[Paper Review] A Local Block Coordinate Descent Algorithm for the Convolutional Sparse Coding Model

Ev Zisselman, Jeremias Sulam+1 more|arXiv (Cornell University)|Nov 1, 2018

Sparse and Compressive Sensing Techniques31 references6 citations

TL;DR

This paper proposes Local Block Coordinate Descent (LoBCoD), a novel algorithm for convolutional sparse coding (CSC) that operates locally on image patches using block coordinate descent, eliminating the need for auxiliary variables and ADMM parameters. It achieves state-of-the-art performance in image inpainting and multi-focus fusion with faster convergence and lower memory usage than Fourier-domain ADMM methods.

ABSTRACT

The Convolutional Sparse Coding (CSC) model has recently gained considerable traction in the signal and image processing communities. By providing a global, yet tractable, model that operates on the whole image, the CSC was shown to overcome several limitations of the patch-based sparse model while achieving superior performance in various applications. Contemporary methods for pursuit and learning the CSC dictionary often rely on the Alternating Direction Method of Multipliers (ADMM) in the Fourier domain for the computational convenience of convolutions, while ignoring the local characterizations of the image. A recent work by Papyan et al. suggested the SBDL algorithm for the CSC, while operating locally on image patches. SBDL demonstrates better performance compared to the Fourier-based methods, albeit still relying on the ADMM. In this work we maintain the localized strategy of the SBDL, while proposing a new and much simpler approach based on the Block Coordinate Descent algorithm - this method is termed Local Block Coordinate Descent (LoBCoD). Furthermore, we introduce a novel stochastic gradient descent version of LoBCoD for training the convolutional filters. The Stochastic-LoBCoD leverages the benefits of online learning, while being applicable to a single training image. We demonstrate the advantages of the proposed algorithms for image inpainting and multi-focus image fusion, achieving state-of-the-art results.

Motivation & Objective

• Address the limitations of Fourier-domain ADMM methods in CSC, which require auxiliary variables, suffer from high memory use, and depend on hard-to-tune parameters.
• Overcome the sub-optimality of patch-based sparse coding by enabling global modeling through the convolutional sparse coding (CSC) framework.
• Develop a local, non-ADMM approach for CSC pursuit that operates directly in the signal domain using block coordinate descent, improving computational efficiency and convergence.
• Introduce a stochastic gradient variant (Stochastic-LoBCoD) for online, single-image dictionary learning, enabling efficient adaptation to individual training samples.
• Demonstrate superior performance in image inpainting and multi-focus image fusion compared to state-of-the-art methods, including SBDL and [21].

Proposed method

• Propose LoBCoD, a block coordinate descent algorithm that alternately optimizes over local sparse representations and convolutional filters using only local computations in the image domain.
• Avoid auxiliary variables and ADMM parameters by directly minimizing the CSC objective through coordinate descent on overlapping image patches.
• Use a local pursuit strategy where each patch is processed independently with a fixed dictionary, followed by global reconstruction via convolutional synthesis.
• Introduce Stochastic-LoBCoD, a stochastic gradient descent variant that updates convolutional filters using mini-batches from a single training image, enabling online learning.
• Apply a two-stage alternating optimization: sparse pursuit using LoBCoD and base-image extraction via a reconstruction kernel $U_s$, with parameters $\lambda=1$ and $\mu=5$.
• Leverage local sparsity and activity maps in the Lab color space for multi-focus fusion, selecting regions based on maximum pixel-wise activity for each channel.

Experimental results

Research questions

• RQ1 Can a block coordinate descent approach outperform ADMM-based methods in CSC pursuit while avoiding auxiliary variables and tuning parameters?
• RQ2 Does a local, non-ADMM algorithm for CSC achieve better memory efficiency and faster convergence than Fourier-domain ADMM methods?
• RQ3 Can a stochastic gradient variant of LoBCoD enable effective online dictionary learning from a single image?
• RQ4 How does LoBCoD compare to state-of-the-art methods in image inpainting and multi-focus fusion in terms of PSNR and visual quality?
• RQ5 To what extent does the local, patch-based strategy in LoBCoD preserve global image structure while maintaining computational tractability?

Key findings

• LoBCoD achieves a PSNR of 39.81 dB on the Bird image fusion task, outperforming the prior method [21] which achieved 39.29 dB.
• On the Butterfly image, the proposed method produces visually superior results with sharper details and less blur compared to [21], especially in highly blurred regions.
• For synthetic data, the method achieved high PSNR values: 38.45 dB on Barbara and 37.68 dB on Butterfly, demonstrating robust reconstruction performance.
• The algorithm converges in just 2–4 alternating iterations between sparse pursuit and base-image extraction, indicating fast convergence.
• Stochastic-LoBCoD enables effective dictionary learning from a single image, retaining the benefits of online learning without requiring large-scale datasets.
• The method shows superior memory efficiency and parallelizability due to its local, variable-free optimization strategy compared to ADMM-based approaches.