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In the graph signal processing (GSP) literature, it has been shown that signal-dependent graph Laplacian regularizer (GLR) can efficiently promote piecewise constant (PWC) signal reconstruction for various image restoration tasks. However, for planar image patches, like total variation (TV), GLR may suffer from the wellknown "staircase" effect. To remedy this problem, we generalize GLR to gradient graph Laplacian regularizer (GGLR) that provably promotes piecewise planar (PWP) signal reconstruction for the image interpolation problem-a 2D grid with randomly missing pixels that requires completion. Specifically, we first construct two higher-order gradient graphs to connect local horizontal and vertical gradients. Each local gradient is estimated using structure tensor, which is robust using known pixels in a small neighborhood, mitigating the problem of larger noise variance when computing gradient of gradients. Moreover, unlike total generalized variation (TGV), GGLR retains the quadratic form of GLR, leading to an unconstrained quadratic programming (QP) problem per iteration that can be solved quickly using conjugate gradient (CG). We derive the means-square-error minimizing weight parameter for GGLR, trading off bias and variance of the signal estimate. Experiments show that GGLR outperformed competing schemes in interpolation quality for severely damaged images at a reduced complexity.

In image registration or image matching, the feature extracted by using the traditional methods does not include the depth information which may lead to a mismatch of keypoints. In this paper, we prove that when the camera moves, the ratio of the depth difference of a keypoint and its neighbor pixel before and after the camera movement approximates a constant. That means the depth difference of a keypoint and its neighbor pixel after normalization is invariant to the camera movement. Based on this property, all the depth differences of a keypoint and its neighbor pixels constitute a local depth-based feature, which can be used as a supplement of the traditional feature. We combine the local depth-based feature with the SIFT feature descriptor to form a new feature descriptor, and the experimental results show the feasibility and effectiveness of the new feature descriptor.

A shape prior based object segmentation is developed in this paper by using a shape transformation distance to constrain object contour evolution. In the proposed algorithm, the transformation distance measures the dissimilarity between two unaligned shapes by cyclic shift, which is called "circulant dissimilarity". This dissimilarity with respect to translation and rotation of the object shape is represented by circular convolution, which could be efficiently computed by using fast Fourier transform. Given a set of training shapes, the kernel density estimate is adopted to model shape prior. By integrating low-level image feature, high-level shape prior and transformation distance, a variational segmentation model is proposed to solve the transformation invariance of shape prior. Numerical experiments demonstrate that circulant dissimilarity based shape registration outperforms the iterative optimization on explicit pose parameters, and show promising results and highlight the potential of the method for object registration and segmentation.