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Apple Computer Inc. v. Burst.com, Inc.

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IEEE TRANSACTIONS ON COMMUNICATIONS, V O L . COM-32, NO. 3,MARCH 1984

Scene Adaptive Coder
WEN-HSIUNG CHEN,
MEMBER, IEEE,-%D

WILLIAM K . PRATT, SENIOR MEMBER, IEEE

Abstract-An efficient single-pass adaptive bandwidth compression technique using the discrete cosine transform is described. The coding process involves a simple thresholding and normalization operation on the transform coefficients. Adaptivity is achieved by using a rate buffer for channel rate equalization. The buffer status and input rate are monitored to generate a feedback normalization factor. Excellent results are demonstrated for codine of color imaees at 0.4 bitshixel corresponding to real-time color television transmission over a 1.5 Mbit/s channel.

,

11. COSINE TRANSFORM REPRESENTATION
L

The two-dimensiona~ discrete cosine transform of a sequence f(i,k ) for i, k = 0 , 1, ..., N - 1, can be defined as 61 F(u,U) =

4C(U)C(U)

N-lN-'
j=o

N2
cos

cc
k=O

f(l> k )

I. INTRODUCTION
RANSFORM image coding, developed about 1.5 years ago, has been proven to be an efficient means of image coding' [ 1 ] - [ 6 ] . In the basic transform image coding concept, an image is divided into small blocks of pixels, and each block undergoes a two-dimensional transformation to produce an equal-sized array of transform coefficients. Among various transforms investigated for image coding applications, the cosine transform has emerged as the best candidate from the standpoint of compression factor and ease of implementation [ 71-[ 11 1. With the basic system, the array of transform coefficients is quantized and coded using a zonal coding strategy [ 31 ; the lowest spatial frequency coefficients, which generally possess the greatest energy, are quantized most finely, and the highest spatial frequency coefficients are quantized coarsely., Binary codes are assigned t o the quantization levels, and the code words are assembled in a buffer for transmission. At the receiver. inverse processes occur t o decode the received bit stream, 'and t o inierse transform the quantized transform coefficients t o reconstruct a block of pixels. The basic transform image coding concept, previously described, performs well on most natural scenes. A pixel coding rate of about 1.5 bits/pixel is achievable, with no apparent visual degradation. T o achieve lower coding rates, without increasing coding error, it is necessary to adaptively quantize transform coefficients so that those blocks of coefficients containing large amounts of energy are allocated more quantization levels and code bits than low energy blocks. In almost all adaptive transform coding designs to date, transforms are computed, and transform energy is measured or estimated on a first pass through the image. This information is then utilized t o determine the quantization levels and code words for a second pass [ 9 ] . With this scheme, compression factors can be reduced by a factor of t w o or more as compared t o nonadaptive coding. The practical difficulties are the memory required for the second pass and the complexity of the quantization algorithm. Both these problems are eliminated in the scene adaptive coder described in this paper. The scene adaptive coder is a single-pass adaptive coder of relative simplicity. The following sections describe the coding scheme and present subjective and quantitative performance evaluations. Paper approved by the Editor for Communication Theory of the IEEE Communications Society for publication after presentationat the International Conference on Communications, Philadelphia, PA, J u n e 1981. Manuscript received September 2, 1982; revised July 8, 1983. W. Chen is with Compression Labs, Inc., San Jose, CA 95131. W.K.Pratt is with VICOM Systems, Inc., San Jose, CA 95131.

2N

for

'3

= o,

C(w) =

I7
3

.'

N - 1, where
for w = 4 f o r w = 1,2;-.N-l.

1

The inverse transform is given by

f(i, k , =

xx .[
u=o

N-IN-1

u=o
(2i

c(")c(u)F(L',

'1

cos

cos i(2k ;)UT]

for j , k = 0, 1, -., N - 1. Among the class of transform possessing fast computational algorithms, the cosine transform has a superior energy compaction property [ 6 ] - [ 9 ] . The following sections present some other properties of the cosine transform, which are useful t o the subsequent discussion.

A . Statistical Description of D C T Coefficients Let the pixel array f(j, k ) represent a sample of a random process with zero mean represented in two's complement format over an integer range -M < f(j, k ) < (M - 1). The probability density of the cosine transform coefficients F(u, u ) has been modeled by a number of functions [3], 1121. Among them, the Laplacian density has been shown to provide the best fit [ 121. This function can be written as

'

,

(3)
where a(u, u ) denotes the standard deviation of a coefficient.

B. Coefficient Bound The maximum coefficient value for the cosine transform can be derived from (1) as
Fmax(O,O> = 2fmax and
16
' T f m a x <Frnax(u,

n

u> Q 2fmax

0090-6778/84/0300-0225$01.OO 0 1984 IEEE

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where f m a x is the maximum value

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TRANSACTIONS COMMUNICATIONS, IEEE ON

VOL. COM-32, NO. 3,

0). If the transform is performed in a vector length 0f.N 16, = A . Cosine Transform then Referring to the block diagram in Fig. 1, the original image Fmax (0,O) = 2fmax ( 5 4 is firstpartitionedinto16 X 16pixelblocks.Eachblock of data is then cosine transformed as defined by (1). The resultant u)= F,ax(u, (5b) 1.628fmax. transform coefficients F(u, u ) are stored in a register according to the zigzag scan of Fig. 2. Scanning the data in this fashion C. Mean Square Error Representation minimizes the usage of runlength codes during the subsequent Themeansquarequantizationerrorbetweenanoriginal coding process. image f(j, k ) and its reconstructed imagef(j, k ) can be written B: Thresholding as The transform coefficients in the register undergo a threshF ( 0 , 0), that old process in which all the coefficients, except are below the threshold are set to zero, and those coefficients above threshold subtracted the the are by threshold. This results in The unitary property of the cosine transform allows one to express the MSE in the transform domain as

k ) and Fmax the maximum value for the coefficient is

of the discrete array

F(u, lowing sections describe the coding algorithm in greater detail.

transformed to reconstruct the output pixel block. The fol-

which reduces to

where T is the threshold. Fig. 3 shows a plot of the percentage of coefficients below a threshold versus the threshold for the As demonoriginalimagesexhibitedinFigs.8(a)and9(a). strated, more than 90 percent of the coefficients have absolute magnitude of less than a value of 3, even though the maximum

coefficient value could be as large as 1.628 fmax [see 5(b)]. Therefore, the thresholding process indicated in (10) will set a * p(x; u , u ) dx (8) majorportionofcoefficientstozero,andtherebylimitthe where p(x; u , u ) is the probability density function,D , is a Set numberofcoefficientstobequantized.Thevalue of the of decision levels, and x , is asetofreconstruction levels. threshold varies with respect to the globally desired bit rate. With Laplacian modeling of the probability density function, However, it can be adjusted locally on a block-to-block basis if desired. as represented in(3), the MSE becomes

C. Normalization and Quantization
The thresholdsubtractedtransformcoefficients F T ( u , U) are scaled by a feedback normalization factor D from the output rate buffer according to the relation

This result has been verified by computer simulation.

111. SCENEADAPTIVE CODER
Fig. 1 contains a block diagram of the scene adaptive coder. In operation, the input image undergoes a cosine transform in 16 X 16 pixel blocks. An initial threshold is established, and those transform coefficients whose magnitudes greater are than the threshold are scaled according to a feedback parameter from the output rate buffer. The scaled coefficients are quantized, Huffman coded, and fed into the rate buffer. The rate buffer operates with a variable rate input, dependent upon the instantaneous image energy, and a constant channel output rate. The buffer status (fullness) and input rate are monitored to generate the coefficient scaling factor. At the receiver, the received fixed rate data are fed to a rate buffer that generates Huffmancodewordsatavariableratetothedecoder.The decoded transform coefficients are then inverse normalized by the feedback parameter, added to the threshold, and inverse

The scaling process adjusts the range of the coefficients such thatadesirednumber of codebitscan be usedduringthe coding process. The quantization process is simply a floating point to inNo decision reconstruction and teger roundoff conversion. tables are required. Therefore, there is a significant simplification and saving for a hardware implementation. Because many of the threshold subtracted coefficients areof fractional value, the roundoff process will set some of the coefficients to zero and leave only a limited number of significant coefficients t o beamplitudecoded.Thequantizedcoefficientscannowbe represented as
~ T N ( U , = integer u) part

of [ F T N ( u ,u )

+ 0.51.(12)

.

It should be noted that a lower bound has to be set for the normalization factor in order to introduce meaningful transform coefficients to the coder. This lower bound is dependent upon how accurately the cosine transform is computed. Generallyspeaking,settingtheminimumvalueof D to unity is sufficientformost of thecompressionapplications.Inthis case, the worst-case quantization error can be obtained from = n - 0.5, D , = n 0.5, and X , = n. (9) by letting D,-

+

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'-JET

FIGHTER

Fig. 2. Thus

Zigzagscanof

cosine transform coefficients.

MSE = ;
4

1 N-IN-1

u=o

u=o

x

e ( u , u>
Fig. 3.
0.10

1

2 3 COEFFICIENT THRESHOLD

208

Distribution of cosine transform coefficients.

1/12
0.08

+ l)n]exp

[

-fi(n

0.5) -~ 4 L 1 , u>
4 >
W

+

Fig. 4 shows the functional relationship between e(u, u ) and a(u, u). As can be seen, e(u, u ) is always less than 1/12. (Note:

I1

0.06

0.04

1/12 is the quantization error for a uniform density.) Therefore, the MSE represented by (13) is always less t h a n N 2 / 4 8 . For N = 16, this MSE is less than 16/3, which corresponds to anormalizederror of 0.0082percent.This MSE alsocorreof sponds to a peak-to-peak signal-to-quantization-noise ratio more than 40.9 dB, which is relatively insignificant. Thecoefficient F ( 0 , 0) intheupperleft-handcorner of each luminance transform block is proportional to the average luminance of thatblock.Becauseblock-to-blockluminance variationsresultingfromquantization of F ( 0 , 0) areeasily discerned visually, F ( 0 , 0) is linearly quantized and coded with a 9 bit code. As for the other nonzero coefficients, their magnitudes coded an are by amplitude lookup table, the and addresses of the coefficients are coded using a runlength lookup table. amplitude runlength The and lookup tables are simply Huffman codes derived from the histograms of typical

0.02

D.Coding

0

Fig. 4. Quantization error e(u, u ) versus coefficient standard deviation u(u, u ) at normalization factor of unity and coding threshold of zero. (Probability density function of the coefficient is assumed to be Laplacian.) transform coefficients. As demonstrated by the histograms of of Fig. 5 , the domination of low amplitudes and short runs zero-valued coefficients indicates that both Huffman tables are relatively insensitive to the type of input images and the de-

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 3, MARCH 1 9 8 4

TABLE I
HUFFMAN CODE TABLE FOR COEFFICIENT AMPLITUDE IN ABSOLUTE VALUE
NUMBER OF CODE BITS

AMPLITUDE
10,000 5.000
c
L Y

HUFFRAN CODES

.

1 2 3

w

".
Y

u

I

0 0

1.000 500

4 5 6 7

1 3 4 5 5 .
6

1 001 0111 00001
,

2 r
L

8 9 1 0
11

7 7

01101 011001 0000001
0110001

a
8
E 6+a
4
3

8

12
13 EOE RL PREFIX

00000001 01100001 000001+8 BITS

00000000 ' 01100000

0001 010

length of the table can be represented by
I
I

I

I

I

1

I

1

3

AMPLITUDEN I

5

7

ABSOLUTE VALUE

9

11

.13

15

L R = ( N 2 - 1)

(a)

50,000

1

10.000
5.000
"7

i

t

u
bY

z

;1.000
8
Y

! $
E

500

4

l10L . :1

3

RUNS OF CONSECUTIVE ZEROS 5 7 9 1

113

15

where N is the transform block size. The subtraction of unity is because the dc coefficient in individual blocks is separately encoded.Foratransformblock sizeof 16 x 16pixels,the length is 255.Inpractice,thelength of bothtablescanbe shortened to less than`30 entries by assigning Huffman codes toonlythelowamplitudecoefficientsandshortrunlengths (using a length elsewhere). fixed code Again, to due the domination of the low amplitude coefficients and short runlengths, the loss of coding efficiency is insignificant. Tables I and I1 show typical truncated Huffman code tables fortheamplitudeandrunlength,respectively.Itshouldbe noted that the amplitude codes in Table I include an "end of EOB block (EOB)" code and a "runlength prefix" code. The code is used to terminate coding of the block as soon as the last significant coefficient of the block is coded. The runlength prefixcode is required in order to distinguish the runlength code from the amplitude code. It should be noted that there are many ways to improve the coding efficiency of the coder. One way is to cut down the number of runs in the runlength coder. This can be accomplishedbyskipping single isolatedcoefficientswithabsolute magnitude of one, or by introducing an amplitude code for the isolated coefficients with zero amplitude. The coding improvement is generally quite significant; this is especiaIly true if t h e average coding rate is low.

Fig. 5. (a) Histogram of cosine transform coefficients obtained with threshold of zero and normalization factor of one. (b) Histogram of runs of consecutive zero counts obtained with threshold of zero and normalization factor of one. sired bit rate. This suggests that only two predetermined tables are needed for the coding process. The length of the Huffman table for the amplitude codes is afunctionofthenormalization,factorandthetransform bound represented (5b). a by Fortwo-dimensional cosine transform of 16 X 16 pixels, the maximum length is

(b)

E. Rate Buffer
The rate buffer in the SAC performs channel rate equalization. The buffer has a variable rate data input and a constant dataoutput.Thedifferentialsaremonitoredfromblockto block, and the status is converted into a scaling factor that is fed back to the normalizer. The buffer always forces the coder to adjust to the local coding variations, while ensuring global performance at a desired level. The general operation of a rate buffer is well documented [ 131, [ 141. The specific method used in this paper is described as follows. Let B ( m ) represent the number of bits into the rate buffer for mth the block let and $(m) represent normalized the buffer status at the end of the mth block (-0.5 < S ( m ) < 0.5). Then, B ( m ) and $ ( m ) can be written as

where D m i n is the minimum allowable normalization factor. For If,,, I and Dmin equal to 128 and 1, respectively, the length will be 208. As for the runlength of zero counts, the

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H U F F M A N CODE TABLE FOR THE N U M B E R O F CONSECUTIVE

TABLE I1

ZERO-VALUED COEFFICIENTS
RUN-LENGTH

I

NUMBER OF CODE BITS

1

HUFFMAN

CODE

into the buffer. However, there is no guarantee that the buffer will not underflow if a minimum allowable normalization factor is set t o a fixed value. Therefore, the buffer status has to be constantly monitored and, if the status is closer to -0.5, fill bits must be introduced into the channel.

2
3 4

I

7 8 9 10 1 1 12
15 1 6
17

5 6

3 3 4 4 5 5

5
5

5
6

;;
19 20

6
6 6

18

6 6 6
7 7

21 22
23 24

7
7 7

7

25 26 27 28

a
a

7 7

8 8

518

101 011 0101 0011 01000 10010 01001 10001 10011 001000 100000 001010 001001 100001 000011 001011 0000000 0000100 0000010 0001110 0000001 0000101 0000011 0001111 00011000 00011010 00011001 00011011 00010+8 BITS

IV. SCENE ADAPTIVECODING OF

@LOR

IMAGES

Fig.7containsablockdiagram of acolorimagecoding systembasedonthesceneadaptivecoder.Inthissystem, a color image, represented by tristimulus signals R ( j , k ) , G ( j , k ) , B ( j , k ) , is firstconvertedto a newthree-dimensionalspace defined by

S(m) = S(m
where

-

1)

+ [ B ( t n )- 256Rl I,

where Y ( j , k ) is the luminance signal and Z ( j , k ) and Q ( j , k ) are chrominance signals. Thisconversioncompactsmost of the signal energy into the Y plane such that more efficient coding can be accomplished [ 151. The I and Q chrominance planes are spatially averaged and `subsampled by a factor of 4 to 1 in both the horizontal and vertical directions. The luminance and subsampled chrominance images then are partitioned into 16 X 16 pixel blocks and coded by the SAC in the order of 3 2 Y , two I , and two Q sequences. At thereceiver,the receivedcodebitsaredecoded.Inversecosinetransformand inversecoordinateconversionsarethenperformed to reconstructthesourcetristimulus signals. Theinversecoordinate (17)conversion is describedby
r(j,

[ f i ~ ~ u)], quantized coefficients (u,

of the mth block, as

Hi.)
K L

defined in (1 2) Huffman coding function rateaverage coding size. buffer rate

G(j, k )

i'] [
=

1.000 0.956 1.000 -0.272

-0.647

::l1:]

[Y(j, k)]

Z(j, k )

.

(21)

B(j, k )

-1.106 1.000

Q(i,k )

The buffer status S(p) is used to select instantaneous an normalization factor D ( m ) according t o an empirically determined"normalizationfactorversusstatus"curve.Thisrelationship is described by

V. SIMULATION RESULTS

Computer simulations have been conducted evaluate to theperformance of the scene adaptive coder. original The test images shown in Figs. 8(a) and 9(a) are of size 5 1 2 X 5 12 pixelswitheachred,green,andbluetristimulusvalueuniformly quantized to 8 bits/pixel. Figs. 8(b) and 9(b) show the reconstructedimagesatacombinedaveragebitrate of 0.4 bits/pixel.Thisratecorrespondstoachannelbandwidth of In order to smooth out this instantaneous normalization factor1.5 Mbits a frame/s for15 intraframe coding system. The such that the desired normalization factor does not fluctuate excellent reconstruction of the images is clearly demonstrated. too much, a recursive filtering process is applied t o generate Table I11 tabulatesthe average mean square error between the original and the reconstructed images. Also included in the table is thepeak'signal-to-noiseratioforthereconstructed D ( m ) = cD(m - 1) ( 1 - c ) b ( m ) 9) (1 image.

+

where c is a constant with value less than unity. The desired operating conditions for the rate buffer algorithmare:a)thefeedbacknormalizationfactor isas stable as possible; b) the buffer status is able t o converge rapidly and stay asclose to the half full position ( S ( m ) = 0) as possible. Both these conditions may be satisfied using the above set of of thenormalization equations.Fig. 6 showstypicalvalues factor and-buffer status as a function of block indexes for the images shown in Figs. 8(a> and 9(a). The rate buffer can be guaranteed not to overflow. This is because the `normalization factor can get very large within a few blocks of operation, and effectively limit the data going

VI. SUMMARY
The scene adaptive coder described herein encodes cosine transform coefficients in a simple. manner. The coding process involves only thresholdjng, normalization, roundoff, and rate bufferequalization.Theperformance of thecoder is quite good in terms of mean square error and subjective evaluation. Because the coding process is dependent upon the instantaneous coefficient content inside the block and the accumulated rate buffer content, it is well suited for intraframe coding of moving images. Compression At Labs, the Inc., co'der has been implemented real-time with hardware code to NTSC color video at a channel rate 1.5 Mbits/s. of

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'F

. ........................... . ............ . ......... ...... . . . . . . . . . ......

.......

/

JET FIGHTER

I

0`

BLOCK INDEX

(a)

-0.4 -0.5

I
Fig. 6 .

BLOCK INDEX

(b) Buffer status for the last 114 blocks of images shown in Figs. 8(a) and 9(a).

-

SCENE ADAPTIVE DECODING

F(u.\)

INVERSE F ( ~ )BLOCK , COSINE + DEMULTIXFORM PLEXING

-

h

Y(J.K)-R(J.K)
I ( J * K ) INVERSE

MEllpaA~~m

-

COORDINATE CONVERSION

11 m *1

2(J,K) -

h

2(J,K)

%J,K)

Pig. 7.

Cosine transform color image coding/decoding system.

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Fig. 9.

(b) Cosinetransformsceneadaptivecoding.(a)Original.(b)Reconstructed image at 0.4 bits/pixel.

Fig. 8.

Cosinetransformsceneadaptivecoding.(a)Original.(b)Reconstructed image at 0.4 bits/pixel.

TABLE 111 MEAN SQUARE ERROR BETWEEN ORIGINAL AND RECONSTRUCTED IMAGES AT0.4 BITS/PIXEL; MEAN SQUARE ERROR IS COMPUTED AT THE INPUT O F CODER AND THE OUTPUT O F DECODER WITHfmax(j, k) NORMALIZED TO 1

0.0359% 0.0521%

34.45 DE 32.84 DB

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ACKNOWLEDGMENT
The authors wish t o express their sincere gratitude to Dr. A. Tescher for his valuable discussions and suggestions in the area of rate buffer stabjlizations.

REFERENCES
H.C.AndrewsandW. K . Pratt,"Fouriertransformcoding of images," in Proc.Hawaii Inr. Conf. Svst. Sci., Jan.1968, pp 677-678. W. K. Pratt, J. Kane.and H. C. Andrews."Hadamardtransform image coding." Proc. IEEE. vol. 57. pp.58-68, Jan. 1969. W. K . Pratt. W. H. Chen, and R. Welch. "Slant transform image vol. COM-22,,pp. 1075-1093, coding," IEEETrans.Commun., Aug.1974. A. Habibi and P. A . Wintz, "Image coding by linear transformaIEEETrans.Commun.Technol., tion block and quantization." vol. COM-19. pp. 5 0 6 2 . Feb. 1971. NewYork: Wiley-lnterW. K. Pratt, DigitalImageProcessing. science,1978. N . Ahmed, T. Natarjan,and K. R.Rao."Discretecosinetransform." IEEE Trans. Comput.. vol. C-23, pp. 90-93. Jan. 1974. M . Hamidiand J . Pearl, "Comparison of thecosineand Fowler transforms of Markov-Isignals," IEEETrans.Acousr.,Speech, Signal Processing, vol. ASSP-24. pp. 428429, Oct. 1976. A.K.Jain."Asinusoidal family of unitarytransforms," IEEE Trans. Pattern Anal. Mach. I n t e l l . , vol. PAMI-I. Oct. 1979. W. H. Chen and C. .H. Smith, "Adaptive coding of monochrome IEEETrans.Commun.. vol. COM-25. pp. andcolorimages," 1285-1292,Nov.1977. W. H. Chen. C. H. Smith, and S . Fralick, "A fast computational IEEE Trans. Comalgorithm for the discrete cosine transform," m u n . , vol. COM-25.pp. 1004-1009, Sept.1977. R. M. Haralick, "A storage efficient way to implement the discrete cosine transform." fEEE Trans. Comput..vol. C-25, pp. 764-765, July1976. R.C.Reininger and J.D.Gibson."Distributions of the twodimensional DCT coefficients for images."IEEE Trans. Commun., vol. COM-31, pp. 835-839. June 1983. A. Tescher. "Rate adaptive communication," in Proc. Nut. Telecommun. Conj., 1978. -, "A dual transform coding algorithm," in Proc. Nut. Telecommun. C o n j . . 1980.

*
William K. Pratt (S'57-M'61-SM'75) received the B.S. degree in electricalengineeringfrom BradleyUniversity.Peoria, 1L. in1959andthe M . S . and Ph.D. degrees in electrical engineering
from the University of Southern California, Los Angeles, in 1961 and 1965. respectively. He received Master's Doctoral and FellowshipsfromHughesAircraftCompanyand was employed there from 1959 to 1965. He became an Assistant Professor of ElectricalEngineeringat the University of Southern California in 1965, an Associate Professor in 1969. and a Full Professor in 1975. At U.S.C. he was the Director of the Image Processing Institute and of the Engineering Computer Laboratory. He is now the President and Chairman of the Board of Vicom Systems, Inc., San Jose, CA. I n 1976 Dr. Pratt was awarded a Gugenheim Fellowship for research and image analysis techniques. He is a member of Sigma Tau, Omicron Delta Kappa, and Sigma Xi.
. I

W . K. Pratt, "Spatialtransformcoding o f colorimages,'' lEEE Trans. Commun. Technol.. vol. COM-19, pp. 980-992. Dec. 1971.

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--

\

(b) Fig. 5. Cosine transform scene adaptive coding. (a) Original; (b) Reconstructed image at 0.4 bits/pixel.

*

Cosine transform ,scene adaptive coding. (a) Original. (b) Reconstructed image at 0.4 bits/pixel.

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