Elsevier

Applied Soft Computing

Volume 33, August 2015, Pages 77-85
Applied Soft Computing

Sparse Double-Geometric Nonlocal Mean image recovery via steerable kernel

https://doi.org/10.1016/j.asoc.2015.04.004Get rights and content

Abstract

It has been proved that the geometric information in images produce most stimulus of human eyes, which is important for the image recovery. However, the local geometric structure in images are too diverse to be accurately captured. Recent decade has witnessed a flourish of biological inspired algorithms in the image recovery. In this paper, inspired the adaptive and sparse characteristic of visual perception of humans, we advance an adaptive steerable kernel based Sparse Double-Geometric Nonlocal Mean (SDGNLM) denoising algorithm by exploring the local geometric information in both the “neighbor location” and “similarity measure”. In our method, a steerable kernel is employed to reveal the local geometric information of pixels, and sparse assumption of neighbors is cast on pixels to achieve more accurate image recovery. Moreover, a weighted sparse optimization algorithm is proposed to find and weight neighbors having the similar characteristics with each pixel. Some experiments are taken on some benchmark natural images, and the experimental results demonstrate its superiorities to NLM algorithm and its variants, in terms of both visual results and numerical guidelines.

Introduction

Image recovery has found wide applications in computer vision and signal processing, including image denoising, image super-resolution, image deblurring, image impainting and so on, whose tasks are to recover images from corrupted or implement observations. In order to solve these ill-posed problems, many efforts have done on imposing some additional priors on the recovery, such as the smooth, self-similarity of images and so on. Recent decade has witnessed a flourish of biological inspired algorithms in the image recovery, where the characteristics of visual organization are explored to acquire more efficient and high-quality image recovery. It has been proved that vision neurons are very sensitive to the geometric information, such as edges, contours and textures in images. That is, the geometric information in images produce most stimulus of eyes, so the geometric regularity of images is important for recovery quality. Consequently, in the image recovery the geometric regularities along singularity of edges or contours should be emphasized. However, the local geometric structure in images is too diverse to be accurately captured. In this paper we advance an adaptive sparse recovery algorithm for image denoising, which can adaptively capture the local geometric information via steerable kernel and Nonlocal Means (NLM) filtering.

Removing unknown additive noises from measured corrupted images has received much attention in the past fifty years [21], [22]. Initial noise reduction approaches focus on imposing various smoothness assumptions on the recovered images [1], [2], [3], [4], [5]. Recently, there are some popular approaches based on NLM filtering, sparse representation, learning dictionary or structural clustering [6], [7], [9], [10], [11]. Among them, NLM algorithm is attracting increasing interests, which explores the nonlocal self-similarity of image patches and utilizes the weighted average of all the pixels in a nonlocal region to recover images. Although several NLM algorithms explores the nonlocal similarity prior [12], [13], [15], [16], [17], it is very likely that the participation of all the pixels in a nonlocal region will have a negative effect on the result, when the local characteristic of pixels used for average is dissimilar to the one to be restored.

In the following we illustrate the necessity of introducing the “geometric” information into NLM algorithm. Fig. 1 plots a nonlocal neighborhood region of one pixel i to be recovered. In the original NLM algorithm [6], Gaussian kernel is adopted to weight the neighborhood patches of patch 0. Consequently the patches 3 and 4 will have a larger weight decrease than patches 1 and 2, because they are nearer to the pixel i. However, patches 1 and 2 have more similar geometric structure with that of patch 0, and the center pixels of patches 1 and 2 should contribute much more to pixel i than patches 3 and 4. An adjustment way is to consider the geometric structure of pixels in weighting neighbors in addition to the similarity between patches. That is, patches that have the same geometric structure to patch 0 will have a large weight in averaging pixels, and we call these patches as the “geometric neighbor”. In the paper we name this geometric prior as the local geometric information in the “neighbor location”. Another geometric prior we should explore in weighting pixels is to measure “geometric similarity” of patches, and an example is shown in Fig. 1. Although patch 3 is more similar to patch 0 than patch 4 according to the measure of Euclidean distance, the center pixel of patch 3 should have smaller contributions to the recovery of pixel i than that of patch 4, because patch 4 is more geometrically similar to patch 0. Therefore, geometric weights should cast on different pixels by taking the geometric information into account. We call this prior as the local geometric information in the “similarity measure”.

In this paper, we incorporate the local geometric information in both the “neighbor location” and “similarity measure” into weighting neighbors in NLM, and propose a Double-Geometric NLM (DGNLM) denoising algorithm via steerable kernels, where both the “geometric neighbors” and “geometric similarity” are explored to find more relevant pixels. Moreover, an assumption that only a small number of patches are relevant to the image recovery, is adopted, to advance a Sparse DGNLM (SDGNLM) algorithm for reducing noises. Compared with the available improved NLM algorithms, our proposed SDGNLM has the following characteristics: (1) The local geometric information in both the “neighbor location” and “similarity measure” is explored in weighting pixels, (2) Sparse assumption of related neighbors is cast on the image patches, to make concise and accurate recovery possible.

The rest of this paper is organized as follows. In Section 2, we introduce the NLM algorithm, steerable kernel, sparse neighborhood and our proposed DGNLM and SDGNLM. In Section 3, some experiments are taken to illustrate the efficiency of our proposed SDGNLM method. Finally some conclusions are drawn in Section 4.

Section snippets

Sparse Double-Geometric Nonlocal Mean Denoising algorithm via steerable kernel

In this section we firstly discuss the original NLM algorithm and construction of steerable kernels, and then detail the double-geometric information exploration via steerable kernel in our proposed DGNLM. Finally the SDGNLM denoising algorithm is described in detail.

Experimental results

In this section, several experiments are taken on some natural images to investigate the performance of our method. We take eleven benchmark images shown in Fig. 4 (including the natural images of human, building, animals, houses, and some cartoon images), as the example to demonstrate the efficiency of DGNLM. In Fig. 4, the first three images are of size 256 × 256 and the rest are of size 512 × 512. In the test, the denoising results of the following methods are compared: (1) original NLM

Conclusions

By exploring the local geometric characteristics of pixels in determining “neighbor location” and “similarity measure”, in this paper we propose a Double-Geometric Nonlocal Mean Denoising (DGNLM) algorithm to find and weight neighbors having the similar local characteristics with each pixel. Moreover, sparse assumption of neighbors is cast on pixels to further improve the image recovery result. The location of neighbors is reduced to a weighted sparse optimization task, to acquire an adaptive

Acknowledgements

The authors would like to thank the anonymous reviewers for their very helpful comments and suggestions that greatly improve this paper. This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2013CB329402, the Major Research Plan of the National Natural Science Foundation of China (Nos. 91438103, 91438201), the fundamental research funds for the Central Universities under Grant No. BDY021429, Huawei Innovation Research Program, the Kunshan

References (22)

  • P. Perona et al.

    Scale-space and edge detection using anisotropic diffusion

    IEEE Trans. Pattern Anal. Mach. Intell.

    (1990)
  • L. Rudin et al.

    Total variation based image restoration with free local constraints

  • S. Mallat

    A Wavelet Tour of Signal Processing

    (1999)
  • J. Portilla et al.

    Image denoising using scale mixtures of Gaussians in the wavelet domain

    IEEE Trans. Image Process.

    (2003)
  • S. Roth et al.

    Fields of experts: a framework for learning image priors

  • A. Buades et al.

    A non-local algorithm for image denoising

  • M. Elad et al.

    Image denoising via sparse and redundant representations over learned dictionaries

    IEEE Trans. Image Process.

    (2006)
  • Z. Wang et al.

    Image quality assessment: from error visibility to structural similarity

    IEEE Trans. Image Process.

    (2004)
  • K. Dabov et al.

    Image denoising by sparse 3D transform domain collaborative filtering

    IEEE Trans. Image Process.

    (2007)
  • J. Mairal et al.

    Non-local sparse models for image restoration

    IEEE CVPR

    (2009)
  • W. Dong et al.

    Sparsity-based image denoising via dictionary learning and structural clustering

    IEEE CVPR

    (2011)
  • View full text