One-Dimensional Regularization with Discontinuities

Regularization is equivalent to fitting smoothing splines to the data so that efficient and reliable numerical algorithms exist for finding solutions. however, the results exhibit poor performance along edges and boundaries. To cope with such anomalies, a more general class of smoothing splines that preserve corners and discontinuities is studied. Cubic splines are investigated in detail, since they are easy to implement and produce smooth curves near all data points except those marked as discontinuities or creases. A discrete regularization method is introduced to locate corners and discontinuities in the data points before the continuous regularization is applied. >

[1]  Edmund Taylor Whittaker.  On a New Method of Graduation , 1922, Proceedings of the Edinburgh Mathematical Society.

[2]  I J Schoenberg,et al.  SPLINE FUNCTIONS AND THE PROBLEM OF GRADUATION. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[3]  C. Reinsch.  Smoothing by spline functions , 1967 .

[4]  J. L. Walsh,et al.  The theory of splines and their applications , 1969 .

[5]  J. Uhlig.  C. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems. (Series in Automatic Computation) XI + 148 S. Englewood Cliffs, N.J. 1967. Prentice-Hall, Inc. Preis geb. 54 s. net , 1972 .

[6]  L. Schumaker.  Fitting surfaces to scattered data , 1976 .

[7]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[8]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[9]  F. Utreras.  Cross-validation techniques for smoothing spline functions in one or two dimensions , 1979 .

[10]  G. Wahba.  Convergence rates of "thin plate" smoothing splines wihen the data are noisy , 1979 .

[11]  M. Powell,et al.  Approximation theory and methods , 1984 .

[12]  Katsushi Ikeuchi,et al.  Numerical Shape from Shading and Occluding Boundaries , 1981, Artif. Intell..

[13]  Eric L. W. Grimson,et al.  From Images to Surfaces: A Computational Study of the Human Early Visual System , 1981 .

[14]  Ellen C. Hildreth,et al.  Measurement of Visual Motion , 1984 .

[15]  Demetri Terzopoulos,et al.  Multiresolution computation of visible-surface representations , 1984 .

[16]  G. Wahba.  Surface fitting with scattered noisy data on Euclidean D-space and on the sphere , 1984 .

[17]  David Lee,et al.  Optimal algorithms for image understanding: Current status and future plans , 1985, J. Complex..

[18]  Tomaso Poggio,et al.  Computational vision and regularization theory , 1985, Nature.

[19]  Tomaso A. Poggio.  Early vision: From computational structure to algorithms and parallel hardware , 1985, Comput. Vis. Graph. Image Process..

[20]  W. Eric L. Grimson,et al.  Discontinuity detection for visual surface reconstruction , 1985, Comput. Vis. Graph. Image Process..

[21]  Demetri Terzopoulos,et al.  Regularization of Inverse Visual Problems Involving Discontinuities , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[22]  David Lee,et al.  Fast multiplication of a recursive block Toeplitz matrix by a vector and its application , 1986, J. Complex..