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首页> 外文期刊>Applied Mathematical Modelling >Compensated convexity on bounded domains, mixed Moreau envelopes and computational methods
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Compensated convexity on bounded domains, mixed Moreau envelopes and computational methods

机译：有界域的补偿凸性，混合莫鲁信封和计算方法

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摘要

Compensated convex transforms have been introduced for extended real valued functions defined over 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 Ω of 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 Ω. 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.
机译：已引入补偿凸形变换，以便在R〜n上定义的扩展实值函数。在其应用于图像处理，插值和形状询问时，其中一个涉及在有界域上定义的函数，所以函数在数据域的边界处与其变换相吻合。在本文中，我们通过对整个空间的功能的特定扩展来引入R〜N中限定的函数的局部补偿凸形变换，并通过混合的危急莫鲁建立与全局定义的补偿凸面变换的关系信封。我们发现这种延伸的补偿凸面变换与静电闭合中的局部补偿凸面变换一致。我们还提出了一种用于计算MOREAU信封的数值方案，根据原始功能的规律性建立具有收敛速度的方案的收敛性。我们估计了计算离散莫鲁集信封所需的迭代次数。然后，我们将局部补偿凸形变换应用于图像处理和形状询问。将结果与通过使用基于计算凸面转换的原始定义计算凸包孔而获得的结果进行比较。

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