Improved image deblurring with anti-reflective boundary conditions and re-blurring

Abstract

Anti-reflective boundary conditions (BCs) have been introduced recently in connection with fast deblurring algorithms. In the noise free case, it has been shown that they substantially reduce artefacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, reflective BCs) and lead to O(n2log(n)) arithmetic operations, where n2 is the size of the image. In the one-dimensional case, for noisy data, we proposed a successful approach called re-blurring: more specifically, when the PSF is symmetric, the normal equations product ATA is replaced by A2, where A is the blurring operator (see Donatelli et al 2005 Inverse Problems 21 169–82). Our present goal is to extend the re-blurring idea to nonsymmetric point spread functions (PSFs) in two dimensions. In this more general framework, suitable for real applications, the new proposal is to replace AT by A′ in the normal equations, where A′ is the blurring matrix related to the current BCs with PSF rotated by 180°. We notice that, although with zero Dirichlet and periodic BCs the re-blurring approach is equivalent to the normal equations scheme, since there A′ = AT, the novelty concerns both reflective BCs and anti-reflective BCs, where in general A′ ≠ AT. We show that the re-blurring with anti-reflective BCs is computationally convenient and leads to a large reduction of the ringing effects arising in classical deblurring schemes. A wide set of numerical experiments concerning 2D images and nonsymmetric PSFs confirms the effectiveness of our proposal.

High-quality motion deblurring from a single image

“…The most common technique is Richardson-Lucy (RL) deconvolution [1974], which computes the latent image with the assumption that its pixel intensities conform to a Poisson distribution. Donatelli et al [2006] use a PDE-based model to recover a latent image with reduced ringing by incorporating an anti-reflective boundary condition and a re-blurring step. Several approaches proposed in the signal processing community solve the deconvolution problem in the wavelet domain or the frequency domain [Neelamani et al 2004]; many of these papers lack experiments in de-blurring real photographs, and few of them attempt to model error in the estimated kernel.…”

Antireflective Boundary Conditions for Deblurring Problems

“…The relative restoration error (RRE) is f − f 2 / f 2 , where f is the computed approximation of the true image f. The signal-to-noise ratio (SNR) is computed as 20log 10 g b 2 / ν 2 , where g b is the blurred image without noise and ν is the noise vector [32]. [19,33,46]), where the advantage on some classes of images, in terms of the restored image quality, of the application of AR-BCs has been emphasized. Here we present only a 2D image deblurring example with Gaussian blur and various levels of white Gaussian noise.…”

Block-iterative Richardson-Lucy methods for image deblurring

“…One group of methods imposes certain conditions on pixels in − . Examples include periodic, reflective, and anti-reflective boundary conditions [23][24][25][26][27].…”

“…This suggestion is based on the observation that the free boundary condition successfully suppresses boundary artifacts for arbitrarily shaped images; note that periodic, reflective, and anti-reflective boundary conditions can be applied to rectangularshaped images only. For details, see [23][24][25][26].…”