Abstract
Mapping functions forward is required in image warping and other signal processing applications. The problem is described as follows: specify an integer d ≥ 1, a compact domain D ⊂ R d, lattices L 1,L 2 ⊂ R d, and a deformation function F : D → R d that is continuously differentiable and maps D one-to-one onto F(D). Corresponding to a function J : F(D) → R, define the function I = J ○ F. The forward mapping problem consists of estimating values of J on L 2 ⋂ F(D), from the values of I and F on L 1 ⋂ D. Forward mapping is difficult, because it involves approximation from scattered data (values of I ○ F -1 on the set F(L 1 $#x22C2; D)), whereas backward mapping (computing I from J) is much easier because it involves approximation from regular data (values ofJ on L 2 ⋂ D). We develop a fast algorithm that approximates J by an orthonormal expansion, using scaling functions related to Daubechies wavelet bases. Two techniques for approximating the expansion coefficients are described and numerical results for a one dimensional problem are used to illustrate the second technique. In contrast to conventional scattered data interpolation algorithms, the complexity of our algorithm is linear in the number of samples.
Similar content being viewed by others
References
A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, “Stationary Subdivision,” Memoirs of the American Mathematical Society, vol. 93, 1991, pp. 1–186.
A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 45, 1992, pp. 485–560.
L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, “Optical data processing and filtering systems,” IRE Transactions on Information Theory, vol. IT-6, 1960, pp. 384–400.
W. Dahmen and C. A. Micchelli, “On stationary subdivision and the construction of compactly supported orthonormal wavelets,” Numerical Mathematics, vol. 94, 1990, pp. 69–89.
I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, 1988, pp. 909–996.
I. Daubechies, Ten Lectures on Wavelets, Philadelphia: Society for Industrial and Applied Mathematics, 1992.
R. Glowinski, W. Lawton, M. Ravachol, and E. Tenenbaum, “Wavelet solution of linear and nonlinear elliptic, parabolic, and hyperbolic problems in one space dimension,” in R. Glowinski and A. Lichnewsky (eds.), Proceedings of the Ninth International Conference on Computing Methods in Applied Sciences and Engineering, Paris, France, 1990. SIAM, Philadelphia.
R. C. Gonzales and R. E. Woods, Digital Image Processing, Reading, MA: Addison-Wesley, 1992.
J. L. Horner, Optical Signal Processing, New York: Academic Press, 1987.
C. L. Lawson, “Software for C1 Surface Interpolation,” Mathematical Software III, J. R. Rice (ed.), London: Academic Press, 1977, pp. 161–194.
W. Lawton, “A new polar Fourier transform for computer-aided tomography and spotlight synthetic aperture radar,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 36, no. 6, 1988, pp. 931–933.
W. Lawton, “Necessary and sufficient conditions for constructing orthonormal wavelet bases,” Journal of Mathematical Physics, vol. 32, no. 1, 1991, pp. 57–61.
W. Lawton, “Multilevel properties of the wavelet-Galerkin operator,” Journal of Mathematical Physics, vol. 32, no. 6, 1991, pp. 1440–1443.
W. Lawton, “Application of complex-valued wavelets to subband decomposition,” IEEE Transactions on Signal Processing, vol. 41, no. 12, 1993, pp. 3566–3568.
W. Lawton, S. L. Lee, and Z. Shen, “Characterization of compactly supported refinable splines,” Advances in Computational Mathematics, vol. 3, 1995, pp. 137–145.
R. Franke and G. Nielson, “Scattered Data Interpolation and Applications: A Tutorial and Survey,” H. Hagen and D. Roller (eds.), Geometric Modelling: Method and Applications, New York: Springer-Verlag, 1991, pp. 131–159.
Z. C. Li, T. D. Bui, Y. Y. Tang and C. Y. Suen, Computer Transformations of Digital Images and Patterns, Singapore: World Scientific, 1989.
W. K. Pratt, Digital Image Processing, New York: John Wiley, 1978.
D. Ruprecht and H. Müller, “Image warping with scattered data interpolation methods,” in H. Hagen and D. Roller (eds.), Geometric modelling: Methods and applications, New York: Springer-Verlag, 1991.
S. D. Riemenschneider and Z. Shen, “General interpolation on the lattice hZs: Compactly supported fundamental solutions,” Numerische Mathematik, to appear.
G. Strang and G. Fix, “A Fourier analysis of the finite element variational method,” in G. Geymonat (ed.), Constructive Aspects of Functional Analysis, C.I.M.E., 1973, pp. 739–840.
G. Wohlberg and T. E. Boult, “Digital Image Warping,” IEEE Computer Society Press, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lawton, W. A Fast Algorithm to Map Functions Forward. Multidimensional Systems and Signal Processing 8, 219–227 (1997). https://doi.org/10.1023/A:1008233310373
Issue Date:
DOI: https://doi.org/10.1023/A:1008233310373