Patch-Ordering-Based Wavelet Frame and Its Use in Inverse Problems

In our previous work, we have introduced a redundant tree-based wavelet transform (RTBWT), originally designed to represent functions defined on high dimensional data clouds and graphs. We have further shown that RTBWT can be used as a highly effective image-adaptive redundant transform that operates on an image using orderings of its overlapped patches. The resulting transform is robust to corruptions in the image, and thus able to efficiently represent the unknown target image even when it is calculated from its corrupted version. In this paper, we utilize this redundant transform as a powerful sparsity-promoting regularizer in inverse problems in image processing. We show that the image representation obtained with this transform is a frame expansion, and derive the analysis and synthesis operators associated with it. We explore the use of this frame operators to image denoising and deblurring, and demonstrate in both these cases state-of-the-art results.