提出可精确控制误差的总变差近似算子，提升图像重建效率
Closed-Form Approximation of the Total Variation Proximal Operator
- 基于光滑化思路构造闭式近似算子，理论保证其有效性
- 误差可由缩放参数完全控制，在去噪与CT重建中表现稳定
- 适合需要高效、可解释正则化的图像恢复研究者
总变差（TV）是图像反问题中广泛使用的正则化函数，特别适用于分块常数结构的图像。TV正则化优化通常采用近端方法求解，但受限于缺乏TV函数近端算子的闭式表达。已有研究提出一种近似方法，但其理论精度未被深入探讨。本文通过多项新理论贡献填补这一空白：证明该近似对应某个凸函数的近端算子；等价于对光滑化后的TV执行梯度下降步；并能完整表征和控制其误差，误差大小由缩放参数决定。我们在图像去噪与稀疏视角计算机断层扫描（CT）图像重建任务上实验验证了理论结果的有效性。
原文摘要 · Abstract (English)
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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