We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source and freely available.

我们提出了一种从少量测量值稳定重建一类二进制图像的新方法。我们考虑的图像是代数域（即定义为二元多项式零位点的域）的特征函数，并且我们假定每个图像仅知道有限的一组均匀样本。可以根据与一组图像力矩相关的线性方程式来设置该问题的解决方案。但是，力矩对噪声的敏感性使数值解非常不稳定。为了导出鲁棒的图像恢复算法，我们用二元Bernstein多项式表示代数多项式和相应的图像矩，并应用生成多项式的可精炼采样内核。这种方法对于噪声是鲁棒的，计算速度快并且易于实现。我们通过大量的数值实验说明了从嘈杂样本中重建算法的性能。我们的代码是开源发布的，可以免费获得。