Picture Engineering pp 75-95 | Cite as

# Shape Design, Representation, and Restoration with Splines

## Abstract

The purpose of the present paper is to demonstrate several applications of interpolation and approximation methods such as splines to picture engineering In particular, three types of applications, i.e., shape design, shape representation, and shape restoration are discussed. Shape design is concerned with construction of visually beautiful curves and interactive modification of the curves. The method of B-splines gives a physically natural (minimum curvature] curve which is locally modifiable, since it is a linear combination of basis functions (B-splines) having local support. Bezier’s method generates a smooth curve (polynomial) by means of Bezier polygon and is appropriate for interactive use, although a local modification may propagate throughut the whole interval. Shape representation approximates a given irregular shape, where a method suitable for use depends on a particular way in which data are given. For surface approximation Coons methods are used as a standard tool, where an entire surface is subdivided into pieses whose boundary data are used to construct an approximating surface for the corresponding piece. When the boundary data are given at a selected set of points, tensor product of splines has been known to be usable. For surfaces which can be adequately described in terms of one-parameter family of curves, the technique known as lofting is applicable. If two families of parametric curves must be mixed to define a given surface, one may effectively use boolean sum approximation. Shape restoration is an approximation of an existing object, where the data are noisy or incomplete. In this case an approximation will depend on smoothing property and the nature of data given. Moreover, in some cases, irregularly distributed data points must be incorporated. In the image restoration the convolution property of B-splines is useful for the restoration of space- invariant degradations. As an additional example, we mention the pattern generation of air pollution, where spline under tension is prefered to the ordinary cubic spline in view of the accuracy of the approximation.

## Keywords

Tensor Product Point Spread Function Image Restoration Shape Design Smoothing Spline## Preview

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## References

- 1.S. C. Wu, J. F. Abel, D. P. Geenberg: Comm. ACM, 10, 703–712 (1977)CrossRefGoogle Scholar
- 2.J. H. Ahlberg, E. N. Nilson, J. L. Walsh: The Theory of Splines and Their Applications, (Academic Press, New York, 1967)zbMATHGoogle Scholar
- 3.K. Hatano: Johoshori (Journal of Information Processing Society of Japan), 1, 19–27 (1981)Google Scholar
- 4.R. S. Varga: Functional Analysis and Approximation Theory in Numerical Analysis, (SIAM Publications, Philadelphia, Pennsylvania, 1971)zbMATHCrossRefGoogle Scholar
- 5.T. Lyche, R. Winther: J. Approx. Th. 266–279 (1979)Google Scholar
- 6.D. G. Schweikert: J. Math. Physics, 312–317 (1966)Google Scholar
- 7.S. D. Curry, I. J. Schoenbert: Bull. Amer. Math. Soc. 1114 (1947)Google Scholar
- 8.C. de Boor: A Practical Guide to Splines, (Springer, New York, 1978)zbMATHCrossRefGoogle Scholar
- 9.W. K. Giloi: Interactive Computer Graphics — Data Structures, Algorithms, Languages, (Prentice-Hall, Englewood Cliffs, N. J., 1978)Google Scholar
- 10.S. D. Conte, C. de Boor: Elementary Numerical Analysis — an Algorithmic Approach, 2nd ed. (McGraw-Hill, New York, 1972)zbMATHGoogle Scholar
- 11.P. E. Bezier: Emploi de Machines a Commande Numeri que, (Masson et Cie, Paris, 1970), Translated by D. R. Forrest, A. F. Pankhurstas P. F. Bezier: Numerical Control Mathematics and Applications, (Wiley, 1972)Google Scholar
- 12.W. J. Gordon, R. F. Riesenfeld: B-spline Curves and Surfaces, In Computer- Aided Geometric Design, R. E. Barnhill and R. F. Riesenfeld, eds. (Academic Press, New York, 1974)Google Scholar
- 13.D. F. Rogers, J. A. Adams: Mathematical Elements for Computer Graphics, (McGraw-Hill, New York, 1976)Google Scholar
- 14.A. R. Forrest: Computer Graphics and Image Processing, 1, 4, 341–359 (1972)MathSciNetCrossRefGoogle Scholar
- 15.A. Coons: M.I.T., MAC-TR-41 (1967)Google Scholar
- 16.W. J. Gordon: J. Math. Mech. 18, 931–952 (1969)MathSciNetzbMATHGoogle Scholar
- 17.I. J. Schoenberg: Proc. Nat. Acad. Sci., 52, 947–950 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
- C. H. Reinsch: Numer. Math., 10, 177–183 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
- 19.H. S. Hou, C. Andrews: IEEE Trans. Computers, C-26, 9, 856–873 (1977)CrossRefGoogle Scholar
- 20.A. Rosenfeld, A. C. Kak: Digital Picture Processing, (Academic Press, New York, 1976)Google Scholar
- 21.M. J. Peyrovian, A. A. Sawchuk: Appl. Opt. 4, 660–666 (1978)CrossRefGoogle Scholar
- 22.M. Fujiwara, K. Oi, J. Shindo: Symposium in Environmental and Sanitary Engineering, (in Japanese) Kyoto Univ. Aug. (1981)Google Scholar
- 23.Oonishi: J. Oceanographical Soc. Japan (in Japanese) 259–264 (1975)Google Scholar
- 24.A. K. Cline: Atmospheric Technology, NCAR, 3, 60–65 (1973)Google Scholar
- 25.M. Shinohara, M. Naito: Kankyo Joho Shori (J. Information in Environmental Science, in Japanese), 9, 4, 62–64 (1980)Google Scholar
- 26.H. Akima: ACM Trans. Math. Software, 4, 2, 148–159 (1978)zbMATHCrossRefGoogle Scholar
- 27.W. R. Goodin, G. J. McRae, J. H. Seinfeld: J. Appl. Meteor. 18, 761–771 (1979)CrossRefGoogle Scholar