digital image procesing - communications and signal processingtania/teaching/dip 2014/klt.pdf ·...
TRANSCRIPT
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
,
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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.
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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
Karhunen-Loeve Transform
• You can demonstrate very easily that:
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l.dimensiona be then would vectors The
. size
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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
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myAx
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x
TK myAx ˆ
is ˆ vector original theoftion reconstruc The
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correspond which rseigenvecto thefrom matrix a form We
ingcorrespond its from vectorsoriginal t thereconstruc To
x
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Inverse Karhunen-Loeve Transform
x
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myAx
AA
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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.
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)