Meir Barzohar

Rank Estimation and Redundancy Reduction of High-Dimensional Noisy Signals With Preservation of Rare Vectors

In this paper, we address the problem of redundancy-reduction of high-dimensional noisy signals that may contain anomaly (rare) vectors, which we wish to preserve. For example, when applying redundancy reduction techniques to hyperspectral images, it is essential to preserve anomaly pixels for target detection purposes. Since rare-vectors contribute weakly to the -norm of the signal as compared to the noise, -based criteria are unsatisfactory for obtaining a good representation of these vectors. The proposed approach combines and norms for both signal-subspace and rank determination and considers two aspects: One aspect deals with signal-subspace estimation aiming to minimize the maximum of data-residual -norms, denoted as , for a given rank conjecture. The other determines whether the rank conjecture is valid for the obtained signal-subspace by applying Extreme Value Theory results to model the distribution of the noise -norm. These two operations are performed alternately using a suboptimal greedy algorithm, which makes the proposed approach practically plausible. The algorithm was applied on both synthetically simulated data and on a real hyperspectral image producing better results than common -based methods.

Stable fitting of 2D curves and 3D surfaces by implicit polynomials

This work deals with fitting 2D and 3D implicit polynomials (IPs) to 2D curves and 3D surfaces, respectively. The zero-set of the polynomial is determined by the IP coefficients and describes the data. The polynomial fitting algorithms proposed in this paper aim at reducing the sensitivity of the polynomial to coefficient errors. Errors in coefficient values may be the result of numerical calculations, when solving the fitting problem or due to coefficient quantization. It is demonstrated that the effect of reducing this sensitivity also improves the fitting tightness and stability of the proposed two algorithms in fitting noisy data, as compared to existing algorithms like the well-known 3L and gradient-one algorithms. The development of the proposed algorithms is based on an analysis of the sensitivity of the zero-set to small coefficient changes and on minimizing a bound on the maximal error for one algorithm and minimizing the error variance for the second. Simulation results show that the proposed algorithms provide a significant reduction in fitting errors, particularly when fitting noisy data of complex shapes with high order polynomials, as compared to the performance obtained by the above mentioned existing algorithms.

Local-global background modeling for anomaly detection in hyperspectral images

In this paper, we address the problem of unsupervised detection of anomalies in hyperspectral images. Our proposed method is based on a novel statistical background modeling approach that combines local and global approaches. The local-global background model has the ability to adapt to all nuances of the background process like local approaches but avoids over-fitting due to a too high number of degrees of freedom, which produces a high false alarm rate. This is done by constraining the local background models to be interrelated. The results strongly prove the effectiveness of the proposed algorithm. We experimentally show that our localglobal algorithm performs better than several other global or local anomaly detection techniques, such as the well known RX or its Gaussian Mixture version (GMRX).

Recognition of 3D Objects Based on Implicit Polynomials

Closed-form expressions for a new set of 3D rotation invariants that are linear, quadratic, and angular combinations of implicit polynomial (IP) coefficients are developed. Based on these invariants, we propose a 3D object recognition method that outperforms recognition based on IP fitting after pose estimation, and the MPEG-7 SSD technique.

Using implicit polynomials for image compression

Implicit polynomials (IF) are being used to represent 2D curves and 3D surfaces. The zero-set of a 2D implicit polynomial of the form of p(x,y)=a/sub 1/x/sup N/+a/sub 2/x/sup N-1/y+...+ a/sub r/=0 can be used to describe data points making up a 2D curve. A similar 3D polynomial-p(x,y,z)=0 describes data points on a 3D surface. We describe a way to restrict the zero-set of the fitted polynomial so that correct restoration of the data from the polynomials coefficients will be achieved.

Non-local means denoising using a content-based search region and dissimilarity kernel

The Non-Local Means (NLM) denoising algorithm uses a weighted average of pixels, within a defined search region in an image, to estimate the noise-free pixel value. The search region is usually a rectangular neighborhood, centered at the pixel of interest, which may include pixels whose original gray value do not match the value of the original central pixel. Consequently, their participation in the averaging process degrades denoising performance. To eliminate their effect, researchers suggest creating an adaptive search region which excludes those dissimilar pixels. In this paper, we present a novel model-based method which defines a set of similar pixels, from the initial search region, using the statistical distribution of the dissimilarity measure. Moreover, to enhance the denoising, our method also adaptively assigns one of two dissimilarity kernels to each pixel, based on its local features. Experimental results show that the proposed algorithm has better performance than the original one in terms of PSNR, SSIM, and visual quality and is found to be more efficient than other examined approaches.

Anomaly Preserving

In this paper, we address the problem of redundancy reduction of high-dimensional noisy signals that may contain anomaly (rare) vectors, which we wish to preserve. Since anomaly data vectors contribute weakly to the l2-norm of the signal as compared to the noise, l2 -based criteria are unsatisfactory for obtaining a good representation of these vectors. As a remedy, a new approach, named Min-Max-SVD (MX-SVD) was recently proposed for signal-subspace estimation by attempting to minimize the maximum of data-residual l2-norms, denoted as l2,l and designed to represent well both abundant and anomaly measurements. However, the MX-SVD algorithm is greedy and only approximately minimizes the proposed l2,l-norm of the residuals. In this paper we develop an optimal algorithm for the minization of the l2,l-norm of data misrepresentation residuals, which we call Maximum Orthogonal complements Optimal Subspace Estimation (MOOSE). The optimization is performed via a natural conjugate gradient learning approach carried out on the set of n dimensional subspaces in IRm, m ≫ n, which is a Grassmann manifold. The results of applying MOOSE, MX-SVD, and l2- based approaches are demonstrated both on simulated and real hyperspectral data.

\ell _{\scriptscriptstyle 2,\infty }

The use of 3D point clouds is currently of much interest. One of the cornerstones of 3D point cloud research and applications is point cloud registration. Given two point clouds, the goal of registration is aligning them in a common coordinate system. In particular, we seek in this work to align a sparse and noisy local point cloud, created from a single stereo pair of images, to a dense and large-scale global point cloud, representing an urban outdoors environment. The common approach of keypoint-based registration, tends to fail due to the sparsity and low quality of the stereo local cloud. We propose here a new approach. It consists of the creation of a dictionary of much smaller clouds using a grid of synthetic viewpoints over the dense global cloud. We then perform registration via an efficient dictionary search. Our approach shows promising results on data acquired in an urban environment.

-Optimal Dimensionality Reduction Over a Grassmann Manifold

Presents an approach to contour representation and coding. It consists of an improved fitting of high-degree (4/sup th/ to 18/sup th/) implicit polynomials (IPs) to the contour which is robust to coefficient quantization. The proposed approach to solve the fitting problem is a modification of the 3L linear solution developed by Lei et al. (1997) and is more robust to noise and to coefficient quantization. We use an analytic approach to limit the maximal fitting error between each data point and the zero-set generated by the quantized polynomial coefficients. We than show that consideration of the quantization error (which led to a specific sensitivity criterion) also brought about a significant improvement in fitting IPs to noisy data, as compared to the 3L algorithm.

Point cloud registration using a viewpoint dictionary

In this paper we refer to the correlation analysis between the Non-Local Means (NLM) dissimilarity elements within a given search region. This analysis is required for a more accurate determination of a model-based adaptive search region that we introduced earlier. We explore three levels of correlation according to the degree of patches overlap and explain how this analysis can be used in our model-based NLM approach.