A Novel Tensor Ring Sparsity Measurement for Image Completion | JoVE Visualize

Area of Science:

Data Science
Applied Mathematics
Computer Vision

Background:

Sparse modeling and matrix rank are effective for image recovery but struggle with multi-dimensional data.
Tensor decomposition (TD) generalizes matrix methods for multi-dimensional data, but its rank doesn't directly correlate with data sparsity.
Existing low-rank tensor modeling methods have limitations in directly quantifying tensor sparsity.

Purpose of the Study:

To introduce a novel Tensor Ring Sparsity Measurement (TRSM) for accurately quantifying sparsity in multi-dimensional tensor data.
To provide a unified interpretation of sparsity measurement, analogous to matrix rank, using tensor ring representations.
To enhance the practical applicability of TRSM through nonconvex relaxation and apply it to tensor completion problems.

Main Methods:

Developed a Tensor Ring Sparsity Measurement (TRSM) based on the tensor ring (TR) Kronecker basis representation.
Proposed an efficient computation method for TRSM using the product of ranks of mode-2 unfolded TR-cores.
Incorporated the folded-concave penalty (a nonconvex relaxation of the minimax concave penalty) to improve TRSM performance.
Extended TRSM to tensor completion and employed the alternating direction method of multipliers (ADMM) for solving.

Main Results:

TRSM provides a direct and unified measure of tensor sparsity, analogous to matrix rank.
The proposed method achieves efficient computation of tensor sparsity using TR representations.
Experiments on image and video data completion demonstrate the effectiveness of the TRSM-based approach.
The integration of nonconvex penalties enhances the practical performance of the sparsity measurement.

Conclusions:

TRSM offers a significant advancement in measuring sparsity for multi-dimensional data, overcoming limitations of matrix-based approaches.
The proposed method and its application to tensor completion show promising results for image and video recovery.
This work establishes a stronger theoretical link between tensor decomposition ranks and data sparsity, paving the way for improved multi-dimensional data analysis.