Low Rank Tensor Completion With Poisson Observations

Poisson observations for videos are important models in video processing and computer vision. In this paper, we study the third-order tensor completion problem with Poisson observations. The main aim is to recover a tensor based on a small number of its Poisson observation entries. A existing matrix-based method may be applied to this problem via the matricized version of the tensor. However, this method does not leverage on the global low-rankness of a tensor and may be substantially suboptimal. Our approach is to consider the maximum likelihood estimate of the Poisson distribution, and utilize the Kullback-Leibler divergence for the data-fitting term to measure the observations and the underlying tensor. Moreover, we propose to employ a transformed tensor nuclear norm ball constraint and a bounded constraint of each entry, where the transformed tensor nuclear norm is used to get a lower transformed multi-rank tensor with suitable unitary transformation matrices. We show that the upper bound of the error of the estimator of the proposed model is less than that of the existing matrix-based method. Also an information theoretic lower error bound is established. An alternating direction method of multipliers is developed to solve the resulting convex optimization model. Extensive numerical experiments on synthetic data and real-world datasets are presented to demonstrate the effectiveness of our proposed model compared with existing tensor completion methods.

1 Introduction

Poisson observations are dependent on the number of photons recorded in the observed images and videos [1], where light sources emit photons following a Poisson distribution. An important task in image systems is to restore the images from Poisson observations, including astronomical images, positron emission tomography, hyperspectral images, and X-ray computed tomography [2], [3], [4]. These datasets are multi-dimensional arrays, which allow us to take advantage of the inherent correlations. However, the observations, which follow Poison distributions, are incomplete in a variety of applications due to sensor working conditions. The problem of low-rank tensor completion with Poisson observations is to recover a nonnegative and bounded tensor from a subset of entries, where the observed entries follow Poisson distributions.