Near-Optimal Regularization Parameters for Applications in Computer Vision Changjiang Yang, Ramani Duraiswami and Larry Davis Computer Vision Laboratory University of Maryalnd College Park, MD 20742, USA Abstract Computer vision requires the solution of many ill-posed problems such as optical flow, structure from motion, shape from shading, surface reconstruction, image restoration and edge detection. Regularization is a popular method to solve ill-posed problems, in which the solution is sought by minimization of a sum of two weighted terms, one measur- ing the error arising from the ill-posed model, the other in- dicating the distance between the solution and some class of solutions chosen on the basis of prior knowledge (smooth- ness, or other prior information). One of important issues in regularization is choosing optimal weight(or regulariza- tion parameter). Existing methods for choosing regular- ization parameters either require the prior information on noise in the data, or are heuristic graphical methods. In this work we apply a new method for choosing near-optimal regularization parameters by approximately minimizing the distance between the true solution and the family of regu- larized solutions. We demonstrate the effectiveness of this approach for the regularization on two examples: edge de- tection and image restoration. 1. Introduction Computer vision consists of the problems such as edge detection, motion estimation, surface reconstruction, and shape from shading, that aim at recovering the physical properties of surfaces in 3D from 2D images. As pointed out by Poggio et al. [11, 1] many problems of computer vi- sion are ill-posed in the sense of Hadamard, for which at least one of the conditions of existence, uniqueness or con- tinuity of the solution are violated. The regularization technique is a popular method to transform the original ill-posed problem to a well-posed one. The basic idea of regularization is to find an opti- mal approximation of the exact solution from a family of approximate solutions depending on a positive parameter called regularization parameter. The regularization param- eter controls the degree of regularity and the closeness of the solution to the data. The choice of the value of the regularization parameter is a crucial and difficult problem in the theory of regulariza- tion. Several methods have been developed to find the reg- ularization parameter. The discrepancy principle requires a precise estimate of the energy of the noise. The estimate of the regularization parameter is the value such that the dis- crepancy of the corresponding regularized solution is just equal to the energy of the noise. Another approach proposed by Miller [8] assumes that one has both a bound on the en- ergy and a bound on the discrepancy of the unknown object. An approximate solution can be found from the intersection of the permissible regions of the two bounds. The meth- ods considered previously require knowledge of the noise level. On the other hand, based on stochastic assumptions, generalized cross-validation (GCV) tries to choose the reg- ularization parameter from the data itself using statistical methods [2]. The L-curve method, introduced by Hansen and O’Leary [6], is a graphical method that does not require information about the noise. The corner of the L-curve cor- responds to the best compromise between approximation er- ror and the noise-propagation error. All these regularization parameter choosing methods are either computationally intensive (as they require solution of the problem for many values of the regularization parame- ter), or graphically motivated (thus needing human interpre- tation). Recently a near-optimal method was proposed by O’Leary [9]. This method is distinguished by the fact that without the priori information about the noise, it chooses a near optimal regularization parameter which approximately minimizes the distance from the noise-free solution to the family of the regularized solutions. Compared with other methods, the computation overhead is relatively low. In this work we applied the algorithm to regularization of vision problems for determining the regularization parame- ter. We also extend the algorithm to the problems with gen- eral regularization term. The rest of this paper is organized as follows. In section 2 the regularization for some appli- cations in computer vision are described. In section 3 we