_______________________________________________________________ Fast Nonconvex Nonsmooth Minimization Methods (FNNMM) (2010) _______________________________________________________________ 1. Introduction __________________ In the paper "Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction" [1], we considered the degradation model g = h * f + n, where f is the original image, h is the convolution kernel, * is the convolution operator, n is the noise, and g is the blurry and noisy image. Here we assumed that both g and h are known where f is unknown. To recover f, we proposed to use nonconvex and nonsmooth regularization. Nonconvex and nonsmooth minimization has better restoration results over convex minimization, however, it can be computationally costly [2]. In [1], we considered two fast numerical schemes to overcome the problem based on previous works [3], [4] respectively: Algorithm 1: alternating minimization algorithm based on fitting to f Algorithm 2: alternating minimization algorithm based on fitting to Df This software provides a demo on how to use these numerical schemes to restore blurry and noisy images. 2. How to use _________________ A) To get started with demoFNNMM.m, you can run the program directly. A restoration process will be demonstrated accordingly. You can also change the following settings for specific experiments: 1. "algorithm" You need to specify which algortihm you are going to use at the begining. If you want to choose the Algorithm 1 in [1], you can set "algorithm = 1;" in the m file. If you want to choose the Algorithm 2, you can set "algorithm = 2;". Otherwise, both of the algorithms will be chosen. 2. "Input image" Specify the input image. This software is only for graylevel square image. By default it is a two circle image. 3. "Degradation parameters" For the convolution kernel, you can change the kernel type "kernel_type", the kernel size "kernel_size", and the kernel standard deviation "kernel_std" .By default we set it as a gaussian kernel of size 7 by 7 with SD 1.5. For the noise model, it is set to zero-mean additive gaussian noise. You can change the noise level by changing "noise_std". By default the noise level is 0.05. 4. "Regularization parameters" Two parameters "alpha_ep" and "beta" can be tuned for different noise levels and images. By default they are 0.5 and 0.015 respectively. After fix the settings and run the program, the corresponding results will be shown in the Matlab command window, including the observed PSNR, the restored PSNR, and the CPU time. Also 3 figures will be displayed, including the original image, the observed image, and the restored image. B) You can also run the functions "FNNMM1.m" (Algorithm 1) and "FNNMM2.m" (Algorithm 2) directly: function u = FNNMM1(g, h, alpha_ep, beta) function f = FNNMM2(g, h, alpha_ep, beta) The inputs of these functions are the observed image "g", the convolution kernel "h", and the regularization parameters "alpha_ep" and "beta". After the nonconvex and nonsmooth minimzations, restored images "u" and "f" or a message of failure will be output. The programming structures of these functions are following the algorithms in [1]. 3. Contact information _________________________ If there is any suggestion or comment, please feel free to contact the following authors: Mila Nikolova: Centre de Mathematiques et de Leurs Applications (nikolova@cmla.ens-cachan.fr) Michael K. Ng: Centre for Mathematical Imaging and Vision and Department of Mathematics, Hong Kong Baptist University (mng@math.hkbu.edu.hk) ChiPan Tam : Centre for Mathematical Imaging and Vision and Department of Mathematics, Hong Kong Baptist University (cptam@math.hkbu.edu.hk) 4. Copyright ________________ This work is offered without any warranty either expressed or implied; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE! This is a freeware and can be freely used, distributed, and modified. 5. Reference: ________________ [1] M. Nikolova, M. K. Ng, and C. Tam, "Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction", IEEE Transactions on Image Processing, to appear. [2] M. Nikolova, M. K. Ng, S. Zhang, and W. Ching, "Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization," SIAM Journal on Imaging Sciences, 1(1), 2008, pp. 2-25. [3] Y. Huang, M. K. Ng, and Y. Wen, "A fast total variation minimization method for image restoration", SIAM Journal Multiscale Modeling and Simulation, 7(2), 2008, pp. 774-795. [4] Y. Wang, J. Yang, W. Yin, and Y. Zhang, "A new alternating minimization algorithm for total variation image reconstruction", SIAM Journal on Imaging Sciences, 1(3), 2008, pp.248-272.