1 shows a typical method of obtaining projections. Each horizontal line shown in

10.14 Write the observation model of (10.68) as v(x, YJ � !./? -1 w = h(x, y) 0 f(x, y) + 1J(X, y)

where h(x , y) � H (t i . b) = H P (t, 0) � hp (s,

8) and 11 � .cA' -1 v, whose

Problems Chap. 10

A (�, 0) = H; Sp[IHP SP + l�IS;1r1 . Implement this filter in the Radon transform domain, as

shown in Fig. 10.9, to arrive at the filter 12 Ap = l�IAP.

10.15 Compare the operation counts of the Fourier method with the convolution/filter back­ projection methods. Assume N x N image size with aN projections, a� constant.

10.16 (Radon inversion formula for divergent rays) a. Starting with the inverse Radon transform in polar coordinates, show that the

reconstructed object from fan-beam geometry projections b (CT, 13) can be written as

13) _ ab(CT, 13)

J,2"J" oCT a13 dCT dl3

41'T2 o -.., r cos( CT + 13 - 4>) -R sin CT where

fi P (r ' <j>) = -

JCTl

5. -y.

b. Rewrite the preceding result as a generalized convolution back-projection result, called the Radon inversion formula for divergent rays, as

fi P ( r,

<j>) = - 1

41'T2 dCT dl3 -� CT - CT where

_1 r cos(13 - <j>) CT = tan

R + r sm l3 - 4> ·c ) sin(l3 - <j>)]2}112 > O Show that

p � {[r cos(l3 - <j>)]2 + [R + r

CT1

and p correspond to a ray (CT', 13) that goes through the object at location (r,

4>) and p is the distance between the source and (r, 4> ) . The inner integral in the above Radon inversion formula is the Hilbert transform of ljl( CT, ·, ·) and the outer integral is analogous to back projection. c. Develop a practical reconstruction algorithm by replacing the Hilbert transform by

a bandlimited filter, as in the case of parallel beam geometry.

10.17 (Two-stage reconstruction in three dimensions) a. Referring to Fig. 10.19, rotate the x- and y-axes by an angle <j>, that is, let x ' = x cos <j> + y sin<j>, y ' = -x sin <j> + y cos <j>, and obtain

Jf

g (s, <j>, 6) = g (s,

' dz where/<1> and g are the two-dimensional Radon transforms off (with z constant) and

a) =

t<1> (x ', z)S(x ' sin 0 +z cos 0 - s ) dx

f<1> (with <j> constant), respectively, that is,

b. Develop the block diagram for a digital implementation of the two-stage reconstruction algorithm.

472 Image Reconstruction from Projections Chap. 1 0

BIBLIOGRAPHY

Section 10.1 For image formation models of CT, PET, MRI and overview of computerized

tomography:

1. IEEE Trans. Nucl. Sci. Special Issue on topics related to image reconstruction. NS-21, no. 3 (1974) ; NS-26, no. 2 (April 1979) ; NS-27, no. 3 (June 1980).

2. IEEE Trans. Biomed. Engineering. Special Issue on computerized medical imaging. BME-28, no. 2 (February 1981).

3. Proc. IEEE. Special Issue on Computerized Tomography. 71, no. 3 (March 1983).

4. A. C. Kak. "Image Reconstruction from Projections," in M. P. Ekstrom (ed.). Digital Image Processing Techniques. New York: Academic Press, 1984, pp. 1 1 1-171.

5. G. T. Herman (ed.). Image Reconstruction from Projections. Topics in Applied Physics, vol. 32. New York: Springer-Verlag, 1979.

6. G. T. Herman. Image Reconstruction from Projections-The Fundamentals of Com­ puterized Tomography. New York: Academic Press, 1980.

7. H. J. Scudder. "Introduction to Computer Aided Tomography." Proc. IEEE 66, no. 6 (June 1978). 8. Z. H. Cho, H. S. Kim, H. B. Song, and J. Cumming. "Fourier Transform Nuclear

Magnetic Resonance Tomographic Imaging." Proc. IEEE, 70, no. 10 (October 1982):

1152-1173. 9. W. S. Hinshaw and A. H. Lent. "An Introduction to NMR Imaging," in [3] .

10. D. C. Munson, Jr., J. O'Brien, K. W. Jenkins. "A Tomographic Formulation of Spot­

light Mode Synthetic Aperture Radar." Proc. IEEE, 71, (August 1983): 917-925 .

Literature on image reconstruction also appears in other journals such as: J. Com­ put. Asst. Torno., Science, Brit. J. Radio/., J. Magn. Reson. Medicine, Comput.

Biol. Med., and Medical Physics.

11. J. L. C. Sanz, E. B. Hinkle, A. K. Jain. Radon and Projection Transform-Based Machine Vision: Algorithms, A Pipeline Architecture, and Industrial Applications, Berlin: Springer-Verlag, (1988). Also see, Journal of Parallel and Distributed Computing, vol. 4, no. 1 (Feb. 1987) : 45-78.

Sections 10.2-1 0.5 Fundamentals of Radon transform theory appear in several of the above references,

such as [ 4-7] , and: 12. J. Radon. "Uber die Bestimmung van Funktionen durch ihre Integralwerte Tangs

gewisser Mannigfaltigkeiten" (On the determination of functions from their integrals along certain manifolds). Bertichte Saechsiche Akad. Wissenschaften (Leipzig), Math. Phys. Klass

69, (1917): 262-277.

13. D. Ludwig. "The Radon Transform on Euclidean Space." Commun. Pure Appl. Math.

19, (1966): 49-81 . Bibliog raphy

Chap. 1 0 473

14. D. E. Kuhl and R. Q. Edwards. "Image Separation Isotope Scanning," Radiology 80, no. 4 (1963): 653-662.

15. P. F. C. Gilbert. "The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy: II. Direct Methods." Proc. Roy.

Soc. London Ser. B, vol. 182, (1972): 89-102.

16. P. R. Smith, T. M. Peters, and R. H. T. Bates. "Image Reconstruction from Finite Number of Projections." J. Phys. A: Math Nucl. Gen. 6, (1973): 361-382. Also see New Zealand J. Sci.,

14, (1971): 883-896.

17. S. R. Deans. The Radon Transform and Some of Its Applications. New York: Wiley, 1983.

Section 1 0.6 For convolution/filter back-projection algorithms, simulations, and related details:

18. S. W. Rowland, in [5], pp. 9-79.

19. G. N. Ramachandran and A. V. Lakshminarayanan. "Three-Dimensional Recon­ struction from Radiographs and Electron Micrographs: II. Application of convolutions instead of Fourier Transforms." Proc. Nat. Acad. Sci. ,

68 (1971): 2236-2240. Also see Indian J. Pure Appl. Phys. 9 (1971): 997-1003.

20. R. N. Bracewell and A. C. Riddle. "Inversion of Fan-Beam Scans in Radio Astronomy." Astrophys. J. 150 (1967): 427-437.

21. L. A. Shepp and B. F. Logan. "The Fourier Reconstruction of a Head Section." IEEE Trans. Nucl. Sci. NS-21, no. 3 (1974): 21-43.

Sections 10.7-10.8 Results on Radon transform of random fields were introduced in

22. A. K. Jain and S. Ansari. "Radon Transform Theory for Random Fields and Image Reconstruction From Noisy Projections." Proc. ICASSP, San Diego, 1984.

23. A. K. Jain. "Digital Image Processing: Problems and Methods," in T. Kailath (ed.), Modern Signal Processing. Washington: Hemisphere Publishing Corp., 1985.

For reconstruction from noisy projections see the above references and:

24. Z. Cho and J. Burger. "Construction, Restoration, and Enhancement of 2- and 3-Dimensional Images," IEEE Trans. Nucl. Sci. NS-24, no. 2 (April 1977): 886-895.

25. E. T. Tsui and T. F. Budinger. "A Stochastic Filter for Transverse Section Recon­ struction." IEEE Trans. Nucl. Sci. NS-26, no. 2 (April 1979): 2687-2690.

Section 10.9

26. R. N. Bracewell. "Strip Integration in Radio Astronomy." Aust. J. Phys.

9 (1956): 198-217.

27. R. A. Crowther, D. J. Derosier, and A. Klug. "The Reconstruction of a Three­ Dimensional Structure from Projections and Its Application to Electron Microscopy."

474 Image Reconstruction from Projections Chap. 10

Proc. Roy. Soc. London Ser. A, vol. 317 (1970): 319-340. Also see Nature (London) 217 (1968): 130-134.

28. G. N. Ramachandran. "Reconstruction of Substance from Shadow: I. Mathematical Theory with Application to Three Dimensional Radiology and Electron Microscopy." Proc. Indian Acad. Sci. 74 (1971): 14-24.

29. R. M. Merseau and A. V. Oppenheim. "Digital Reconstruction of Multidimensional Signals From Their Projections." Proc. IEEE 62, (1974): 1319-1332.

30. R. F. King and P. R. Moran. "Unified Description of NMR Imaging Data Collection Strategies and Reconstruction." Medical Physics 11, no. 1 (1984): 1-14.

Sections 1 0. 1 0-10.13 For fan-beam reconstruction theory, see [6, 7] and Horn in [l(iii), pp. 1616-1623] .

For algebraic techniques and ART algorithms, see [5, 6] and:

31. S. Kaczmarz. "Angenii.herte Auflosung von Systemen Linearer Gleichungen." Bull. Acad. Polan. Sci. Lett. A. 35 , (1937): pp. 355-357.

32. R. Gordon, "A Tutorial on ART (Algebraic Reconstruction Techniques)." IEEE Trans. Nucl. Sci. NS-21, (1974): 78.

33. P. F. C. Gilbert. "Iterative Methods for the Reconstruction of Three-Dimensional Ob­ jects from Projections." J. Theor. Biol. 36 (1972): 105-117.

34. G. T. Herman, A. Lent, and S. W. Rowland. "ART: Mathematics and Applications," J. Theor. Biol. 42 (1973) : 1-32.

35. A. M. Cormack. "Representation of a Function by its Line Integrals with Some Radiological Applications." J. Appl. Phys. 34 (1963): 2722-2727. Also see Part II, J. Appl. Phys. 35 (1964): 2908-2913.

For other applications of Radon transform and its extensions:

36. M. Bernfield. "CHIRP Doppler Radar." Proc. IEEE, vol. 72, no. 4 (April 1984): 540-541 .

37. J. Raviv, J. F. Greenleaf, and G. T. Herman (eds.). Computer Aided Tomography and Ultrasonics in Medicine. Amsterdam: North-Holland, 1979.

Bibliog raphy Chap. 10 475