The nth partial sums of a classical Fourier series have large oscillations near the jump discontinuities. This behaviour is the well-known Gibbs phenomenon. Recently, the inverse polynomial reconstruction method (IPRM) has been successfully implemented to reconstruct piecewise smooth functions by reducing the effects of the Gibbs phenomenon for Fourier series. This paper addresses the 2-D fractional Fourier series (FrFS) using the same approach used with the 1-D fractional Fourier series and finds that the Gibbs phenomenon will be observed in 1-D and 2-D fractional Fourier series expansions for functions at a jump discontinuity. The existing IPRM for resolution of the Gibbs phenomenon for 1-D and 2-D FrFS appears to be the same as that used for Fourier series. The proof of convergence provides theoretical basis for both 1-D and 2-D IPRM to remove Gibbs phenomenon. Several numerical examples are investigated. The results indicate that the IPRM method completely eliminates the Gibbs phenomenon and gives exact reconstruction results.

A novel superpixel segmentation approach driven by uniform mixture model with spatially constrained (UMMS) is proposed. Under this algorithm, each observation, i.e. pixel is first represented as a five-dimensional vector which consists of colour in CLELAB space and position information. And then, we define a new uniform distribution through adding pixel position, so that this distribution can describe each pixel in input image. Applied weighted 1-Norm to difference between pixels and mean to control the compactness of superpixel. In addition, an effective parameter estimation scheme is introduced to reduce computational complexity. Specifically, the invariant prior probability and parameter range restrict the locality of superpixels, and the robust mean optimization technique ensures the accuracy of superpixel boundaries. Finally, each defined uniform distribution is associated with a superpixel and the proposed UMMS successfully implements superpixel segmentation. The experiments on BSDS500 dataset verify that UMMS outperforms most of the state-of-the-art approaches in terms of segmentation accuracy, regularity, and rapidity.

Finding the correspondence between two images of the same object or scene is an active research field in computer vision. This paper develops a rapid and effective Content-based Superpixel Image matching and Stitching (CSIS) scheme, which utilizes the content of superpixel through multi-features fusion technique. Unlike popular keypoint-based matching method, our approach proposes a superpixel internal feature-based scheme to implement image matching. In the beginning, we make use of a novel superpixel generation algorithm based on content-based feature representation, named Content-based Superpixel Segmentation (CSS) algorithm. Superpixels are generated in terms of a new distance metric using color, spatial, and gradient feature information. It is developed to balance the compactness and the boundary adherence of resulted superpixels. Then, we calculate the entropy of each superpixel for separating some superpixels with significant characteristics. Next, for each selected superpixel, its multi-features descriptor is generated by extracting and fusing local features of the selected superpixel itself. Finally, we compare the matching features of candidate superpixels and their own neighborhoods to estimate the correspondence between two images. We evaluated superpixel matching and image stitching on complex and deformable surfaces using our superpixel region descriptors, and the results show that new method is effective in matching accuracy and execution speed.