Improving the Stability of the Recovery of Algebraic Curves via Bernstein Basis Polynomials and Neural Networks

Abstract

We present new methods for the stable reconstruction of a class of binary images from sparse measurements. The images that we consider are characteristic functions of algebraic shapes, that is, interiors of zero sets of bivariate polynomials, and we assume that we only know a finite set of samples of these images. A solution to this problem can be formulated in terms of a system of linear equations of moments. Although it was shown in the literature that one can improve the stability of the reconstruction by increasing the number of moments, the recovery of an algebraic shape remains unstable in the sense that small errors in the computation of the moments may have a catastrophic impact on the recovery algorithm. To address this numerical and theoretical instability, we introduce a novel approach where we represent bivariate polynomials and moments in terms of Bernstein basis polynomials and use them in combination with a polynomial-reproducing, refinable sampling kernel. We show that this is approach is very robust, straightforward to implement, and fast to compute. We also address the same reconstruction problem using an alternative approach that combines a convolutional neural network with a model-based constraint supported by our prior theoretical study. This approach also yields very competitive results and is even more robust to noise. We illustrate the performance of our algorithms on noisy samples through extensive experiments. Our code is publicly accessible on GitHub at github.com/wjmolina/AlgebraicCurves.

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Keywords

algebraic, curve, shape, polynomial, bivariate, kernel, neural, network, spline, Bernstein, moment

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