The Application of Approximation and Complexity Theory Methods to the Solution of Computer Vision Problems

We survey aspects of approximation and complexity theory and their application to the numerous computer vision problems that require an approximate solution because only partial information is available. We consider ill-posed computer vision problems and the methods that can be employed towards reformulating them as well-posed. We are particularly interested in the surface reconstruction problem that is encountered in the construction of the 2 1/2-D sketch, and which has been formulated and solved using different methods. We apply regularization theory, information-based complexity, and other methods to the solution of this problem. Finally, the shape from shadows problem is formulated and the optimal error algorithm is constructed and analyzed

The Application of Approximation and Complexity Theory Methods to the Solution of Computer Vision Problems

We survey aspects of approximation and complexity theory and their application to the numerous computer vision problems that require an approximate solution because only partial information is available. We consider ill-posed computer vision problems and the methods that can be employed towards reformulating them as well-posed. We are particularly interested in the surface reconstruction problem that is encountered in the construction of the 2 1/2-D sketch, and which has been formulated and solved using different methods. We apply regularization theory, information-based complexity, and other methods to the solution of this problem. Finally, the shape from shadows problem is formulated and the optimal error algorithm is constructed and analyzed

B-splines, Pólya curves, and duality

AbstractLocal duality between B-splines and Pólya curves is examined, mostly from the viewpoint of computer-aided geometric design. Certain known results for the two curve types are shown to be related. A few new results for Pólya curves and a curve scheme related to B-splines also follow from these investigations

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

International audienceThe verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq's logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples

Combined parametric and worst case circuit analysis via Taylor models

This paper proposes a novel paradigm to generate a parameterized model of the response of linear circuits with the inclusion of worst case bounds. The methodology leverages the so-called Taylor models and represents parameter-dependent responses in terms of a multivariate Taylor polynomial, in conjunction with an interval remainder accounting for the approximation error. The Taylor model representation is propagated from input parameters to circuit responses through a suitable redefinition of the basic operations, such as addition, multiplication or matrix inversion, that are involved in the circuit solution. Specifically, the remainder is propagated in a conservative way based on the theory of interval analysis. While the polynomial part provides an accurate, analytical and parametric representation of the response as a function of the selected design parameters, the complementary information on the remainder error yields a conservative, yet tight, estimation of the worst case bounds. Specific and novel solutions are proposed to implement complex-valued matrix operations and to overcome well-known issues in the state-of-the-art Taylor model theory, like the determination of the upper and lower bound of the multivariate polynomial part. The proposed framework is applied to the frequency-domain analysis of linear circuits. An in-depth discussion of the fundamental theory is complemented by a selection of relevant examples aimed at illustrating the technique and demonstrating its feasibility and strength