Compensated convex transforms have been introduced for extended real valued functions defined over ${\mathbb{R}}^n$. In their application to image processing, interpolation and shape interrogation, where one deals with functions defined over a bounded domain, the implicit assumption was made that the function coincides with its transform at the boundary of the data domain. In this paper, we introduce local compensated convex transforms for functions defined in bounded open convex subsets $\Omega$ of ${\mathbb{R}}^n$ by making specific extensions of the function to the whole space, and establish their relations to globally defined compensated convex transforms via the mixed critical Moreau envelopes. We find that the compensated convex transforms of such extensions coincide with the local compensated convex transforms in the closure of $\Omega$. We also propose a numerical scheme for computing Moreau envelopes, establishing convergence of the scheme with the rate of convergence depending on the regularity of the original function. We give an estimate of the number of iterations needed for computing the discrete Moreau envelope. We then apply the local compensated convex transforms to image processing and shape interrogation. Our results are compared with those obtained by using schemes based on computing the convex envelope from the original definition of compensated convex transforms.

已针对在上定义的扩展实值函数引入了补偿凸变换。 ${\mathbb{[R}}^{ñ}$。在将它们应用于在有界域上定义的函数的图像处理，插值和形状询问的应用中，隐式假定函数与数据域边界处的变换一致。在本文中，我们介绍了在有界开放凸子集中定义的函数的局部补偿凸变换。$\Omega$ 的 ${\mathbb{[R}}^{ñ}$通过将函数特定扩展到整个空间，并通过混合的临界莫罗包络建立它们与全局定义的补偿凸变换的关系。我们发现这种扩展的补偿凸变换与局部闭合补偿凸变换一致。$\Omega$。我们还提出了一种用于计算Moreau包络的数值方案，根据原始函数的正则性确定方案的收敛速度。我们给出了计算离散莫罗包络所需的迭代次数的估计。然后，我们将局部补偿的凸变换应用于图像处理和形状查询。我们的结果与通过使用基于补偿凸变换的原始定义计算凸包络的方案所获得的结果进行了比较。