1 .9 Coding of Two-tone Images 551 1 .9 Coding of Two-tone Images 551

Sec. 1 1 .9 Coding of Two-tone Images 551 Sec. 1 1 .9 Coding of Two-tone Images 551

k --+-- 0 0 2

t t t· t t

Prediction errors

State S0

Run

State S2

Figure

11.48 TUH method of predictive coding. The run length 1. of state s. means a prediction error has occurred after state s. has repeated

I, times.

due to large values of N. Experimentally, 4 to 7 pixel predictors have been found to

be adequate. Corresponding to the prediction rule of (11. 126), the minimized prediction error is

2N

Pe =

2: Pk min(qk> 1- qk)

k= I

If the random sequence u (m, n) is Markovian with respect to the prediction window W, then the run lengths for each state Sk are independent. Hence, the prediction­

error run lengths for each state (Fig. 11.47) can be coded by the truncated Huffman code, for example. This method has been called the Technical University of Hannover (TUH) code [le, p. 1425] .

Adaptive Predictors Adaptive predictors are useful in practice because the image data is generally

nonstationary. In general, any pattern classifier or a discriminant function could be used as a predictor. A simple classifier is a linear learning machine or adaptive

threshold logic unit (TLU), which calculates the threshold qk as a linear functional of the states of the pixels in the prediction window. Another type of pattern classifier is

a network of TLUs called layered machines and includes piecewise linear discrimi­ nant functions and the so-called cx-perceptron. A practical adaptive predictor uses a counter Ck of L bits for each state [43]. The counter runs from 0 to 2L - 1. The adaptive prediction rule is

- {l '

' - 0, if Ck < 2L-I (11.128) After prediction of a pixel has been performed, the counter is updated as

if C k u 2:::2L - l (m n)

Ck = {

min (Ck + l , 2L - 1), if u (m, n ) = l

552 Image Data Compression Chap. 11

TABLE 1 1 . 1 5

Compression Ratios of Different Binary Coding Algorithms One-dimensional codes

Two-dimensional codes Normal or high resolution

High resolution Truncated Modified RAC TUH

Normal resolution

code code READ Document code

Huffman Huffman code code

K=4 K=4 code, K = 2 K=4 K = 4 code, K = 4 1 13.62 17.28 16.53 16.67 20.32 15.71 19.68 24.66 19.77

11.10 14.31 16.45 14.55 The value L = 3 has been found to yield minimum prediction error for a typical

printed page. Comparison of Algorithms Table 11.15 shows a comparison of the compression ratios achievable by different

algorithms. The compression ratios for one-dimensional codes are independent of the vertical resolution. At normal resolution the two-dimensional codes improve the compression by only 10 to 30% over the modified Huffman code. At high resolution the improvements are 40 to 60% and are significant enough to warrant the use of these algorithms. Among the two-dimensional codes, the TUH predictive code is superior to the relative address techniques, especially for text information.

However, the latter are simpler to code. The CCITT READ code, which is a modification of the RAC, performs somewhat better.

Other Methods Algorithms that utilize higher-level information, such as whether the image con­

tains a known type (or font) of characters or graphics, line drawings, and the like, can be designed to obtain very high-compression ratios. For example, in the case of

printed text limited to the 128 ASCII characters, for instance, each character can

be coded by 7 bits. The coding technique would require a character recognition algorithm. Likewise, line drawings can be efficiently coded by boundary-following algorithms, such as chain codes, line segments, or splines. Algorithms discussed here are not directly useful for halftone images because the image area has been modulated by pseudorandom noise and thresholded thereafter. In all these cases special preprocessing and segmentation is required to code the data efficiently.

1 1 .10 COLOR AND MULTISPECTRAL IMAGE CODING Data compression techniques discussed so far can be generalized to color and

multispectral images, as shown in Fig. 11 .49. Each pixel is represented by a p x 1 Sec. 1 1 . 1 0

Color and Multispectral Image Coding

Component coder

R"

G Coordinate

8 coder Component

Figure 11.49 Component coding of color images. For multispectral images the input vector has a dimension greater than or equal to 2.

vector. For example, in the case of color, the input is a 3 x 1 vector containing the R, G,

B components. This vector is transformed to another coordinate system, where each component can be processed by an independent spatial coder. In coding color images, consideration should be given to the facts that (1) the luminance component (Y) has higher bandwidth than the chrominance compo­ nents (/, Q ) or ( U, V) and (2) the color-difference metric is non-Euclidean in these coordinates, that is, equal noise power in different color components is perceived

differently. In practical image coding schemes, the lower-bandwidth chrominance signals are sampled at correspondingly lower rates. Typically, the I and Q signals are sampled at one-third and one-sixth of the sampling rate of the luminance signal. Use of color-distance metric(s) is possible but has not been used in practical systems primarily because of the complexity of the color vision model (see Chapter 3).

An alternate method of coding color images is by processing the composite color signal. This is useful in broadcast applications, where it is desired to manage only one signal. However, since the luminance and color signals are not in the same

frequency band, the foregoing monochrome image coding techniques are not very efficient if applied directly. Typically, the composite signal is sampled at 3.fsc (the lowest integer multiple of subcarrier frequency above the Nyquist rate) or 4/"' and

............._ _ _ · - · -· ---- · - -­ D4 C4 84 A4

83 A3 03 C3 B� A;

--·-·-·-·-·-·-- -

(a) 3fsc - sampling (b) 4fsc - sampling Figure

11.50 Subcarrier phase relationships in sampled NTSC signal. Pixels hav­ ing the same labels have the same subcarrier phase.

Image Data Compression Chap. 11 Image Data Compression Chap. 11

relationships of neighboring pixels in the sampled NTSC signal. Due to the presence of the modulated subcarrier, higher-order predictors are required for DPCM of composite signals. Table 11.16 gives examples of predictors for predictive coding of the NTSC signal.

Table 1 1 . 17 lists the practical bit rates achievable by different coding algo­ rithms for broadcast quality reproduced images. These results show that the chromi­ nance components can be coded by as few as ! bit per pixel (via adaptive transform coding) to 1 bit per pixel (via DPCM).

Due to the flexibility in the design of coders, component coding performs somewhat better than composite coding and may well become the preferred choice with the advent of digital television.

For multispectral images, the input data is generally KL transformed in the temporal direction to obtain the principal components (see Section 7.6). Each

TABLE 1 1 . 1 6

Predictors for DPCM of Composite NTSC Signal. z- 1 = 1 pixel delay,

1 line delay, z -252N = 1 field delay, p (leak) :s 1 . Sampling

z -N =

rate Predictor, P(z)

!z -1 + pz-3 (1 - !z-1)

h -1 _ pz -262N (l - !z -1)

Hz -1 - z -2 + z -3) + Mz -4 + z -N (z-2 + z2)]

h -1 + pz-262N (l - h -1)

TABLE 1 1 .17

Typical Performance of Component Coding Algorithms on Color Images

Rate Average

per component rate Method

Components

bits/component/pixel bits/pixel PCM

Raw data

u•, v•, w•

Color space quantizer

1024 color cells 10

8 12 DPCM One-step predictor

Y, l, Q

subsampled I, Q

2 to 3 3 to 4.5 Transform (cosine, slant)

Y, I, Q

subsampled I, Q

Y, l, Q

No subsampling

Y (1 .75 to 2) 2.5 to 3 (0.75 to 1) I, Q

Y, J, Q

Same as above with

Variable 1 to 2

Y, U, V adaptive classification

Sec. 1 1 . 1 0 Color and Mu ltispectral Image Coding

TABLE 1 1 .18

Summary of Image Data Compression Methods

Typical average rates

Method

bits/pixel

Comments

Zero-memory methods

Simple to implement.

PCM

6-8

Contrast quantization

4-5

Pseudorandom noise--quantization

4-5

Line interlace

Dot interlace

2-4

Predictive coding Delta modulation

1 Performance poorer than DPCM, oversample data for improvement. Intraframe DPCM

Predictive methods are generally simple to implement, but sensitive Intraframe adaptive DPCM

2-3

to data statistics. Adaptive techniques improve performance lnterframe conditional-replenishment

1-2

substantially. Channel error effects are cumulative and visibly lnterframe DPCM

1-2

1-1.5

degrade image quality.

Interframe adaptive DPCM

0.5-1

Transform coding Intraframe

Achieve high performance, small sensitivity to fluctuation in data Intraframe adaptive

1-1 .5

statistics, channel and quantization errors distributed over the lnterframe

0.5-1

image block. Easy to provide channel protection. Hardware lnterframe adaptive

0.5-1

0. 1-0.5

complexity is high.

Hybrid coding Intraframe

Achieve performance close to transform coding at moderate rates Intraframe adaptive

1-2

(0.5 to 1 bit/pixel). Complexity lies midway between transform lnterframe

0.5-1 .5

0.5-1

coding and DPCM.

Interframe adaptive

0.25-0.5

Color image coding Intraframe

1-3

The above techniques are applicable.

Interframe

0.25-1

Two-tone image coding 1-D Methods

Distortionless coding. Higher compression achievable by pattern 2-D Methods

0.06-0.2

0.03-0.2

recognition techniques.

component is independently coded by a two-dimensional coder. An alternative is to identify a finit«<.._number of clusters in a suitable feature space. Each multispectral pixel is represented by the information pertaining to (e.g., the centroid of) the cluster to which it belongs. Usually the fidelity criterion is the classification (rather than mean square) accuracy of the encoded data [45, 46].

1 1 . 1 1 SUMMARY Image data compression techniques are of significant practical interest. We have

considered a large number of compression algorithms that are available for imple­ mentation. We conclude this chapter by providing a summary of these methods in Table 1 1 . 18.

PROBLEMS

11.1* For an 8-bit integer image of your choice, determine the Nth-order prediction error

field EN (m, n) � u (m, n) - uN (m, n) where uN (m, n) is the best mean square causal predictor based on the N nearest neighbors of u (m, n ). Truncate uN (m, n) to the nearest integer and calculate the entropy of EN (m, n) from its histogram for N =

0, 1 , 2, 3, 4, 5. Using these as estimates for the Nth-order entropies, calculate the achievable compression. 11.2 The output of a binary source is to be coded in blocks of M samples. If the successive

outputs are independent and identically distributed with p = 0.95 (for a 0), find the Huffman codes for M = 1 , 2, 3, 4 and calculate their efficiencies.

11.3 For the AR sequence of (11.25), the predictor for feedforward predictive (Fig

11 .6) is chosen as u (n) � pu (n - 1). The prediction error sequence e(n)

u (n) - u (n) is quantized using B bits. Show that in the steady state,

£(18u (n)i2] = � = <T �f(B)

(J"

1-p

where <T � = <T � (1 - p2)f(B ) is the mean square quantization error of e(n). Hence the feedforward predictive coder cannot perform better than DPCM because the pre­ ceding result shows its mean square error is precisely the same as in PCM. This result happens to be true for arbitrary stationary sequences utilizing arbitrary linear pre­ dictors. A possible instance where the feedforward predictive coder may be preferred over DPCM is in the distortionless case, where the quantizer is replaced by an entropy coder. The two coders will perform identically, but the feedforward predic­ tive coder will have a somewhat simpler hardware implementation.

11.4 (Delta modulation analysis) For delta modulation of the AR sequence of (11.25), write the prediction error as e(n) = e(n) - (1 - p)u (n - 1) + 8e (n - 1). Assuming a

1-bit Lloyd-Max quantizer and 8e (n) to be an uncorrelated sequence, show that <T ; (n) = 2(1 - p)<T � + (2p - l)<T ; (n - 1)/(1)

from which (11.26) follows after finding the steady-state value of <T � l<T ; (n ). Problems

Chap. 1 1 557

11.5 Consider images with power spectral density function

S(z1 , z2) - -

A (z1 , z2)A(z1 ,z2 ) _1 -1 A (z1 ,z2) � 1 - P(z1 ,z2)

where A (z1 , z2) is the minimum-variance, causal, prediction-error filter and 132 is the variance of the prediction error. Show that the DPCM algorithm discussed in the text

takes the form shown in Fig. Pll.5, where H(z1 , z2) = l/A (z1 , z2). If the channel adds independent white noise of variance CT; , show that the total noise in the

reconstructed output would be CT� = CT� + (CT; CT�)/132, where CT� � E[(u(m, n))2].

r - - - - - - - - - - - - - - -, H(z1 , z2) I I

u(m, n)

e·

I i/"(m, n)

III

A(z1 , z2) II

Channel

L ___________ _J I

noise

Coder Decoder

Figure Pll.5

11.6 (DPCM analysis) For DPCM of the AR sequence of (11.25), write the prediction error as e(n) = E(n) + p8e(n - 1). Assuming 8e(n) to be an uncorrelated sequence,

show that the steady-state distortion due to DPCM is

D � - E[8e2 (n)] = CT2 (n)f(B) • = CT� (l - p2)f(B)

l - p2f(B) For p =

0.95, plot the normalized distortion DICT� as a function of bit rate B for B =

1, 2, 3, 4 for a Laplacian density quantizer and compare it with PCM. 11.7* For a 512 x 512 image of your choice, design DPCM coders using mean square predictors of orders up to four. Implement the coders for B = 3 and compare the

reconstructed images visually as well as on the basis of their mean square errors and entropies.

11.8 a. Using the transform coefficient variances given in Table 5.2 and the Shannon quantizer based rate distortion formulas (11.61) to (11.63), compare the distor­

tion versus rate curves for the various transforms. (Hint: An easy way is to arrange CT f in decreasing order and Jet 0= CT f,

j = 0, . . . , N - 1 and plot Di � l/N

}2': ::/ CT� versus Ri � l/2N }:,� = 0 log2 CT VCT f).

b. Compare the cosine transform R versus D function when the bit allocation is determined first by truncating the real numbers obtained via (11.61) to nearest integers and second, using the integer bit allocation algorithm.

11.9 (Whitening transform versus unitary transform) An N x 1 vector u with covariance R = {plm -nl} is transformed as v = Lu, where L is a lower triangular (nonunitary)

matlix whose elements are

558 Image Data Compression Chap. 1 1 558 Image Data Compression Chap. 1 1

b. If v (k) is quantized using nk bits, show the reproduced vector u· � L -i v· has the

mean square distortion a Wo ( (1 - p2N) k)

= 1 Wk � 1- p2 N -p - 2 '

c. For N = 15, p = 0.95, and f(x) = Z-2..r, find the optimum rate versus distortion function of this coder and compare it with that of the cosine transform coder.

From these results conclude that it is more advantageous to replace the (usually slow) KL transform by a fast unitary transform rather than a fast decorre­ lating transform.

11.10 (Transform coding versus DPCM) Suppose a sequence of length N has a causal representation

n- 1

u (n) = u (n) + E(n) �

2: a (n, k)u(k) + E(n),

0 s nsN-1

k-0

where u (n) is the optimum linear predictor of u (n) and E(n) is an uncorrelated sequence of variance 13� . Writing this in vector notation as Lu = £, E(O) � u (0), where L is an N xN unit lower triangular matrix, it can be shown that R, the covariance

N-1

matrix of u,

satisfies IRI = TI 13� . If u (n) is DPCM coded, then for small levels of dis- k-o tortion, the average minimum achievable rate is

D < min {13n k a. Show that at small distortion levels, the above results imply

2Nk-o - log2 D = - 2N 2

log2 IR I - - log2 D = RKL

that is, the average minimum rate achievable by KL transform coding is lower than that of DPCM.

b. Suppose the Markov sequence of (11 .25) with cr � =

1, 135 = 1, 13� = (1 - p2) for

1 snsN- 1, is DPCM coded and the steady state exists for n�1. Using the results of Problem 11.6, with/(B) = Z-28, show that the number of bits B required to achieve a distortion level D is given by

B =2 og2 p ( ) , ls n sN- 1 .

I 1 2 (1 - p2)

For the initial sample E(O), the same distortion level is achieved by using � log2 (1/D ) bits. From these and (11.69) show that

log2 p + ( ------n- RvPcM = ZN log2 D + 2 (1 - p) )

1 1 (N - 1) �

D< (1 - p)

(1 + p) Calculate this difference for N = 16, p= 0.95, and D = 0 . 01 , and conclude that

at low levels of distortion the performance of KL transform and DPCM coders Problems

Chap. 1 1 559 Chap. 1 1 559

11.11 For the separable covariance model used in Example 11.5, with p = 0.95, plot and compare the R versus D performances of (a) various transform coders for 16 x 16 size block utilizing Shannon quantizers (Hint: Use the data of Table 5.2.) and (b) N x N block cosine transform coders with N = 2", n = l, 2, . . . , 8. (Hint: Use eq.

(P5.2S-2).]

11.12 Plot and compare the R versus D curves for 16 x 16 block transform coding of images modeled by the nonseparable exponential covariance function

using the discrete, Fourier, cosine, sine, Hadamard, slant, and Haar transforms. (Hint: Use results of Problem P5.29 to calculate transform domain variances.)

11.13* Implement the zonal transform coding algorithm of Section 11.5 on 16 x 16 blocks of an image of your choice. Compare your results for average rates of 0.5, 1.0, and 2.0

bits per pixel using the cosine transform or any other transform of your choice. 11.14* Develop a chart of adaptive transform coding algorithms containing details of the algorithms and their relative merits and complexities. Implement your favorite of

these and compare it with the 16 x 16 block cosine transform coding algorithm.

11.15 The motivation for hybrid coding comes from the following example. Suppose an N x N image u(m, n) has the autocorrelation function r(k, /) = plkl + l'I.

a. If each column of the image transformed as Vn = «l>u n, where «I> is the KLT of Un, then show that the autocorrelation of Vn, that is, E[vn (k)v". (k')] = Ak pl" - n'I &( k - k ' ). What are «I> and Ak?

b. This means the transformed image is uncorrelated across the rows and show that the pixels along each row can be modeled by the first order AR process of (11.84) with a (k) = p, b (k) = 1, and cr; (k) = (1 - p2)>.k .

11.16 For images having separable covariance function with p = 0.95, find the optimum pairs n, k (n) for DPCM transmission over a noisy channel with p. = 0.001 employing

the optimum mean square predictor. (Hint: �2 = (1 - p2) 2 .)

11.17 Let the transition probabilities qo = p (Oll) and q1 = p (llO) be given. Assuming all the runs to be independent, their probabilities can be written as

I ;::: 1, i = O(white), l(black) a. Show that the average run lengths and entropies of white and black runs are

µ; = liq; and H; = (-1/q;)[q; log2q; + (1 - q;) log2 (1 - q,)]. Hence the achievable compression ratio is (Ho Palµ,o + H1 P1 /µ,1), where P; = q;l(qo + q1), i = 0, 1 are the

a priori probabilities of white and black pixels. b. Suppose each run length is coded in blocks of m-bit words, each word represent­

ing the M - 1 run lengths in the interval [kM, (k + l)M - 1], M = 2m, k = 0, 1, . . . , and a block terminator code. Hence the average number of bits used for white and black runs will be m L;=0 (k + l)P[kM :5 /; :5 (k + l)M - 1], i = 0, 1. What is the compression achieved? Show how to select M to maximize it.

560 Image Data Compression Chap. 11

BIBLIOGRAPHY

Section 1 1 .1 Data compression has been a topic of immense interest in digital image processing.

Several special issues and review papers have been devoted to this. For details and extended bibliographies:

55, no. 3 (March 1967), (b) IEEE Commun. Tech. COM-19, no. 6, part I (December 1971), (c) IEEE Trans. Commun. COM-25, no. 11 (November 1977), (d) Proc. IEEE

1. Special Issues (a) Proc. IEEE

68, no. 7 (July 1980), (e) IEEE Trans. Commun. COM-29 (December 1981), (f) Proc. IEEE

73, no. 2 (February 1985).

2. T. S. Huang and 0. J. Tretiak (eds.). Picture Bandwidth Compression. New York: Gordon and Breach, 1972.

3. L. D. Davisson and R. M. Gray (eds.). Data Compression. Benchmark Papers in Elec­ trical Engineering and Computer Science, Stroudsberg, Penn.: Dowden Hunchinson & Ross, Inc., 1976.

4. W. K. Pratt (ed.). Image Transmission Techniques. New York: Academic Press, 1979.

5. A. N. Netravali and J. 0. Limb. "Picture Coding: A Review." Proc. IEEE 68, no. 3 (March 1980): 366-406.

6. A. K. Jain. "Image Data Compression: A Review." Proc. IEEE 69, no. 3 (March 1981): 349-389.

7. N. S. Jayant and P. Noll. Digital Coding of Waveforms. Englewood Cliffs, N.J.: Prentice­ Hall, 1984.

8. A. K. Jain, P. M. Farrelle, and V. R. Algazi. "Image Data Compression." In Digital

Image Processing Techniques, M. P. Ekstrom, ed. New York: Academic Press, 1984.

9. E. Dubois, B. Prasada, and M. S. Sabri. "Image Sequence Coding." In Image Sequence Analysis, T. S. Huang, (ed.) New York: Springer-Verlag, 1981, pp. 229-288.

Section 1 1 .2 For entropy coding, Huffman coding, run-length coding, arithmetic coding, vector

quantization, and related results of this section see papers in (3] and:

10. J. Rissanen and G. Langdon. "Arithmetic Codie." IBM J. Res. Develop. 23, (March

1979): 149-162. Also see IEEE Trans. Comm. COM-29, no. 6 (June 1981): 858-867.

11. J. W. Schwartz and R. C. Barker. "Bit-Plane Encoding: A Technique for Source Encoding." IEEE Trans. Aerospace Electron. Syst. AES-2, no. 4 (July 1966): 385-392.

12. A. Gersho. "On the Structure of Vector Quantizers." IEEE Trans. Inform. Theory IT-28 (March 1982): 157-165. Also see vol. IT-25 (July 1979): 373-380.

Section 1 1 .3 For some early work on predictive coding, Delta modulation and DPCM see Oliver,

Harrison, O'Neal, and others in Bell Systems Technical Journal issues of July 1952, May-June 1966, and December 1972. For more recent work:

Bibliography Chap. 11

13. R. Steele. Delta Modulation Systems. New York: John Wiley, 1975. 14. I. M. Paz, G.C. Collins and B. H. Batson. "A Tri-State Delta Modulator for Run-Length

Encoding of Video." Proc. National Telecomm. Conf Dallas, Texas, vol. I (November 1976): 6.3-1-6.3-6.

For adaptive delta modulation algorithms and applications to image trans­ mission, see [7], Cutler (pp. 898-906), Song, et al. (pp. 1033-1044) in [lb] , Lei et al. in [le]. For more recent work on DPCM of two-dimensional images, see Musmann in [4] , Habibi (pp. 948-956) in [lb], Sharma et al. in [le],

15. J. B. O'Neal, Jr. "Differential Pulse-Code Modulation (DPCM) with Entropy Coding." IEEE Trans. Inform. Theory IT-21, no. 2 (March 1976): 169-174. Also see vol. IT-23 (November 1977): 697-707.

16. V. R. Algazi and J. T. De Witte. "Theoretical Performance of Entropy Coded DPCM." IEEE Trans. Commun. COM-30, no. 5 (May 1982): 1088-1095.

17. J. W. Modestino and V. Bhaskaran. "Robust Two-Dimensional Tree Encoding of Images." IEEE Trans. Commun. COM-29, no. 12 (December 1981): 1786-1798.

18. A. N. Netravali. "On Quantizers for DPCM Coding of Picture Signals." IEEE Trans. Inform. Theory IT-23 (May 1977): 360-370. Also see Proc. IEEE

65 (April 1977): 536-548.

For adaptive DPCM, see Zschunk (pp. 1295-1302) and Habibi (pp. 1275- 1284) in [le], Jain and Wang [32] , and:

19. L. H. Zetterberg, S. Ericsson, C. Couturier. "DPCM Picture Coding with Two­ Dimensional Control of Adaptive Quantization." IEEE Trans. Commun. COM-32, no. 4 (April 1984): 457-462.

20. H. M. Hang and J. W. Woods. "Predictive Vector Quantization of Images," IEEE Trans. Commun., (1985).

Section 1 1 .4 For results related to the optimality of the KL transform, see Chapter 5 and the

bibliography of that chapter. For the optimality of KL transform coding and bit allocations, see [6] , [32] , and:

21. A. Segall. "Bit Allocation and Encoding for Vector Sources." IEEE Trans. Inform. Theory IT-22, no. 2 (March 1976): 162-169.

Section 1 1 .5 For early work on transform coding and subsequent developments and examples of

different transforms and algorithms, see Pratt and Andrews (pp. 515-554), Woods and Huang (pp. 555-573) in [2] , and:

22. A. Habibi and P. A. Wintz. "Image Coding by Linear Transformation and Block Quanti­ zation." IEEE Trans. Commun. Tech. COM-19, no. 1 (February 1971): 50-63.

562 Image Data Compression Chap. 11

23. P. A. Wintz. "Transform Picture Coding." Proc. IEEE 60, no. 7 (July 1972): 809-823.

24. W. K. Pratt, W. H. Chen, and L. R. Welch. "Slant Transform Image Coding." IEEE Trans. Commun. COM-22, no. 8 (August 1974): 1075-1093.

25. K. R. Rao, M. A. Narasimhan, and K. Revuluri. "Image Data Processing by Hadamard­ Haar Transforms." IEEE Trans. Computers C-23, no. 9 (September 1975): 888-896.

The concepts of fast KL transform and recursive block coding were introduced in [26 and Ref 17, Ch 5). For details and extensions see [ 6), Meiri et al. (pp. 1728-1735) in [le) , Jain et al. in [8) , and:

26. A. K. Jain. "A Fast Karhunen-Loeve Transform for a Class of Random Processes." IEEE Trans. Commun. COM-24 (September 1976): 1023-1029.

27. A. K. Jain and P. M. Farrelle. "Recursive Block Coding." Sixteenth Annual Asilomar Conference on Circuits, Systems, and Computers, November 1982. Also see P. M. Farrelle and A. K. Jain. IEEE Trans. Commun. (February 1986), and P. M. Farrelle, Ph.D. Dissertation, U.C. Davis, 1988.

For results on two-source coding, adaptive transform coding, and the like, see Yan and Sakrison (pp. 1315-1322) in [le) , Jain and Wang [32), Tasto and Wintz (pp. 956-972) in [lb) , Graham (pp. 336-346) in [la) , Chen and Smith in [le), and:

28. W. F. Schreiber, C. F. Knapp, and N. D. Kay. "Synthetic Highs: An Experimental TV Bandwidth Reduction System." J. Soc. Motion Picture and Television Engineers 68 (August 1959): 525-537.

29. V. R. Algazi and D. J. Sakrison. "Encoding of a Counting Rate Source with Orthogonal Functions." Computer Processing in Communications, N.Y.: Polytechnic Institute of Brooklyn, 1969, pp. 85-100.

30. J. L. Mannos and D. J. Sakrison. "The Effects of a Visual Fidelity Criterion on the Encoding of Images." IEEE Trans. Inform. Theory IT-20 (July 1974): 525-536.

Section 1 1 .6 Hybrid coding principle, its analysis and relationship with semicausal models, and

its applications can be found in [6, 8), Jones (Chapter 5) in [4), and:

31. A. Habibi. "Hybrid Coding of Pictorial Data." IEEE Trans. Commun. COM-22 (May 1974): 614-626.

32. A. K. Jain and S. H. Wang. "Stochastic Image Models and Hybrid Coding," Final Report, NOSC contract N00953-77-C-003MJE, Department of Electrical Engineering, SUNY Buffalo, New York, October 1977. Also see Technical Report #SIPL-79-6, Signal

and Image Processing Laboratory, ECE Dept., University of California at Davis, September 1979.

33. R. W. Means, E. H. Wrench and H. J. Whitehouse. "Image Transmission via Spread Spectrum Techniques." ARPA Quarterly Technical Reports ARPA-QR6, QR8 Naval Ocean Systems Center, San Diego, Calif., January-December 1975.

34. A. K. Jain. "Advances in Mathematical Models for Image Processing." Proc. IEEE 69 (March 1981): 502-528.

Bibliography Chap. 11 563

Section 1 1 .7 For interframe predictive coding, see [5, 6, 8] , Haskell et al. (pp. 1339-1348) in [le],

Haskell (Chapter 6) in [4], and: 35. J. C. Candy, et al. "Transmitting Television as Clusters of Frame-to-Frame Differences."

Bell Syst. Tech. J. 50 (August 1971): 1889-1917. 36. J. R. Jain and A. K. Jain. "Displacement Measurement and Its Application in Inter­

frame Image Coding." IEEE Trans. Comm. COM-29 (December 1981): 1799-1808.

37. A. N. Netravali and J. D. Robbins. "Motion Compensated Television Coding-Part I," Bell Syst. Tech. J. (March 1979): 631-670.

Interframe hybrid and transform coding techniques are discussed in [5, 6, 8, 9], Roese et al. (pp. 1329-1338), Natrajan and Ahmed (pp. 1323-1329) in [le], and:

38. J. A. Stuller and A. N. Netravali. "Transform Domain Motion Estimation," Bell Syst. Tech. J. (September 1979): 1623-1702. Also see pages 1703-1718 of the same issue for

application to coding. 39. J. R. Jain and A. K. Jain. "lnterframe Adaptive Data Compression Techniques for

Images." Tech. Rept., Signal and Image Processing Laboratory, ECE Department, University of California at Davis, August 1979.

40. J. 0. Limb and J. A. Murphy. "Measuring the Speed of Moving Objects from Television Signals," IEEE Traris. Commun. COM-23 (April 1975): 474-478.

41. C. Cafforio and F. Rocca. "Methods for Measuring Small Displacements of Television Images." IEEE Trans. Inform. Theory IT-22 (September 1976): 573-579.

Section 1 1 .8 For the optimal mean square encoding and decoding results and their application,

we follow [6, 39] and: 42. G. A. Wolf. "The Optimum Mean Square Estimate for Decoding Binary Block Codes."

Ph.D. Thesis, University of Wisconsin at Madison, 1973. Also see G. A. Wolf and R. Redinbo. IEEE Trans. Inform. Theory IT-20 (May 1974): 344-351.

For extended bibliography, see [6, 8]. Section 1 1 .9 [lf] is devoted to coding of two tone images. Details of CCITT standards and

various algorithms are available here. Some other useful references are Arps (pp. 222-276) in [4], Huang in [le, 2] , Musmann and Preuss in [le], and:

43. H. Kobayashi and L. R. Bahl. "Image Data Compression by Predictive Coding I: Prediction Algorithms." and "II: Encoding Algorithms." IBM J. Res. Dev.

18, no. 2 (March 1974): 164-179.

564 Image Data Compression Chap. 1 1

44. T. S. Huang and A. B. S. Hussain. "Facsimile Coding by Skipping White." IEEE Trans. Commun. COM-23, no. 12 (December 1975): 1452-1466.

Section 1 1 .10 Color and multispectral coding techniques discussed here have been discussed by

Limb et al. in [le] , Pratt in [lb] , and: 45. F. E. Hilbert. "Cluster Compression Algorithm, A Joint Clustering/ Data Compression

Concept." JPL Publication 77-43, Jet Propulsion Laboratory, Pasadena, Calif., December 1, 1977.

46. J. N. Gupta and P. A. Wintz. "A Boundary Finding Algorithm and its Applications." IEEE Trans. Circuits and Systems CAS-22 (April 1976): 351-362.

Section 1 1 . 1 1 Several techniques not discussed in this chapter include nonuniform sampling tech­

niques combined with interpolation (such as using splines), use of singular value decompositions, autoregressive (AR) model synthesis, and the like. Summary dis­ cussion and the relevant sources of these and other useful methods are given in

(6, 8] .

Bibliography Chap. 1 1

565