Additive Tensor Decomposition Considering Structural Data Information

Abstract

Tensor data with rich structural information become increasingly important in process modeling, monitoring, and diagnosis in manufacturing medical and other applications. Here structural information is referred to the information of tensor components such as sparsity, smoothness, low-rank, and piecewise constancy. To reveal useful information from tensor data, we propose to decompose the tensor into the summation of multiple components based on their different structural information. In this article, we provide a new definition of structural information in tensor data. We then propose an additive tensor decomposition (ATD) framework to extract useful information from tensor data. This framework specifies a high dimensional optimization problem to obtain the components with distinct structural information. An alternating direction method of multipliers (ADMM) algorithm is proposed to solve it, which is highly parallelable and thus suitable for the proposed optimization problem. Two simulation examples and a real case study in medical image analysis illustrate the versatility and effectiveness of the ATD framework. Note to Practitioners—This article was motivated by a real case in medical imaging: extracting aortic valve calcification (AVC) regions from the tensor data obtained from computed tomography (CT) image series of the aortic region. The main objective is to decompose image series into multiple components corresponding to tissues, calcium deposition, and error. Similar needs are pervasive in other medical image analysis applications as well as the image-based modeling, monitoring, and diagnosis of industrial processes and systems. Existing methods fail to incorporate a detailed description of the properties of image series that reflect the physical understanding of the system in both the spatial and temporal domains. In this article, we provide a systematic description of the properties of image series and use them to develop a decomposition framework. It is applicable to various applications and can generate more accurate and interpretable results.