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Transform coding
Transform coding is the second approach to exploiting
redundancy by using scalar quantization with linear
preprocessing.
The source samples are collected into a vector that is linearly
transformed (multiplied by a transform matrix) and the
resulting coefficients are scalar quantized.
Notice that each coefficient can be quantized by different
quantizer. The method was introduced in 1956 by Kramer
and Mathews and then popularized in 1962-1963 by
Huang and Schultheiss.
It was developed for coding images and video. More
recently, transform coding has also been widely used in
high-fidelity audio coding.
Transform coding
1x
2x
Nx
T
1y
2y
Ny
1q
2q
Nq
1T
1x
2x
Nx
2y
Ny
1y
Transform coding
A typical discrete linear transform is a decomposition of the
input discrete-time signal over a system of basis functions.
Harmonic functions are often used as basis functions. In this
case coefficients represent intensities of the corresponding
harmonics.
Transformations of this type are called conversion to the
frequency domain.
Let x be an input column vector of dimension then linear Ntransform of can be expressed as follows x
,xy T
where T is NN transform matrix.
Properties of transforms
Usually it is required that transform would have the following
properties:
Localization of the essential part of the signal energy in a
small number of coefficients. After quantization we can
exclude from consideration the least informative coefficients .
Coefficients should be uncorrelated. In this case scalar
quantization followed by symbol-by-symbol variable-length
coding provides close to
Transform should be orthonormal. If this property holds then
the MSE introduced by quantizing the transform coefficients
coincides with MSE in the input vector. Preserves H(D)!
Low computational complexity. It is desirable to use the
separable transforms.
)(DR ).(DH
Properties of transforms
A transform is called the orthonormal if
,*1 TT (6.1)
where T is a matrix with complex elements. If is real T
then (6.1) reduces to the condition .1 TTT (6.2)
(6.2) is equivalent to the following relation for rows of
,iT t Ni ,...,2,1
,ij
T
ji tt
where
ji
jiij
,0
,1 is Kronecker’s delta function.
Properties of transforms
Vectors it are the orthonormal basis vectors of transform .T
For orthonormal transform
,xy T ,1yyx
TTT .
1
N
i
T
iiy tx
The input vector can be represented as a weighted sum
of basis vectors, are transform coefficients.
x
iy
The orthonormal transform preserves the signal energy
.1 1
22
N
i
N
j
ji yx
This property is known as the discrete Parseval’s theorem
N
j
j
TTTTTTTN
i
T
i yTTTTTTx1
21
1
2 )()( yyyyyyyyxx
Properties of transforms
Noncorrelatedness of transform coefficients.
This property implies that the transform coefficients ,iyNi ,...,2,1 satisfy the condition
,ijijjii yyyy ji,
where i is a variance of ,iy denotes mathematical
expectation. For simplicity we assume .0yx
Localization of most part of signal energy in a small number of
transform coefficients.
Let
Nyy ...,1 be sorted in such a manner that
....21 N
Properties of transforms Assume that only first ,pN 10 p coefficients are
transmitted. The receiver uses truncated vector T
pNyy )0,...,0,,...,(ˆ1y to reconstruct .ˆˆˆ 1
yyxTTT
The MSE occurred when we replace by x x is
xxxx ˆˆ1
ˆ1
1
2 TN
i
iiN
xxN
N
j
jj
TTTT yyN
TTTTN 1
2)ˆ(1
ˆˆ1
yyyy
N
pNj
j
N
pNj
jN
yEN 11
2 .11
(6.3)
We would like to find the orthonormal transform which
minimizes the error (6.3)
Properties of transforms
Low computational complexity.
Any orthonormal transform preserves the achievable rate-
distortion function !!! )(DH
To apply nonseparable 2-D transform we rearrange
input matrix into a vector with components X
and multiply it by the transform matrix of size
x 2N
2T 22 NN
xy 2T
For separable transform the matrix of the transform
coefficients can be obtained as NN
TTXTY
The matrix is the Kronecker product of
two matrices of a 1-D transform
22 NN 2TNN T
We reduce the computational complexity to instead of 32N4N
The Karhunen-Loeve transform
The most efficient in terms of listed properties is the
Karhunen-Loeve transform.
•This transform is orthonormal.
•Its coefficients are uncorrelated.
•The KL transform minimizes (among all orthonormal
transforms) the MSE (6.3) occured because of rejecting
transform coefficients with small variances.
KL transform is optimal in terms of localization signal
energy and it maximizes the number of transform
coefficients which are insignificant and might be
quantized to 0
The Karhunen-Loeve transform
Let KLT be the matrix of KL transform.
The covariance matrix of the input vector
x is expressed
via covariance matrix of transform coefficients as
.KL
TT
KL
T TTR yyxx
Multiplying by we obtain
T
KLT
TT
KL
T
KL TRT yy
Since coefficients are uncorrelated
,
00
0...0
001
N
T
yy
where is the variance of i .iy
The Karhunen-Loeve transform
,T
KLii
T
KLiR tt Ni ,...,1
Thus the basis vectors of the Karhunen-Loeve transform are
eigenvectors of the covariance matrix normalized to
satisfy
R.ij
T
KLjKLi tt
The variances of the transform coefficients are
eigenvalues of
i.R
Since is symmetric and positive definite matrix (
for any nonzero ) then eigenvalues are real and
positive. As the result the basis vectors and the transform
coefficients of KL transform are real.
R
0TAxx
x
The Karhunen-Loeve transform
It can be shown that among all possible orthonormal transforms
applied to stationary vectors of dimension the KL
transform minimizes the MSE (6.3) occurred due to truncation,
that is, the KL transform is optimal in terms of localization
signal energy.
Equation (6.3) for the KL transform has the form
N
.11
11
2
N
pNj
j
N
pNj
jN
yEN
The main shortcoming of the KL transform is that its basis
functions depend on the transformed signal. We have to store
not only quantized transform coefficients but also the basis
functions which can require many more bits for storing than
the quantized coefficients.
The discrete Fourier transform
The DFT is the counterpart of the continuous Fourier transform.
It is defined for the discrete-time signals. The transformed
signal represents samples of the signal spectrum.
Let )( snTx be input sequence then the DFT of is )( snTx
,)()(1
0
sTjknN
n
s enTxkX
.10 Nk
The inverse transform is defined as follows
,)(1
)(1
0
sTjknN
k
s ekXN
nTx
,10 Nn
where 2 / sNT is the base frequency of the transform
or the distance between samples of the signal spectrum.
(6.4)
(6.5)
The discrete Fourier transform
Notice that (6.4) determines a periodical sequence of
numbers with period .N
Expressions (6.4),(6.5) can be rewritten in the form
,)()(1
0
N
n
kn
NWnxkX 10 Nk
,)(1
)(1
0
kn
N
N
k
WkXN
nx
,10 Nn
./2 Nj
N eW where
In the matrix form (6.4), (6.5) can be rewritten as
,xX FT ,1 *1
XXx FF TN
T
The discrete Fourier transform
,))1(),...,1(),0(( TNXXX X
,))1(),...,1(),0(( TNxxx x
)1)(1()2)(1(1
)1)(2()2)(2(2
12
...1
...1
...............
...1
11...11
NN
N
NN
N
N
N
NN
N
NN
N
N
N
N
N
N
NN
F
WWW
WWW
WWW
T
is the transform matrix. The basis functions are powers of
./2 Nj
N eW
The discrete Fourier transform
Since
*1 1FF T
NT the DFT is orthogonal transform. It is
easy to normalize the DFT in order to obtain the orthonormal
transform. For this purpose we should use factors in
the forward and in the inverse transform instead of using factor N/1
N/1 in the inverse transform.
The 2-dimensional DFT is defined as follows 2 21 1 ( )
0 0
( , ) ( , )N M j kn lm
N M
n m
k l n m e
X x
,10 Nk
,10 Ml
),( mnx is the element of the input matrix
,x
),( lkX is
the element of .X
The discrete Fourier transform
Since
2 22 21 1 1 1
0 0 0 0
( , ) ( , ) ( , )M N N Mjkn jknjlm jlm
N NM M
m n n m
k l n m e e n m e e
X x x
the 2-dimensional DFT can be split into two 1-dimensional
transforms, that is, the DFT is a separable transform.
Linearity: )()()()( nybDFTnxaDFTnbynaxDFT
Circular convolution: Let and be DFTs of )(kX
)(nx
)(kY
and )(ny , respectively. Then the inverse transform of
the product
)()( kYkX is
.)()(1
)(1
0
2
N
k
klN
j
ekYkXN
lv
The discrete Fourier transform
By inserting definitions for )(kX )(kYand we obtain
.)()(1
)(
21
0
1
0
21
0
2kl
NjN
k
N
m
kmN
jN
n
knN
j
eemyenxN
lv
Changing the order of summation we get
.)()(1
)(1
0
1
0
1
0
)(2
N
n
N
m
N
k
mnlN
kj
emynxN
lv
The sum in brackets is equal to 0 for all and m n except
Nnlm mod)( for which it is equal to .N
1
0
1
0
)mod)(()()mod)(()()(N
n
N
n
NnlxnyNnlynxlv
is circular or periodical convolution.
Example
Let
( ) (2, 3, 1)x n and ( ) (1, 2, 4)y n We extend
)(nx
periodically (…, 2, 3, 1, 2, 3, 1, 2, 3, 1,…)
Then is determined as follows )(lv
16341221)1()2()2()1()0()0()0( xyxyxyv
11142231)2()2()0()1()1()0()1( xyxyxyv
15243211)0()2()1()1()2()0()2( xyxyxyv
The periodical sequence is reversed in time and
multiplied by the corresponding term in Then the
reversed is shifted by 1 sample to the right and again
multiplied by generating the new sample of
)(nx).(ny
)(nx)(ny ).(lv
)(nx
Example
The periodical reversed )(nx
1 3 2 1 3 2 1 3 2 1 3 2 )(ny 1 2 4
The reversed shifted to the right )(nx
2 1 3 2 1 3 2 1 3 2 1 3 2
3 2 1 3 2 1 3 2 1 3 2 1 3
The discrete Fourier transform
lk
NWkXNlnxDFT )())mod)(((
That is displacement of l samples from the end to the beginning
of )(nx is equivalent to multiplying DFT of this sequence
by exp{ 2 / }j lk N
Example.
Let )(nx be equal to 1 2 3 4 5 3 7 4 and its DFT be equal to
)(kX then is 3 7 4 1 2 3 4 5 and its DFT
8mod)3( nx
is determined as ( )exp{ 2 3 /8}X k j k
The discrete Fourier transform
In general case the DFT and IDFT require approximately
2N additions and multiplications of complex numbers. 2N
There exist the so-called fast algorithms which allow to reduce
the computational complexity of DFT to NN 2log
operations.
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