Digital Image Procesing - Communications and signal processingtania/teaching/DIP 2014/KLT.pdf ·...
Transcript of Digital Image Procesing - Communications and signal processingtania/teaching/DIP 2014/KLT.pdf ·...
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Digital Image Procesing
The Karhunen-Loeve Transform (KLT) in Image Processing
DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON
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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
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eigenvalue the to
ingcorrespondmatrix the of reigenvecto an called isvector The Ce
eeC
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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
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Example: x vectors could be pixel values
in several spectral bands (channels)
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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
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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
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Karhunen-Loeve Transform
• You can demonstrate very easily that:
n
y
Txy
C
AACC
yE
00
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00
0
2
1
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is ˆ vector original theoftion reconstruc The
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. size
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ingcorrespond its from vectorsoriginal t thereconstruc To
x
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yx
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Inverse Karhunen-Loeve Transform
x
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TK myAx ˆ
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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
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yx
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Inverse Karhunen-Loeve Transform
x
T
T
myAx
AA
1
x
TK myAx ˆ
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Mean squared error of approximate reconstruction
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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.
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Example: x vectors could be pixel values
in several spectral bands (channels)
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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.
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(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
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Example: Principal Components (cont.)
Original images (channels) Six principal components
after KL transform
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Example: Original Images (left)
and Principal Components (right)