Elastica Models for Color Image Regularization

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, we can think of the image spatial coordinates could as a parameterization 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 three-dimensional color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, we proposed a color elastica model, which minimizes both the surface area and the 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 the introduction of new vector- and matrix-valued variables. Then, the minimization problems are converted to 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.