利用帧系数的L1保真度消除乘性噪声
Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients
摘要：
我们解决了被乘性噪声污染的图像的去噪，例如斑点噪声。解决这些问题的经典方法是滤波、统计（贝叶斯）方法、变分方法和将乘性噪声转换为加性噪声（使用对数函数）的方法、小波基或帧中的对数图像数据的系数的收缩，以及使用指数函数将结果变换回来。我们提出了一种由几个阶段组成的方法：我们使用对数图像数据，并对其曲波变换应用合理的次优硬阈值;然后我们应用变分方法，其中我们最小化由$\ell^1$数据拟合到阈值系数和图像域中的全变差正则化（TV）项组成的专门标准;所恢复的图像是所获得的最小值的指数，以保持原始图像的平均值的方式加权。我们的恢复图像结合了收缩和变分方法的优点，并避免了他们的主要缺点。对于最小化阶段，我们提出了一个适当适应的快速最小化方案的基础上道格拉斯-Rachford分裂。证明了我们的专业标准的极小化的存在，我们证明了最小化方案的收敛性。所得的数值结果优于主要的替代方法。
英文摘要：
We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods.
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