Binomial Convolutions and Derivatives Estimation from Noisy Discretizations

Abstract

Abstract. We present a new method to estimate derivatives of digitized functions. Even with noisy data, this approach is convergent and can be computed by using only the arithmetic operations. Moreover, higher order derivatives can also be estimated. To deal with parametrized curves, we introduce a new notion which solves the problem of correspondence between the parametrization of a continuous curve and the pixels numbering of a discrete object.

Normals and Curvature Estimation for Digital Surfaces Based on Convolutions

“…Although the theoretical work done in [14] shows that higher orders estimates may be computed in a similar way for 1D digitized functions, it seems from our first experiments that the precision is not so good when using on-surface convolution.…”

“…For this purpose, we have used digitized spheres and tori with several radii and measured the average angular error between the estimated and the exact normal vectors for all surfels. In these experiments, we use a number n of convolution iterations inspired by the result of [14] (Theorem 1) for continuous functions from R to R known through their digitizations. Following the latter result, if h is the width of the pixels used for the digitization process, then a convergence at rate h 2 3 for the estimation of the first derivative of the function may be obtained by using a convolution mask with a width w = ⌊h − 4 3 ⌋.…”

“…Figure 7 shows the estimated and exact Gaussian curvatures for all the surfels sorted according to their increasing exact Gaussian curvatures. The size of the averaging mask to be used was set according to the result of [14] about higher order derivatives (Theorem 3). Actually, we divided the estimation process in two steps.…”

“…Then, we use two orthogonal differentiation operators on the resulting centers to estimate partial derivatives, and by a cross product we obtain normal vectors. As previously mentioned, we follow the criterion given in [14] to set the optimal number of iterations depending on the digitization step (voxel size).…”

“…We eventually use the iterated averaging followed by convolutions with differentiation masks to estimate partial derivatives and then normal vectors over a surface. We provide an heuristics based on [14] for an optimal mask size and show results. …”