School of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada.
IEEE Trans Image Process. 2011 May;20(5):1281-99. doi: 10.1109/TIP.2010.2090532. Epub 2010 Nov 1.
The problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications. A particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill-conditioned operator. In such a situation, the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties. Such a priori information--commonly referred to as simply priors--is essential for image restoration, rendering it stable and robust to noise. Moreover, using the priors makes the recovered images exhibit some plausible features of their original counterpart. Particularly, if the original image is known to be a piecewise smooth function, one of the standard priors used in this case is defined by the Rudin-Osher-Fatemi model, which results in total variation (TV) based image restoration. The current arsenal of algorithms for TV-based image restoration is vast. In this present paper, a different approach to the solution of the problem is proposed based upon the method of iterative shrinkage (aka iterated thresholding). In the proposed method, the TV-based image restoration is performed through a recursive application of two simple procedures, viz. linear filtering and soft thresholding. Therefore, the method can be identified as belonging to the group of first-order algorithms which are efficient in dealing with images of relatively large sizes. Another valuable feature of the proposed method consists in its working directly with the TV functional, rather then with its smoothed versions. Moreover, the method provides a single solution for both isotropic and anisotropic definitions of the TV functional, thereby establishing a useful connection between the two formulae. Finally, a number of standard examples of image deblurring are demonstrated, in which the proposed method can provide restoration results of superior quality as compared to the case of sparse-wavelet deconvolution.
从降质测量中恢复数字图像的问题在许多实际重要的应用中起着核心作用。当退化现象由病态算子建模时，这个问题的一个特别具有挑战性的实例出现了。在这种情况下，噪声的存在使得在没有使用关于其特性的某些先验信息的情况下，无法恢复感兴趣的图像的有价值的近似值。这种先验信息——通常简称为先验信息——对于图像恢复是必不可少的，使其对噪声稳定且具有鲁棒性。此外，使用先验信息可以使恢复的图像表现出与其原始图像的一些合理特征。特别是，如果原始图像已知是分段光滑函数，则在这种情况下使用的标准先验之一由鲁丁-奥舍-法蒂米模型定义，这导致基于全变差（TV）的图像恢复。基于 TV 的图像恢复的当前算法库是巨大的。在本研究中，提出了一种基于迭代收缩（又名迭代阈值）方法的解决方案。在提出的方法中，通过递归应用两个简单的过程，即线性滤波和软阈值处理，来进行基于 TV 的图像恢复。因此，该方法可以被确定为属于一阶算法组，该算法组在处理相对较大尺寸的图像时效率很高。所提出方法的另一个有价值的特征是它直接处理 TV 函数，而不是其平滑版本。此外，该方法为各向同性和各向异性 TV 函数定义提供了单一的解决方案，从而在两个公式之间建立了有用的联系。最后，展示了一些标准的图像去模糊示例，在这些示例中，与稀疏小波反卷积相比，所提出的方法可以提供质量更高的恢复结果。
