Constrained least squares problems arise in a variety of applications, and many iterative methods are already available to compute their solutions. This paper proposes a new efficient approach to solve nonnegative linear least squares problems. The associated KKT conditions are leveraged to form an adaptively preconditioned linear system, which is then solved by a flexible Krylov subspace method. The new method can be easily applied to image reconstruction problems affected by both Gaussian and Poisson noise, where the components of the solution represent nonnegative intensities. Theoretical insight is given, and numerical experiments and comparisons are displayed in order to validate the new method, which delivers results of equal or better quality than many state-of-the-art methods for nonnegative least squares solvers, with a significant speedup.
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. The projection method of solving a set of linear equations was used to restore linearly degraded images. The advantages of this method are that it always converges, it can readily make use of a priori information about the image, and it does not need excessive computation time. Experimental results indicate that this method could potentially achieve superresolution.