Armin Eftekhari

Robust-SL0 for stable sparse representation in noisy settings

In the last few years, we have witnessed an explosion in applications of sparse representation, the majority of which share the need for finding sparse solutions of underdetermined systems of linear equations (USLEs). Based on recently proposed smoothed ℓ0-norm (SL0), we develop a noise-tolerant algorithm for sparse representation, namely Robust-SL0, enjoying the same computational advantages of SL0, while demonstrating remarkable robustness against noise. The proposed algorithm is developed by adopting the corresponding optimization problem for noisy settings, followed by theoretically-justified approximation to reduce the complexity. Stability properties of Robust-SL0 are rigorously analyzed, both analytically and experimentally, revealing a remarkable improvement in performance over SL0 and other competing algorithms, in the presence of noise.

The Restricted Isometry Property for block diagonal matrices

In compressive sensing (CS), the Restricted Isometry Property (RIP) is a powerful condition on measurement operators which ensures robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. Early papers in CS showed that Gaussian random matrices satisfy the RIP with high probability, but such matrices are usually undesirable in practical applications due to storage limitations, computational considerations, or the mismatch of such matrices with certain measurement architectures. To alleviate some or all of these difficulties, recent research efforts have focused on structured random matrices. In this paper, we study block diagonal measurement matrices where each block on the main diagonal is itself a Gaussian random matrix. The main result of this paper shows that such matrices can indeed satisfy the RIP but that the requisite number of measurements depends on the coherence of the basis in which the signals are sparse. In the best case—for signals that are sparse in the frequency domain—these matrices perform nearly as well as dense Gaussian random matrices despite having many fewer nonzero entries.

Block-wise 2D kernel PCA/LDA for face recognition

Direct extension of (2D) matrix-based linear subspace algorithms to kernel-induced feature space is computationally intractable and also fails to exploit local characteristics of input data. In this letter, we develop a 2D generalized framework which integrates the concept of kernel machines with 2D principal component analysis (PCA) and 2D linear discriminant analysis (LDA). In order to remedy the mentioned drawbacks, we propose a block-wise approach based on the assumption that data is multi-modally distributed in so-called block manifolds. Proposed methods, namely block-wise 2D kernel PCA (B2D-KPCA) and block-wise 2D generalized discriminant analysis (B2D-GDA), attempt to find local nonlinear subspace projections in each block manifold or alternatively search for linear subspace projections in kernel space associated with each blockset. Experimental results on ORL face database attests to the reliability of the proposed block-wise approach compared with related published methods.

Matched Filtering From Limited Frequency Samples

In this paper, we study a simple correlation-based strategy for estimating the unknown delay and amplitude of a signal based on a small number of noisy, randomly chosen frequency-domain samples. We model the output of this “compressive matched filter” as a random process whose mean equals the scaled, shifted autocorrelation function of the template signal. Using tools from the theory of empirical processes, we prove that the expected maximum deviation of this process from its mean decreases sharply as the number of measurements increases, and we also derive a probabilistic tail bound on the maximum deviation. Putting all of this together, we bound the minimum number of measurements required to guarantee that the empirical maximum of this random process occurs sufficiently close to the true peak of its mean function. We conclude that for broad classes of signals, this compressive matched filter will successfully estimate the unknown delay (with high probability and within a prescribed tolerance) using a number of random frequency-domain samples that scales inversely with the signal-to-noise ratio and only logarithmically in the observation bandwidth and the possible range of delays.

Endocardial boundary extraction in left ventricular echocardiographic images using fast and adaptive B-spline snake algorithm

PurposeA fast and robust algorithm was developed for automatic segmentation of the left ventricular endocardial boundary in echocardiographic images. The method was applied to calculate left ventricular volume and ejection fraction estimation.MethodsA fast adaptive B-spline snake algorithm that resolves the computational concerns of conventional active contours and avoids computationally expensive optimizations was developed. A combination of external forces, adaptive node insertion, and multiresolution strategy was incorporated in the proposed algorithm. Boundary extraction with area and volume estimation in left ventricular echocardiographic images was implemented using the B-spline snake algorithm. The method was implemented in MATLAB and 50 medical images were used to evaluate the algorithm performance. Experimental validation was done using a database of echocardiographic images that had been manually evaluated by experts.ResultsComparison of methods demonstrates significant improvement over conventional algorithms using the adaptive B-spline technique. Moreover, our method reached a reasonable agreement with the results obtained manually by experts. The accuracy of boundary detection was calculated with Dice’s coefficient equation (91.13%), and the average computational time was 1.24 s in a PC implementation.ConclusionIn sum, the proposed method achieves satisfactory results with low computational complexity. This algorithm provides a robust and feasible technique for echocardiographic image segmentation. Suggestions for future improvements of the method are provided.

Two-dimensional random projection

As an alternative to adaptive nonlinear schemes for dimensionality reduction, linear random projection has recently proved to be a reliable means for high-dimensional data processing. Widespread application of conventional random projection in the context of image analysis is, however, mainly impeded by excessive computational and memory requirements. In this paper, a two-dimensional random projection scheme is considered as a remedy to this problem, and the associated key notion of concentration of measure is closely studied. It is then applied in the contexts of image classification and sparse image reconstruction. Finally, theoretical results are validated within a comprehensive set of experiments with synthetic and real images.

Greed is super: A new iterative method for super-resolution

We present a new greedy algorithm for super-resolution. Given the low-frequency part of the spectrum of a sequence of impulses, our objective is to estimate their positions. The backbone of our work is the fundamental work of Slepian et al. involving discrete prolate spheroidal wave functions and their unique properties. By its greedy nature, our work differs from the approach of Candès et al. based on convex optimization. By its use of prolate functions, our work also differs from the greedy algorithm presented by Fannjiang et al.

A near-stationary subspace for ridge approximation

Abstract Response surfaces are common surrogates for expensive computer simulations in engineering analysis. However, the cost of fitting an accurate response surface increases exponentially as the number of model inputs increases, which leaves response surface construction intractable for high-dimensional, nonlinear models. We describe ridge approximation for fitting response surfaces in several variables. A ridge function is constant along several directions in its domain, so fitting occurs on the coordinates of a low-dimensional subspace of the input space. We review essential theory for ridge approximation – e.g., the best mean-squared approximation and an optimal low-dimensional subspace – and we prove that the gradient-based active subspace is near-stationary for the least-squares problem that defines an optimal subspace. Motivated by the theory, we propose a computational heuristic that uses an estimated active subspace as an initial guess for a ridge approximation fitting problem. We show a simple example where the heuristic fails, which reveals a type of function for which the proposed approach is inappropriate. We then propose a simple alternating heuristic for fitting a ridge function, and we demonstrate the effectiveness of the active subspace initial guess applied to an airfoil model of drag as a function of its 18 shape parameters.

Fast and automatic LV mass calculation from echocardiographic images via B-spline snake model and markov random fields

Left ventricular (LV) mass has several important diagnostic and indicative implications. In this paper, a fast and accurate technique for detection of inner and outer boundaries of LV and, consequently, calculation of LV mass from apical 4-chamber echocardiographic images is presented. For detection of the inner boundary, a modified B-spline snake is proposed, which relies merely on image intensity and obviates the need for computationally-demanding image forces. The outer boundary is then obtained using a Markov random fields model in the neighborhood of the estimated inner border. Experimental validation of the proposed technique demonstrates remarkable improvement over conventional algorithms.

k/K-Nearest Neighborhood Criterion for Improving Locally Linear Embedding

Spectral manifold learning techniques have recently found extensive applications in machine vision. The common strategy of spectral algorithms for manifold learning is exploiting the local relationships in a symmetric adjacency graph, which is typically constructed using k-nearest neighborhood (k-NN) criterion. In this paper, with our focus on locally linear embedding as a powerful and well-known spectral technique, shortcomings of k-NN for construction of the adjacency graph are first illustrated, and then a new criterion, namely k/K-nearest neighborhood (k/K-NN) is introduced to overcome these drawbacks. The proposed criterion involves finding the sparsest representation of each sample in the dataset, and is realized by modifying Robust-SL0, a recently proposed algorithm for sparse approximate representation. k/K-NN criterion gives rise to a modified spectral manifold learning technique, namely Sparse-LLE, which demonstrates remarkable improvement over conventional LLE through our experiments.