Many mathematical imaging problems are posed as non-convex optimization problems. When numerically tractable global optimization procedures are not available, one is often interested in testing ex post facto whether or not a locally convergent algorithm has found the globally optimal solution. When the problem is formulated in terms of maximizing the likelihood function under a statistical model for the measurements, one can construct a statistical test that a local maximum is in fact the global maximum. A one-sided test is proposed for the case that the statistical model is a member of the generalized location family of probability distributions, a condition often satisfied in imaging and other inverse problems. We propose a general method for improving the accuracy of the test by reparameterizing the likelihood function to embed its domain into a higher-dimensional parameter space. We show that the proposed global maximum testing method results in improved accuracy and reduced computation for a physically motivated joint-inverse problem arising in camera-blur estimation.

中文翻译：

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使用重新参数化嵌入来测试局部可能性的全局最优性

许多数学成像问题被提出为非凸优化问题。当数值上难以处理的全局优化程序不可用时，人们通常会对事后测试是否有局部收敛算法找到了全局最优解感兴趣。当根据用于测量的统计模型下的似然函数最大化来表述问题时，可以构建统计检验，以证明局部最大值实际上是全局最大值。对于统计模型是概率分布的广义位置族的成员，在成像中经常满足的条件和其他反问题的情况，提出了一种单面检验。我们提出了一种通过重新参数化似然函数以将其域嵌入到高维参数空间中来提高测试准确性的通用方法。我们表明，提出的全局最大测试方法可提高照相机模糊估计中产生的物理动机联合逆问题的准确性，并减少计算量。