Stable recovery of planar regions with algebraic boundaries in Bernstein form
Stable recovery of planar regions with algebraic boundaries in Bernstein form
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DOI：
10.1007/s10444-021-09843-0
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发表时间：
2021-02
影响因子：
1.7
通讯作者：
C. Conti;M. Cotronei;D. Labate;Wilfredo Molina
中科院分区：
文献类型：
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作者：
C. Conti;M. Cotronei;D. Labate;Wilfredo Molina
我们提出了一种新的方法稳定重建一类二值图像从少量的测量。我们考虑的图像是代数域的特征函数，即定义为二元多项式的零轨迹的域，并且我们假设每个图像只知道有限的均匀样本集。这样的问题的解决方案可以建立在与一组图像矩相关联的线性方程方面。然而，矩对噪声的敏感性使得数值解非常不稳定。为了得到一个强大的图像恢复算法，我们表示代数多项式和相应的图像矩的二元伯恩斯坦多项式和应用多项式生成，可细化采样内核。这种方法对噪声具有鲁棒性，计算速度快，易于实现。我们通过大量的数值实验说明了我们的重建算法从噪声样本的性能。我们的代码是开源的，免费提供。
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.