PIECEWISE UNIFORM SWITCHED VECTOR QUANTIZATION OF THE MEMORYLESS TWO
DIMENSIONAL LAPLACIAN SOURCE
Zoran H. Peric, Ivana Lj. Tosic
Faculty of Electronic Engineering, University of Nis, Serbia, Yugoslavia
e-mail peric@elfak.ni.ac.yu
abstract: In this paper we will present a simple and complete
asymptotical analysis of an optimal piecewise uniform
quantization of two-dimensional memoryless Laplacian source
with the respect to distortion (D) i.e. the mean-square error
(MSE). Piecewise uniform quantization consists of L different
uniform vector quantizers. Uniform quantizer optimality
conditions and all main equations for number of optimum
number of output points and optimal number of levels for each
partition are presented (using rectangular cells). We derive the
optimal granular distortion
D gopt (i) for each partition in a
closed form.
Switched quantization is used in order to: give higher quality
by increasing signal-to-quantization-noise-ratio (SQNR) in a
wide range of signal volumes (variances) or to decrease
necessary sample rate.
Key words: piecewise uniform quantization, switched
quantization, distortion
Introduction
The use of digital representation for audio, speech,
images and video is rapidly growing with the exploding use of
computers and multimedia computer applications. To provide
a more efficient representation of data, many compression
algorithms have been developed, and in the heart of all these
algorithms is quantization. The concept of quantization is a
mapping of a large set of amplitudes of infinite precision to a
smaller finite set of values, as shown in the Figure 1(a). Vector
quantization is simply an extension of the scalar quantization
to multidimensional spaces; that is, a vector quantizer operates
on vectors (blocks of samples) instead on scalars.
integers
real numbers
4
3
2
1
0
4
3
2
1
0
unquantize d samples
quantized samples
(a)
codevectors
block into vectors
4
3
2
1
0
codevector
indicies
000 001 010 011 100
unquantized samples
codebook with 2-D codevectors
(b)
Figure 1 Illustration of (a) scalar and (b) vector quantization
Quantizers play an important role in the theory and
practice of modern-day signal processing. The asymptotic
optimal quantization problem, even for the simplest case uniform scalar quantization, is very actual nowadays, [1,2].
They do consider the problem of finding the optimal
maximum amplitude, so-called, support region for scalar
quantizers by minimization of the total distortion D, which is a
combination of granular (Dg) and overload (Do) distortion,
D  D g  Do . Extensive results have been developed on
scalar quantization but more on vector quantization. The
simplest vector quantization is two-dimensional vector
quantization.
The analisys of vector quantizer for arbitrary
distribution of the source signal was given in paper [3]. The
authors derived the expression for the optimum granular
distortion and optimum number of output points. However,
they didn't prove the optimality of the proposed solutions.
Also, they didn't define the partition of the multidimensional
space into subregions. In paper [4], the expressions for the
optimum number of output points are derived, however the
proposed partitioning of the multidimensional space for
memoryless Laplacian source doesn't consider the geometry of
the multidimensional source. In paper [5], vector quantizers of
Laplacian and Gaussian sources were analyzed. The proposed
solution for the quantization of memoryless Laplacian source,
unlike in [5], takes into consideration the geometry of the
source, however, the proposed vector quantizer design
procedure is too complicated and unpractical.
In this paper we will give a systematic analysis of
piecewise uniform vector quantizer (PUQ) of Laplacian
memoryless source. We will give a general and simple way to
design a piecewise uniform vector quantizer. We will derive
the optimum number of output points and the optimality of the
proposed solutions will be proved. The goal of this paper is to
solve a quantization problem in a case of PUQ and to find
corresponding support region. It is done by analytical
optimization of the granular distortion and numerical
optimization of the total distortion. If the distortion is
measured by squared error, D becomes the mean squared error
(MSE). Distortion mean-squared error (MSE i.e. quantization
noise) is used as the criterion for optimization.
The MSE of a two-dimensional vector source
x  ( x1 , x2 ) , where xi are zero-mean statistically
independent Laplacian random variables of variance  , is
commonly used for the transform coefficients of speech or
imagery. The first approximation to the long-time-averaged
probability density function (pdf) of amplitudes is provided by
2
Laplacian model [7,p32]. Waveforms are sometimes
represented in terms of adjacent-sample differences. The pdf of
the difference signal for an image waveform follows the
Laplacian function [7,p33]. Laplace source is a model for
speech [8,p384]. Consider two independent identically
distributed Laplace random variables (x1,x2) with the zero
mean and unity variance. To simplify the vector quantizer, the
Helmert transformation is applied on the source vector giving
contours with constant probability densities. The
transformation is defined
u
1
2
x
1
as:
r
1
2
x
1
 x2

,
 x 2 . In this paper, quantizers are designed
and analysed under additional constraint – each scalar
quantizer is a uniform one.
PUQ consists of L optimal uniform vector quantizers.
More precisely, our quantizer divides the input plane into L
partitions and every partition is further subdivided into Li
( 1  i  L ) subpartitions. Every concentric subpartition can be
subdivided in four equivalent regions, i.e. J-th subpartition in
signal plane is allowed to have p ij ( 1  i  L, 1  j  Li )
cells. We perform two-steps optimization: 1) distortion
optimization ( Di ) in every partition under the constraint
Li
4 p ij  N i
and
2) optimization of the total granular
j 1
distortion Dg 
C , which is the codebook. The codebook has
a positive integer number of code vectors (denoted by y i ) that
the finite subset
defines the codebook size, denoted by N. The bit rate R
associated with the VQ depends on N and the vector dimension
k. Since the bit rate is the number of bits per sample,
(1)
R  (log 2 N ) / k
In contravention to basic scalar quantization which is
fixed-rate, for VQ is natural to have fractional bit rates such as
½, ¾, etc. The decoding process is very simple and requires
only a table (codebook) lookup, but the encoding procedure is
complex and involves finding a best matching code vector,
using a distortion measure as a criterion. The most common
distortion measure is mean squared error, given by:
k
d ( x, y )  ( x  y ) t ( x  y )  ( x[l ]  y[l ]) 2 ,
(2)
l 1
where x[l ] and y[l ] are the elements of the vector x and
y , respectively.

ENCODER
E
x
search
I
output index
x
DECODER
reconstructed
D
vector
lookup
x=y
i
i
000
i
encoder
codebook
001
010
000
decoder
codebook
001
indices
010
L
D
i 1
i
which achieves the optimal number of
L
points N i on each partition under the constraint
N
i 1
i
 N.
In this work we will design a piecewise uniform vector
quantizer for optimal compression function. We will perform
analytical optimisation of the granular distortion and numerical
optimization of the total distortion using rectangular cells.
The switching quantization aims are to improve the
quality of the signal-to-noise ratio in the wide range of the
signal average power (i.e. variance) or to decrease the sample
rate. The switching quantization is adaptive quantization for
memoryless sources and it is aplicable only if adaptation is
performed on the basis of the signal average power, what was
just done in this paper. As an input source, we will consider
memoryless Laplacian source.
Basic notes on VQ
The conceptual notion of VQ is illustrated in Figure 1
(b). These blocks of samples are represented by code vectors
and stored in a codebook- a process called encoding. A block
diagram of the encoder is shown in Figure 2. The encoder 
k
performs a mapping from k-dimensional space R to the
index set I , and the decoder D maps the index set I into
111
codevectors
y
111
Figure 2 Block diagram of a VQ encoder and decoder.
It is convenient to view the operation of a vector
quantizer geometrically, using our intuition for the case of two
or three-dimensional space. Thus, a 2-dimensional quantizer
assigns any input point in the plane to one of a particular set of
N points or locations in the plane. The plane is divided into N
partition cells, as shown in the Fig.3.(a), and the dots represent
code vectors, one in each cell. A unique partitioning of the
space is defined by the encoding procedure, and optimized for
a given input source, to give the best performance. Now
consider quantizing our fictional input source with scalar
quantization at an equivalent bit rate. The cells implied by
using a scalar quantizer for the input source are shown in the
Fig.3(b). Notice that each cell is constrained to be rectangular,
and some cells are forced to be placed in regions where the
input source may not be significantly populated. These
observations lead to two immediately recognizable advantages
of VQ over scalar quantization. First, VQ provides greater
freedom to control the shapes of the cells to achieve more
efficient tilings of the space. This property is often called cell
shape gain. Second, VQ allows a greater number of cells to be
concentrated in the regions where the source has the greatest
density,
which
reduces
the
average
u
distortion.
m
ri
ri+1
ri
mi,j
ri
i,j(i)
i, j
uijk
, uˆ i , j , k 
ui, j,k+1
r
2mi,j
ri,j
(a)
ri,j+1
(b)
Figure 3 Illustration of the partition cell associated with VQ
and scalar quantization: (a) partition cells for a 2D VQ; (b)
partition cells corresponding to scalar quantization.
Figure 4 Two-dimensional space partitioning.
The initial expression for granular distortion is:
2
2
  r  mi, j   u  uˆ i, j,k  
L Li pi , j ri , j 1 ui , j , k 1
D g  4  
i 1 j 1 k 1 ri , j ui , j , k
Piecewise uniform vector quantizer design

1

(6)
2r
e  drdu
Joint pdf function of two independent, identically
distributed Laplace random variables (x1 , x2 ) with zero mean
is given with the following expression:
2  x1  x 2 
1 

f1,2  x1 , x 2  
e
(3)
2
Rectangular cell dimensions are:
After applying the Helmert transformation [9,10]:
 i  ri 1  ri , i 
2
1
r
2
x
1
 x2  , u 
1
2
x
1
 x2
we get probability density function:
2r
1 
f (r , u ) 
2 2

(4)
The otput point coordinates are given by the equations:
mi , j 
ri , j 1  ri , j
2
uˆ i , j ,k 
and
u i , j ,k  u i , j ,k 1
2
ri , j  ri , j 1
i
and  ij 
pi , j
Li
ri , j  ri  j   i , i  0,  , L , j  0, , Li
(7)
(8)
(9)
rmax . To determine the
boundary values of every concentric domain, denoted as ri ,
The range of the quantizer is
e 
(5)
In two-dimensional ru system the pdf function given
by equation (5) represents square line. The square surface
(0, rmax ) representing dynamic range of a two dimensional
quantizer, can be partitioned into L concentric domains as
shown in Fig. 1. In the case of nonuniform vector quantization,
these concentric domains are of unequal width. The number of
output points in each domain is denoted by N i , where
N  i 1 N i represents the total number of output points.
L
Every concentric domain can be further partitioned into
Li
concentric subdomains of equal width. Every subdomain is
divided into four regions each containing p i , j rectangular
cells. An output point is placed in the centre of each cell.
Coordinates of the kth output point in jth subregion of the ith
region in ru coordinate system are m i , j , uˆ i , j , k .

2 2

for the case of nonuniform vector quantization we will perform
segmentation and linearization of the optimal compress
function, given by the following expresion:

h(r )  rmax
e
1
r
2
1
1
 rmax
2
e 
(10)
1
The method for linearization of compression function,
named first derivate segmentation, was selected based on the
analysis performed in [11]. The principle used in this method
is to do a uniform segmentation of first derivate of
compression function, and find corresponding ri points by
substituting uniformly distributed
derivate function.
Total numer of output points is:
h values in inverse first
L
N   Ni
i 1
(11)
where N i is the number of output points in the ith domain.
We can also write:
Li
Ni
4
 pi , j 
j 1
Function
L
D g   D g i 
i 1
Li pi , j ri , j 1 ui , j , k 1
D g i   4  
 
j 1k 1 ri , j ui , j , k

1
r  m
i, j
2  u  uˆi, j,k 2 
(14)
2r
e  drdu
2 2
After integration over u and reordering equation (14) becomes:
2r
Li ri , j 1
2 
2

D g i     r  mi, j ri, j 1  ri , j
e
dr 

2
j 1  ri , j
(15)


ri , j 1



ri, j 1  ri, j 3
12 p i2, j
ri , j

2r 

e  dr 

2

2
ri, j 1  ri, j  2mi, j .
From equation (7) it follows that
When we substitute this in equation (15) we get:
2r
Li  ri , j 1
1 
2

D g i  
r  mi, j 4mi , j
e
dr 
2


j 1 ri , j

  
ri , j 1


ri , j

(13)
mi2, j
3 pi2, j


1

4mi , j
e
dr 
2




of
.
Li
 4mi, j
j 1
Li
  4mi, j
j 1
1

2
1
2


e

ri , j 1

mi2, j
 r  m 2 dr 
dr
i, j
 3 p 2 
 
i, j
ri , j

 ri , j
2mi , j



j 1
(20)
After differentiating J with respect to pi , j , and equalizing the
derivate with zero we get:
4mi3, j
J
0
P0 mi, j    0
(21)
pi, j
3 pi3, j
From the preceding equation we can write the following:
4mi3, j
4 f 0 mi, j  i
3
(22)
pi , j 
P0 mi, j  mi, j 3

3

2 2mi, j 
  P mi, j
  2mi, j i 
12 3 pi2, j 
j 1

Li
where P

pi, j
j 1
we can eliminate

mi, j  denotes the probability:

3




3
from equation (22) in equation (12):


4 f 0 mi, j  i
N
 i
3
4
(23)

by substituting:
L
4 i

mi, j 3 4 f 0 mi, j  i
N i j 1



(24)
in equation (22):
pi , j 
 3 mi2, j

 i 
i 
 12 3 pi2, j



(19)
J  D g i     pi , j
 mi, j 3
2 m i , j  ri , j 1
e

Li

D g i  
(18)
We'll start from the following equation:
If we substitute
Li
exp  2r /   is constant over  i .
In that case we can substitute exp  2r /   with
exp  2mi, j /  . Equation (16) can be now written as:


mi , j
e
2
2mi , j
minimum granular distortion defined by the equation (17).
Because we are designing an optimal quantizer for one value
of variance  0 , in calculating pi , j we will use  0 instead
3
2r
2
2

By using the Langrangian multipliers we can obtain the
optimum number of cells in one region pi , j , which yields the
(16)
We will now assume that


f mi, j 
Dg (i) is:
where

 
e
2
f mi, j  is defined as:
P mi, j   i f mi, j   i
(12)
Equation (12) can be written as:


2


N i mi, j 3 g 0 mi, j
4 Li
 mi, k 3 g 0 mi, k


(25)
k 1
wher g 0 mi, j denotes the function:

(17)



g 0 mi, j 
1

2mi , j

0
(26)
e
2
0
If we multily numerator and denominator with  i , we can
approximate the sum by the integral:


N mi, j 3 g 0 mi, j   i
pi , j  i
4 ri 1
 r  3 g 0 r  dr
L
(27)
J  D g N i     N i
i 1
(34)
After differentiating (33) with respect to Ni, and for  0 we
get:
ri
pi, j from equation (27) in equation (17) we
By substituting
get:
D g i  
2i
Li
 mi, j g 0 (mi, j )i 
3L2i j 1
(28)
Li
64 L2i
g (mi, j )

I  (i ) 2 mi, j

2 2 0
2/3 i
3N i  i
g
(
m
)
j 1
0 i, j
After approximating the sum by the integral, we can rewrite
(28) as:
64 L2i
2
(29)
D g i  
I i  
I  i 2 I (i )
2
2 2 0
3Li
3N i 


The functions I 0 i  , I (i ) and I i are defined as:
ri 1
I 0 i   r  3 g 0 r dr
ri
I i  
ri 1
 r g
ri
I i  
g (r )
0 (r )
2/3
dr
(30)
(31)
(32)
ri
Substituting the expression for Li from equation (31) in
equation (29), Dg(i) becomes:
Dg i  

14
k 1
I 0 (i ) 
I  i  I i 
14
3
0
(37)
0
Now, we can callculate the optimum granular distortion of
uniform piecewise vector quantizer as:
L

D g i  
 I  (i ) 
  0 
I (i )
i 1 
 0 

 I  (i )

8
I (i )
I 0 (i ) 0  I (i )  0  I (i )  (33)


Ni
I 0 (i )
 I 0 (i )

The optimum number of output points in the ith subdomain is
obtained using Lagrangian multipliers:
3/ 4
8
3N
 I 0 (i) 3 I 0 (i)1/ 4 
L
i 1
1/ 4
 I (i) 
 I (i)   0 
 I 0 (i) 
(38)

 I (i)


We can calculate the overload distortion as:
p L , LL  u L , j
D0  4
r  m L, L L 2  u  uˆ L, j 2
j 1 rmax u L , j
   

where I 0 (i) is defined as:
ri 1
 r  g 0 r dr
8
3N
L
domain:
I 0 i  
 I 0 k 3 I 0 k 
L
i 1
ri
After differentiating Dg from equation (29) with respect to Li,
and for  0 , we obtain the optimum number subdomains in ith
64 I 0 i 3


Dg 
 r  g r dr
Liopt   4

 I  (i )

I (i )
  0  I (i )  0  I (i ) 
 I (i )

I 0 (i )
 0

ri 1
I 0 i N i2
(35)
Using the condition (11) we can eliminate  from the
equation (35):
14
I 0 i 3 I 0 i 
(36)
Ni  N
L
1
4
I 0 k 3 I 0 k 
k 1
Finally, after substituting the expression for optimum number
of output points from equation (36) into equation (33) we can
write:
D g (i ) 

16 I 0 i  I 0 i I 0 i 
3
Ni 
1
2 2

 


(39)
2r
e  drdu
After some calculation, we get:
2r



mL, LL  max
2
Do 
e  2 rmax
 rmax 2 2  4 mL, LL 
2

(40)


  3  2mL, LL   2mL2, L   

L

From equations (31) and (33), we can calculate the total
distortion for one dimension as:
2mL2, L
L
2
3 p L, L
L
D
1
( D g  Do )
2
(41)
30
Numerical results
25
The results are shown in Figure 5 for two values of
 0 (two different quantizers): 0dB and -10dB, and for bitrate
of R=7.5. For comparison, dash-dotted lines show SQNR for
scalar quantization for R=8, for a compression function given
with:
r

rmax
1 e
(42)
f (r )  rmax

1 e
where  is a compression factor, with a critical value for
c 
2 rmax
3 0
SQNR[dB]
15
SQNR is a signal-to-quantization noise ratio, given with:
2
(44)
SQNR[dB]  10 log
Duk
We can see from this figure that there is a 0.5 bits gain in the
case of vector quantization. In Figure 6 we have changed
bitrate to R=4.
In Table 1., as an illustration of previous analysis, the
number of rectangular cells in each of four regions in a
particular subdomain, denoted as pi , j , is given. The optimal
number of subdomains in every domain ( Liopt ) is calculated
for L=4 concentric domains and bitrate R=4, giving the value
2.
50
vector (variance=-10dB, 0dB), R=7.5
10
5
scalar
(v=5, 20)
0
-5
-10
-15
0:
(43)
vector (variance=-10dB, 0dB)
20
R=4
-20
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
variance
Figure 6 SQNR for scalar and vector (bitrate=4) quantization
with two optimal quantizers designed for different values of
0
pi , j
1
2
L=4
1
1
3
2
6
7
4
13
13
3
9
11
Table 1 Number of rectangular cells for R=4
For nonstationary inputs a logical scheme is switched
quantization; this consists in providing a bank of B fixed
quantizers, and switching among them as appropriate, in
response to changing input statistics. This scheme is used for
two designed quantizers, as shown in the Figure 7.
40
Q1( · )
SQNR[dB]
30
Q2( · )
·
·
·
x(n)
20
10
Q B (· )
scalar (v=5,20)
R=8
CONDITIONAL
ESTIMATOR
0
-10
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
variance
Figure 5 SQNR for scalar (bitrate=8) and vector (bitrate=7.5)
quantization with two optimal quantizers designed for different
values of
0
Figure 7 Block diagram of switched quantization
Conclusion
The optimization of two-dimensional Laplace source
piecewise nonuniform vector quantization is carried. Simple
expression for granular distortion, number of subdomains and
number of output points in closed form is obtained. The results
obtained by using two vector quantizers optimized for two
different values of  (variance) demonstrate the significant
performance gain over the uniform scalar quantization, giving
a 0.5 bits/sample gain. Memoryless Laplacian source is used,
considering the possible application of this quantizer. The
transform coefficients of DCT (discrete cosine transform)
encoding of speech or imagery are often modeled as Laplacian,
except for the DC coefficient of imagery. By using switched
quantization, with two quantizers in this case, necessary rate is
decreased by 0.5 bits per sample, with compliance of the
appropriate standard. With a larger set of quantizers we could
increase this sample rate even more, thus giving a better
compression quality with higher SQNR.
References
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