Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Abstract

Abstract-We present nonquadratic Hessian-based regularization methods that can be effectively used for image restoration problems in a variational framework. Motivated by the great success of the total-variation (TV) functional, we extend it to also include second-order differential operators. Specifically, we derive second-order regularizers that involve matrix norms of the Hessian operator. The definition of these functionals is based on an alternative interpretation of TV that relies on mixed norms of directional derivatives. We show that the resulting regularizers retain some of the most favorable properties of TV, i.e., convexity, homogeneity, rotation, and translation invariance, while dealing effectively with the staircase effect. We further develop an efficient minimization scheme for the corresponding objective functions. The proposed algorithm is of the iteratively reweighted least-square type and results from a majorization-minimization approach. It relies on a problem-specific preconditioned conjugate gradient method, which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time. We validate the overall proposed regularization framework through deblurring experiments under additive Gaussian noise on standard and biomedical images.

Variational Phase Imaging Using the Transport-of-Intensity Equation

“…In such cases, staircase effect can occur and thus a higher-order regularization should be incorporated into our model [35].…”

Hessian-based regularization for 3-D microscopy image restoration

“…According to (4), TV can be interpreted as a mixed L1-L2 norm where the L1-norm acts on the image domain while the L2-norm acts on the domain specified by the angles (θ, φ) of the directional derivative. As shown below, this new interpretation of the TV functional permits us to extend its definition to higher-order differential operators.…”

“…We minimize (13) following a majorization-minimization (MM) approach [4,6,7]. To develop a MM-based algorithm, we first derive an appropriate quadratic majorizer QR (f ; f ) of our penalty function R (f ).…”

“…In this paper, we extend our prior work [4] to deal with the problem of 3-D image restoration. Specifically, we propose in Section 2 a higher-order non quadratic functional that is a valid extension of the TV semi-norm.…”

“…However, TV regularization cannot be considered as a universal choice irrespectively of the class of images under consideration. In particular, it has been recently shown that Hessian-based regularizers are better adapted for the restoration of biomedical images in the 2-D case [4].…”