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Elastic Principal Graphs and Manifolds and their Practical Applications
 COMPUTING
, 2005
"... Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through “the middle” of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based o ..."
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Cited by 17 (8 self)
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Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through “the middle” of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of standalone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.
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 Industrial Model Predictive Control Technology", Chem. Process Control
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Elastic maps and nets for approximating principal manifolds and their application to microarray data visualization
 In this book
"... Summary. Principal manifolds are defined as lines or surfaces passing through “the middle ” of data distribution. Linear principal manifolds (Principal Components Analysis) are routinely used for dimension reduction, noise filtering and data visualization. Recently, methods for constructing nonline ..."
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Cited by 5 (4 self)
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Summary. Principal manifolds are defined as lines or surfaces passing through “the middle ” of data distribution. Linear principal manifolds (Principal Components Analysis) are routinely used for dimension reduction, noise filtering and data visualization. Recently, methods for constructing nonlinear principal manifolds were proposed, including our elastic maps approach which is based on a physical analogy with elastic membranes. We have developed a general geometric framework for constructing “principal objects ” of various dimensions and topologies with the simplest quadratic form of the smoothness penalty which allows very effective parallel implementations. Our approach is implemented in three programming languages (C++, Java and Delphi) with two graphical user interfaces (VidaExpert and ViMiDa applications). In this paper we overview the method of elastic maps and present in detail one of its major applications: the visualization of microarray data in bioinformatics. We show that the method of elastic maps outperforms linear PCA in terms of data approximation, representation of betweenpoint distance structure, preservation of local point neighborhood and representing point classes in lowdimensional spaces. Key words: elastic maps, principal manifolds, elastic functional, data analysis, data visualization, surface modeling 1
Piecewise C¹ Continuous Surface Reconstruction of Noisy Point Clouds via Local Implicit Quadric Regression
 IEEE VISUALIZATION
, 2003
"... This paper addresses the problem of surface reconstruction of highly noisy point clouds. The surfaces to be reconstructed are assumed to be 2manifolds of piecewise C¹ continuity, with isolated small irregular regions of high curvature, sophisticated local topology or abrupt burst of noise. At each ..."
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This paper addresses the problem of surface reconstruction of highly noisy point clouds. The surfaces to be reconstructed are assumed to be 2manifolds of piecewise C¹ continuity, with isolated small irregular regions of high curvature, sophisticated local topology or abrupt burst of noise. At each sample point, a quadric field is locally fitted via a modified moving least squares method. These locally fitted quadric fields are then blended together to produce a pseudosigned distance field using Shepard's method. We introduce a prioritized front growing scheme in the process of local quadrics fitting. Flatter surface areas tend to grow faster. The already fitted regions will subsequently guide the fitting of those irregular regions in their neighborhood. Several other techniques are also introduced for automatic local fitting model selection and normal alignment.