A fast iterative shrinkage-thresholding algorithm for linear inverse problems. (English) Zbl 1175.94009

Beck, Amir; Teboulle, Marc

SIAM J. Imaging Sci. 2, No. 1, 183-202 (2009).

Summary: We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA which is shown to be faster than ISTA by several orders of magnitude.

MSC:

94A08	Image processing (compression, reconstruction, etc.) in information and communication theory
68U10	Computing methodologies for image processing
65F22	Ill-posedness and regularization problems in numerical linear algebra

Keywords:

iterative shrinkage-thresholding algorithm; deconvolution; linear inverse problem; least squares and \(l_1\) regularization problems; optimal gradient method; global rate of convergence; two-step iterative algorithms; image deblurring

Software:

Regularization tools; FISTA