On the Number of Modes of a Gaussian Mixture

Carreira-Perpiñán, Miguel Á.; Williams, Christopher K. I.

doi:10.1007/3-540-44935-3_44

Cited by 53 publications

(44 citation statements)

References 31 publications

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“…• The number of modes of the homoscedastic Gaussian mixture seldom increases as the scale τ increases (Carreira-Perpinan and Williams, 2003). That is, mode creation is less expected if the Gaussian functions are unequally weighted.…”

Section: Estimation Of Point Cloud Densitymentioning

confidence: 99%

Image Hierarchy in Gaussian Scale Space

Sakai¹,

Narita²,

Komazaki³

et al. 2011

Advances in Imaging and Electron Physics

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We investigate the topological structure of an image and the hierarchical relationship between local and global structures provided by spatial gradients at different levels of scale in the Gaussian scale space. The gradient field curves link stationary points of an image including a local minimum at infinity, and construct the topological structure of the image. The evolution of the topological structure with respect to scale is analysed using pseudograph representation. The hierarchical relationships among the structures at different scales are expressed as trajectories of the stationary points in the scale space, which we call stationary curves. Each top-point of the local extremum curve generically has a specific gradient field curve, which we call the antidirectional figure-flow curve. The antidirectional figure-flow curve connects between the top-point and another local extremum to which the toppoint is subordinate. A point at infinity can also be connected to the toppoints of local minimum curves. These hierarchical relationships among the stationary points are expressed as a tree. This tree expresses a hierarchical structure of dominant parts. We clarify the graphical grammar for the construction of this tree in the Gaussian scale space. Furthermore, we show a combinatorial structure of singular points in the Gaussian scale space using conformal mapping from Euclidean space to the spherical surface. We define segment edges as a zero-crossing set in the Gaussian scale space using the singular points. An image in the Gaussian scale space is the convolution of the image and the Gaussian kernel. The Gaussian kernel of an appropriate variance is a typical presmoothing operator for segmentation. The variance is heuristically selected using statistics of images such as the noise distribution in images. The variance of the kernel is determined using the singular-point configuration in the Gaussian scale space, since singular points in the Gaussian scale space allow the extraction of the dominant parts of an image. This scale selection strategy derives the hierarchical structure of the segments. Unsupervised segmentation methods, however, have difficulty in distinguishing valid segments associated with the objects from invalid random segments due to noise. Showing that the number of invalid segments monotonically decreases with increasing scale, we characterise thevalid and invalid segments in the Gaussian scale space. This property allows us to identify the valid segments from coarse to fine, and to prevent under-and oversegmentation. Finally, we develop principal component analysis (PCA) of a point cloud on the basis of scale-space representation of its probability density function (PDF). We explain the geometric features of a point cloud in the Gaussian scale space, and observe reduction of dimensionality with respect to the loss of information. Furthermore, we introduce a hierarchical clustering of the point cloud, and analyse the statistical significance of the clusters and their subspaces. Moreover, ...

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Section: Estimation Of Point Cloud Densitymentioning

confidence: 99%

Image Hierarchy in Gaussian Scale Space

Sakai¹,

Narita²,

Komazaki³

et al. 2011

Advances in Imaging and Electron Physics

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“…In MS, no default parameter is needed to be set at first. It has been reasoned that a Gaussian mixture model can be optimized by fixed-point bound optimization method [35]. This method is also applicable to the surface created by other kernel functions [24].…”

Section: The Improved Msmentioning

confidence: 99%

Multibandwidth Kernel-Based Object Tracking

Dargazany

Soleimani

Ahmadyfard

2010

Advances in Artificial Intelligence

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Object tracking using Mean Shift (MS) has been attracting considerable attention recently. In this paper, we try to deal with one of its shortcoming. Mean shift is designed to find local maxima for tracking objects. Therefore, in large target movement between two consecutive frames, the local and global modes are not the same as previous frames so that Mean Shift tracker may fail in tracking the desired object via localizing the global mode. To overcome this problem, a multibandwidth procedure is proposed to help conventional MS tracker reach the global mode of the density function using any staring points. This gradually smoothening procedure is called Multi Bandwidth Mean Shift (MBMS) which in fact smoothens the Kernel Function through a multiple kernel-based sampling procedure automatically. Since it is important for us to have less computational complexity for real-time applications, we try to decrease the number of iterations to reach the global mode. Based on our results, this proposed version of MS enables us to track an object with the same initial point much faster than conventional MS tracker.

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“…In general, it is a difficult problem to locate the modes of a Gaussian mixture because of the potentially complex ways in which the components can interact [7]. However, for test optimization problems, it is desirable to know the values and locations of the optima of the objective function.…”

Section: A the Basic Modelmentioning

confidence: 99%

A general-purpose tunable landscape generator

Gallagher

Yuan

2006

IEEE Trans. Evol. Computat.

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Abstract-The research literature on metaheuristic and evolutionary computation has proposed a large number of algorithms for the solution of challenging real-world optimization problems. It is often not possible to study theoretically the performance of these algorithms unless significant assumptions are made on either the algorithm itself or the problems to which it is applied, or both. As a consequence, metaheuristics are typically evaluated empirically using a set of test problems. Unfortunately, relatively little attention has been given to the development of methodologies and tools for the large-scale empirical evaluation and/or comparison of metaheuristics.In this paper, we propose a landscape (test-problem) generator that can be used to generate optimization problem instances for continuous, bound-constrained optimization problems. The landscape generator is parameterized by a small number of parameters, and the values of these parameters have a direct and intuitive interpretation in terms of the geometric features of the landscapes that they produce. An experimental space is defined over algorithms and problems, via a tuple of parameters for any specified algorithm and problem class (here determined by the landscape generator). An experiment is then clearly specified as a point in this space, in a way that is analogous to other areas of experimental algorithmics, and more generally in experimental design. Experimental results are presented, demonstrating the use of the landscape generator. In particular, we analyze some simple, continuous estimation of distribution algorithms, and gain new insights into the behavior of these algorithms using the landscape generator.Index Terms-Continuous optimization, empirical algorithm analysis, estimation of distribution algorithm, test-problem generator.

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On the Number of Modes of a Gaussian Mixture

Cited by 53 publications

References 31 publications

Image Hierarchy in Gaussian Scale Space

Image Hierarchy in Gaussian Scale Space

Multibandwidth Kernel-Based Object Tracking

A general-purpose tunable landscape generator

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