Evaluating sums of multivariate Gaussians is a common computational task in computer vision and pattern recognition, including in the general and powerful kernel density estimation technique. The quadratic computational complexity of the summation is a significant barrier to the scalability of this algorithm to practical applications. The fast Gauss transform (FGT) has successfully accelerated the kernel density estimation to linear running time for lowdimensional problems. Unfortunately, the cost of a direct extension of the FGT to higher-dimensional problems grows exponentially with dimension, making it impractical for dimensions above 3. We develop an improved fast Gauss transform to efficiently estimate sums of Gaussians in higher dimensions, where a new multivariate expansion scheme and an adaptive space subdivision technique dramatically improve the performance. The improved FGT has been applied to the mean shift algorithm achieving linear computational complexity. Experimental results demonstrate the efficiency and effectiveness of our algorithm. 1

Abstract—In this paper, we present a new information-theoretic approach to image segmentation. We cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities, subject to a constraint on the total length of the region boundaries. We assume that the probability densities associated with the image pixel intensities within each region are completely unknown a priori, and we formulate the problem based on nonparametric density estimates. Due to the nonparametric structure, our method does not require the image regions to have a particular type of probability distribution and does not require the extraction and use of a particular statistic. We solve the information-theoretic optimization problem by deriving the associated gradient flows and applying curve evolution techniques. We use level-set methods to implement the resulting evolution. The experimental results based on both synthetic and real images demonstrate that the proposed technique can solve a variety of challenging image segmentation problems. Futhermore, our method, which does not require any training, performs as good as methods based on training. Index Terms—Curve evolution, image segmentation, information theory, level-set methods, nonparametric density estimation.

The study of many vision problems is reduced to the estimation of a probability density function from observations. Kernel density estimation techniques are quite general and powerful methods for this problem, but have a significant disadvantage in that they are computationally intensive. In this paper we explore the use of kernel density estimation with the fast gauss transform (FGT) for problems in vision. The FGT allows the summation of a mixture of M Gaussians at N evaluation points in O(M + N) timeasopposedtoO(MN)time for a naive evaluation, and can be used to considerably speed up kernel density estimation. We present applications of the technique to problems from image segmentation and tracking, and show that the algorithm allows application of advanced statistical techniques to solve practical vision problems in real time with today’s computers. 1

Modeling the color distribution of a homogeneous region is used extensively for object tracking and recognition applications. The color distribution of an object represents a feature that is robust to partial occlusion, scaling and object deformation. A variety of parametric and non-parametric statistical techniques have been used to model color distributions. In this paper we present a non-parametric color modeling approach based on kernel density estimation as well as a computational framework for efficient density estimation. Theoretically, our approach is general since kernel density estimators can converge to any density shape with sufficient samples. Therefore, this approach is suitable to model the color distribution of regions with patterns and mixture of colors. Since kernel density estimation techniques are computationally expensive, the paper introduces the use of the Fast Gauss Transform for efficient computation of the color densities. We show that this approach can be used successfully for color-based segmentation of body parts as well as segmentation of multiple people under occlusion. 1

In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as summation of Gaussians. Though this operation usually requires O(MN) work when there are M summations to compute and the number of terms appearing in each summation is N, we can reduce the amount of work to O(M + N) by using a technique called fast Gauss transform. In this paper, we apply this technique to the multinomial method and the stochastic mesh method, and show by numerical experiments how it can speed up these methods considerably, both for the Black-Scholes model and Merton’s lognormal jump-diffusion model. We also propose some extensions to apply the fast Gauss transform to Kou’s double-exponential jump-diffusion model and Heston’s stochastic volatility model. 1

The fast Gauss transform proposed by Greengard and Strain reduces the computational complexity of the evaluation of the sum of N Gaussians at M points in d dimensions from O(MN) to O(M + N). However, the constant factor in O(M + N) grows exponentially with increasing dimensionality d, which makes the algorithm impractical in higher dimensions. In this paper we present an improved fast Gauss transform where the constant factor is reduced to asymptotically polynomial order. The reduction is based on a new multivariate Taylor expansion scheme combined with the space subdivision using the k-center algorithm. The complexity analysis and error bound are presented which helps to determine parameters automatically given a desired precision to which the sum must be evaluated. We present numerical results on the performance of our algorithm and provide numerical verification of the corresponding error estimate.

We construct a fast algorithm for the computation of discrete Gauss transforms with complex parameters, capable of dealing with non equispaced points. Our algorithm is based on the fast Fourier transform at non equispaced knots and requires only O(N) arithmetic operations. Key words and phrases. Gauss transform; Unequally spaced Fourier transforms; Fast algorithms; Chirped Gaussian; NFFT

Abstract 2 This thesis presents two related bodies of work. The first is about methods for speeding up inference in graphical models, and the second is an application of the graphical model framework to the beat tracking problem in sampled music. Graphical models have become ubiquitous modelling tools; they are commonly used in computer vision, bioinformatics, coding theory, and speech recognition, and are central to many machine learning techniques. Graphical models allow statistical independence relationships between random variables to be expressed in a flexible, powerful, and intuitive manner. Given observations, there are standard algorithms to compute probability distributions over unknown states (marginals) or to find the most likely configuration (maximum a posteriori, MAP, state). However, if each node in the graphical model has N states, then these computations cost O � N 2 �. This is a particular concern when dealing with continuous (or large discrete) state spaces. In such state spaces, Monte Carlo methods are of great use; these methods typically

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Gaussian processes (GPs) provide a flexible framework for probabilistic regression. The necessary computations involve standard matrix operations. There have been several attempts to accelerate these operations based on fast kernel matrix-vector multiplications. By focussing on the simplest GP computation, corresponding to test-time predictions in kernel ridge regression, we conclude that simple approximations based on clusterings in a kd-tree can never work well for simple regression problems. Analytical expansions can provide speedups, but current implementations are limited to the squared-exponential kernel and low-dimensional problems. We discuss future directions. 1.