Large-Scale Inverse Problems in Imaging

Abstract

Large-scale inverse problems arise in a variety of significant applications in image processing, and efficient regularization methods are needed to compute meaningful solutions. This chapter surveys three common mathematical models including a linear model, a separable nonlinear model, and a general nonlinear model. Techniques for regularization and large-scale implementations are considered, with particular focus on algorithms and computations that can exploit structure in the problem. Examples from image deconvolution, multi-frame blind deconvolution, and tomosynthesis illustrate the potential of these algorithms. Much progress has been made in the field of large-scale inverse problems, but many challenges still remain for future research.

Flexible Krylov Methods for Edge Enhancement in Imaging

“…Common and somewhat basic choices for the penalty term include and with , corresponding to standard and generalized Tikhonov regularization, respectively. Although such choices reduce ( 2 ) to a quadratic problem, two drawbacks arise when 2-norm regularization is applied to solve inverse problems in imaging, where A is typically unstructured and large-scale: firstly, an iterative solver must be employed to compute (see [ 1 , 4 , 5 , 8 ] and the references therein); secondly, may be inherently over-smoothed and therefore unsuitable when edge information should be accurately recovered (see [ 2 ]). To overcome the second drawback, one should resort to functionals involving some q -(quasi)norm, , and solve ( 2 ) using appropriate optimization methods: there is a rich body of literature about this, and we point to [ 9 ] for a recent survey.…”

Flexible GMRES for total variation regularization

“…consecutive ones), and e ∈ R N is unknown Gaussian white noise. Systems like this typically arise when discretizing inverse problems, which are central in many applications (such as astronomical and biomedical imaging, see [11,18] and the references therein). This paper mainly deals with signal deconvolution (deblurring) problems, where x ∈ R N is the unknown sharp signal we wish to recover, and b is the measured (blurred and noisy) signal.…”

“…Unfortunately, when considering large-scale problems whose associated coefficient matrix A may not have an exploitable structure or may not be explicitly stored, one cannot assume the GSVD to be available. In this setting iterative regularization methods are the only option, i.e., one can either solve the Tikhonov-regularized problem (1.2) iteratively, or apply an iterative solver to the original system (1.1) and terminate the iterations early (see [4,11,16,18] and the references therein).…”

Regularization by inexact Krylov methods with applications to blind deblurring

“…Although (1.3) has a closed-form solution, when dealing with a large-scale and unstructured A, and without prior knowledge of a suitable value of λ, computing a good z λ would potentially involve repeatedly applying a (matrix-free) iterative solver for LS problems, one for each considered value of λ. In this framework, many Krylov methods are successfully applied to either (1.1) (i.e., as stand-alone solvers that regularize by early termination of the iterations), or (1.3) (in a so-called hybrid fashion, i.e., by combining projection onto Krylov subspaces of increasing dimensions and Tikhonov regularization, with the possibility of efficiently and adaptively choosing λ as the iterations progress); see, for instance, [1,4,13,14] and the references therein. Among the Krylov methods routinely used for regularization, the mathematically equivalent LSQR and CGLS methods are arguably among the the most popular ones, as their theoretical properties and practical performance are generally well-understood; see [16,20].…”

“…Problems like (1.4), (1.5) arise in a variety of signal and image processing tasks, such as instrument calibration [5,15] and super-resolution [3], just to name a few. In this paper we are particularly interested in spatially invariant blind deblurring [4], where b is a blurred and noisy image (reshaped as a vector) and A(y) encodes information about a parametric blur (i.e., defined by the unknown parameters y true ) that corrupts every pixel of the unperturbed unknown image x true (reshaped as a vector). To mitigate the complexity of problem (1.4) one can take advantage of separability, as the objective function in (1.4) is linear in x.…”