论文标题
线性逆问题非负约束：优化者的奇异性
Linear inverse problems with nonnegativity constraints: singularity of optimisers
论文作者
论文摘要
我们查看与线性逆问题相关的优化问题中的连续解决方案$ y = ax $，具有非阴性约束$ x \ geq 0 $。我们专注于噪声模型通过一般差异导致最大似然估计的情况，该差异涵盖了广泛的常见噪声统计数据，例如高斯和泊松。考虑到$ x $是重建的域上的ra度量，我们表现出一般的奇异结果。在对应于$ y \ notin \ {{ax} \ mid {x \ geq 0} \} $的高噪声状态下，在差异和操作员$ a $上的关键假设下，任何优化器都具有相对于Lebesgue度量的单数。因此，我们提供了一个解释，说明为什么任何可能成功解决优化问题的算法都会导致图像分辨率变得更细时，将导致令人难以置信的尖峰图像，这是文献中有充分记录的现象。我们以几个受医学成像启发的数值示例来说明这些结果。
We look at continuum solutions in optimisation problems associated to linear inverse problems $y = Ax$ with non-negativity constraint $x \geq 0$. We focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which covers a wide range of common noise statistics such as Gaussian and Poisson. Considering $x$ as a Radon measure over the domain on which the reconstruction is taking place, we show a general singularity result. In the high noise regime corresponding to $y \notin \{{Ax}\mid{x \geq 0}\}$ and under a key assumption on the divergence as well as on the operator $A$, any optimiser has a singular part with respect to the Lebesgue measure. We hence provide an explanation as to why any possible algorithm successfully solving the optimisation problem will lead to undesirably spiky-looking images when the image resolution gets finer, a phenomenon well documented in the literature. We illustrate these results with several numerical examples inspired by medical imaging.
