Amit Bermanis

3-D Symmetry Detection and Analysis Using the Pseudo-polar Fourier Transform

Symmetry detection and analysis in 3D images is a fundamental task in a gamut of scientific fields such as computer vision, medical imaging and pattern recognition to name a few. In this work, we present a computational approach to 3D symmetry detection and analysis. Our analysis is conducted in the Fourier domain using the pseudo-polar Fourier transform. The pseudo-polar representation enables to efficiently and accurately analyze angular volumetric properties such as rotational symmetries. Our algorithm is based on the analysis of the angular correspondence rate of the given volume and its rotated and rotated-inverted replicas in their pseudo-polar representations. We also derive a novel rigorous analysis of the inherent constraints of 3D symmetries via groups-theory based analysis. Thus, our algorithm starts by detecting the rotational symmetry group of a given volume, and the rigorous analysis results pave the way to detect the rest of the symmetries. The complexity of the algorithm is O(N3log (N)), where N×N×N is the volumetric size in each direction. This complexity is independent of the number of the detected symmetries. We experimentally verified our approach by applying it to synthetic as well as real 3D objects.

Accelerating Particle Filter Using Randomized Multiscale and Fast Multipole Type Methods

Particle filter is a powerful tool for state tracking using non-linear observations. We present a multiscale based method that accelerates the tracking computation by particle filters. Unlike the conventional way, which calculates weights over all particles in each cycle of the algorithm, we sample a small subset from the source particles using matrix decomposition methods. Then, we apply a function extension algorithm that uses a particle subset to recover the density function for all the rest of the particles not included in the chosen subset. The computational effort is substantial especially when multiple objects are tracked concurrently. The proposed algorithm significantly reduces the computational load. By using the Fast Gaussian Transform, the complexity of the particle selection step is reduced to a linear time in n and k, where n is the number of particles and k is the number of particles in the selected subset. We demonstrate our method on both simulated and on real data such as object tracking in video sequences.

Accelerating Particle filter using multiscale methods

We present a method that accelerates the Particle Filter computation. Particle Filter is a powerful method for tracking the state of a target based on non-linear observations. Unlike the conventional way of calculating weights over all particles in each run, we sample a small subset of the particles using matrix decomposition methods, followed by a novel function extension algorithm to recover the density function of all particles. This significantly reduces the computational load where the measurement computation is substantial, as often happens, for example, when tracking targets in videos. We demonstrate our method on both simulated data and real data (videos).

Learning from patches by efficient spectral decomposition of a structured kernel

We present a kernel based method that learns from a small neighborhoods (patches) of multidimensional data points. This method is based on spectral decomposition of a large structured kernel accompanied by an out-of-sample extension method. In many cases, the performance of a spectral based learning mechanism is limited due to the use of a distance metric among the multidimensional data points in the kernel construction. Recently, different distance metrics have been proposed that are based on a spectral decomposition of an appropriate kernel prior to the application of learning mechanisms. The diffusion distance metric is a typical example where a distance metric is computed by incorporating the relation of a single measurement to the entire input dataset. A method, which is called patch-to-tensor embedding (PTE), generalizes the diffusion distance metric that incorporates matrix similarity relations into the kernel construction that replaces its scalar entries with matrices. The use of multidimensional similarities in PTE based spectral decomposition results in a bigger kernel that significantly increases its computational complexity. In this paper, we propose an efficient dictionary construction that approximates the oversized PTE kernel and its associated spectral decomposition. It is supplemented by providing an out-of-sample extension for vector fields. Furthermore, the approximation error is analyzed and the advantages of the proposed dictionary construction are demonstrated on several image processing tasks.

Dictionary construction for patch-to-tensor embedding

The incorporation of matrix relation, which can encompass multidimensional similarities between local neighborhoods of points in the manifold, can improve kernel based data analysis. However, the utilization of multidimensional similarities results in a larger kernel and hence the computational cost of the corresponding spectral decomposition increases dramatically. In this paper, we propose dictionary construction to approximate the kernel in this case and its respected embedding. The proposed dictionary construction is demonstrated on a relevant example of a super kernel that is based on the utilization of the diffusion maps kernel together with linear-projection operators between tangent spaces of the manifold.