PSF匹配核的正则化技术。I.核基的选择
Regularization Techniques for PSF-Matching Kernels. I. Choice of Kernel Basis
摘要：
我们回顾了当前的方法，用于构建PSF匹配内核的图像减法或coaddition的目的。这种方法使用一系列基函数上的核的线性分解。这些基函数的正确选择是匹配的效率和有效性的基础-所选择的基应该使用合理少量的形状来表示基础信号，和/或具有最小数量的用户可调节的调谐参数。我们研究的方法，其基础包括多个高斯-厄米多项式，以及由三角函数组成的形式自由的基础。从delta函数导出的核函数表现得更有表现力，这并不令人惊讶;它们能够采用更一般的形状，并且在高斯和方法失败的情况下表现得更好。然而，由于它的许多自由度（核大小允许的最大数量），这个基础往往过拟合的问题，并产生噪声内核具有较大的方差。我们引入了一种新的技术来正则化这些三角函数核的解决方案，它弥合了三角函数核的一般性和高斯核的紧致性之间的差距。通过这种正则化，我们能够创建一般的内核解决方案，表示PSF匹配内核的内在形状，只有一个自由度，正则化lambda的强度。lambda的作用是有效地将所得差分图像中的方差与内核本身中的方差进行交换。我们研究在选择lambda值时的考虑因素，包括统计风险估计和解决方案预测相邻区域解决方案的能力。这两者都表明λ在0.1和1.0之间的中等强度，尽管这种优化可能依赖于数据集。
英文摘要：
We review current methods for building PSF-matching kernels for the purposes of image subtraction or coaddition. Such methods use a linear decomposition of the kernel on a series of basis functions. The correct choice of these basis functions is fundamental to the efficiency and effectiveness of the matching - the chosen bases should represent the underlying signal using a reasonably small number of shapes, and/or have a minimum number of user-adjustable tuning parameters. We examine methods whose bases comprise multiple Gauss-Hermite polynomials, as well as a form free basis composed of delta-functions. Kernels derived from delta-functions are unsurprisingly shown to be more expressive; they are able to take more general shapes and perform better in situations where sum-of-Gaussian methods are known to fail. However, due to its many degrees of freedom (the maximum number allowed by the kernel size) this basis tends to overfit the problem, and yields noisy kernels having large variance. We introduce a new technique to regularize these delta-function kernel solutions, which bridges the gap between the generality of delta-function kernels, and the compactness of sum-of-Gaussian kernels. Through this regularization we are able to create general kernel solutions that represent the intrinsic shape of the PSF-matching kernel with only one degree of freedom, the strength of the regularization lambda. The role of lambda is effectively to exchange variance in the resulting difference image with variance in the kernel itself. We examine considerations in choosing the value of lambda, including statistical risk estimators and the ability of the solution to predict solutions for adjacent areas. Both of these suggest moderate strengths of lambda between 0.1 and 1.0, although this optimization is likely dataset dependent.
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