Recovery of Point Clouds on Surfaces: Application to Image Reconstruction

TL;DR

This paper presents a novel framework for recovering points on smooth surfaces in high-dimensional spaces, with applications to dynamic imaging, using annihilation relations, nuclear norm minimization, and an iterative kernel-based algorithm.

Contribution

It introduces a new method combining annihilation relations and nuclear norm minimization for surface point recovery, along with an iterative kernel trick algorithm for high-dimensional data.

Findings

Effective recovery of surface points from noisy, undersampled data

Application to dynamic imaging such as cardiac and breathing data

Framework as a continuous domain alternative to discrete graph signal processing

Abstract

We introduce a framework for the recovery of points on a smooth surface in high-dimensional space, with application to dynamic imaging. We assume the surface to be the zero-level set of a bandlimited function. We show that the exponential maps of the points on the surface satisfy annihilation relations, implying that they lie in a finite dimensional subspace. We rely on nuclear norm minimization of the maps to recover the points from noisy and undersampled measurements. Since this direct approach suffers from the curse of dimensionality, we introduce an iterative reweighted algorithm that uses the "kernel trick". The resulting algorithm has similarities to iterative algorithms used in graph signal processing (GSP); this framework can be seen as a continuous domain alternative to discrete GSP theory. The use of the algorithm in recovering free breathing and ungated cardiac data shows the potential of this framework in practical applications.