Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage
Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage
复制标题
DOI：
10.1109/83.661182
复制
发表时间：
1998-03-01
影响因子：
10.6
通讯作者：
Lucier, BJ
中科院分区：
文献类型：
--
作者：
Chambolle, A;DeVore, RA;Lucier, BJ
本文探讨了基于小波变换的图像处理算法和变分问题之间的关系。算法是作为变分问题的精确或近似极小值导出的;特别是，我们证明了小波收缩可以被认为是以下问题的精确极小值：给定一个定义在正方形I上的图像F，在Besov空间B-1（1）（L-1（I））中，在所有g上最小化平行于F-g平行于（L2（I））（2）+ lambda平行于g平行于B-1（1）（（L1（I）。我们使用L-2（I）中的非线性小波图像压缩理论，通过应用于i.i.d.污染的图像的小波收缩来获得精确的噪声去除误差界，均值为零，高斯噪声。一个新的信号噪声比（SNR），我们声称更准确地反映了图像中的噪声的视觉感知，在这个推导中出现，我们提出了广泛的计算，支持假设，近最佳的收缩参数，如果一个人知道，（或可以估计）关于图像F的仅两个参数：最大的α，其中F <$B-q（alpha）（L-q（I）），1/q = alpha/2 + 1/2，平行于F的范数平行于B-q（alpha）（L-q（I））。理论和实验结果表明，我们的收缩参数的选择产生均匀更好的结果比多诺霍和约翰斯通的VisuShrink程序，但是，一个例子表明，多诺霍和约翰斯通的SureShrink方法，它使用不同的收缩参数为每个二元水平，实现比我们的程序更低的误差。
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-1(1) (L-1(I)) the functional parallel to F-g parallel to(L2(I))(2) + lambda parallel to g parallel to B-1(1) ((L1(I))). We use the theory of nonlinear wavelet image compression in L-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 alpha for which F epsilon B-q(alpha) (L-q(I)), 1/q = alpha/2 + 1/2, and the norm parallel to F parallel to B-q(alpha)(L-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 SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves lower error than our procedure.