Efficient Sum of Outer Products Dictionary Learning (SOUP-DIL) and Its Application to Inverse Problems

Abstract

The sparsity of signals in a transform domain or dictionary has been exploited in applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise compared to analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. This paper exploits the ideas that drive algorithms such as K-SVD, and investigates in detail efficient methods for aggregate sparsity penalized dictionary learning by first approximating the data with a sum of sparse rank-one matrices (outer products) and then using a block coordinate descent approach to estimate the unknowns. The resulting block coordinate descent algorithms involve efficient closed-form solutions. Furthermore, we consider the problem of dictionary-blind image reconstruction, and propose novel and efficient algorithms for adaptive image reconstruction using block coordinate descent and sum of outer products methodologies. We provide a convergence study of the algorithms for dictionary learning and dictionary-blind image reconstruction. Our numerical experiments show the promising performance and speed-ups provided by the proposed methods over previous schemes in sparse data representation and compressed sensing-based image reconstruction.

Accepted to IEEE Transactions on Computational Imaging. This paper also cites experimental results reported in arXiv:1511.08842

References in corpus (5)

• Exact Recovery of Sparsely-Used Dictionaries
• Low-rank and Adaptive Sparse Signal (LASSI) Models for Highly Accelerated Dynamic Imaging
• Data-Driven Learning of a Union of Sparsifying Transforms Model for Blind Compressed Sensing
• Efficient Sum of Outer Products Dictionary Learning (SOUP-DIL) and Its Application to Inverse Problems
• Efficient Sum of Outer Products Dictionary Learning (SOUP-DIL) - The ${\ell }_0$ Method

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• CNN-Based Projected Gradient Descent for Consistent Image Reconstruction
• Image Reconstruction: From Sparsity to Data-adaptive Methods and Machine Learning
• Convolutional Dictionary Learning: Acceleration and Convergence
• Transform Learning for Magnetic Resonance Image Reconstruction: From Model-based Learning to Building Neural Networks
• Low-rank and Adaptive Sparse Signal (LASSI) Models for Highly Accelerated Dynamic Imaging
• Wasserstein GANs for MR Imaging: from Paired to Unpaired Training
• Efficient Sum of Outer Products Dictionary Learning (SOUP-DIL) and Its Application to Inverse Problems
• Optimization methods for MR image reconstruction (long version)
• Multi-Channel Convolutional Analysis Operator Learning for Dual-Energy CT Reconstruction
• Adaptive Local Neighborhood-based Neural Networks for MR Image Reconstruction from Undersampled Data