A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling

IF 2.1 3区数学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

SIAM Journal on Imaging Sciences Pub Date : 2023-03-30 DOI:10.1137/22m1496141

Dmitrij Sitenko, Bastian Boll, Christoph Schnörr

引用次数: 1

Abstract

This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as a nonlocal reparametrization of the assignment flow approach that was introduced in [J. Math. Imaging Vision, 58 (2017), pp. 211–238]. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference of convex (DC) functions decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.

查看原文本刊更多论文

图像标注的非局部图pde和高阶几何积分

本文提出了一种新的非局部偏差分方程(G-PDE)，用于标记图上度量数据。G-PDE是在文献[J]中引入的分配流方法的非局部再参数化。数学。影像视觉，58 (2017)，pp. 211-238。由于这种参数化，数值求解G-PDE被证明等同于计算关于非凸势的黎曼梯度流。我们设计了一种熵正则化的凸差分(DC)函数分解，并证明了积分分配流的基本几何欧拉格式等价于用已建立的DC规划格式求解G-PDE。此外，几何积分的观点揭示了利用驱动分配流的矢量场的高阶信息来设计一种新的加速直流规划方案的基本方法。对两种格式进行了详细的收敛性分析，并通过数值实验加以说明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Imaging Sciences COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE-COMPUTER SCIENCE, SOFTWARE ENGINEERING

CiteScore

3.80

自引率

4.80%

发文量

审稿时长

>12 weeks

期刊介绍： SIAM Journal on Imaging Sciences (SIIMS) covers all areas of imaging sciences, broadly interpreted. It includes image formation, image processing, image analysis, image interpretation and understanding, imaging-related machine learning, and inverse problems in imaging; leading to applications to diverse areas in science, medicine, engineering, and other fields. The journal’s scope is meant to be broad enough to include areas now organized under the terms image processing, image analysis, computer graphics, computer vision, visual machine learning, and visualization. Formal approaches, at the level of mathematics and/or computations, as well as state-of-the-art practical results, are expected from manuscripts published in SIIMS. SIIMS is mathematically and computationally based, and offers a unique forum to highlight the commonality of methodology, models, and algorithms among diverse application areas of imaging sciences. SIIMS provides a broad authoritative source for fundamental results in imaging sciences, with a unique combination of mathematics and applications. SIIMS covers a broad range of areas, including but not limited to image formation, image processing, image analysis, computer graphics, computer vision, visualization, image understanding, pattern analysis, machine intelligence, remote sensing, geoscience, signal processing, medical and biomedical imaging, and seismic imaging. The fundamental mathematical theories addressing imaging problems covered by SIIMS include, but are not limited to, harmonic analysis, partial differential equations, differential geometry, numerical analysis, information theory, learning, optimization, statistics, and probability. Research papers that innovate both in the fundamentals and in the applications are especially welcome. SIIMS focuses on conceptually new ideas, methods, and fundamentals as applied to all aspects of imaging sciences.