Image Analysis Using Multigrid Relaxation Methods

Image analysis problems, posed mathematically as variational principles or as partial differential equations, are amenable to numerical solution by relaxation algorithms that are local, iterative, and often parallel. Although they are well suited structurally for implementation on massively parallel, locally interconnected computational architectures, such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements. Hence, they converge extremely slowly when confronted by the large representations of early vision. Application of multigrid methods can overcome this drawback, as we showed in previous work on 3-D surface reconstruction. In this paper, we develop multiresolution iterative algorithms for computing lightness, shape-from-shading, and optical flow, and we examine the efficiency of these algorithms using synthetic image inputs. The multigrid methodology that we describe is broadly applicable in early vision. Notably, it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill-posed image analysis problems.

[1]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[2]  R. P. Fedorenko A relaxation method for solving elliptic difference equations , 1962 .

[3]  N. Bakhvalov On the convergence of a relaxation method with natural constraints on the elliptic operator , 1966 .

[4]  E. Land,et al.  Lightness and retinex theory. , 1971, Journal of the Optical Society of America.

[5]  A. Brandt Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems , 1973 .

[6]  Berthold K. P. Horn,et al.  Determining lightness from an image , 1974, Comput. Graph. Image Process..

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

[8]  Azriel Rosenfeld,et al.  Scene Labeling by Relaxation Operations , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[9]  M. Blumer,et al.  Azaarenes in Recent Marine Sediments , 1977, Science.

[10]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[11]  R. Nicolaides On the ² convergence of an algorithm for solving finite element equations , 1977 .

[12]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[13]  Janette Atkinson,et al.  Channels in Vision: Basic Aspects , 1978 .

[14]  Shimon Ullman,et al.  Relaxation and constrained optimization by local processes , 1979 .

[15]  Kenneth E. Batcher,et al.  Design of a Massively Parallel Processor , 1980, IEEE Transactions on Computers.

[16]  P. Hemker Introduction to multigrid methods , 1981 .

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

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

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

[20]  Grahame B. Smith The Recovery of Surface Orientation from Image Irradiance , 1982 .

[21]  Steven W. Zucker,et al.  On the Foundations of Relaxation Labeling Processes , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[22]  R. Bajcsy,et al.  A computerized system for the elastic matching of deformed radiographic images to idealized atlas images. , 1983, Journal of computer assisted tomography.

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

[24]  Geoffrey E. Hinton,et al.  Parallel visual computation , 1983, Nature.

[25]  W. Eric L. Grimson,et al.  An implementation of a computational theory of visual surface interpolation , 1983, Comput. Vis. Graph. Image Process..

[26]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[27]  F. Glazer Multilevel Relaxation in Low-Level Computer Vision , 1984 .

[28]  Azriel Rosenfeld,et al.  Multiresolution image processing and analysis , 1984 .

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

[30]  T. Poggio,et al.  Ill-Posed Problems and Regularization Analysis in Early Vision , 1984 .

[31]  Alan L. Yuille,et al.  An Extremum Principle for Shape from Contour , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[32]  Ellen C. Hildreth,et al.  Computations Underlying the Measurement of Visual Motion , 1984, Artif. Intell..

[33]  Takeo Kanade,et al.  Adapting optical-flow to measure object motion in reflectance and x-ray image sequences (abstract only) , 1984, COMG.

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

[35]  A. Blake,et al.  On Lightness Computation in Mondrian World , 1985 .

[36]  W. Daniel Hillis,et al.  The connection machine , 1985 .

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

[38]  Michael J. Brooks,et al.  The variational approach to shape from shading , 1986, Comput. Vis. Graph. Image Process..

[39]  Hans-Hellmut Nagel,et al.  An Investigation of Smoothness Constraints for the Estimation of Displacement Vector Fields from Image Sequences , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[40]  Tomaso Poggio,et al.  Cooperative computation of stereo disparity , 1988 .