Revisiting the Regularizers in Blind Image Deblurring With a New One
Wenze Shao
Abstract
Image deblurring and its counterpart blind problem are undoubtedly two fundamental tasks in computational imaging and computer vision. Interestingly, deterministic edge-preserving regularization for maximum-a-posteriori (MAP) based non-blind image deblurring has been largely made clear 25 years ago. As for the blind task, the state-of-the-art MAP-based approaches seem to also reach a consensus on the characteristic of deterministic image regularization, i.e., formulated in an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> composite style or termed as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> style, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> is often a discriminative term such as dark channels-based sparsity regularization. However, with a modeling perspective as such, non-blind and blind deblurring are entirely disconnected from each other. Additionally, because <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> are motivated very differently in general, it is not easy in practice to derive an efficient numerical scheme. In fact, since the prosperity of modern blind deblurring 15 years ago, a physically intuitive yet practically effective and efficient regularization has been always desired. In this paper, representative deterministic image regularization terms in MAP-based blind deblurring are firstly revisited, with an emphasis on their differences from edge-preserving regularization for non-blind deblurring. Inspired by existing robust losses in the statistical and deep learning literature, an insightful conjecture is then made. That is, deterministic image regularization for blind deblurring can be naively formulated using a type of redescending potential functions (RDP), and interestingly, a RDP-induced blind deblurring regularization term is actually the 1 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rst</sup> -order derivative of a nonconvex edge-preserving regularization for non-blind image deblurring. An intimate relationship in regularization is therefore established between the two problems, differing much from the mainstream modeling perspective on blind deblurring. Via above principle analysis, the conjecture is demonstrated on benchmark deblurring problems in the final, accompanied with comparisons against several top-performing <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> style methods. We note that, the rationality and practicality of the RDP-induced regularization is particularly highlighted here, aiming to open up an alternative line of possibility for modeling blind deblurring.