## Patch-dictionary method for whole image recoveryYangyang Xu and Wotao Yin## BackgroundVarious algorithms have been proposed for dictionary learning such as KSVD and the online dictionary learning method. Among those for image processing, many use We address a simple yet open issue regarding ## Notation and our method We shall recover an image from itscorrupted measurement , where denotes a linear operator, and is some noise. Depending on the applications, can take different forms. Let denote the -th patch ( is a linear operator) and assume its has an (approximately) sparse representation under a given dictionary , i.e., where is an (approximately) sparse vector. When a set of patches together covers every pixel of , we can represent by where is the adjoint of , and is an operator that averages the overlapping patches. Now, we take as a set of covering but Two examples of non-overlapping patches that cover a whole image
To recover from , we solve the following model where is a weight vector. The model () can be solved by many convex optimization methods, for example, YALL1 in our numerical experiements. When a set of non-overlapping patches is used in (), the solution sometimes bears the grid artifact. An effective strategy to avoid this artifact is to solve multiple instances of () with 's that arrange their patches in different ways, like the two examples above though we use up to five different, and then take the average of the solutions. Of course, the different instants of () can be solved in parallel. It is important to use non-overlapping patches since this limits the number of free variables in () and improves solution quality (despite the possible grid artifact). If all the overlapping patches are used, there will be far more free variables. ## Selected numerical resultsLEFT: image denoising results by solving () with all the overlapping patches (PSNR = 26.98); RIGHT: the same with just one set of non-overlapping patches (PSNR = 30.57)
We solve *five*instances of () with 's that arrange their patches in different ways. Out of the five recovered images , the one with the highest PSNR is denoted by . We also compute , which is the running average of the first recovered images. The table below reports their PSNRs. Image PSNRs improve as more average is taken.
The quality of dictionary plays a vital role in image recovery. We tested recovering images from their compressive circulant measurements and report their PSNRs below. The four columns in each group, from left to right, correspond to: the solution to () with the learned dictionary , that with the discrete cosine transform (DCT) , that with the adaptively updated dictionary (see Sec. 2.2 of our report or ), the solution to a total variation model. The winner is the adpatively updated dictionary.
More numerical results, as well as the technical details, can be found in our report. ## Citation
## References. M. Aharon, M. Elad, and A. Bruckstein. . J. Mairal, F. Bach, J. Ponce, and G. Sapiro. . M. Elad, and M. Aharon. « Back |