Elastica Models for Color Image Regularization

Abstract

The choice of a proper regularization measure plays an important role in the field of image processing. One classical approach treats color images as two dimensional surfaces embedded in a five dimensional spatial-chromatic space. In this case, a natural regularization term arises as the image surface area. Choosing the chromatic coordinates as dominating over the spatial ones, the image spatial coordinates could be thought of as a paramterization of the image surface manifold in a three dimensional color space. Minimizing the area of the image manifold leads to the Beltrami flow or mean curvature flow of the image surface in the 3D color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, the authors proposed a color elastica model, which minimizes both the surface area and elastica of the image manifold. In this paper, we propose to modify the color elastica and introduce two new models for color image regularization. The revised measures are motivated by the relations between the color elastica model, Euler's elastica model and the total variation model for gray level images. Compared to our previous color elastica model, the new models are direct extensions of Euler's elastica model to color images. The proposed models are nonlinear and challenging to minimize. To overcome this difficulty, two operator-splitting methods are suggested. Specifically, nonlinearities are decoupled by introducing new vector-and matrix-valued variables. Then, the minimization problems are converted to solving initial value problems which are time-discretized by operator splitting. Each subproblem, after splitting either, has a closed-form solution or can be solved efficiently. The effectiveness and advantages of the proposed models are demonstrated by comprehensive experiments. The benefits of incorporating the elastica of the image surface as regularization terms compared to common alternatives are empirically validated.

Three-Dimensional Surface Reconstruction from Point Clouds Using Euler’s Elastica Regularization

“…On the other hand, Euler's elastica term has long been used in image processing, especially in image segmentation [24,25] and denoising [26]. Using Euler's elastica as a regularization term to address issues such as the staircase effect and contrast loss in loworder models is currently a popular topic [27,28]; however, due to the challenges associated with its application in point cloud reconstruction, there is currently a lack of research in this specific area.…”

An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Ampère Equation

“…All variables will be updated in an alternative fashion, where each subproblem either has an explicit solution or can be solved efficiently. The operator-splitting method has been applied to numerically solving PDEs [28,37], image processing [39,15,38,40], surface reconstruction [32], inverse problems [27], obstacle problems [41], and computational fluid dynamics [8,7]. We refer readers to monographs [29,30] for detailed discussions on operatorsplitting methods.…”

An efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère equation

“…All variables will be updated in an alternative fashion, where each subproblem either has an explicit solution or can be solved efficiently. The operator-splitting method has been applied to numerically solving PDEs [30,39], image processing [16,40,41,42], surface reconstruction [34], inverse problems [29], obstacle problems [43], and computational fluid dynamics [7,8]. We refer readers to monographs [31,32] for detailed discussions on operator-splitting methods.…”