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- An introduction to source-coding
- Transform coding
- The optimal orthogonal transform
In this module, we establish that for transform coding, the optimum orthogonal tranform is the Karhunen-Loeve Transform (KLT). Related properties are also discussed.
- Ignoring the effect of transform choice on uniform-quantizer efficiency
γ
y ,
Gain Over PCM suggests that TC reconstruction
error can be minimized by choosing the orthogonal transform
T that minimizes the product of coefficient variances.
(Recall that orthogonal transforms preserve the arithmetic averageof coefficient variances.)
Eigen-analysis of autocorrelation matrices
Say that
R is the
autocorrelation matrix of a
real-valued, wide-sense stationary, discrete time stochastic process.The following properties are often useful:
-
R is symmetric and Toeplitz. (A symmetric matrix obeys
, while a Toeplitz matrix has equal elements on
all diagonals.)
-
R is positive semidefinite or PSD. (PSD means that
for any real-valued
x .)
-
R has an eigen-decomposition
where
V is an orthogonal matrix (
)
whose columns are eigenvectors
of
R :
and
Λ is a diagonal matrix whose elements
are the eigenvalues
of
R :
- The eigenvectors
of
R are real-valued
and non-negative.
- The product of the eigenvectors equals the determinant
(
)
and the sum of the eigenvalues equals the trace(
).
-
The KLT: Using the outer product,
Using
to denote the
diagonal element
of a matrix
A , matrix theory implies
thus minimization of
would occur if equality
could be established above.Say that the eigen-decomposition of the autocorrelation matrix of
,
which we now denote by
R
x , is
for orthogonal eigenvector matrix
V
x and diagonal eigenvalue
matrix
Λ
x .
Then choosing
,
otherwise known as the
Karhunen-Loeve transform (KLT),
results in the desired property:
To summarize:
- the orthogonal transformation matrix
T minimizing
reconstruction error variance has rows equal to the eigenvectorsof the input's
autocorrelation matrix,
- the variances of the optimal-transform outputs
are equal to the eigenvalues of the input
autocorrelation matrix, and
- the optimal-transform outputs
are uncorrelated. (Why? Note the zero-valued off-diagonal elements
of
.)
- Note that the presence of mutually uncorrelated transform coefficients
supports our approach of quantizing each transform output independentlyof the others.
Klt coder
Recall
Example 1 from "Background and Motivation" with Gaussian input having
The eigenvalues of
R
x can be determined from the
characteristic equation
:
The eigenvector
v
0 corresponding to eigenvalue
solves
.
Using the notation
and
,
Similarly,
yields
For orthonormality,
Thus the KLT is given by
.
Using the KLT and optimal bit allocation, the error reduction
relative to PCM is
since
for Gaussian
.
This value equals 0.6 when
, and 0.98 when
(compare to
Figure 2 from "Background and Motivation" ).
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Source:
OpenStax, An introduction to source-coding: quantization, dpcm, transform coding, and sub-band coding. OpenStax CNX. Sep 25, 2009 Download for free at http://cnx.org/content/col11121/1.2
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