Directional Total Variation Based Image Deconvolution with Unknown Boundaries

Like many other imaging inverse problems, image deconvolution suffers from ill-posedness and needs for an adequate regularization. Total variation (TV) is an effective regularizer; hence, frequently used in such problems. Various anisotropic alternatives to isotropic TV have also been proposed to capture different characteristics in the image. Directional total variation (DTV) is such an instance, which is convex, has the ability to capture the smooth boundaries as conventional TV does, and also handles the directional dominance by enforcing piecewice constancy through a direction. In this paper, we solve the deconvolution problem under DTV regularization, by using simple forward-backward splitting machinery. Besides, there are two bottlenecks of the deconvolution problem, that need to be addressed; one is the computational load revealed due to matrix inversions, second is the unknown boundary conditions (BCs). We tackle with the former one by switching to the frequency domain using fast Fourier transform (FFT), and the latter one by iteratively estimating a boundary zone to surrounder the blurred image by plugging a recently proposed framework into our algorithm. The proposed approach is evaluated in terms of the reconstruction quality and the speed. The results are compared to a very recent TV-based deconvolution algorithm, which uses a “partial” alternating direction method of multipliers (ADMM) as the optimization tool, by also plugging the same framework to cope with the unknown BCs.