IMAGE DECONVOLUTION USING MULTIGRID NATURAL IMAGE PRIOR AND ITS APPLICATIONS Tingbo Hou 1 , Sen Wang 2 , Hong Qin 1 , Rodney L. Miller 2 1 Department of Computer Science, Stony Brook University, USA 2 Kodak Research Laboratories, Eastman Kodak Company, USA {thou, qin}@cs.sunysb.edu {sen.wang, rodney.miller}@kodak.com ABSTRACT The natural image prior has been shown as a powerful tool for image deblurring in recent work, while its performance against noise and various applications have not been thor- oughly studied. In this paper, we present a multigrid natural image prior for image deconvolution that enhances its robust- ness against noise, and furnish three applications of image deconvolution using this prior: deblurring, super-resolution and denoising. The prior is based on a remarkable property of natural images that derivatives with different resolutions are subject to the same heavy-tailed distribution with a spatial factor. It can serve in both blind and non-blind deconvolu- tions. The performances of the proposed prior in different applications are demonstrated by corresponding experimental results. Index Terms— Image deblurring, natural image prior, super-resolution, image denoising 1. INTRODUCTION Image deconvolution is a common and important problem with consistently intensive attentions in image processing and computational photography. While people from different fields have different focuses on this problem, we are mainly interested by the recent progress in image deblurring us- ing the natural image prior [1, 2, 3, 4], which refers to the heavy-tailed distribution of image gradient magnitudes. Nev- ertheless, it still remains challenging and hard to understand in many respects [5]. Recently, the natural image prior has been successfully applied to a variety of applications in image processing and computational photography. The intrinsic property of this prior is that, most pixels of natural images have very small gradient magnitudes, while only a few pixels have large gra- dient magnitudes. The strength of this prior lies in its remark- able consistency over various types of natural images. This prior penalizes pixels with great gradient magnitudes in such This work was performed at Kodak Research Laboratories when the first author worked as a research intern. a way to reduce ringing and preserve sharp edges. There are some representations appearing in the previous work [1, 2, 3], where we found that the sparse prior [2] is concise and effec- tive, which is also referred as the hyper-Laplacian prior in [4]. Due to a recent evaluation of deconvolution algorithms [5], the sparse prior has been shown to achieve the best perfor- mance in the non-blind deconvolution process. In this paper, we propose a new multigrid prior that em- bodies the hyper-Laplacian priors in multi-resolutions, which inherits the advantage of natural image prior on artifacts con- trol, and enhances the robustness of deconvolution against noise. We apply this prior to deconvolution-based methods working on three problems: deblurring, super-resolution and denoising. In the experiments, we compare our method only with deconvolution-based methods, as we are focusing on im- prove the performance of image deconvolution. An exception can be found in the experiment of denoising, since the convo- lution kernel here is degenerated to a delta function. 2. DECONVOLUTION WITH MULTIGRID NATURAL IMAGE PRIOR 2.1. Multigrid natural image prior Natural images have an intrinsic property on the statistics of their gradient magnitudes, i.e., the “heavy-tailed” distribu- tion. In our observation, magnitudes of multi-resolution gra- dients are also subject to this property. Fig. 1 shows two natu- ral images and their statistical responses of three derivative fil- ters with different resolutions, which are selected from 1-ring (red), 2-ring (blue) and 3-ring (green) neighbors of the central pixel (black). It illustrates that the responses of derivative fil- ters within a certain size of local neighborhood (7 × 7 in this example) have the similar distributions. Moreover, the tails of response distribution become lighter when the distance of derivative filter increases, which indicates that the space vari- ation affects the distribution of filter outputs. Inspired by the above observations, we propose a new prior in image deconvolution based on a series of derivative filters: {F d } with (d =1, ..., n w ) which computes the multi- grid discrete derivatives of a pixel in its w × w neighborhood,