Radial Hilbert derivative-based total variation penalty for image reconstruction

Gradient-based penalties such as total variation (TV) and its variants are popular in computational imaging literature due to their simultaneous noise suppression and edge preservation properties. The functional (or Euler-Lagrange) derivatives of these penalty functions involve a combination of gradient and divergence operations that are typically evaluated using the finite difference approximation along the row and column directions of the image matrix. Repeated evaluation of functional derivatives as required in iterative image reconstruction algorithms can cause faint grid-like artifacts, and the Fourier spectrum of the reconstructed image is observed to develop spurious energy bands. We propose a TV-like penalty using the Laguerre-Gaussian radial Hilbert transform, which is known to provide isotropic edge enhancement. The functional derivative of this penalty function does not have a simple analytical form but can be evaluated using the readily available automatic differentiation tools. Numerical illustrations for image denoising and deconvolution using the proposed penalty function with gradient descent-type iterations show that the reconstructed images retain their natural texture, and their Fourier spectrum does not show undesirable structures. The proposed penalty function is simple to implement and can be used generically in computational imaging algorithms. Furthermore, we demonstrate that the proposed approach is not limited just to the TV penalty and can be systematically extended to a broader class of gradient-based regularizers.