Asymptotic analysis of nonuniform polar quantization Текст научной статьи по специальности «Математика»

MATEMATH^HE TA KOMn'^TEPHE MO^EËraBAHHH

YAK 681.32

Zoran H. Peric, Danijela R. Aleksic

ASYMPTOTIC ANALYSIS OF NONUNIFORM POLAR QUANTIZATION

The motivation for this work is maintaining high accuracy of phase information that is required for some applications such as interferometry and polarimetry, polar quantization techniques as well as their applications in areas such as computer holography, discrete Fourier transform encoding, and image processing. In this paper the simple and complete asymptotically analysis is given for a nonuniform polar quantizer with respect to the mean-square error (MSE) i. e. granular distortion (Dg). The equation for Dg°Pt is given in a closed form. The goal of this paper is solving the quantization problem in case of nonuniform polar quantizer and finding the corresponding support region. We also gave the conditions for optimum of the polar quantizer and optimal compressor function. The construction procedure is given for iid Gaussian

1 INTRODUCTION

Polar quantization techniques as well as their applications in areas such as computer holography, discrete Furrier transform encoding, image processing and communications have been studied extensively in the literature. Synthetic Aperture Radars (SARs) images can be represented in the polar format (i. e., magnitude and phase components) [4]. In the case of MSE quantization of a symmetric two-dimensional source, polar quantization gives the best result in the field of the implementation [4]. The motivation behind this work is to maintain high accuracy of phase information that is required for some applications such as interferometry and polarimetry, without loosing massive amounts of magnitude information [4].

One of the most important results in polar quantization are given by Swaszek and Ku who derived the asymptotically Unrestricted Polar Quantization (UPQ) [6]. Swaszek and Ku gave an asymptotic solution for this problem without a mathematical proof of the optimum and using, sometimes, quite hard approximations, which limit the application. Polar quantization consists of separate but uniform magnitude and phase quantization, on N levels, so that rectangular coordinates of the source (x,y) are transformed into the polar coordinates in the following form: r=(x2+y2)1/2, where r represents magnitude and ^ is

phase:

-l y

tan

n + tan

n + tan 1 —

ly

2n + tan 1 —

The asymptotic optimal quantization problem, even for the simplest case - uniform scalar quantization, is actually nowadays [2]. In [8] the analysis of scalar quantization is done in order to determine the optimal maximal amplitude.

Swaszek and Ku [6] didn't consider the problem of finding the optimal maximal amplitude, so-called, support region. The approximation given by Swaszek and Ku for the asymptotically Unrestricted Polar Quantization (UPQ) [6]:

1

ri+i — mi " m — r =■

2Lg , {m,

This approximation for i=L is:

1

rL +1 - mL

m, — r, = ■

2Lg ' \mLJ

for I, II, III and IV quadrant.

is not correct for Unrestricted Polar Quantization because rL+1 — mL ^ ^ . That is the elementary reason for introducing support region ( rmax ), where rmax is restricted for the analysis of scalar quantization which is based on using compressor function g .

The support region for scalar quantizers has been found in [8] by minimization of the total distortion D, which is a combination of granular (Dg ) and overload (Do) distortions, D= Dg + Do. The goal of this paper is solving the quantization problem in the case of nonuniform polar quantizer and finding the corresponding support region. It is done by analytical optimization of the granular distortion and numerical optimization of the total distortion.

In the paper Peric and Stefanovic [9] analyses are given for optimal asymptotic uniform polar quantization. In this paper the simple and complete asymptotical analyses are given for a nonuniform polar quantizer with respect to the mean-square error (MSE) i.e. granular distortion (Dg ).

We consider D as a function of the vector P= (Pi )1:3<L whose elements are numbers of phase quantization levels at the each magnitude level. Said by different words, each concentric ring in quantization pattern is allowed to have a different number of partitions in the phase quantizer (Pi) when r is in the ¿-th magnitude ring. One Restricted Polar Quantization (RPQ) must satisfy the constraint

L

yi Pi = N in order to use all of N regions for the quan-

i=i

tization. We prove the existence of one minimum and derive the expression for evaluating Pp (r,m) for fixed

x

x

x

x

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values of reconstruction levels, decision levels and number of levels L. We also gave the conditions for optimum of the polar quantizer, optimal compressor function and optimal numbers of levels. We derive Dgp in a closed form.

We also gave the example of quantizer constructing for a Gaussian source. This case has the importance because of using Gaussian quantizer on an arbitrary source; we can take advantage of the central limit theorem and the known structure of an optimal scalar quantizer for a Gaussian random variable to encode a general process by first filtering it in order to produce an approximately Gaussian density, scalar-quantizing the result, and then inverse-filtering to recover the original [5]. Since each value in the complex "image" is derived from linear combinations of echo data, the central limit theorem can be invoked to assert the probability density function of the real and imaginary components of each pixel value in the complex image. Probability density functions are Gaussian [4].

2 CONDITIONS FOR OPTIMALITY AND DESIGN OF NONUNIFORM POLAR QUANTIZER

The transformed probability density function for the

L P, j 1+1

1 J* _ f(r)

. With-

D = 1 ££ J fc + m -2m cos(>-V,, ^drd? .(6)

i=i j=i

. L p 'i+1 )

D = 2EE J J[r2+m -2rmi c0^-^j>]fnr'drd?+

i=1 j=1 hj r

1 PL §L,j+1 ~ , )

+ J \[r2 + mL - 2rmL cos(§-yLJ.(7)

j=1 <?Lj rL+1

We integrated (2) by ^ , and get the equation for granular distortion:

1 L ri+1

Dg (Pj,---, PL ) = J[r2 + m,2 - 2rmt sin c(—)]f(r)dr ,(8)

i=1 r, i

(where in sine (x)=sin (x)/x); (2) we use: sin(x) = 1 - 1X 2 +E(x),

6

L ri+l

D„

J[(r - m, )2 + rjm--fïfridr . (9)

dDs

From: -— = 0 we can find mt as:

dm,

Gaussian source is f(r A) = .

7 V 2no2 2n

out loosing generality we assume that variance is: o2 = 1 .

We consider nonuniform polar quantizer with L magnitude levels and Pi phase reconstruction points at magnitude reconstruction level mi, 1 < i < L . In order to minimize the distortion we proceed as follows.

First we partition the magnitude range [0, rL+^ into magnitude rings by L+1 decision levels r=(ri,..., rL+1) and

(0=r1<r2<...<rL<rL + 1= rmax).

The magnitude reconstruction levels m=(mi,..., mL) obviously satisfy (0<mi<m2<.<mL). Next we partition each magnitude ring into Pi phase subdivisions. Let and + i be two phase decision levels, and let be j-th phase reconstruction level for the i-th magnitude ring, 1 < j < Pi. Then <(,., j = (j - 1)2n / N,j = 1,2,..., N, + 1 ,

and V,j = (2 j - 1)n / Mt .

The distortion D for UPQ (rL+1=x ) is [6]:. Total distortion D, for RPQ (rL+1= rmax ) is a combination of granulation and overload distortions D=Dg+Do:

'¿+1 'i

1 -1 ~T

6 P2

As final result, we find approximation for m, as:

1+i -1

m, = •

2

(10)

(1)

We can obtain from High Resolution Theory [1] that high values for R and critical values for Pi satisfy given approximation.

The equation for Dg is obtained by using High Resolution Theory [1].

D = y f (m, )A3 + y mf n2 f (m, )A, s Z-i 24 " ¿of

,=1

, =1

6P 2

(12)

We prove that the problem of minimizing the Dg (P) is a convex programming problem. Function Dg (P) is convex if its Hessian matrix is the positive semidefinite one [3, p27]. From we obtain:

dDs

2n2

dp 6(P )■

d 2 Ds

mi2 f(mi)Ai ,

dP dP,

1 J

(P, )4 0 , * j

d 2 D,

2 f (m, )A,,, = j

=>

d P, d Pj

■> 0

(13)

it follows that Dg (P) is a convex function of P.

x

r

m=

2

2

K

MATEMATH^HE TA KOMn'ÜTEPHE MOÂEËÛBAHHH

The optimization problem for polar quantizer can be formulated in this way: it is necessary to find partial derivations of Dg (P). Than, we can use the equation:

J = Dg + Pi, where X represents Lagrange multiplier. From = 0 we obtain:

dPi

asymptotical analysis as it is suggested: from the condition

dDg dL

= 0 we came to the optimal solution for Lopt:

T _ r 4i_

^opt 'ma^l 4n 2 j 3

hN 2

(18)

dJ 2n2

dp 6(P )

-m2 f (m, )Ai +X ,

and finally:

The optimal granular distortion is:

D>' = 6NJ^ ■

(19)

Piopt _ N L

3¡ mj f (m, )A,.

£ 3m22f (m; )A j

1 < / < L

j=1

(14)

The approximation given by Swaszek and Ku for the asymptotically Unrestricted Polar Quantization (UPQ) [6]:

We can obtain g(r) like in [6] by using Holder's ine-qual ity:

g(r) = (rmax )fLdr) ( j dr) (20)

and

1

rL +1 - mL a mL - rL =

2Lg ' {mL>

(15)

is not correct for Unrestricted Polar Quantization because rL+1 — mL ^ x . That is the elementary reason for introducing support region ( rmax ), where rmax is restricted for the scalar quantization analysis, which is based on using compressor function g.

r

We replaced Ai = —max , where g is compressor

Lg(mi)

function, and approximate the sums by integrals ( Ai ~ dr ), and we get Pi as:

max

Df = 6N( jVTT(T)dr)2 . (21)

0

For rmax = ^ RPQ is transformed in UPQ and we get same distortion as in [6]. Example:

We compared results for Gaussian source. Numbers of magnitude levels and reconstruction points, reconstruction points and decision levels are calculated by using [6]:,

P

Nrmax3mi f (m, V g'(m, )

L jVr2f(r)(g'(r))2dr

(16)

As final result, we find the equation for granular distortion:

L _

N

1/2

^ -1/ 4 f 1/ 4 (s)ds

422k

(js1/2 f1/2 (s)ds)1

/2

_ V^Nl/ 2 fl/4(m, )m,3 /4

popt M

/2

(js1/2 f1/2 (s)ds)1

(22)

(23)

2 rmax

r

D _ max I

_ f (r)

24L2 J (g'(r))2

-dr +

2 t2 rmax

n L , f3/ 2

_ g-1[(i - 1)/ L], 1 < i < L;

_ g-'[(2i - 1)/ 2L], 1 < i < L '

6N2r2

* ' max Q

( J*^r2/"(r)(g,(r))2dr)3 = g(r) is a compressor function given by:

-h +■

22 n L

24L2 0 ' 6N2 rmax

(17)

g(r) _ (jWds) / (j;

f (s)

ds) ■

(24)

(25)

The function Dg(L) is convex of L, because

32 D r2

g _ ' max

dL

In +-

4L4 0 ' 3N2^

I . The optimal number of

levels problem can be solved analytically only for the

Method presented in the paper [6] cann't be applied for some values of N and numbers of level L. For number of

level L, the total number of points is in the range ( [N^ -), N1 = 2(round (L) — 0.5)2 , N2 = 2(round (L) + 0.5)2 .

0

r

0

Q

0

r

m

+

r

r

3

j

s

Q

0

2

n

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This follows from the fact that r and m are equal for any N in the range (N1-N2), and since Popt is dependent

L

of m, N and introduced approximations, then ^ P, = N

i=1

will not be satisfied. In addition, for some values of N

L

from the former range, we cannot reach ^Tp = N .

i =1

For Gaussian source is:

2

L = VN / 2 , Pi = V^N1 / 2 mt exp(-.

For estimation of Pi we gave following approxima-i=1

tion: we found the total number of points [6] as:

L L __2 .

^p = VSNmi exp(-m2) ^

, =1

, =1

8 A,

round(L)^. Nm, exp(- ^^)A, ^ V 2 ' 4 !

IN r 2

round(L)-\— r exp(--)dr =

V 2 J 4

0

= roun d(L)4ÎN = M

§1 =11,84; and for N=N2=264 => L=11,^T P. =254,48,

i=1

§2 =9,52.

For Piopt=round (Pi) we can't satisfy constraint

L

^ Piopt = N . The difference cann't be compensated by i =1

rounding \_Pi J or |"p. .

For N=256 we get following results: 10 = 0,3882 ; I = 3,4527 ; Lopt = 11 ; D= 0,0082 (see Table 2.

We get eleven values for Pi by rounding, but eight of them are different from values in [6].

The optimal numbers of phase levels in each ring were found by evaluating Pi at the L magnitude decision levels, again taking the nearest integer values. If a specific value of N is desired, the Pi values can be adjusted (rounded up

or down) to sum to N exactly, in order to minimize Dg . Exact optimal value for rmax is obtained by repeating our optimization method for different rmax and choosing the values for which D=Dg +Do is minimal.

Table 1

We considered the most critical values for N= |~Nj ] and N= |_N2J where S, = |N - M| . (see Table 1.)

By Swaszek and Ku [6] for each L=const, m and r are

L

equal. For N=N^221=^=11^ P. =232,84, and

L Ln 2J §1 §2

11 221 264 10,26 11,24

23 1013 1104 22,26 23,25

45 3961 4140 44,25 45,25

91 16381 16744 90,25 91,25

181 65161 65884 180,25 181,25

Table 2

i=1

riopt miopt Ai [6] Ai opt Pi [6] Popt Preal

0 0,1135 0,2278 0,2272 3,2192 3 3,2685

0,2273 0,3419 0,2312 0,2302 9,5798 10 9,7209

0,4576 0,5749 0,2376 0,2364 15,6913 16 15,9147

0,6941 0,8159 0,2481 0,2465 21,3603 22 21,6616

0,9409 1,0695 0,2638 0,2617 26,3733 27 26,7468

1,2028 1,3417 0,2867 0,2841 30,466 31 30,9102

1,4873 1,6413 0,3215 0,3176 33,3000 34 33,8161

1,8058 1,9834 0,377 0,3709 34,4023 35 34,9966

2,1782 2,3963 0,4771 0,4650 33,0337 34 33,7276

2,6473 2,9488 0,7112 0,6726 27,8550 29 28,6924

3,3379 3,9219 1,5513 15,33 16 16,5446

МАТЕМАТИЧНЕ ТА КОМП'ЮТЕРНЕ МОДЕЛЮВАННЯ

3 CONCLUSION

REFERENCES

Swaszek and Ku gave an asymptotic solution for unrestricted nonuniform polar quantization without a mathematical proof of the optimum and using, sometimes, quite hard approximations, which limit the application. We gave elementary reasons for consideration of Restricted Polar Quantization. In this paper the simple and complete asymptotical optimal analysis is given for constructing nonuniform restricted polar quantizer. We also gave the conditions for optimality of the nonuniform polar quantizer. We gave an equation for optimal number of points for different levels and also, optimal number of levels. The

equation for D°°pt is given in a closed form. We gave the

asymptotically optimized equations that can be used for each N. The solutions for these equations always satisfy

the constraint:

J^Popt = N.

1. Gersho A. and Gray R. M. "Vector Quantization and Signal Compression", Boston M. A. Kluwer 1992

2. Hui D. , Neuhoff D.L. , "Asymptotic Analysis of Optimal Fixed-Rate Uniform Scalar Quantization," IEEE Transaction on Information Theory, vol.47, pp. 957-977, March 2001.

3. Himmelblau D. M. , Applied Nonlinear Programming, McGraw-Hill, Inc., USA, 1972.

4. Arslan F.T. "Adaptive Bit Rate Allocation in Compression of SAR Images with JPEG2000", The University of Arizona, USA, 2001.

5. K. Popat and K. Zeger, "Robust quantization of memoryless sources using dispersive FIR filters," IEEE Trans. Commun., vol. 40, pp. 1670-1674, Nov. 1992

6. Swaszek P. F. , T. W. Ku, "Asymptotic Performance of Unrestricted Polar Quantizer", IEEE Transactions on Information Theory, vol. 32, pp. 330-333, 1986.

7. Gray R.M. and Neuhoff D.L., "Quantization", IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2325-2384, October 1998.

8. S. Na, D.L. Neuhoff, "On the Support of MSE-Optimal, Fixed-Rate Scalar Quantizers" IEEE Transaction on Information Theory, vol.47, pp. 2972-2982, November 2001.

9. Z. H. Peric, M. C. Stefanovic, "Asymptotic Analysis of Optimal Uniform Polar Quantization" International Journal of Electronics and Communications, vol.56, pp. 345347,2002

Надшшла 08.08.2003

Повышение точности фазовой информации, требуемой в

интерферометрии и поляриметрии и их приложениях в

компьютерной обработке сигналов.

1=1

УДК 681.5

А.Е. Архипов, Е.А. Бабенко

СМЕЩЕНИЕ ОЦЕНОК ДИСПЕРСИИ ОЦЕНОК, ПОЛУЧАЕМЫХ С ПОМОЩЬЮ СКОЛЬЗЯЩЕГО КОНТРОЛЯ

В работе показано, что оценка дисперсии оценки метода наименьших квадратов, получаемая с помощью варьирования выборки по методу скользящего контроля, может быть сколь-угодно сильно смещённой, причём относительное смещение всегда бесконечно возрастает с ростом объёма выборки.

ВВЕДЕНИЕ

Варьирование выборки применяется при решении задач идентификации (см., например, [1-9]), обучения распознаванию образов, интерпретации результатов косвенных экспериментов ([2, 3]) и других задач, сводящихся к задаче минимизации среднего риска по эмпирическим данным. Одним из стандартных применений варьирования выборки является построение псевдовыборок, статистически однородных с исходной выборкой и применяемых для оценки качества процедуры идентификации: мерой качества, наряду с прочими показателями, является близость значений оценок параметров, полученных на разных псевдовыборках. Действительно, устойчивая процедура идентификации должна на близких (псевдо)выборках давать близкие оценки параметров. Для реализации такого метода оценки качества необходимы надёжные методы генерации псевдовыборок с требуемыми свойствами.

Различными специалистами разработаны и применяются разнообразные по своим принципам и свойствам методы варьирования выборки, позволяющие с той или иной точностью решать определённые прикладные задачи, возникающие в инженерном деле, медицине, биологии, социальной, экономической, экологической и других предметных областях. При этом в литературе крайне редки сведения, касающиеся свойств самих методов и достоверности результатов, получаемых с их помощью. С целью сокращения данного пробела в настоящей работе показано, что оценка дисперсии оценки метода наименьших квадратов (МНК), получаемая с помощью варьирования выборки по методу скользящего контроля, может быть смещённой, причём относительное смещение всегда бесконечно возрастает с ростом объёма выборки.

Особый интерес это явление представляет в связи с изложенным в [3] результатом о несмещённости оценки среднеквадратичного риска (в некотором случае), получаемой по методу скользящего контроля.

Для простоты выкладок рассмотрим однопараметри-ческую модель

г = х а + е .

Шум е считаем центрированным, с независимыми между собой и с х компонентами, Пе = О2. Компоненты случайного вектора х также считаем независимыми.

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