全变分近端算子的闭式近似
Closed-Form Approximation of the Total Variation Proximal Operator
AI总结：
本文提出并理论分析了全变分近端算子的闭式近似，证明其对应某个凸函数的近端算子且等价于平滑TV上的一步梯度下降，并在图像去噪和稀疏视角CT重建中验证了其有效性。
摘要：
全变分（TV）是一种广泛用于正则化成像逆问题的函数，特别适用于底层结构为分段常数的图像。TV正则化优化问题通常使用近端方法求解，但其应用方式受到TV函数近端算子缺乏闭式表达式的限制。此前已有人提出TV近端算子的闭式近似，但其精度尚未在理论上得到详细探讨。我们通过做出若干新的理论贡献来填补这一空白，证明该近似可导出某个凸函数的近端算子，等价于在TV的平滑版本上进行一步梯度下降，并且其误差可以通过其缩放参数被完全刻画和控制。我们在图像去噪和稀疏视角计算机断层扫描（CT）图像重建上实验验证了我们的理论结果。
英文摘要：
Total variation (TV) is a widely used function for regularizing imaging inverse problems that is particularly appropriate for images whose underlying structure is piecewise constant. TV regularized optimization problems are typically solved using proximal methods, but the way in which they are applied is constrained by the absence of a closed-form expression for the proximal operator of the TV function. A closed-form approximation of the TV proximal operator has previously been proposed, but its accuracy was not theoretically explored in detail. We address this gap by making several new theoretical contributions, proving that the approximation leads to a proximal operator of some convex function, it is equivalent to a gradient descent step on a smoothed version of TV, and that its error can be fully characterized and controlled with its scaling parameter. We experimentally validate our theoretical results on image denoising and sparse-view computed tomography (CT) image reconstruction.
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