[论文解读] A fast patch-dictionary method for whole image recovery

Yangyang Xu, Wotao Yin|arXiv (Cornell University)|Aug 16, 2014

Sparse and Compressive Sensing Techniques参考文献 17被引用 13

一句话总结

本文提出了一种用于全局成像任务（如压缩感知和医学成像）中全图像恢复的快速、非重叠块字典方法。通过将图像划分为非重叠的块子集，独立地对每个子集进行稀疏恢复，并对结果进行平均，该方法在减少过拟合和计算时间的同时，实现了最先进水平的PSNR性能，并通过块邻近梯度方法实现了全局收敛和加速字典学习。

ABSTRACT

Various algorithms have been proposed for dictionary learning. Among those for image processing, many use image patches to form dictionaries. This paper focuses on whole-image recovery from corrupted linear measurements. We address the open issue of representing an image by overlapping patches: the overlapping leads to an excessive number of dictionary coefficients to determine. With very few exceptions, this issue has limited the applications of image-patch methods to the local kind of tasks such as denoising, inpainting, cartoon-texture decomposition, super-resolution, and image deblurring, for which one can process a few patches at a time. Our focus is global imaging tasks such as compressive sensing and medical image recovery, where the whole image is encoded together, making it either impossible or very ineffective to update a few patches at a time. Our strategy is to divide the sparse recovery into multiple subproblems, each of which handles a subset of non-overlapping patches, and then the results of the subproblems are averaged to yield the final recovery. This simple strategy is surprisingly effective in terms of both quality and speed. In addition, we accelerate computation of the learned dictionary by applying a recent block proximal-gradient method, which not only has a lower per-iteration complexity but also takes fewer iterations to converge, compared to the current state-of-the-art. We also establish that our algorithm globally converges to a stationary point. Numerical results on synthetic data demonstrate that our algorithm can recover a more faithful dictionary than two state-of-the-art methods. Combining our whole-image recovery and dictionary-learning methods, we numerically simulate image inpainting, compressive sensing recovery, and deblurring. Our recovery is more faithful than those of a total variation method and a method based on overlapping patches.

研究动机与目标

• 为解决在使用重叠图像块进行全图像恢复时面临的过拟合和高计算成本问题。
• 在压缩感知和医学成像等任务中实现高效且准确的全局图像恢复，其中逐块处理方法无效。
• 通过使用非重叠块子集而非重叠块来减少稀疏编码中的自由变量数量。
• 通过采用每步成本更低、收敛更快的块邻近梯度法，加速字典学习。
• 建立所提算法在非凸稀疏恢复问题中全局收敛至驻点的理论保证。

提出的方法

• 该方法将图像划分为多个非重叠且覆盖完整的块子集，以减少自由变量并避免过拟合。
• 对每个子集，通过ℓ₁最小化独立执行稀疏恢复，从而获得全图像重建结果。
• 对来自不同块子集的多个重建结果进行平均，以消除网格伪影并提高保真度。
• 采用块邻近梯度算法高效学习字典，降低每步计算成本并加速收敛。
• 该算法交替更新稀疏系数和字典，确保全局收敛至驻点。
• 该方法避免一次性处理所有重叠块，从而支持并行计算并具备可扩展性。

实验结果

研究问题

• RQ1 非重叠块子集能否有效用于减少全图像恢复中的过拟合和计算成本？
• RQ2 对多个非重叠块子集的重建结果进行平均是否能提高图像保真度并消除网格伪影？
• RQ3 与最先进方法相比，用于字典学习的块邻近梯度法是否能实现更快的收敛速度和更低的每步计算成本？
• RQ4 所提算法在非凸稀疏恢复问题中是否能全局收敛至驻点？
• RQ5 在压缩感知和去模糊任务中，该方法与总变差（TV）方法及基于重叠块的方法相比，在PSNR和速度方面表现如何？

主要发现

• 在压缩感知中，该方法的PSNR高于总变差（TV）方法和基于重叠块的方法，当采样率为10%、噪声为1%时，PSNR最高达到37.76。
• 在图像去模糊任务中，该方法在1%噪声条件下对“Plane”图像的PSNR达到35.35，优于基于重叠块的方法和TV方法。
• 在30%采样率下，使用学习字典的该方法在“Lena”图像上的PSNR达到35.44，优于基于DCT的方法和TV方法。
• 仅使用五个非重叠块子集进行平均即可有效消除网格伪影并提升重建质量。
• 与现有方法相比，用于字典学习的块邻近梯度法收敛更快，且每步计算复杂度更低。
• 该算法全局收敛至驻点，为其稳定性与可靠性提供了理论保障。