Assuming the cepstrum c(m, n) is absolutely summable, prove the stability conditions for the causal and semicausal models.

6.20 Assuming the cepstrum c(m, n) is absolutely summable, prove the stability conditions for the causal and semicausal models.

6.21 a. Show that the KL transform of any periodic random field that is also stationary is the two-dimensional (unitary) DFT.

b. Suppose the finite-order causal, semicausal, and noncausal MVRs given by (6.70) are defined on a periodic grid with period (M, N). Show that the SDF of these random fields is given by

( w, -

)( 8 w2 - ) ,

M- 1 N -

S (wi , w2) =

k�o 1�0

What is S (k, I)? Sec. 6

Problems 229

BIBLIOGRAPHY

Section 6.1 For a survey of mathematical models and their relevance in image processing and

related bibliography:

1. A. K. Jain. "Advances in mathematical models for image processing." Proceedings IEEE

69, no. 5 (May 1981): 502-528. Sections 6.2-6.4

Further discussions on spectral factorization and state variable models, ARMA models, and so on, are available in:

2. A. V. Oppenheim and R. W. Schafer. Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall, 1975.

3. A. H. Jazwinsky. Stochastic Processes and Filtering Theory. New York: Academic Press, 1970, pp. 70-92.

4. P. Whittle. Prediction and Regulation by Linear Least-Squares Methods. London: English University Press, 1954.

5. N. Wiener. Extrapolation, Interpolation and Smoothing of Stationary Time Series. New York: John Wiley, 1949.

6. C. L. Rino. "Factorization of Spectra by Discrete Fourier Transforms." IEEE Transactions on Information Theory IT-16 (July 1970): 484-485.

7. J. Makhoul. "Linear Prediction: A Tutorial Review." Proceedings IEEE 63 (April 1975): 561-580.

8. IEEE Trans. Auto. Contr. Special Issue on System Identification and Time Series Analysis. T. Kailath, D. 0. Mayne,and R. K. Mehra (eds.), Vol. AC-19, December 1974.

9. N. E. Nahi and T. Assefi. "Bayesian Recursive Image Estimation." IEEE Trans. Comput. (Short Notes) C-21 (July 1972): 734-738.

10. S. R. Powell and L. M. Silverman. "Modeling of Two Dimensional Covariance Functions with Application to Image Restoration." IEEE Trans. Auto. Contr. AC-19 (February 1974): 8-12.

11. R. P. Roesser. "A Discrete State Space Model for Linear Image Processing." IEEE Trans. Auto. Contr. AC-20 (February 1975): 1-10.

12. E. Fornasini and G. Marchesini. "State Space Realization Theory of Two-Dimensional Filters." IEEE Trans. Auto. Contr. AC-21 (August 1976): 484-492.

Section 6.5 Noncausal representations and fast KL transforms for discrete random processes

are discussed in [1, 21] and: 13. A. K. Jain. "A Fast Karhunen Loeve Transform for a Class of Random Processes."

IEEE Trans. Comm. COM-24 (September 1976): 1023-1029. Also see IEEE Trans. Comput. C-25 (November 1977): 1065-1071.

230 Image Representation by Stochastic Models Chap. 6

Sections 6.6 and 6.7 Here we follow [ 1]. The linear prediction models discussed here can also be gen­

eralized to nonstationary random fields [1]. For more on random fields:

14. P. Whittle. "On Stationary Processes in the Plane." Biometrika 41 (1954): 434-449. 15. T. L. Marzetta. "A Linear Prediction Approach to Two-Dimensional Spectral Factoriza­

tion and Spectral Estimation." Ph.D. Thesis, Department Electrical Engineering and Computer Science, MIT, February 1978.

16. S. Ranganath and A. K. Jain. "Two-Dimensional Linear Prediction Models Part I: Spectral Factorization and Realization." IEEE Trans. ASSP ASSP-33, no. 1 (February 1985): 280-299. Also see S. Ranganath. "Two-Dimensional Spectral Factorization,

Spectral Estimation and Applications in Image Processing." Ph.D. Dissertation, Department Electrical and Computer Engineering, UC Davis, March 1983.

17. A. K. Jain and S. Ranganath. "Two-Dimensional Linear Prediction Models and Spectral Estimation," Ch. 7 in Advances in Computer Vision and Image Processing. (T. S. Huang, ed.). Vol. 2, Greeenwich, Conn.: JAi Press Inc. , 1986, pp. 333-372.

18. R. Chellappa. "Two-Dimensional Discrete Gaussian Markov Random Field Models for Image Processing," in Progress in Pattern Recognition, L. Kanai and A. Rosenfeld (eds). Vol. 2, New York, N.Y.: North Holland, 1985, pp. 79-112.

19. J. W. Woods. "Two-Dimensional Discrete Markov Fields." IEEE Trans. Inform. Theory IT-18 (March 1972): 232-240.

For stability of two-dimensional systems: 20. D. Goodman. "Some Stability Properties of Two-dimensional Linear Shift Invariant

Filters," IEEE Trans. Cir. Sys. CAS-24 (April 1977): 201-208. The relationship between the three types of prediction models and partial differ­

ential equations is discussed in: 21. A. K. Jain. "Partial Differential Equations and Finite Difference Methods in Image

Processing, Part I-Image Representation." J. Optimization Theory and Appl. 23, no. 1 (September 1977): 65-91. Also see IEEE Trans. Auto. Control AC-23 (October 1978): 817-834.

Section 6.8 Here we follow [1] and have applied the method'of [6] and:

22. M. P. Ekstrom and J. W. Woods. "Two-Dimensional Spectral Factorization with Application in Recursive Digital Filtering." IEEE Trans. on Acoust. Speech and Signal Processing ASSP-24 (April 1976): GllS-128.

Section 6.9 For fast KL transform decomposition in two dimensions, stochastic decoupling, and

related results see [13] , [21], and: Sec. 6

Bibliog raphy 231

23. S. H. Wang. "Applications of Stochastic Models for Image Data Compression." Ph.D. Dissertation, Department of Electrical Engineering, SUNY Buffalo, September 1979. Also Technical Report SIPL-79-6, Signal and Image Processing Laboratory, Department Electrical and Computer Engineering, UC Davis, September 1979.

24. A. K. Jain. "A Fast Karhunen-Loeve Transform for Recursive Filtering of Images Corrupted by White and Colored Noise." IEEE Trans. Comput. C-26 (June 1977): 560-571.

232 Image Representation by Stochastic Models Chap. 6