Abstract

Many problems in low-level vision are ill-posed in the sense that their solution does not exist, is not unique, or does not depend continuously on the data. Following the ideas of Poggio and Torre¹, we approach such problems by using the Tikhonov and Arsenin² regularization. That means we find the solution that is as consistent as possible with the data and also as smooth as possible. Smoothing the solution tends to blur boundaries and makes it harder to recognize regions where there are sharp changes in real-world variables. We approach this problem by assuming the error (i.e., the amount of inconsistency of the solution with the data) of nearby points is correlated. The error is modeled as a Levy stable (e.g., Gaussian) blur of a white noise signal. We must first deblur this error. The regularization condition we use tries to minimize the deblurred error and also maintain smoothness. Our condition causes the derivatives of the solution to track the derivatives of the data thus preserving discontinuities. We also observe that the usual smoothness condition constrains only the low-order derivatives of the solution. We constrain all the derivatives. Our procedures require us to perform numerical differentiation of the data, a highly unstable operation. In the course of discussing how to deal with this problem, we develop nonlinear filters for edge detection. Much previous work on discontinuous variation can be viewed theoretically in terms of the concepts we introduce.

© 1987 Optical Society of America