A $$\varGamma $$ -Convergence Result and An Off-the-Grid Charge Algorithm for Curve Reconstruction in Inverse Problems

IF 1.3 4区数学 Q4 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE Journal of Mathematical Imaging and Vision Pub Date : 2024-05-07 DOI:10.1007/s10851-024-01190-1

Bastien Laville, Laure Blanc-Féraud, Gilles Aubert

引用次数: 0

Abstract

Several numerical algorithms have been developed in the literature and employed for curves reconstruction. However, these techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. Conversely, off-the-grid (or gridless) optimisation does not rely on a fine grid and offer a tractable theoretical and numerical framework. In this work, we present a gridless method accounting for the reconstruction of both open and closed curves, based on the latest theoretical development in off-the-grid curve reconstruction. This paper also shows $\varGamma $-convergence results of the discretised surrogate functional towards the continuous energy we coined CROC.

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用于逆问题曲线重构的 $$varGamma $$ 收敛结果和离网充电算法

文献中已经开发了几种数值算法，并用于曲线重建。然而，这些技术都是在离散环境下开发的，即超分辨图像是在比观测图像更精细的网格上定义的。相反，非网格（或无网格）优化不依赖于精细网格，并提供了一个可操作的理论和数值框架。在这项工作中，我们基于离网格曲线重建的最新理论发展，提出了一种无网格方法，可用于重建开放和封闭曲线。本文还展示了离散代用函数向我们称之为 CROC 的连续能量的收敛结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Imaging and Vision 工程技术-计算机：人工智能

CiteScore

4.30

自引率

5.00%

发文量

审稿时长

3.3 months

期刊介绍： The Journal of Mathematical Imaging and Vision is a technical journal publishing important new developments in mathematical imaging. The journal publishes research articles, invited papers, and expository articles. Current developments in new image processing hardware, the advent of multisensor data fusion, and rapid advances in vision research have led to an explosive growth in the interdisciplinary field of imaging science. This growth has resulted in the development of highly sophisticated mathematical models and theories. The journal emphasizes the role of mathematics as a rigorous basis for imaging science. This provides a sound alternative to present journals in this area. Contributions are judged on the basis of mathematical content. Articles may be physically speculative but need to be mathematically sound. Emphasis is placed on innovative or established mathematical techniques applied to vision and imaging problems in a novel way, as well as new developments and problems in mathematics arising from these applications. The scope of the journal includes: computational models of vision; imaging algebra and mathematical morphology mathematical methods in reconstruction, compactification, and coding filter theory probabilistic, statistical, geometric, topological, and fractal techniques and models in imaging science inverse optics wave theory. Specific application areas of interest include, but are not limited to: all aspects of image formation and representation medical, biological, industrial, geophysical, astronomical and military imaging image analysis and image understanding parallel and distributed computing computer vision architecture design.

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