Ill-posed problems in early vision

Mathematical results on ill-posed and ill-conditioned problems are reviewed and the formal aspects of regularization theory in the linear case are introduced. Specific topics in early vision and their regularization are then analyzed rigorously, characterizing existence, uniqueness, and stability of solutions. A fundamental difficulty that arises in almost every vision problem is scale, that is, the resolution at which to operate. Methods that have been proposed to deal with the problem include scale-space techniques that consider the behavior of the result across a continuum of scales. From the point of view of regulation theory, the concept of scale is related quite directly to the regularization parameter lambda . It suggested that methods used to obtained the optimal value of lambda may provide, either directly or after suitable modification, the optimal scale associated with the specific instance of certain problems. >

[1]  J. Hadamard.  Sur les problemes aux derive espartielles et leur signification physique , 1902 .

[2]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[3]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[4]  R. Courant,et al.  Methods of Mathematical Physics, Vol. I , 1954 .

[5]  R. Thom.  Quelques propriétés globales des variétés différentiables , 1954 .

[6]  Fritz John,et al.  Continuous dependence on data for solutions of partial differential equations with a prescribed bound , 1960 .

[7]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[8]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

[9]  V. Ivanov,et al.  The approximate solution of operator equations of the first kind , 1966 .

[10]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

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

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

[13]  Berthold K. P. Horn.  SHAPE FROM SHADING: A METHOD FOR OBTAINING THE SHAPE OF A SMOOTH OPAQUE OBJECT FROM ONE VIEW , 1970 .

[14]  Thomas O. Binford,et al.  On Boundary Detection , 1970 .

[15]  K. Miller.  Least Squares Methods for Ill-Posed Problems with a Prescribed Bound , 1970 .

[16]  M. Nashed.  Generalized Inverses, Normal Solvability, and Iteration for Singular Operator Equations , 1971 .

[17]  Valentin F. Turchin,et al.  The use of mathematical-statistics methods in the solution of incorrectly posed problems , 1971 .

[18]  M. Nashed.  Differentiability and Related Properties of Nonlinear Operators: Some Aspects of the Role of Differentials in Nonlinear Functional Analysis , 1971 .

[19]  M. Z. Nashed,et al.  On the Convergence of the Conjugate Gradient Method for Singular Linear Operator Equations , 1972 .

[20]  J. Franklin.  On Tikhonov’s method for ill-posed problems , 1974 .

[21]  Berthold K. P. Horn.  Obtaining shape from shading information , 1989 .

[22]  Jean Duchon,et al.  Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces , 1976 .

[23]  Tomaso Poggio,et al.  A Theory of Human Stereo Vision , 1977 .

[24]  G. Wahba.  Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy , 1977 .

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

[26]  Fred M. Dickey,et al.  An Optimal Frequency Domain Filter for Edge Detection in Digital Pictures , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  B K Horn,et al.  Calculating the reflectance map. , 1979, Applied optics.

[28]  Bruno O. Shubert,et al.  Random variables and stochastic processes , 1979 .

[29]  T. Poggio,et al.  A computational theory of human stereo vision , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[30]  G. Wahba,et al.  Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation , 1980 .

[31]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[32]  G. Wahba.  Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems with Noisy Data. , 1980 .

[33]  Paul Virilio,et al.  Vision machine , 1981, Nature.

[34]  Berthold K. P. Horn,et al.  Hill shading and the reflectance map , 1981, Proceedings of the IEEE.

[35]  Katsushi Ikeuchi,et al.  Determining Surface Orientations of Specular Surfaces by Using the Photometric Stereo Method , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[36]  Berthold K. P. Horn,et al.  Determining Optical Flow , 1981, Other Conferences.

[37]  A. Roger,et al.  Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem , 1981 .

[38]  Michael Brady.  MIT Progress in Understanding Images , 1982 .

[39]  Demetri Terzopoulos.  Multi-Level Reconstruction of Visual Surfaces: Variational Principles and Finite Element Representations , 1982 .

[40]  W E Grimson,et al.  A computational theory of visual surface interpolation. , 1982, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[41]  Alan L. Yuille,et al.  The Smoothest Velocity Field and Token Matching , 1983 .

[42]  J. Canny.  Finding Edges and Lines in Images , 1983 .

[43]  Demetri Terzopoulos,et al.  Multilevel computational processes for visual surface reconstruction , 1983, Comput. Vis. Graph. Image Process..

[44]  J. Marroquín.  Surface Reconstruction Preserving Discontinuities , 1984 .

[45]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[46]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[47]  E. Hildreth.  The computation of the velocity field , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[48]  V. A. Morozov,et al.  Methods for Solving Incorrectly Posed Problems , 1984 .

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

[50]  Alan L. Yuille.  The Smoothest Velocity Field Token Matching Schemes , 1984, ECAI.

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

[52]  Tomaso Poggio.  Integrating vision modules with coupled MRFs , 1985 .

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

[54]  Mario Bertero,et al.  Linear inverse problems with discrete data. I. General formulation and singular system analysis , 1985 .

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

[56]  John R. Kender,et al.  Visual Surface Reconstruction Using Sparse Depth Data , 1986, CVPR 1986.

[57]  Andrew Blake,et al.  Weak Continuity Constraints Generate Uniform Scale-Space Descriptions of Plane Curves , 1986, ECAI.

[58]  Tomaso A. Poggio,et al.  On Edge Detection , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[59]  M. Bertero.  Regularization methods for linear inverse problems , 1986 .

[60]  Iterative inversion of experimental data in weighted spaces , 1986 .

[61]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[62]  Tomaso A. Poggio,et al.  An Optimal Scale for Edge Detection , 1988, IJCAI.

[63]  Tomaso Poggio,et al.  Probabilistic Solution of Ill-Posed Problems in Computational Vision , 1987 .

[64]  L. Payne,et al.  Improperly Posed Problems in Partial Differential Equations , 1987 .

[65]  T. Poggio,et al.  Visual Integration and Detection of Discontinuities: The Key Role of Intensity Edges , 1987 .

[66]  David Lee,et al.  Some computational aspects of low-level computer vision , 1988, Proc. IEEE.

[67]  David Lee,et al.  One-Dimensional Regularization with Discontinuities , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[68]  A. Verri,et al.  Mathematical properties of the 2-D motion field: from singular points to motion parameters , 1989, [1989] Proceedings. Workshop on Visual Motion.

[69]  Tomaso A. Poggio,et al.  Motion Field and Optical Flow: Qualitative Properties , 1989, IEEE Trans. Pattern Anal. Mach. Intell..