Journal of Mathematical Imaging and Vision

In this paper, we propose a computational framework to incorporate regularization terms used in regularity based variational methods into least squares based methods. In the regularity based variational approach, the image is a result of the competition between the fidelity term and a regularity term, while in the least squares based approach the image is computed as a minimizer to a constrained least squares problem. The total variation minimizing denoising scheme is an exemplary scheme of the former approach with the total variation term as the regularity term, while the moving least squares method is an exemplary scheme of the latter approach. Both approaches have appeared in the literature of image processing independently. By putting schemes from both approaches into a single framework, the resulting scheme benefits from the advantageous properties of both parties. As an example, in this paper, we propose a new denoising scheme, where the total variation minimizing term is adopted by the moving least squares method. The proposed scheme is based on splitting methods, since they make it possible to express the minimization problem as a linear system. In this paper, we employed the split Bregman scheme for its simplicity. The resulting denoising scheme overcomes the drawbacks of both schemes, i.e., the staircase artifact in the total variation minimizing based denoising and the noisy artifact in the moving least squares based denoising method. The proposed computational framework can be utilized to put various combinations of both approaches with different properties together.

In view of the exceptional ability of curvature in connecting missing edges and structures, we propose novel sparse reconstruction models via the Euler’s elastica energy. In particular, we firstly extend the Euler’s elastica regularity into the nonlocal formulation to fully take the advantages of the pattern redundancy and structural similarity in image data. Due to its non-convexity, non-smoothness and nonlinearity, we regard both local and nonlocal elastica functional as the weighted total variation for a good trade-off between the runtime complexity and performance. The splitting techniques and alternating direction method of multipliers (ADMM) are used to achieve efficient algorithms, the convergence of which is also discussed under certain assumptions. The weighting function occurred in our model can be well estimated according to the local approach. Numerical experiments demonstrate that our nonlocal elastica model achieves the state-of-the-art reconstruction results for different sampling patterns and sampling ratios, especially when the sampling rate is extremely low.

Preserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

Linear scale-space is considered to be a modern bottom-up tool in computer vision. The American and European vision community, however, is unaware of the fact that it has already been axiomatically derived in 1959 in a Japanese paper by Taizo Iijima. This result formed the starting point of vast linear scale-space research in Japan ranging from various axiomatic derivations over deep structure analysis to applications to optical character recognition. Since the outcomes of these activities are unknown to western scale-space researchers, we give an overview of the contribution to the development of linear scale-space theories and analyses. In particular, we review four Japanese axiomatic approaches that substantiate linear scale-space theories proposed between 1959 and 1981. By juxtaposing them to ten American or European axiomatics, we present an overview of the state-of-the-art in Gaussian scale-space axiomatics. Furthermore, we show that many techniques for analysing linear scale-space have also been pioneered by Japanese researchers.

The paper presents an effective, robust and geometrically invariants, collection of contours or boundaries base local binary pattern (LBP) for binary object shape retrieval and classification. The contours segmentation or deformations of an object is a preprocessing step of shape retrieval and classification that segment the binary object shape in a shape-preserving sequence of contours segment using a coordination number shape segmentation approach. The proposed local binary pattern extracts the minimum decimal value corresponding to the pattern of object contour points for each and every contours segment. It is one of the most important features in content-based image retrieval. At the matching stage, we find Euclidean distance between eigenvalues of correlation coefficient of Hu’s seven moments corresponding to each contour segment for given two objects. The LBP pattern corresponding to the image contour provides excellent power, which is demonstrated by excellent retrieval performance on several popular shape benchmarks, including MPEG-7 CE-Shape-1 dataset and Kimia’s dataset. Experimental results obtained from popular databases demonstrate that the proposed linear binary pattern can achieve comparably better results than existing algorithms.

Transport-based distances, such as the Wasserstein distance and earth mover’s distance, have been shown to be an effective tool in signal and image analysis. The success of transport-based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However, these distances require the signal to be non-negative and normalised. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here, we study a transport-based distance, called the $$TL^p$$ distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multichannelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multichannelled non-positive one-dimensional signals and two-dimensional images, and colour transfer.