Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded Space

TL;DR

This paper introduces a novel tensor completion method that employs a multi-way delay-embedding transform and Tucker-based low-rank tensor factorization to recover missing slices, especially when traditional methods fail.

Contribution

It extends delay embedding to tensors and combines it with low-rank tensor factorization for effective missing slice recovery in incomplete tensor data.

Findings

Successfully recovered missing slices in color images

Effective in functional magnetic resonance imaging data

Outperformed traditional nuclear-norm and total variation methods

Abstract

Let us consider a case where all of the elements in some continuous slices are missing in tensor data. In this case, the nuclear-norm and total variation regularization methods usually fail to recover the missing elements. The key problem is capturing some delay/shift-invariant structure. In this study, we consider a low-rank model in an embedded space of a tensor. For this purpose, we extend a delay embedding for a time series to a "multi-way delay-embedding transform" for a tensor, which takes a given incomplete tensor as the input and outputs a higher-order incomplete Hankel tensor. The higher-order tensor is then recovered by Tucker-based low-rank tensor factorization. Finally, an estimated tensor can be obtained by using the inverse multi-way delay embedding transform of the recovered higher-order tensor. Our experiments showed that the proposed method successfully recovered missing slices for some color images and functional magnetic resonance images.