Intrinsic Stabilizers of Planar Curves

Regularization o ers a powerful framework for signal recon struction by enforcing weak constraints through the use of stabilizers Stabilizers are functionals measuring the degree of smoothness of a sur face The nature of those functionals constrains the properties of the reconstructed signal In this paper we rst analyze the invariance of stabilizers with respect to size transformation and their ability to con trol scale at which the smoothness is evaluated Tikhonov stabilizers are widely used in computer vision even though they do not incorporate any notion of scale and may result in serious shape distortion We rst introduce an extension of Tikhonov stabilizers that o ers natural scale control of regularity We then introduce the intrinsic stabilizers for pla nar curves that apply smoothness constraints on the curvature pro le instead of the parameter space Introduction Most tasks in computer vision can be described as inferring geometric and phys ical properties of three dimensional objects from two dimensional images A characteristic of those inverse problems is their ill posed nature PT Assump tions about the scene such as smoothness or shape must be made to retain the best solution within the range of prior knowledge Regularization transforms an ill posed problem into a well posed minimization problem by constraining the solution to belong to a set of allowed functions If the problem is formalized as A d where A is an operator describing the image formation process and d is a function describing the data extracted from the image then the regularized problem consists in minimizing a functional of the form BTT E S D kP k kA dk kP k evaluates the smoothness of the solution and is called a stabilizing functional or stabilizers kA dk evaluates the distance between the solution to the data The regularizing parameter weights the relative importance of smoothness with respect to the closeness of t Variational principles involving smoothness constraints are widely used in computer vision ranging from surface reconstruction BK segmentation with active contours KWT and surfaces DHI b Geometric modeling primitives such as splines under tension Sch Beta Spline BT proposed in computer aided design are derived from variational principles similar to the interpolation approach of regularization In this paper we rst analyze the di erent smoothness measures with regard to ve criteria of invariance Then we extend the notion of stabilizing functionals to di erential stabilizers by transforming the variational principle of equation into the problem of solving a di erential equation Finally we propose a generalization of Tikhonov stabilizers that provides both spatial control of the smoothness constraint and intrinsic shape formulation