Weighted gradient domain image processing problems and their iterative solutions
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Weighted gradient domain image processing problems and their iterative solutions Jung Gap Kuk1 and Nam Ik Cho2*
Abstract This article explores an energy function and its minimization for the weighted gradient domain image processing, where variable weights are applied to the data term of conventional function for attaining better results in some applications. To be specific, larger weights are given to the regions where original pixel values need to be kept unchanged, like strong edge regions in the case of image sharpening application or high contrast regions when fusing multi-exposure images. In the literatures, it is shown that the solution to a constant weight problem can be efficiently obtained in the frequency domain without iterations, whereas the function with the varying weights can be minimized by solving a large sparse linear equation or by iterative methods such as conjugate gradient or preconditioned conjugate gradient (PCG) methods. In addition to introducing weighted gradient domain image processing problems, we also proposed a new approach to finding an efficient preconditioning matrix for this problem, which greatly reduces the condition number of the system matrix and thus reduces the number of iterations for the PCG process to reach the solution. We show that the system matrix for the constant weight problem is an appropriate preconditioner, in the sense that a sub-problem in the PCG is efficiently solved by the FFT and also it ensures the convergent splitting of the system matrix. For the simulation and experiments on some applications, it is shown that the proposed method requires less iteration, memory, and CPU time. 1 Introduction Since human visual system (HVS) is sensitive to the intensity changes, processing an image in the gradient domain often produces subjectively better results than the conventional intensity domain processing. Specifically, the gradient domain approach has been successfully applied to high dynamic range (HDR) imaging [1], image stitching [2-5], filtering [6], alignment [7], matching [8], etc. In the case of image stitching problems which include seamless cloning [2,5] and image composite for panoramic view [3,4], the gradient domain processing is considered the state-of-the-art method. The gradient domain method is basically matching the gradients with priors, and the first step is to generate a targeting gradient image from the input or assume a gradient profile that meets the given purposes or specifications [6,9]. Then the output image that corresponds to the targeting gradients is generated. In this process, since the *Correspondence: [email protected] 2 Department of Electrical and Computer Engineering and INMC, Seoul National University, Seoul, Korea Full list of author information is available at the end of the article
gradient is usually non-integrable, the output cannot be obtained by the direct integration of gradients. Instead, an image whose gradient is close to the targeting gradient is obtained. To be precise, for the given gradient g(x, y), th
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