Fast Nonconvex Nonsmooth Minimization Methods for Image Restoration and Reconstruction

• Published in: IEEE transactions on image processing Vol. 19; no. 12; pp. 3073 - 3088
• Main Authors: Nikolova, Mila, Ng, Michael K, Chi-Pan Tam
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-12-2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Computer Simulation; Continuation methods; Exact sciences and technology; fast Fourier transform; Fast Fourier transforms; Fourier transforms; Image Interpretation, Computer-Assisted; Image processing; Image Processing, Computer-Assisted - methods; Image reconstruction; Image restoration; Information, signal and communications theory; Mathematics; Minimization; Minimization methods; nonconvex nonsmooth global minimization; nonconvex nonsmooth regularization; Optical distortion; Optical imaging; Optimization; Pattern Recognition, Automated - methods; Permission; Production; Reconstruction; Regularization; Renovating; Signal processing; Telecommunications and information theory; total variation; Performance evaluation; Image processing; Fast Fourier transformation; nonconvex nonsmooth regularization; Theoretical study; Image restoration; Image reconstruction; Continuation methods; fast Fourier transform; Predictor corrector method; total variation; Fast algorithm; nonconvex nonsmooth global minimization
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2010.2052275

Nonconvex nonsmooth regularization has advantages over convex regularization for restoring images with neat edges. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonconvex nonsmooth minimization. In this paper, we deal with nonconvex nonsmooth minimization methods for image restoration and reconstruction. Our theoretical results show that the solution of the nonconvex nonsmooth minimization problem is composed of constant regions surrounded by closed contours and neat edges. The main goal of this paper is to develop fast minimization algorithms to solve the nonconvex nonsmooth minimization problem. Our experimental results show that the effectiveness and efficiency of the proposed algorithms.