Image Reconstruction by Minimizing Curvatures on Image Surface
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Image Reconstruction by Minimizing Curvatures on Image Surface Qiuxiang Zhong1 · Ke Yin2 · Yuping Duan1 Received: 8 January 2020 / Accepted: 31 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The curvature regularities are well-known for providing strong priors in the continuity of edges, which have been applied to a wide range of applications in image processing and computer vision. However, these models are usually non-convex, non-smooth, and highly nonlinear, the first-order optimality condition of which are high-order partial differential equations. Thus, numerical computation is extremely challenging. In this paper, we estimate the discrete mean curvature and Gaussian curvature on the local 3 × 3 stencil, based on the fundamental forms in differential geometry. By minimizing certain functions of curvatures over the image surface, it yields a kind of weighted image surface minimization problem, which can be efficiently solved by the alternating direction method of multipliers. Numerical experiments on image restoration and inpainting are implemented to demonstrate the effectiveness and superiority of the proposed curvature-based model compared to state-ofthe-art variational approches. Keywords Image reconstruction · Image surface · Differential geometry · Curvature regularization · Mean curvature · Gaussian curvature
1 Introduction
model becomes
Curves and surfaces are important geometric elements in image processing and analysis, which can be ideally measured by quantities such as arc length, area and curvatures [12]. Let u : → R be an image defined on an open-bounded subset ⊂ R2 with Lipschitz continuous boundary. For each gray level λ, by taking the length energy as the curve model, it yields the well-known total variation (TV) regularization [42]
E(u) =
TV(u) =
∞
−∞
Length(Γλ )dλ =
|∇u|dx,
(1)
where Γλ = {x ∈ |u(x) = λ}. Curvatures can depict the amount of a curve from being straight as in the case of a line or a surface deviating from being a flat plane, which have also been applied to various image processing tasks [1,23,43]. Suppose we take the curvature curve model, then the image
B
Yuping Duan [email protected]
1
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2
Center for Mathematical Sciences, Huazhong Univeristy of Science and Technology, Wuhan, Hubei 430070, China
g(κ)|∇u|dx =
∇u g ∇ · |∇u|dx, (2) |∇u|
where g(κ) = 1 + ακ 2 , α > 0, yielding the Euler’s elastica image model [36]. Nitzberg, Mumford and Shiota [39] observed that line energies such as Euler’s elastica can be used as regularization for the completion of missing contours in images by providing strong priors for the continuity of edges. Since then, Euler’s elastica has been successfully used for image denoising [47,56], inpainting [35,45,54], segmentation [3,18,61], segmentation with depth [22,59] and illusory contour [28]. However, the numerical minimization of Euler’s elastica is highly challenging due to
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