(FDs for a polygon curve) For a continuous curve u(t) � X1 (t) + jx2 (t) with
a. (FDs for a polygon curve) For a continuous curve u(t) � X1 (t) + jx2 (t) with
period T, the FDs are the Fourier series coefficients a(k) = (l!T) f[ u(t) exp(-j2'1Tkt/T) dt. If the object boundary is a polygon whose vertices are repre
sented by phasors Vk, k = 0, 1, . . . , m - 1, show that the FDs are given by
T m a(k) = •
- ( k)2 L (b;- 1 - b;) exp(-J2'1Tkt;/T) 2'1T
Figure P9.12
b. (Line patterns) If the given curve is a line pattern then a closed contour can be 424
Image Analysis and Computer Vision Chap. 9 Image Analysis and Computer Vision Chap. 9
a (k) = a (-k)e -ikC2"'n13
for some 13. If the trace begins at t= 0 at one of the endpoints of the pattern, then
13 = 0. Show how this property may be used to skeletonize a shape. c. The area A enclosed by the outer boundary of a surface is given by
In terms of FDs show that A = = - 2: la (k)i2 kn.
Verify this result for surface area of a line pattern.
k = ':JO
9.13 (Properties of AR models) a. Prove the translation, scaling, and rotation properties of AR model parameters listed in Table 9. 9. b. Show a closed boundary can lie reconstructed from the AR model residuals e; ( n ) by inverting a circulant matrix. c. Find the relation between AR model features and FDs of closed boundaries.
9.14* Scan and digitize the ASCII characters and find their medial axis transforms. Develop any alternative practical thinning algorithm to reduce printed characters to line
shapes. 9.15 Compare the complexity of printed character recognition algorithms based on (a)
template matching, (b) Fourier descriptors, and (c) moment matching.
9.16 (Matched filtering) Write the matched filter output SNR as
2 SNR = 1rJ:JGS�12)[S�112 Uexp{-j(w1 mo + W2 no)}dw1 d w 2 l / s f fJGl2 .. dw1 dw2
where G and U are Fourier transforms of g(m, n), u (m, n), respectively. Apply the Schwartz inequality to show that SNR is maximized only when (9.132) is satisfied within a scaling constant that can be set to unity. What is the maximum value of SNR?
9.17 If J.l.k denotes the mean vector of class k prototypes, show that the decision rule: llx - J.1.kll2 < llx - J.L;i12, i t- k :::} x E Sk, gives a linear discriminant with a k = 2J.Lk, bk = -11 .... k112•
9.18 Find the decision tree of Example 9.11 if an object class with z (l) = 15, z(2) = 30 is added to the list of prototypes. 9.19*
A printed circuit board can be modeled as a network of pathways that either inerge into other paths as terminate into a node. Develop a vision system for isolating defects such as breaks (open circuits) and leaks (short circuits) in the pathways. Discuss and develop practical preprocessing, segmentation, and recognition algorithms for your
system.
BIBLIOGRAPHY
Section 9.1-9.3 Some general references on feature extraction, image analysis and computer vision
are: Bibliography
Chap. 9 425
1. R. 0. Duda and P. E. Hart. Pattern Recognition and Scene Analysis. New York: John Wiley, 1973.
2. A. Rosenfeld and A. C. Kak. Digital Picture Processing. New York: Academic Press, 1976. Also see Vols. I and II, 1982.
3. D. H. Ballard and C. M. Brown. Computer Vision. Englewood Cliffs, N.J.: Prentice Hall, 1982.
4. B. S. Lipkin and A. Rosenfeld (eds.). Picture Processing and Psychopictorics. New York: Academic Press, 1970.
5. J. K. Aggarwal, R. 0. Duda and A. Rosenfeld (eds.). Computer Methods in Image Analysis. Los Angeles: IEEE Computer Society, 1977.
Additional literature on image analysis may be found in several texts referred to in Chapter I, in journals such as Computer Graphics and Image Processing, Pattern Recognition, IEEE Trans. Pattern Analysis and Machine Intelligence, and in the proceedings of conferences and workshops such as IEEE Conferences on Pattern Recognition Image Processing, Computer Vision and Pattern Recognition, International Joint Conference Pattern Recognition, and the like.
Section 9.4 Edge detection is a problem of fundamental importance in image analysis. Different
edge detection techniques discussed here follow from: 6. J. M. S. Prewitt. "Object Enhancement and Extraction," in [4].
7. L. S. Davis. "A Survey of Edge Detection Techniques." Computer Graphics and Image Processing, vol. 4, pp. 248-270, 1975.
8. A. Rosenfeld and M. Thurston. "Edge and Curve Detection for Visual Scene Analysis," in [5]. Also see Vol. C-21, no. 7, (July 1972): 677-715.
9. L. G. Roberts. "Machine Perception of Three-Dimensional Solids," in [SJ . 10. R. Kirsch. "Computer Determination of the Constituent Structure in Biological Im
ages." Compt. Biomed. Res. 4, no. 3 (1971): 315-328.
11. G. S. Robinson. "Edge Detection by Compass Gradient Masks." Comput. Graphics Image Proc. 6 (1977): 492-501.
12. W. Frei and C. C. Chen. "Fast Boundary Detection: A Generalization and a New Algorithm." IEEE Trans. Computers
26, no. 2 (October 1977): 988-998.
B 270 (1980): 187-217.
13. D. Marr and E. C. Hildreth. "Theory of Edge Detection." Proc. R. Soc. Lond.
14. R. M. Haralick. "Zero Crossing of Second Directional Derivative Edge Detector." Robot Vision (A. Rosenfeld, ed.). SPIE 336, (1982): 91-96.
15. M. Beucke!. "An Operator Which Locates Edges in Digitized Pictures." J. ACM 18, no. 1 (January 1971): 113-125. Also see J. ACM
20, no. 4 (October 1973): 634-647. 16. A. K. Jain and S. Ranganath. "Image Restoration and Edge Extraction Based on 2-D
Stochastic Models." Proc. ICASSP-82, Paris, May 1982.
17. W. K. Pratt. Digital Image Processing. New York: Wiley Interscience, 1978, p. 497.
426 Image Analysis and Computer Vision Chap. 9
Section 9.5 For various types of edge linkage rules, contour-following, boundary detection
techniques, dynamic programming, and the like, we follow:
18. R. Nevatia. "Locating Object Boundaries in Textured Environments." IEEE Trans. Comput. C-25 (November 1976): 1170-1180.
19. A. Martelli. "Edge Detection Using Heuristic Search Methods." Comp. Graphics Image Proc. 1 (August 1972): 169-182. Also see Martelli in [5].
20. G. P. Ashkar and J. W. Modestino. "The Contour Extraction Problem with Biomedical Applications." Comp. Graphics Image Proc. 7 (1978): 331-355.
21. J. M. Lester, H. A. Williams, B. A. Weintraub, and J. F. Brenner. "Two Graph Searching Techniques for Boundary Finding in White Blood Cell Images." Comp. Biol. Med. 8 (1978): 293-308.
22. R. E. Bellman and S. Dreyfus. Applied Dynamic Programming. Princeton, N.J.: Prince ton University Press, 1962.
23. U. Montanari. "On the Optimal Detection of Curves in Noisy Pictures." Commun. ACM 14 (May 1971): 335-345.
24. P. V. C. Hough. "Method and Means of Recognizing Complex Patterns." U.S. Patent 3,069,654, 1962. 25. R. Cederberg. "Chain-Link Coding and Segmentation for Raster Scan Devices." Computer Graphics and Image Proc.
10, (1979): 224-234.
Section 9.6 For chain codes, its generalizations, and run-length coding based segmentation
approaches we follow:
26. H. Freeman. "Computer Processing of Line Drawing Images." Computer Surveys 6 (March 1974): 57-98. Also see Freeman in [5] and J. A. Saghri and H. Freeman in IEEE Trans. PAM/ (September 1981): 533-539.
The theory of B -splines is well documented in the literature. For its applica tions in computer graphics:
27. W. J. Gordon and R. F. Riesenfeld. "B-spline Curves and Surfaces," in R. E. Barnhill and R. F. Riesenfeld (eds.), Computer Aided Geometric Design, New York: Academic Press, 1974, pp. 95-126.
28. B. A. Barsky and D. P. Greenberg. "Determining a Set of B-spline Control Vertices to Generate an Interpolating Surface." Computer Graphics and Image Proc.
14 (1980): 203-226. 29. D. Paglieroni and A. K. Jain. "A Control Point Theory for Boundary Representation and Matching." Proc. ICASSP, Vol. 4, pp. 1851-1854, Tampa, Fla. 1985.
30. D. Hoffman, "The Interpretation of Visual Illusions." Scientific American, Dec. 1983, pp. 151-162.
Bibliography Chap. 9 427
Fourier Descriptors have been applied for shape analysis of closed curves and hand-printed characters. For details see Granlund in [5] and:
31. C. T. Zahn and R. S. Roskies. "Fourier Descriptors for Plane Closed Curves." IEEE Trans. Computers C-21 (March 1972): 269-281.
32. E. Persoon and K. S. Fu. "Shape Discrimination Using Fourier Descriptors." IEEE Trans. Sys. Man, Cybern. SMC-7 (March 1977): 170-179.
For theory of AR models for boundary representation, we follow: 33. R. L. Kashyap and R. Chellappa. "Stochastic Models for Closed Boundary Analysis:
Representation and Reconstruction." IEEE Trans. Inform. Theory IT-27 (September 1981): 627-637.
Section 9.7 Further details on quad-trees and medial axis transform:
34. H. Samet. "Region Representation: Quadtrees from Boundary Codes." Comm. ACM 23 (March 1980): 163-170.
Section 9.8 For basic theory of moments and its applications:
35. M. K. Hu. "Visual Pattern Recognition by Moment Invariants," in [5].
36. M. R. Teague. "Image Analysis via the General Theory of Moments." J. of Optical Society of America
70, no. 8 (August 1980): 920-930.
37. G. B. Gurevich. Foundations of the Theory of Algebraic Invariants. Groningen, The Netherlands: P. Noordhoff, 1964.
38. D. Casasent and D. Psaltis. "Hybrid Processor to Compute Invariant Moments for Pattern Recognition." J. Optical Society of America 5, no. 9 (September 1980): 395-397.
39. S. Dudani, K. Breeding, and R. McGhee. "Aircraft Identification by Moment In variants." IEEE Trans. on Computers C-26, no. 1 (January 1977): 39-45.
40. R. Wong and E. Hall. "Scene Matchirtg with Moment Invariants." Computer Graphics and Image Processing 8 (1978): 16-24.
41. H. Blum. "A Transformation for Extracting New Descriptions of Shape." Symposium on Models for the Perception of Speech and Visual Form, Cambridge: MIT Press, 1964. 42. E. R. Davies and A. P. Plummer. "Thinning Algorithms: A Critique and a New Method ology." Pattern Recognition
14, (1981): 53-63.
43. D. Rutovitz. "Pattern Recognition." J. of Royal Stat. Soc. 129, no. 66 (1966): 403-420.
44. J. Serra. Images Analysis and Mathematical Morphology. New York: Academic Press, 1982.
45. T. Pavlidis. "Minimum Storage Boundary Tracing Algorithm and Its Application to Automatic Inspection." IEEE Transactions on Sys., Man. and Cybern.
8, no. 1 (January 1978): 66-69.
428 Image Analysis and Computer Vision Chap. 9
Section 9.1 1 For surveys and further details on texture, see Hawkins in (4], Picket in (4], Haralick
et al. in (5], and:
46. P. Brodatz. Textures: A Photographic Album for Artists and Designers. Toronto: Dover Publishing Co., 1966.
47. R. M. Haralick. "Statistical and Structural Approaches to Texture." Proc. IEEE 67 (May 1979): 786-809. Also see Image Texture Analysis, New York: Plenum, 1981.
48. G. G. Lendaris and G. L. Stanley. "Diffraction Pattern Sampling for Automatic Pattern Recognition," in [5].
49. R. P. Kruger, W. B. Thompson, and A. F. Turner. "Computer Diagnosis of Pneumo coniosis." IEEE Trans. Sys. Man. Cybern. SMC-4, (January 1974): 40-49.
50. B. Julesz, et al. "Inability of Humans to Discriminate Between Visual Textures that Agree in Second Order Statistics-Revisited." Perception 2 (1973): 391-405. Also see IRE Trans. Inform. Theory IT-8 (February 1962): 84-92.
51. 0. D. Faugeraus and W. K. Pratt. "Decorrelation Methods of Texture Feature Extrac tion." IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2 (July 1980): 323-332.
52. B. H. McCormick and S. N. Jayaramamurthy. "Time Series Model for Texture Syn thesis." Int. J. Comput. Inform. Sci 3 (1974): 329-343. Also see vol. 4, (1975): 1-38.
53. G. R. Cross and A. K. Jain. "Markov Random Field Texture Models." IEEE Trans. Pattern Anal. Mach. Intell. PAMI-5, no. 1 (January 1983): 25-39.
54. T. Pavlidis. Structural Pattern Recognition. New York: Springer-Verlag, 1977.
55. N. Ahuja and A. Rosenfeld. "Mosaic Models for Textures." IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, no. 1 (January 1981): 1-11.
Section 9.12
56. G. L. Turin. "An Introduction to Matched Filtering." IRE Trans. Inform. Theory (June 1960): 311-329.
57. A. Vander Lugt, F. B. Rotz, and A. Kloester, Jr. "Character Reading by Optical Spatial Filtering." in J. Tippett et al. (eds), Optical and Electro-Optical Information Processing. Cambridge, Mass.: MIT Press, 1965, pp. 125-141. Also see pp. 5-11 in [5].
58. J. R. Jain and A. K. Jain. "Displacement Measurement and Its Application in Inter frame Image Coding." IEEE Trans. Common COM-29 (December 1981): 1799-1808.
59. D. L. Bamea and H. F. Silverman. "A Class of Algorithms for Fast Digital Image Registration." IEEE Trans. Computers (February 1972): 179-186.
Section 9.13, 9.14 Details of classification and clustering techniques may be found in (1] and other
texts on pattern recognition. For decision tree algorithm and other segmentation techniques:
60. C. Rosen et al. "Exploratory Research in Advanced Automation." SRI Technical Report First, Second and Third Reports, NSF Grant GI-38100Xl, SRI Project 2591, Menlo Park, Calif.: SRI, December 1974.
Bibliography Chap. 9 429
61. G. J. Agin and R. 0. Duda. "SRI Vision Research for Advanced Automation." Proc. 2nd. USA Japan Computer Conf., Tokyo, Japan, August 1975, pp. 113-117.
62. H. C. Andrews. Introduction to Mathematical Techniques in Pattern Recognition, New York: John Wiley, 1972. Also see A. B. Coleman and H. C. Andrews, "Image Segmen tation by Clustering." Proc. IEEE
67, no. 5 (May 1979): 773-785.
63. G. Nagy. "State of the Art in Pattern Recognition." Proc. IEEE 5, no. 5 (May 1968): 836-861. 64. G. H. Ball and D. J. Hall. "ISODATA, A Novel Method of Data Analysis and Pattern Classification." International Communication Conference, Philadelphia, June 1966.
65. C. T. Zahn. "Graph-Theoretical Methods for Detecting and Describing Gestalt Clus ters." IEEE Trans. Computers C-20, no. 1 (January 1971): 68-86. 66. J. C. Gower and G. J. S. Ross. "Minimum Spanning Trees, and Single Linkage Cluster Analysis." Appl. Statistics
18, no. 1 (1969): 54-64.
67. M. R. Anderberg. Cluster Analysis for Application. New York: Academic Press, 1971. 68. E. B. Henrichon, Jr. and K. S. Fu. "A Nonparametric Partitioning Procedure for Pattern
Classification." IEEE Trans. Computers C-18, no. 7 (July 1969). 69. I. Kabir. "A Computer Vision System Using Fast, One Pass Algorithms." M.S. Thesis,
University of California at Davis, 1983. 70. G. Hirzinger and K. Landzattel. "A Fast Technique for Segmentation and Recognition of Binary Patterns." IEEE Conference on Pattern Recognition and Image Processing,
1981 . 71. D. W. Paglieroni. "Control Point Algorithms for Contour Processing and Shape Analy sis," Ph.D. Thesis, University of California, Davis, 1986. 72. C. R. Brice and C. L. Fennema. "Scene Analysis Using Regions," in [5].
Section 9.15 For further reading or image understanding research see proceedings of Interna
tional Joint Conference on Artificial Intelligence, DARPA Image Understanding Workshop and the various references cited there. For pc board inspection and rule
based systems: 73. A. Darwish and A. K. Jain. "A Rule Based Approach for Visual Pattern Inspection."
IEEE Trans. Pattern Anal. Mach. Intell. PAMI-10, no. 1 (January 1988): 56-68.
74. J. R. Mandeville. "A Novel Method for Analysis of Printed Circuit Images." IBM J. Res. Dev.
29 (January 1985): 73-86.
430 Image Analysis and Computer Vision Chap. 9