Graph Clustering, Variational Image Segmentation Methods and Hough Transform Scale Detection for Object Measurement in Images

© 2016, Springer Science+Business Media New York. We consider the problem of scale detection in images where a region of interest is present together with a measurement tool (e.g. a ruler). For the segmentation part, we focus on the graph-based method presented in Bertozzi and Flenner (Multiscale Model Simul 10(3):1090–1118, 2012) which reinterprets classical continuous Ginzburg–Landau minimisation models in a totally discrete framework. To overcome the numerical difficulties due to the large size of the images considered, we use matrix completion and splitting techniques. The scale on the measurement tool is detected via a Hough transform-based algorithm. The method is then applied to some measurement tasks arising in real-world applications such as zoology, medicine and archaeology

Graph approximation and generalized Tikhonov regularization for signal deblurring

Given a compact linear operator \K, the (pseudo) inverse \K^\dagger is usually substituted by a family of regularizing operators $\R_\alpha$ which depends on \K itself. Naturally, in the actual computation we are forced to approximate the true continuous operator \K with a discrete operator \K^{(n)} characterized by a finesses discretization parameter $n$, and obtaining then a discretized family of regularizing operators $\R_\alpha^{(n)}$. In general, the numerical scheme applied to discretize \K does not preserve, asymptotically, the full spectrum of \K. In the context of a generalized Tikhonov-type regularization, we show that a graph-based approximation scheme that guarantees, asymptotically, a zero maximum relative spectral error can significantly improve the approximated solutions given by $\R_\alpha^{(n)}$. This approach is combined with a graph based regularization technique with respect to the penalty term

Using Fuzzy Inference system for detection the edges of Musculoskeletal Ultrasound Images

Edge detection in Musculoskeletal Ultrasound Imaging readily allows an ultrasound image to be rendered as a binary image. This facilitates automated measurement of geometric parameters, such as muscle thickness, circumference and cross-sectional area of the tendon. In this work, we introduced a new method of edge detection based on a fuzzy inference system and apply it to the ultrasound image. An anisotropic diffusion filter was used to reduce speckle noise before implementation of the edge detection method, which consists of three characteristic steps. The first step entailed fuzzification, for which three fuzzy membership functions were applied to the image. The parameters of these functions were selected based on an analysis of the standard deviation of grey level intensities in the image. Secondly, 12 fuzzy rules for identifying edges were constructed. Thirdly, defuzzification was carried out using the Takagi-Sugeno method. Furthermore, a reference-based edge measurement was quantitatively determined by comparing edge characteristics with a standard reference. We made two inferences from our observations. Firstly, the ability to automatically identify the important details of a musculoskeletal ultrasound image in a very short time is possible. Secondly, this method is effective compared with other methods

On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures

A novel general framework for the study of $\Gamma$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $\Gamma$-limit of these kind of functionals by knowing the $\Gamma$-limit of the underlining energies. In particular, the interaction between the functionals and the underlining energies results, in the case these latter converge to a non continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented

Large data limit for a phase transition model with the p-Laplacian on point clouds

The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the $p$-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of $\epsilon=\epsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\to \infty$, in the regime where $\epsilon_n\to 0$. The mathematical tool used to address this question is $\Gamma$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter