Dawit Aberra

Surfaces of revolution in n dimensions

Surface area and volume formulas for surfaces of revolution in are given.

The schwarz reflection principle for polyharmonic functions in

A reflection formula for polyharmonic functions in is suggested. The obtained formula generalizes the celebrated Schwarz reflection principle for harmonic functions to polyharmonic functions. We also offer modification of the obtained formula to the case of nonhomogeneous data on a reflecting curve.

Seeded Laplacian in sparse subspace for hyperspectral image classification

Sparse Representation (SR) has received an increasing amount of interest in recent years. It aims to find the sparsest representation of each data capturing high-level semantics among the linear combinations of the base sets in a given dictionary. In order to further improve the classification performance, the joint SR that incorporates interpixel correlation information of neighborhoods has been proposed for image pixel classification. However, joint SR method yields high computational cost. To improve the performance and computation efficiency of SR and joint SR, we propose a seeded Laplacian based on sparse representation (SeedLSR) framework for hyperspectral image classification, where a hypergraph Laplacian explicitly takes into account the local manifold structure of the hyperspectral pixel in a spatial-type weighted graph. Given the training data in a dictionary, SeedLSR algorithm firstly finds the sparse representation of hyperspectral pixels, which is used to define the spectral-type affinity matrix of an undirected graph. Then, using the training data as user-defined seeds, the final classification can be obtained by solving the combination of spectral and spatial hypergraph Laplacian quadratic problem. To assess the efficiency of the proposed SeedLSR method, experiments were performed on the scene data under daylight illumination. Compared with SR algorithm, the classification results vary smoothly along the geodesics of the data manifold.

An Improved Random Walker with Bayes Model for Volumetric Medical Image Segmentation

Random walk (RW) method has been widely used to segment the organ in the volumetric medical image. However, it leads to a very large-scale graph due to a number of nodes equal to a voxel number and inaccurate segmentation because of the unavailability of appropriate initial seed point setting. In addition, the classical RW algorithm was designed for a user to mark a few pixels with an arbitrary number of labels, regardless of the intensity and shape information of the organ. Hence, we propose a prior knowledge-based Bayes random walk framework to segment the volumetric medical image in a slice-by-slice manner. Our strategy is to employ the previous segmented slice to obtain the shape and intensity knowledge of the target organ for the adjacent slice. According to the prior knowledge, the object/background seed points can be dynamically updated for the adjacent slice by combining the narrow band threshold (NBT) method and the organ model with a Gaussian process. Finally, a high-quality image segmentation result can be automatically achieved using Bayes RW algorithm. Comparing our method with conventional RW and state-of-the-art interactive segmentation methods, our results show an improvement in the accuracy for liver segmentation (p < 0.001).

Superpixel sparse representation for target detection in hyperspectral imagery

Sparse Representation (SR) is an effective classification method. Given a set of data vectors, SR aims at finding the sparsest representation of each data vector among the linear combinations of the bases in a given dictionary. In order to further improve the classification performance, the joint SR that incorporates interpixel correlation information of neighborhoods has been proposed for image pixel classification. However, SR and joint SR demand significant amount of computational time and memory, especially when classifying a large number of pixels. To address this issue, we propose a superpixel sparse representation (SSR) algorithm for target detection in hyperspectral imagery. We firstly cluster hyperspectral pixels into nearly uniform hyperspectral superpixels using our proposed patch-based SLIC approach based on their spectral and spatial information. The sparse representations of these superpixels are then obtained by simultaneously decomposing superpixels over a given dictionary consisting of both target and background pixels. The class of a hyperspectral pixel is determined by a competition between its projections on target and background subdictionaries. One key advantage of the proposed superpixel representation algorithm with respect to pixelwise and joint sparse representation algorithms is that it reduces computational cost while still maintaining competitive classification performance. We demonstrate the effectiveness of the proposed SSR algorithm through experiments on target detection in the in-door and out-door scene data under daylight illumination as well as the remote sensing data. Experimental results show that SSR generally outperforms state of the art algorithms both quantitatively and qualitatively.

Noisy Nonlinear Gene Regulatory Networks Analysis Using Ensemble Kalman Filter Based Particle Filter without a Model

In this paper, we propose a novel ensemble Kalman filter based particle filter for gene regulatory networks (GRNs) analysis, which incorporates ensemble Kalman filter into particle filter. New particles generated by particle filter are sampled by ensemble Kalman filter, which can take current measurements into account to predict the system states. This will alleviate the sample degeneracy problem in particle filter. The proposed method is model-free algorithm. Both particle filter and ensemble Kalman filter can be applied when the model is unknown, noisy, and nonlinear. This combination of approaches results in comparable accuracy, efficiency, and robustness. In the GRNs analysis, simulation results show that the proposed ensemble Kalman filter based particle filter performs better than particle filter in identifying dynamics relations among genes.

Selection of principal components based on Fisher discriminant ratio

Principal component analysis transforms a set of possibly correlated variables into uncorrelated variables, and is widely used as a technique of dimensionality reduction and feature extraction. In some applications of dimensionality reduction, the objective is to use a small number of principal components to represent most variation in the data. On the other hand, the main purpose of feature extraction is to facilitate subsequent pattern recognition and machine learning tasks, such as classification. Selecting principal components for classification tasks aims for more than dimensionality reduction. The capability of distinguishing different classes is another major concern. Components that have larger eigenvalues do not necessarily have better distinguishing capabilities. In this paper, we investigate a strategy of selecting principal components based on the Fisher discriminant ratio. The ratio of between class variance to within class variance is calculated for each component, based on which the principal components are selected. The number of relevant components is determined by the classification accuracy. To alleviate overfitting which is common when there are few training data available, we use a cross-validation procedure to determine the number of principal components. The main objective is to select the components that have large Fisher discriminant ratios so that adequate class separability is obtained. The number of selected components is determined by the classification accuracy of the validation data. The selection method is evaluated by face recognition experiments.

A note on the Schwarz reflection principle for polyharmonic functions

Abstract In 1907, E. Study gave a geometric proof of the classical Schwarz reflection principle for harmonic functions. We show that this method can also be used to obtain the more general point-to-point reflection formula for polyharmonic functions in ℝ2. The advantage of this method over others is that it avoids use of Garabedian´s generalized Green´s functions.