3-D adaptive predictive coding of moving images by block segmentation

Signal Processing 4 (1982) 313-321 North-Holland Publishing Company

SHORT

313

COMMUNICATION

3-D ADAPTIVE PREDICTIVE BLOCK SEGMENTATION

CODING

OF

MOVING

IMAGES

BY

Ernesto CONTE, Maurizio LONGO and Giovanni ZARONE lstituto Elettrotecnico, Facoltd* di Ingegneria, Universitd* degli $tudi di Napoli, Napoli, Italy Received 24 May 1981 Revised 22 September 1981

Abstract. The performance of adaptive, linear prediction in frame-to-frame coding of a sequence of videotelephonic images is examined. The dependence of the entropy of the prediction error upon order of the predictor and size of the blocks, in which prediction coefficients are adaptively computed, is elicited. A convenient selection of parameters mentioned above and an optimum strategy for quantizing prediction coefficients are put forth, which reduce transmission rate to less than 3 bits/pel while ensuring exact reproduction of the source. By allowing coding with a fidelity criterion (MSSTE), rates as low as 1.68 bits/pel are shown to be achievable without any perceptible quality degradation. Zusammenfassung. Die Wirkungsweise adaptiver linearer Priidiktion bei der Codierung einer Bildfolge zur Videotelefoniibertragung wird iiberpriJft. Hier wird insbesondere die Abh~ingigkeit der Entropie des Priidiktionsfehlers yore Grad des Pr~idiktors sowie yon der Gr6J]e des Mei3intervalls untersucht, in welchem die Pr~idiktorkoeffizienten adaptiv berechnet werden. Hieraus ergibt sich eine giJnstige Selektion der oben erwiihnten Parameter sowie eine optimale Strategic fiir die Quantisierung der Pr~idiktorkoeffizienten; die 0bertragungsrate kann bis unter 3 bit je Bildpunkt gesenkt werden, wobei die exakte Reproduzierbarkeit der urspriinglichen Bildfolge erhalten bleibt. Wird die Codierung mit Hilfe eines Giitekriteriums (MSSTE) durchgefiihrt, so k6nnen sogar l~lbertragungsraten bis hinunter zu 1,68 bit je Bildpunkt erreicht werden, ohne daJ~ eine wahrnehmbare Qualit~itsverschlechterung eintritt. Rtsum~. Les performances de la prediction lin~aire adaptative dans le codage trame-~-trame d'une s6quence d'images vid~ophoniques sont examinees. La d~pendance de I'entropie de l'erreur de prediction de l'ordre du pr~dicteur et de la dimension des blocs dans lesquels les coefficients de prediction sont calculus adaptativement est ~lucid~e. Une s~lection appropri~e des param~tres pr6c~demment mefitionn~s et une strategic optimum pour la quantification des coefficients de prediction sont propos~es, r~duisant la cadence de transmission ~, moins de 3 bit/pel tout en garantissant la reproduction exacte de la source. En permettant un codage avee un crit6re de fid61it6,des cadences aussi basses que 1.68 bit/pel peuvent ~tre obtenues sans une d~gradation de qualit~ perceptible.

Keywords. Image coding, adaptive prediction, quantization.

1. Introduction A m o n g t h e v a r i o u s c o d i n g t e c h n i q u e s t h a t have b e e n p r o p o s e d for o n e - d i m e n s i o n a l signals, o n e of the m o s t p o w e r f u l , e s p e c i a l l y for s p e e c h , is a d a p t i v e l i n e a r p r e d i c t i v e c o d i n g [1, 2]. However, rare attempts have been made toward a p p l y i n g such a t e c h n i q u e to efficient c o d i n g of 0165-1684/82/0000-0000/$02.75

still a n d m o v i n g images. In fact most of m e t h o d s b a s e d on linear p r e d i c t i o n which h a v e b e e n e n v i s a g e d a n d i n v e s t i g a t e d for i m a g e signals are n o n a d a p t i v e (see for e x a m p l e [ 3 , 4 ] a n d the r e f e r e n c e s listed thereof). W h e r e a s , given that the i m a g e fields a r e s t r o n g l y n o n s t a t i o n a r y , b e t t e r results s h o u l d be e x p e c t e d f r o m a d a p t i v e methods.

O 1982 N o r t h - H o l l a n d

314

E. Conte et al. / Adaptice coding o[ mouing images

Apart from adaptive quantizers, which have been largely investigated, adaptive techniques have been devised recently that can be classified as 'switched predictors'. Here a small set of prediction modes is chosen in advance, each of them performing well if the signal has certain correlation properties; then, based on a statistical analysis, one switches among the prediction modes by selecting the one that locally performs best [5-9]. Another adaptive technique is 'frame replenishment' [10, 11]: images are partitioned into fixed and moving areas by a suitable estimation procedure and only moving areas are transmitted together with some address information. A refinement is the so called 'movement compensation' method, which accounts also for (estimated) direction and intensity of movement in moving areas to obtain a more accurate prediction [12, 13]. Less attention has been as yet devoted toward using a fixed prediction structure with coefficients adaptively changing to fit local activities of image signals. Haskell [14] investigated the performance of such system in reducing the redundancy of a videotelephonic signal made up of two consecutive frames. He found that using a 3-D predictor of order 2 2 - i.e. 22 pels both on the present frame and on the previous one were taken to predict the present p e l - the entropy of the prediction error reduced substantially as compared to the non adaptive case if the two frames were subdivided into blocks and the prediction coefficients were computed and updated at each block. Such result motivated the present work in which the ultimate attainments of an adaptive linear prediction system with 3-D block segmentation of images are investigated by studying how performances are affected by prediction order, size of blocks, quantization of prediction coefficients, quantization of prediction error. The analysis was carried out on a test signal made of a sequence of 16 not interleaved images; they form the central-and the most movingpart, 64 x 64 pels wide, of a sequence of 256 x 256 images digitized at Bell Labs (Judy), each pel being Signal Processing

uniformly quantized at unit steps in the luminance range (0.255). Statistical and spectral properties of this sequence can be found in [15, 16]. Section 2 of the present paper contains a short review of basic adaptive linear prediction techniques and of modifications required in passing to the multidimensional case. Section 3 deals with the influence of block size and prediction order on the first order entropy of prediction error. In Section 4 an optimum strategy for quantizing prediction coefficients is established and the influence of quantization error of prediction coefficients on the overall transmission rate is discussed. As a result a convenient choice of parameters of the coding scheme is proposed. Finally in Section 5 the quantization of the prediction error with a fidelity criterion is discussed and the compression achieved by a class of quantizers known to be optimum for DPCM is evaluated.

2. The linear predictive coding system A basic scheme of a predictive coding system with fixed coefficients is given in Fig. 1; x(n)

~

PRED~ CODER

~(n) I . . . .

J

DECODER

Fig. i. Basicschemeof apredictivecodingscheme. represents the present pel, £ ( n ) = ~ a k r ( n - k ) is an estimate of x (n) based on p previous pels with prediction coefficients ak, k = 1, 2 . . . . . p; r(n) is the reconstructed pel; e ( n ) = x ( n ) - £ ( n ) is the prediction is rearranged in a one-dimensional e(n) in O. Redundancy reduction is achieved by transmitting eq(n) and adding it to r(n) recomputed at the receiver side in a twin predictor. It is easy to check that assuming error-free transmission, the error inherent in this system is just

E. Conte et al. I Adaptive coding of moving t'mages

the quantization error e ( n ) - e q ( n ) [17]. Therefore, if, as in our case, x(n) is integer valued and, if, as we assume up to adverse notice, O is a uniform unit step quantizer, then r(n) is an exact replica of x(n). In a fixed prediction scheme the coefficients ak, k = 1, 2 . . . . . p, are chosen once and for all under a suitable criterion. In order to reduce transmission requirements, the ideal criterion would be to minimize the entropy of the prediction error. However such criterion would result in exceedingly cumbersome calculations and would not be suitable for practical implementation. Instead, using a suboptimum criterion of minimizing the power of the prediction error, or Mean Square Error, MSE, leads to a simpler solution, both analytically and implementationally. In fact, under the minimum MSE criterion, optimum coefficients are given by a system of linear equations, called 'normal equations':

3t5

In the present work the problem has been solved by first ordering pels according to their mean correlation with the present pel, and then selecting for prediction purposes the p most correlated pels. In order to keep the required memory to a practical manageable size, pels to be used for prediction are confined in the present and the previous image. Fig. 2 shows the ordering which results by estimating correlation coefficients as time averages over the whole available sequence. Estimated correlation coefficients rang from 0.998 (for the pel labelled 1 in the figure) to 0.974 (for the pel labelled 23). i-1 i i.1

__.:

:

:

',

:

1_ .

~4 S rl

S-~

J i.Tl,,l.m 112-i _-: ) 1,.1 )

J-tli._.Iill/

I_;.U-i

i

P

Y~ Rkiak = Roi,

i=1,2 ..... p

k=l

where

present image

Fig. 2. Map of correlation coefficients.

Rki = E{x(k )x (i)} is the correlation between x(k) and x(i). This technique is referred as 'linear prediction'. For practical purposes Rk~ can be estimated by a suitable time average. In the one-dimensional case, if x(n) is stationary, then the matrix {Rk~; k = l . . . . ,p; i = 1 . . . . . p} is Toeplitz; which allows for efficient computational methods in solving normal equations [1, 2]. However, in dealing with images, the Toeplitz property is destroyed when the multi-dimensional array of pels used for prediction is rearranged in a one-dimensional sequence in order to apply the linear prediction method. Another difficulty arises in applying linear prediction to images. In fact, unlike the one-dimensional case, where the prediction is based upon the p samples which immediately precede the present sample, in the 2-D and in the 3-D case there is no such natural choice of pels to be used for prediction purposes.

Admittedly this ordering may depend upon the sequence under consideration; however it can be conjectured that, once the prediction order has been fixed, any reasonable selection criterioncorrelation based or distance based-will attain essentially comparable results. Since TV and videotelephonic signals are strongly nonstationary, fixed prediction coefficients are nonoptimum in general with respect to local correlation properties of the same signals. Adaptivity is then to be achieved by subdividing sequences of images into 2-D or 3-D blocks and computing prediction coefficients in each block. Clearly, in order that the receiver be able to properly reconstruct the signal, part of the available channel capacity must be reserved to provide the receiver with periodically changing coefficients. Thus a basic scheme for an adaptive linear predictive coding system is depicted in Fig. 3. VOI. 4, NO. 4, July 1982

E. Conte et al. / Adaptive coding of moving images

316

xtnl

~

COMPUTE {a,} BLOCK BY BLOCK

e ~ t

:÷n~" ~lnj f-------~l r

rln~

1 ~

CODER

"

DECODER

Fig. 3. Basic scheme of an adaptive predictive coding system.

3. Effects of block size and predictor order on entropy of prediction error

4.2

frame difference signal

3.8

The choice of block size is affected by various factors. First, as mentioned above, given the present technological constraints, it seems appropriate to confine 3-D blocks into two successive frames, although this somewhat limits the exploitation of the temporal correlation. With regard to spatial width, blocks should be narrow enough that computed coefficients fit the local correlation properties of the signal; in fact the analysis showed that if such fit is maintained, then increasing the order of predictor reduces the MSE; otherwise, there is no advantage at all in considering predictor order larger than two or three, say, just as it was pointed out in the 2-D case by Habibi [18]. On the other hand, the smaller the block size, the larger the fraction of channel capacity to be reserved for transmitting the prediction coefficients. To investigate the way in which block size affects the overall transmission requirements, we first study the relationship between the block size S and the first order entropy of the prediction error H(e) for various predictor orders p. H(e) has been estimated by the quantity

3.4 30 2.6

where fi is the relative frequency in the whole sequence of the jth value of e (n). Results are given in Fig. 4. For ease of comparison, in Fig. 4(a) a Signal Processing

=13

2.2 1.8

1.4

I

I

I

27

I

25

S

4.0

'••.non

3.6 H(e)

I

213 211 29

(a)

adaptive

3.2

/S=2 9

2.8 2.4 2.0 1.6 I

- E fJ log2.6 i

1=7

H(e)

(b)

2

6

10

p

14

18

22

Fig. 4. Influence of block size (S) and predictor order (p) on entropy of prediction error.

317

E. Conte et aL / Adaptive coding of moving images

few horizontal lines are also drawn, corresponding to the entropy of the frame difference signal and to the entropies of fixed-prediction errors. We note (see also Fig. 4(b)) that, given the memory constraint mentioned above, such nonadaptive prediction modes compare favourably with mere 'time-axis' adaptation (corresponding to S = 64 x 64 x 2 = 2~3); on the other hand 'spatial' adaptation is appealing since the entropy of prediction error decreases steadily as spatial size of blocks becomes finer and finer. However, as we shall see shortly by taking into account the transmission requirements for prediction coefficients, there is no convenience in using block size smaller than those considered in Fig. 4(a). We also note from Fig. 4(a) that the smaller is S, the more effective is the increase of p in reducing the entropy. A more direct picture of the relationship between p and H(e) is offered in Fig. 4(b). Although H(e) decreases while increasing p, the rate of decrease becomes rapidly small enough that, at least for reasonable block size, no advantage is to be expected in using predictor orders larger than 13 or so. This will be confirmed in a quantitative way in next section.

coefficients is kept at a minimum. Since the entropy of the prediction error is monotonically related to the MSE, an equivalent, but mathematically more tractable criterion is that of allocating B bits over the coefficients so as to minimize the increase of MSE over its minimum. In fact, let p be this minimum, attained when exact solution to normal equations are provided. It is easy to check that

p = Roo + ~, akRok k

where the ak's are exact solutions of normal equations. If round-off errors Aak ; k = 1, 2 . . . . . p are introduced, the corresponding increase in MSE is approximately given by

Assuming that each at is uniformly quantized,

Aak is not greater than one half the quantization step. Therefore, if the semi-range of ak is:

Ak

= ~(ak,max -- ale.rain)

and if bk bits are used for quantizing ak, then: [Aak [max= Ak 2 -b~.

4. Quantization properties of prediction coefficients Prediction coefficients must be quantized in order to be transmitted in a digital communication system like that in Fig. 2. After the round-off process inherent in A / D conversion, normal equation cannot be satisfied exactly, so that we expect an increase of MSE, and consequently of H(e), over the value reported in Fig. 4, with unquantized coefficients. Let us assume that a given slice of available channel capacity has been reserved for transmitting the quantized ai's. In order to make optimum use of this slice, we adopt the criterion of allocating the available number of bits B, say, over the coefficients in such a way that the increase of entropy H'(e)-H(e) due to quantization of

Correspondingly, the maximum increase of MSE is

IApI~ = ~. rkAk2 -b~. k

Then, a minimax approach to the solution of our optimization problem is that of minimizing lAp[max with respect to the bk'S, under the constraint Y bk = B . This is a simple problem in constrained minimization [19] and the solution is bl = B + log2 Al[rl[ _ 1 ~ log2 Ak]rk[,

p

pk

k=1,2

.....

p,

b~b~= 1 + ~ log~ IA,, Jr,,I/A, Ir~I). Vol. J,.No. 4. Jul~ 1982

318

E. Conte et al. / Adaptice coding of moving images

In our case the ratio bk/bl turns out to be approximately unity for all k (for example, for p = 7 and B = 7 0 bits it varies between 0.9 and 1.1); therefore a convenient strategy under the given criterion simply amounts to allocate uniformly the available bit resource over the a~'s. Results relative to the effect of quantizing the ak'S with a uniform number of bits on the entropy of the prediction error H'(e) are depicted in Fig. 5,

Fig. 6 the behaviour of R versus b for fixed p and S, and the same versus S for fixed p and b are reported. Note that the minimum is quite broad, and that for all 'good' choices of b, p, and S - i.e. those ensuring an overall rate R close to the m i n i m u m - t h e transmission rate is for the most part devoted to resolve the residual entropy H'(e), and only a small fraction helps in providing the receiver with the updated coefficients.

3.3

3.3 3.1 H'( e

)

I S:29=16 =16~2

3.2

29

0.,

~,7

3.0 2.5

v-9

10

11

12

~ .

8

Fig. 5. Influenceon the entropy of prediction error of prediction coefficientsquantization.

9

p = 13

and achieves a minimum rate R = 2.95 bits/pel (H'(e) = 2.70 bits/pel; bp/S = 0.25 bits/pel). In Sillnal Processing

12

13

4.0

I

P:I3 I

b:lO bits / c o e f f

R 3.6

3.2

2.8 2

TM

212

2TO

2s

26

(b)

the second term in the RHS being the number of bits per pel required to transmit the prediction coefficients. Both components are affected by the chice of p, b, and S, but in an antisymmetric way; in fact, just opposite to the term bp/S, H'(e) decreases increasing p and b while increases increasing S. An optimum choice of b, p, and S, in the sense of minimizing R, has been determined by heuristic search; it is: b = 10 bits/coeff.,

11 b

2 16

R = H'(e)+ bp/S bits/pel,

10

(a) 4.2

where b = B / p is the number of bits used for quantizing each of the ak'S, k = 1, 2 . . . . . p. We note that if 10 bits or more are used for coding each prediction coefficient, then H'(e) is essentially the same as H(e). The overall bit rate required in order that the receiver reconstruct an exact replica of x(n) is:

j J

_~ _.= ..

2.9

oo

b

S = 29 pels,

,

p:13

~ '

8

~D=13

3.1

D:IO 2.7

,

Fig. 6. Plots of the rate R (bits/pel) vs. block size ($) and vs. b.

5. Entropy reducing quantization of prediction error If the request for exact reproduction of the signal is relaxed, then the prediction error can be quantized more coarsely and the overall bit rate reduced correspondingly. The problem is then that of reducing R at its lowest value compatible with the fulfillment of a given fidelity criterion. However, despite many researches in this field, there is no good fidelity criterion or distortion

319

E. Conte et al. / Adaptive coding of moving images

The values of relevant parameters in above formulae, as well as the visual threshold t(s) have been taken from [20, 21]. The performances of a class of quantizers proposed in [20] have been investigated; a typical characteristic of this class is given in Fig. 7; other characteristics are derived therefrom by first affecting each decision level and each output level by a common scale factor and then performing minor changes required by the constraint that such levels be integer valued and that the range (0,255) be fully covered. With the same choice of S, b, and p as before, the rate R can be reduced while keeping the MSSTE to values less than one, which ensures, following [29], an excellent subjective quality of reproduction. For example, R = 2.5 bits/pel corresponds to MSSTE=0.15; and R = 1.8 bits/pel corresponds to MSSTE = 0.60. In Figs. 8(a) and 8(b) we report respectively the 8th picture in the input 8 bits/pel PCM sequence and the same coded at 2.5 bits/pel. Definitely no impairment is perceptible in the comparison. Slight further reduction of R appears possible only at the expense of a substantial increase of MSSTE; for example R = 1.68 bits/pel corresponds to MSSTE = 1.82. However still no picture quality degradation seems to be produced by such compression (Fig. 8(c)).

measure, especially for moving images. For still images Sharma and Netravali [20] put forth a distortion measure quite closely related to subjective picture evaluation, called Mean Square Supra Threshold Error (MSSTE):

M" I Sl

MSSTE = ~

[(s - yi)2 - t2(s)]

Sl-I

i=l

x u ( s - yi - t(s))w(s) ds

where M is the number of quantization intervals, s is the slope of the source signal, y is the slope of the reconstructed signal, w(. ) is the 'visibility function', t(. ) is the 'visual threshold function', u ( . ) is the unit step function. In a first instance we use the MSSTE as a criterion for evaluating the effect of quantization; in fact, although MSSTE was envisaged for still images, the resulting subjective quality would be not worse than that predictable on grounds of said relationship between MSSTE and subjective ranking of still images. To calculate MSSTE we follow an approach by Limb and Rubinstein [21] who expressed w(s) as a ratio p ~ ( s ) / m ( s ) , a > 0 , thus emphasizing the effect on visibility of two components: - a picture dependent one, namely the probability density function p (s) which has been estimated by the corresponding histogram; - a viewer dependent component, namely the masking function re(s), which, after psychophysical experiments, was shown to behave exponentially

0

3 I

2

29

26

The analysis of adaptive linear prediction coding on a sequence of video-telephonic images has

too, m l > 0 .

m(s)=moexp(-mls)

-I

6. Conclusions

6 4,

5

38

33

34

9 7

8

49

43

44

12 !0

II

64

56

57

17 14

15

83

73

7,t

19 106

9a

95

22 20

25 135

120 121

255

Fig. 7. Quantizer characteristic. Vol. 4. No. 4. July 1982

320

E. Conte et al. / Adaptive coding of moving images

Fig. 8. (a) Original picture (PCM, 8 bits/pel), (b) coded picture (2.5 bits/pel, MSSTE = 0.15), (c) coded picture (1.68 bits/pel, MSSTE = 1.82).

pointed out that redundancy is effectively reduced by using high order of predictors and by subdividing the sequence in small 3-D blocks in which predictor coefficients are adaptively computed. However, available algorithms for computing prediction coefficients in the multidimensional case are not as efficient as those envisaged in one dimension, so that advances in this area are required before the technique becomes suitable for real time operation. Moreover, increasing the order of prediction and decreasing the block size require more bit resource, that is, a greater fraction of the available channel capacity, to be reserved for transmission of prediction coefficients. It has been shown that the simple strategy of uniformly distributing this bit resource over any given number of coefficients is optimal in the sense of minimizing the effect of quantizing the coefficients. Using this strategy the overall rate can be compressed down to 2.95 bits/pel while ensuring that the receiver reconstruct exact replicas of the source in the absence of transmission error. Thus it can be conjectured that the above rate is a reasonable upper bound for the entropy of signals in the same class as the one we tested, that is, 'head and shoulders' videotelephonic sequences with 50% moving area. Signal Processing

By allowing transmission with a fidelity criterion the above rate is amenable to further reduction. In fact using a class of quantizers and a distortion measure known to be optimum for DPCM coding of still images, a rate of t.68 bits/pel could be achieved with excellent quality of reproduction even when reconstructed images were tested one at a time. This figure compares favourably with others reported in the literature on predictive image coding [3]. References [1] J. Makhoul, "Linear prediction: a tutorial review", Proc. IEEE, Vol. 63, April 1975, pp. 561-580. [2] J.D. Markel and A.H. Gray, Linear Prediction of Speech, Springer-Verlag, New York, 1976. [3] H.G. Mussmann, "Predictive image coding", Advances in Electronics and Electrophysics, Suppl. 12: Image Transmission Techniques, Academic Press, New York, 1979. [4] A.N. Netravali and J. O. Limb, "'Picture coding: a review", Proc. [EEE, Vol. 68, No. 3, March 1980, pp. 366--406. [5] W. Zshunke, "DPCM picture coding with adaptive prediction", IEEE Trans. Comm., Vol. COM-25, No. 11, Nov. 1977, pp. 1295-1302. [6] I.J. Dukhovich and S. B. O'Neal, "'A three-dimensional spatial nonlinear prediction for television", IEEE Trans. Comm., Vol. COM-26, No. 5, May 1978, pp. 578-583. [7] A.N. Netravali and C.B. Rubinstein, "Luminance adaptive coding of the TV chrominance signal", IEEE Trans. Comm., Vol. COM-27, No. 4, April 1979, pp. 703-710.

E. Conte et al. / Adaptive coding of moving images [8] H. Yasude and H. Kawanishi, "Predictor adaptive DPCM", Proc. SPIE Conf. on Application of Digital Image Processing, Vol. 149, Aug. 1978, pp. 189-195. [9] N.F. Maxemchuck and J. A. Stuller, "An adaptive intraframe DPCM codec based upon non-stationary image model", BSTJ, Vol. 68, No. 6, July-Aug. 1979, pp. 13951412, [10] F. W. Mounts, "Video encoding with conditional picture element replenishment", BSTJ, Vol. 48, No. 7, Sept. 1969, pp. 2545-2554. [11] B.G. Haskell, "Frame replenishment coding of television", Advances in Electronics and Electrophysics, Suppl. 12: Image Transmission Techniques, Academic Press, New York, 1979. [12] S. Brofferio and F. Rocca, "Interframe redundancy reduction of video signals generated by translating objects", IEEE Trans. Comm., Vol. COM-25, No. 4, April 1977, pp. 448--455. [13] A.N. Netravali and J.D. Robbins, "'Motion-compensated television coding: Part 1", BSTJ, Vol. 58, No. 3, March 1979, p. 631. [14] B.G. Haskell, "Entropy measurements for non-adaptive and adaptive frame-to-frame linear predictive coding of videotelephonic signals", BSTJ, Vol. 54, No. 6, July-Aug. 1975, pp. 1973-1986.

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[15] L. Arena and G. Zarone, "'3-D filtering of TV signals", Alta Frequenza, Vol. XLVI, Feb. 1977, pp. 108-116. [16] L. Arena and G. Zarone, "3-D Walsh-Hadamard filtering of video signals", Alta Frequenza, Vol. XLVII, May 1978, pp. 428--432. [17] B.S. Atal and M.R. Shroeder, "Adaptive predictive coding of speech signals", BSTJ, Vol. 49, No. 8, Oct. 1970, pp. 1973-1986. [18] A. Habibi, "Comparison of N-th order DPCM encoder with linear transformations and block-quantization techniques", IEEE Trans. Comm., Vol. COM-29, No. 6, Dec. 1971, pp. 948-956. [19] R.W. Visvanathan and J. Mackoul, "Ouantization properties of transmission parameters in linear predictive systems", IEEE Trans. ASSP, Vol. 24, No. 3, June 1975, pp. 301-311. [20] D. Sharma and A.N. Netravali, "Design of quantizers for DPCM coding of picture signals", IEEE Trans. Comm., Vol. COM-25, No. 11, Nov. 1977, pp. 1267-1274. [21] J. Limb and C. Rubinstein, "On the design of quantizers for DPCM coding: influence of the subjective testing methodology", IEEE Trans. Comm., Vol. COM-26, No. 5, May 1978, pp. 573-578.

Vol. 4, No. 4, July 1982