues and eigenvectors are important for

understanding the KL transform.

Eigenvalues and Eigenvectors

:that such in vector nonzero a is there if of eigenvalue

an called isscalar a thendimension ofmatrix a isIf

,

nReC

nnC

.

eigenvalue the to

ingcorrespondmatrix the of reigenvecto an called isvector The Ce

eeC

Vector population

• Consider a population of random vectors of the following form:

• The quantity

of the image i .

• The population may arise from the formation of the above

vectors for different image pixels.

level)(grey value the representmay ix

nx

x

x

x2

1

Example: x vectors could be pixel values

in several spectral bands (channels)

Mean and Covariance Matrix

• The mean vector of the population is defined as:

• The covariance matrix of the population is defined as:

• For M vectors of a random population, where M is large

enough

• Therefore, the array formed by the Walsh matrix is a real symmetric matrix.

It is easily shown that it has orthogonal columns and rows.

Tn

T

nx xExExEmmmxEm }{}{}{}{ 2121

T

xx mxmxEC

M

kkx x

Mm

1

1

Karhunen-Loeve Transform

• Let A be a matrix whose rows are formed from the eigenvectors of the

covariance matrix C of the population.

• They are ordered so that the first row of A is the eigenvector

corresponding to the largest eigenvalue, and the last row the

eigenvector corresponding to the smallest eigenvalue.

• We define the following transform:

• It is called the Karhunen-Loeve transform.

• The above is again equivalent to

• The array formed by the inverse Walsh matrix is identical to the one

formed by the forward Walsh matrix apart from a multiplicative factor N.

xmxAy

Karhunen-Loeve Transform

• You can demonstrate very easily that:

n

y

Txy

C

AACC

yE

00

00

00

0

2

1

is ˆ vector original theoftion reconstruc The

l.dimensiona be then would vectors The

. size

ofmatrix ation transforma yielding s,eigenvaluelargest theto

correspond which rseigenvecto thefrom matrix a form We

ingcorrespond its from vectorsoriginal t thereconstruc To

x

Ky

nK

K

KA

yx

K

Inverse Karhunen-Loeve Transform

x

T

T

myAx

AA

1

x

TK myAx ˆ

is ˆ vector original theoftion reconstruc The

l.dimensiona be then would vectors The

. size

ofmatrix ation transforma yielding s,eigenvaluelargest theto

correspond which rseigenvecto thefrom matrix a form We

ingcorrespond its from vectorsoriginal t thereconstruc To

x

Ky

nK

K

KA

yx

K

Inverse Karhunen-Loeve Transform

x

T

T

myAx

AA

1

x

TK myAx ˆ

data. available theofon compresssi

achievecan wetransformKL for the of instead usingBy

expression by thegiven is ˆ

t ionreconstruc eapproximat theand tion reconstrucperfect the

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AA

x

x

K

Mean squared error of approximate reconstruction

n

Kjj

K

jj

n

jjxxe

111

2

msˆ

Drawbacks of the KL Transform

Despite its favourable theoretical properties, the KLT is not

used in practice for the following reasons.

• Its basis functions depend on the covariance matrix of the

image, and hence they have to recomputed and

transmitted for every image.

• Perfect decorrelation is not possible, since images can

rarely be modelled as realisations of ergodic fields.

• There are no fast computational algorithms for its

implementation.

Example: x vectors could be pixel values

in several spectral bands (channels)

Example of the KLT: Original images

(Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2nd Edition.

6 spectral images

from an airborne

Scanner.

(Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2nd Edition.

Example: Principal Components

Component

1 3210

2 931.4

3 118.5

4 83.88

5 64.00

6 13.40

Example: Principal Components (cont.)

Original images (channels) Six principal components

after KL transform

Example: Original Images (left)

and Principal Components (right)