DAVID R. COMSTOCK AND JERRY D. GIBSON struction error (MSE) contributed by a channel error in each individual bit. A theoretical expression is given ...
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Hamming Coding of DCT-Compressed Images OverNoisy Channels DAVID R. COMSTOCK AND JERRY D. GIBSON
struction error (MSE) contributed by a channel error in each individual of bit. A theoreticalexpression i s given which allows thenumber rate to be protectedbits to achieve minimum MSE foreachcode computed. By comparing these minima, the best code and bit allocation can be found. Objective and subjective performance results indicate that using the (7, 4) Hamming code to protect the most important 2D-DCT coefficients can substantially improvereconstructedimagequality at a BER of Furthermore,the allocationof 33 out of the256 bits per block to chahnel coding does not noticeablydegrade reconstructed image quality in the absence of channel errors.
I. INTRODUCTION I n t h e tFansmissionofimagesoveranoisychannelusing transform source coding, reconstructed image quality is substantiallydegradedbychannelerrors. As aresult,fornoisy channelapplicationsitisnecessary to correctthechannel errors or to devise methodsforreducingtheeffects of the errors.Effortsinthislattercategoryincludetheworkby Ngan and Steele [ 11, Mitchell and Tabatabai [ 21 , and Reininger and Gibson [ 31 . The research describedin this correspondence is concerned .with the former approach, namely, forward error correction of transmission errors. Previous work on forin conjunction with ward error correcting (FEC) codes used the discrete cosine transform (DCT) over noisy channels has been perfoTmed b yD u r y e a [ 4 ] andModestino,Daut, and Vickers [ 5 1 . Duryea [4], conducted theoretical and simulation studies of threeconvolutionalcodesandthreeblockcodes.Thethree block codes studied were the (3, 1) repetition or majority vote code, the (7, 4) Hamming code, and the (23, 12) Golay code. For hissimulationstudies,Duryeausesabiterrorrateof andconsidersonlytwoerrorprotectionschemes,with the (7, 4) Hamming code. In the first scheme, a (7,4) Hamming code was applied t o all bits, while in the second method, only a 6 by 6 square block of the lowest frequency DCT coefficients were protected by the (7, 4) Hamming code. While the mean squared error was reduced, the quality of the reconstructed image was not clearly improved. Modestino, Daut, and Vickers [ 51 primarily investigate CORvolutionalcodes,althoughtheybrieflyconsideran (8, 4) Hamming code and a (24, 12) Golay code. They consider the Paper approved by the Editor for Signal Processing and Communication Electronics of the IEEE CommunicationsSociety for publication after presentation at the International Conferenceon Communications, Boston, MA, June 1983. Manuscript received July 14, 1982; revised August 2, 1983. This work was supported in part by the Office of NavalResearch Statistics and Probability Program under Contract N00014-81-K-0299. D. R. Comstock was with the Department of Electrical Engineering, Texas He is now with Teledyne A&M University,CollegeStation,TX77843. Geotech, Garland, TX 75041. A&M J. D. Gibson is with the Department of Electrical Engineering, Texas University, College Station, TX 77843.
three options of coding all bits of each coefficient the same, coding each bit of a specified coefficient the same with variation between coefficients, and coding each bit of each coefficientdifferently.Furthermore,theirworkemphasizesrate 1/12 short-constraint length convolutional codes which allow the use of Viterbidecoding(shortconstraintlength)butlimit channel coding flexibility (rate l/n). Their results indicate that for noisy channels there isa distinct advantage t o allocating additional channel bandwidth t o channel coding rather than to source coding. The research described in the correspondence is an extensive study of using Hamming codes with the two-dimensional DCT (2D-DCT) at atransmitteddatarate of 1 bit/pixelovera binary symmetric channel (BSC). .The system configuration of interest is shown in Fig. 1. The (7, 4), (15, ll), and (31, 26) Hamming codes are used to protect the most important bits in each transformed block, where the most important bits are determined by calculatingthemeansquaredreconstruction error contributed by a channel error in each individual bit. A theoreticalexpressionforthemeansquaredreconstruction error isgiven whichallows one to determine the number of protected bits to achieve minimum error for each code rate. Therefore, unlike previous work on joint source-channel coding, we provide a specific theoretical technique for determining how many bits to protect and which bits to protect for each block code. In particular, the “most important” bits to be protected are identified by using quantities called A-factors [ I O ] , which account for the effects of the source distribution, the quantizer design, and the binary word assignment to quantizeroutput levels. The design biterrorrate of interestis Monte Carlo simulation results and reconstructed images are presented to demonstrate the utility of the method. 11. TWO-DIMENSIONAL DCT The monochrome images used for this work consist of 256 by 256 pixels with each pixel represented by an 8 bit word. Thetwo-dimensionalDCT isa populartransformforimage compression at, 1 bit/pixel [ 61 , and it is considered exclusively in this work. The 2D-DCT is defined by
x
N- 1 N- 1
2 F(u, u ) = - C(.)C(U) N ‘
‘‘cos “cos
j=O
[
(2i
f(i, k ) k=O
+ 1)nu 22 N N
1] 1[ 1] (2k
‘Os ‘Os
+ 1)nu 22 N N
for u , u = 0 , 1, ..:, N - 1 , c ( 0 ) = 1/45,and 1, 2, ..., N - 1. The’inverse 2D-DCT is
C(U)
(1)
= 1 for u =
for j , k = 0, 1, *.., N - 1. One advantage of the 2D-DCTis that it can be computed using “fast” algorithms, To use the 2D-DCT in a data compression system, F ( u , u ) in (1) is calculated over an N by N block ( N = 16 for this paper), the lowest energy coefficients are discarded, and the highestenergycoefficientsarequantizedandcoded.The scheme used for bit allocation is due to W&tz and Kurtenbach
0090-6778/84/0700-0856$01.OO 0 1984 IEEE
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--
Binary Channel
1 E
+
I.
(bmary)
Inverse Quonti-
Channel
Decoding
Transform
Fig. 1. Block diagramofimageprocessingsystem with transformsource coding and channel coding.,
TABLE I QUANTIZATION ERRORAND SINGLE BIT A-FACTORS FOR 1-8 BIT GAUSSIANQUANTIZERS Single Bit A-Factors A5
A4
NO.
A3
of B i t s
A2
A1
1
3.63 x 1 0 6
2
1.17 x 1 0 - l1.117 3.532
3
4
3.45 x 10.‘
A6
A7
A8
2.547
3.858 ,3763 1.506
9.50 ,4525 x ,1095 1.828 3.961
5
2.50 x
6
1.04 x
2.072 3.997
.02968,1216 ,5096
4.001 ,01083 ,04333 2.772 ,1733 ,6933
,003227 ,01292 ,2067 x,8267 3.307 4.007 7 ,051683.04 8
8.88 x
4.009 ,9650 3.860
[ 7 ] . l Thecoefficientswerequantizedusingminimummean squared error (MMSE), nonuniform, Gaussian-assumption quantizers for up to 32 levels [8] and using MMSE, Gaussianassumption, uniform quantizers for more than 3 2 levels. The quantizer output levels are represented digitally by the folded binarycode(FBC).Intheabsenceofchannelerrors,this method produces good quality reconstructed images at a rate of 1 bit/pixel.
III. CHANNEL CODING A biterrorrate(BER) ofcausessubstantialdegradation in the 2D-DCT coded images. To reduce or remove these channel error effects, (7, 4), (1 5, 1 l), and (3 1, 26) Hamming codesareinvestigated.Inorder toapplythesecodes,it is ne,cessary t o determine how many bits to protect and which bitsaremostimportant toprotect.Thelatterquestion is answered first by calculating the mean squared reconstruction error contributed by “flipping” each of the 256 bits in a block and then ranking these bits fromlargest t o smallest error. To illustrate the calculation of the required mean squared k(i,j ) reconstruction error, consider the 2D-DCT’coefficient withenergy(meansquaredvalue) 0 2 ( i , j ) , andassumethat bitshavebeen allocated to represent this coefficient. Let F(i, j ) be the M-bit quantized version of F ( i , j ) . The single bit A-factors in Table I, designated A , - A , , denote the normalized reconstruction error average error-in bits 1-8, respectively, To compute the mean squared 1 in the representation of r_econstruction error due tobit F(i,j ) , i t is onlynecessary to multiply A I by 0 2 ( i , j ) . The product is themeansquaredreconstructionerrordueto a
ults. the
e
Themaximumnumber of bitsallocated to eachcoefficientwas limited to eight due to a specific application. This constraintshould not reduce the utility of I
,2413
.Ob031
,01508
.I303774
,0009436
channel-induced binary error in bit 1 of the quantized coefficient F(i, j ) . Similarly, the reconstruction error due to a binary error in bit 2 alone is A 2 0 2 ( i ,j ) . This process is continued for all M bits allocated t o F(i,j ) , and then the procedure is repeatedfortheremaining2D-DCTcoefficients.Only single bit errors are considered. The resulting reconstruction errors are then ranked from largest t o smallest, with the bits corresponding to the largest errors deemed t o be the most important. The calculations required to determine how many bits to protect are slightly more complicated and are developed in the following paragraphs. Since the DCTis an unitary transform, the mean squared error (MSE) for the reconstructed image in the spatial domain is the same as the MSE in the transform domain. The MSE between.an uncoded DCT coefficient F(i, j ) and its received version F(i, j ) is given by
wheretheexpectation is taken over the source and channel probabilitymeasures.Equation(3)canbeseparatedintotwo components~ 2 . . E (2,~) -
[ca2(i,j)+ eq2(i,j)lu2(i,j)
(4)
where ea2 is the MSE contributed by the channel, f a 2 is the mean squared quantization (which, as usual, includes truncation or clipping error), and u2 is the mean squared value of the coefficient. Each of the terms in brackets in (4) is normalized t o 1. Themeansquaredquantizationerror is dependent on the number of bits assigned to the Particular coefficient and the probability density of the coefficient. The eq2 is in variance, Table nonuniform unitI for termgiven
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Gaussian quantizers with 1-5 bits and for unit variance, uniform Gaussian quantizers for 6-8 bits. The MSE due to the channelis given by
0.028C L
e W U
: 0.0266 0 a
rs v)
C 0
r” 0
._ -
0.0252
0
E z 0
where A I is called the A-factor associated with error sequence
.-0
zl [ 101. The A-factoris the average reconstruction error power
:0.0230 0 + +
zl for a given quantizer caused by the digital error sequence and binary code. For’P, < l o p 2 , the probability of two or more independent channel errors in z I is small, so ( 5 ) can be simplified t o
C
’
0.0224 0.0
27.0 65.0 46.0
04.0
Number o f Coded Bits
Fig. 2.
Normalized mean squared error versus number of bits protected by (7,4) Hamming code for “Girl” image with 10 -*error rate.
where the channel errors are assumed independent with equal probab,ility P, and the first ‘M A-factors are defined to corI lists the single bit A-factors respondto single-bit errors. Table fortheGaussianquantizerschosenandtheFBC.Giventhe optimum bit allocation for the 2D-DCT coefficients of a particularimage, (6) andthedatainTable I canbeusedin(4)to .compute the theoretical mean squared reconstruction error for a desired bit error probability (P,)when n o channel coding is used. For those bits protected by channel coding, the probability P,, so the channel MSE expression beof bit error becomes comes r
ea2(i,i> 2 p,,
L
M
AI
I= 1
x
0 0320
+ pe
AI
(7)
l=r+ 1
where r bits are assumed protected and errors with coding are W ‘independent and equally likely. It is noted that for a Gaussian E noiseassumption,the r largestA-factorscorrespondtothe first i- most significant bits in a codeword. However, for other I 1 1 noise distributions, this may not. be true. The proposed 0 0300 22.0 47.0 72.0 96.0 121.0 method, as represented by ( 7 ) , is to protect those bits with the Number o f Coded Bits largestA-factors.FortheHammingcodesconsidered in this paper, Pec can be found by direct enumeration using a cornFig. 3. Normalized mean squared error versus number of bits protected by puter or by a derivation using the standard array of the partic(15, 11) Hammingcode for “Girl”image with lo-* errorrate. ular code of interest [ 11 1 . Since the DCT coefficients are approximately uncorrelated, thetotal MSE for an N by N block is given by of bits protected at design a BER of lo-*. Theminimum Table value in listed of each is curves of these 11. be As can Table from seen 11, performthebest ( 7 , the 4) yields code Et2 = e2(j,O. ( 8 ) ance, optimal the andnumber ofcode “Girl” the for bits to image is 44. Therefore, for the simulation studies, it was deP O ’ ’ i=O cided to protect .44 bits using the (7,.4) Hamming code. Thenormalizedmeansquaredreproductionerror(NMSE) is Since thetotaltransmitteddatarate is fixedat1bithixel, the total error in (8) divided by the sum of the variances of the using 33 bits per block for channel coding reduces the number It 223. tocoding sourceavailable for of bits is important to coefficients. ascertain how much degradation in image quality is imposed The question of how many bits to protect, or equivalently, by this reduction of source coding bits when the channel is how to choose r in (7), involves a tradeoff between bits alloerrorfree.Fig,5(a)showstheoriginal“Girl”image,using 8 catedtosourcecodingandbitsallocatedtochannelcoding with the overall rate constrained to 1 bit/pixel. By using (8) bits/pixel.Fig.5(b)isthereconstructedimageat1bit/pixel with (4) and ( 7 ) , the NMSE as the number of channel coding withnobitsallocatedtochannelcodingandazeroBER, while Fig. 5(c) is the compressed image at 1, bit/pixel with 33 bitsisincreasedcanbecomputed,Figs. 2-4 forthe“Girl” As is evident, image show how the NMSE varies as a function of the number bitsperblockallocatedtochannelcoding.
I
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 7 , JULY 1984
W
$ 0 3
CT
w
E 0
f
I
/ I
1
0.0470
I
I
I \ 0.0454 26 0
202.0
1430
850
Number of Coded h t r
260.0
.
Fig. 4. Normalized mean squared error versus number of bits protected by (31,26) Hamming code for “Girl” image with 10 -* error rate.
TABLE I1 NUMBER OF BITS CODED THAT ACHIEVESMINIMUM NMSE G i r Il m q e No. b i t s coded
r a t e ECode rror rate ~
4/7
10-2
11/15
10-2
26/31
,
44
NMSE 2 . 2 7 3 x 10.‘
66
3.036 x 10.‘
78
4 . 5 4 4 x 10-2
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860 20.0
[
Probability of Bit Error
Fig. 6 . SNR versus probability of bit error for ‘‘Girl’’ image with 44 bits protected per block by (7,4) Hamming code and without channel coding.
(C)
Fig. 7. Worst and best reconstructed noisy “Girl” images without and with (a) Worstcase withoutchannel coding. (b) channel coding (P,= Best case without channel coding. (c) Worst casewith channel coding. (d) Best case with channel coding.
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a BER of l o p 2 . Clearly, the channel coding scheme proposed here provides a substantial, noticeable improvement in reconstructed image quality. The visible errors in Fig. 7(c) are due to the inducement of more than one channel error in the bits of a codeword, which, sincethe(7,4)Hammingcodeis single-errorcorrecting, in Fig. causes a decoding error. There are many other errors 7(c)and(d)whicharenotevidentupon visualinspection since these errors occur in the lower energyDCT coefficients.
V. CONCLUSIONS The results presented in this paper indicate that using the (7, 4) Hamming code to protect the most important 2D-DCT coefficientscansubstantiallyimprovereconstructedimage quality at a BER of lop2. The novelty of the approach lies in the use of the A-factors to identify the “most important” bits to be protected. The A-factors include the effects of the source distribution, the quantizer design, and the binary word assignment t o quantizeroutput levels. A surprisingandimportant result which runs counter to conventional “wisdom” isthattheallocation of 33 outofthe256bitsperblock to channel coding does not noticeably degrade reconstructed image quality in the absence of channel errors. This fact seems t o bedueprimarilytotheproperty of transformcoding systems which “averages” source coding errors over the entire block. Mean squared error proves to be a useful design criterion, even though it is well known that subjective image quality and mean squared error are not always in agreement. Comparisons between theoretical and simulation results indicate that a good estimate of the probability density function of the DCT coefficients is necessary for the theoretical results to be accurate. Thisworkdemonstratesthatthejudiciouscombination .of source and channel coding methods can produce a data compressionsystemwhichhasboththequalityandrobustness necessary for realistic applications.
A One-Stage Look-Ahead Algorithm for Delta Modulators N. SCHEINBERG, E. FERIA, J. BARBA, AND D. L. SCHILLING
Abstract-This paper describes a one-stage look-ahead algorithm for adaptive delta modulators. The algorithm does not require thecalculation of two encoding paths nor does it require the decision circuitry to choose the optimum path.
I. INTRODUCTION Several different adaptive delta modulator (ADM) algorithms havebeendescribedintheliteratureoverthepast15years. Researchers [ 1 ] and[2] haveshownthattheperformance of the.se algorithmscanbeimprovedbyusing“look-ahead encloding” (also referred t o as “delayed decision encoding”). In this paper, wewill present a general technique that can be used with any of these algorithms to effect a one-stage lookahead. The technique presented in this paper has the advantage that it does not require the calculation of two encoding pathsnordoesitrequirethedecisioncircuitry t o choose theoptimumpaththatminimizesamean-squareerroror absolute error criterion. 11. ILLUSTRATION O F ALGORITHM VIA TWO EXAMPLES A. A Delta Modulation Example
Thealgorithm will beexplainedthroughthefollowing example. Let us assume that the delta modulator is the one proposed by Song [ 3 ] . The ADM is describedbytheequations below and Fig. l :
REFERENCES
r21
r31 r41 [5 I
181
[91
K. N. Ngan and R. Steele, “Enhancement of PCM and DPCM images corrupted by transmission errors,” IEEE Trans. Commun., vol. COM-30, pp. 257-265, Jan. 1982. 0. R. Mitchell andA. J. Tabatabai, “Channel error recovery for transform image coding,” IEEE Trans. Commun., vol. COM-29, pp. 1754-1762, Dec. 1981. R. C. Reininger and J. D. Gibson, “Soft decision demodulation and transform coding of images,” IEEE Trans. Commun., vol. COM-31, pp. 572-577, Apr. 1983. R. A.Duryea,“Performance of asourcekhannel encodedimagery transmission system,” M.S. thesis, Air Force Inst. Technol., WrightPatterson AFB, OH, Dec. 1979. J. W. Modestino, D. G. Daut, and A. L. Vickers, “Combined sourcecosine transform,” IEEE channelcoding of imagesusingtheblock Trans. Commun., vol. COM-29, pp. 1261-1274, Sept. 1981. N. Ahmed and K.R. Rao, Orthogonal Transformsfor Digital Signal Processing. New York:Springer-Verlag, 1975. P. A. Wintz and A. J. Kurtenbach, “Waveform error control in PCM telemetry,” IEEE Trans. Inform. Theory, vol. IT-14, pp. 650-661, Sept.1968. J. Max, “Quantizing for minimum distortion,” IRE Trans. Inform. Theory, vol. IT-6, pp. 7-12, Mar. 1960. R. C. Reininger and J. D. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun., vol. COM-31, pp. 835-839, June 1983. N. RydbeckandC. E.Sundberg, “Analysisof digitalerrors in nonlinear PCM systems,” IEEE Trans. Commun., vol. COM-24, pp.59-65, Jan. 1976. A. M. Michelson, “Thecalculation of post-decoding bit-error probabilities for binaryblock codes,” in Conf. Rec.,Nat. Telecommun. Conf., Dallas, TX, Nov. 29-Dec. 1, 1976, pp. 24.3-1-24.34.
where 1 ) $k+ 1 is an estimate of the signal sample x k + ; 2) Xk is the predicted value of Xk+ ; 4) Y, is astepsizeorcorrectioptermthat,whenadded t o x,, yields the estimate ofx,+,,Xk+l ; 4 ) b1 and b , areconstantschosen t o optimizethedelta modulator with typical values of b l = 1 and b , = 0.5; 5) Y, in is theminimumpositivevaluethat I Yk- I is allowed t o have “at time k” when Yk is being formed; note from (2) that the minimum value of I Y k I is Ymin(b, - b 2 ) where b l b2 ; and 6 ) E k is the output of the hard limiter and is the bit sent to the receiver.
>
Paper approved by the Editor for Signal Processing and Communication Electronics of the IEEE Communications Society for publication without oral presenation.ManuscriptreceivedSeptember 26, 1983; revisedJanuary 24, 1984. N. Scheinberg, J . Barba, and D. L. Schilling are with the Department of Electrical Engineering, City College of New York, New York, NY 10031. E. Feria is with the Department of Engineering, College of Staten Island, Staten Island, NY 10301.
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