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Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage

机译:非线性小波图像处理:变化问题,压缩和通过小波收缩去除噪声

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This paper examines the relationship between wavelet-based image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem. Given an image F defined on a square I, minimize over all g in the Besov space B/sub 1//sup 1/(L/sub 1/(I)) the functional |F-g|/sub L2/(I)/sup 2/+/spl lambda/|g|(B/sub 1//sup 1/(L/sub 1(I)/)). We use the theory of nonlinear wavelet image compression in L/sub 2/(I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signal-to-noise ratio (SNR), which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F: the largest /spl alpha/ for which F/spl isin/B/sub q//sup /spl alpha//(L/sub q/(I)),1/q=/spl alpha//2+1/2, and the norm |F|B/sub q//sup /spl alpha//(L/sub q/(I)). Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Donoho and Johnstone's VisuShrink procedure; an example suggests, however, that Donoho and Johnstone's (1994, 1995, 1996) SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves a lower error than our procedure.
机译:本文研究了基于小波的图像处理算法与变分问题之间的关系。算法是变分问题的精确或近似最小化;特别是,我们表明,小波收缩可以被认为是以下问题的确切最小化方法。给定在正方形I上定义的图像F,则将Besov空间B / sub 1 // sup 1 /(L / sub 1 /(I))中的所有g最小化,函数| Fg | / sub L2 /(I)/ sup 2 / + / spl lambda / | g |(B / sub 1 // sup 1 /(L / sub 1(I)/))。我们使用L / sub 2 /(I)中的非线性小波图像压缩理论,通过将小波收缩应用到以i.d.,均值零,高斯噪声破坏的图像上,得出小波收缩的准确误差范围。在这种推导中出现了一种新的信噪比(SNR),我们称其更准确地反映了图像中噪声的视觉感受。我们提出了广泛的计算,以支持以下假设:如果仅知道(或可以估计)关于图像F的两个参数,则可以得出近乎最佳的收缩参数:F / spl为in / B / sub q的最大/ spl alpha / // sup / spl alpha //((L / sub q /(I)),1 / q = / spl alpha // 2 + 1/2,以及范数| F | B / sub q // sup / spl alpha //(L / sub q /(I))。理论和实验结果均表明,与Donoho和Johnstone的VisuShrink程序相比,我们选择的收缩参数可获得更好的结果。但是,有一个例子表明,Donoho和Johnstone(1994,1995,1996)的SureShrink方法(对于每个二进位水平使用不同的收缩参数)实现了比我们的过程更低的误差。

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