Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Fuselier, Edward J.; Wright, Grady B.

doi:10.1137/080730901

Cited by 39 publications

(52 citation statements)

References 37 publications

(74 reference statements)

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“…For example, the goal in Ballester et al (2001) is to recover missing data in images while interpolating the appropriate vector field. Another example is in the utilization of tangential vector fields interpolation on S 2 for modeling atmospheric air flow and oceanic water velocity (Fuselier and Wright 2009).…”

Section: Introductionmentioning

confidence: 99%

Learning from patches by efficient spectral decomposition of a structured kernel

et al. 2015

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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.

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Section: Introductionmentioning

confidence: 99%

Learning from patches by efficient spectral decomposition of a structured kernel

et al. 2015

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“…In other words, we have the dissipative-free vector field u } ðxÞ with concentric structure. See also another Helmholtz-Hodge decomposition of vector fields (Fuselier and Wright 2009). …”

Section: Introduction: Local and Global Ordermentioning

confidence: 99%

Topological Invariance of Biological Development

2013

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A topological inevitability of early developmental events through the use of classical topological concepts is discussed. Topological dynamics of forms and maps in embryo development are presented. Forms of a developing organism such as cell sets and closed surfaces are topological objects. Maps (or mathematical functions) are additional topological constructions in these objects and include polarization, singularities and curvature. Topological visualization allows us to analyze relationships that link local morphogenetic processes and integral developmental structures and also to find stable spatio-temporal topological characteristics that are invariant for a taxonomic group. The application of topological principles reveals a topological imperative: certain topological rules define and direct embryogenesis. A topological stability of embryonic morphogenesis is proposed and a topological scenario of developmental and evolutionary transformations is presented.

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Section: Introductionmentioning

confidence: 99%

“…For example, the goal in [2] is to recover missing data in images utilizing interpolation of the appropriate vector field. Another example is the utilization of tangential vector fields interpolation on S 2 for modeling atmospheric air flow and oceanic water velocity [8].The discussed enrichments increase considerably the kernel size. Kernel size is a limiting factor in the applicability of spectral decomposition based data analysis methods to real problems and considerable efforts have been invested for example in [1], [7] and others in approximating the spectral decomposition operator.…”

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confidence: 99%

Dictionary Construction for Patch-to-Tensor Embedding

Salhov

Wolf

Bermanis

et al. 2012

Advances in Intelligent Data Analysis XI

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The incorporation of a matrix relations, which encompass multidimensional similarities between local neighborhoods of data points in the underlying manifold of a data, improves the utilization of kernel based data analysis methodologies. However, the utilization of multidimensional similarities results in a larger kernel and hence the computational complexity of the corresponding spectral decomposition increases dramatically. In this paper, we propose an efficient approximation to a spectral decomposition of a multidimensional similarity based kernel. Furthermore, we propose a dictionary construction that approximates the oversized kernel in this case and its associated embedding. The performance of the proposed dictionary construction is demonstrated on an example of a super-kernel that utilizes the Diffusion Maps methodology together with linear-projection operators between tangent spaces in the manifold. I. INTRODUCTIONRecent methods for advanced massive high dimensional data analysis utilize a manifold structure on which data points are assumed to lie. This manifold is immersed (or submersed) in an ambient space that is defined by observable parameters. Kernel methods such as k-PCA and Diffusion Maps (DM) [4] have provided good results in analyzing such massive high dimensional data. The defined kernel can be thought of as an adjacency matrix of a graph whose vertices are the data points in the dataset. The analysis of the eigenvalues and the corresponding eigenvectors of this matrix reveals many properties and connections in the graph. These methods are based on the spectral decomposition of a kernel that was designed to incorporate a scalar similarity measure between data points. The resulting embedding of the data points into an Euclidean space preserves the qualities represented by the designed kernel. This approach extends the core of the classical Multi-Dimensional Scaling (MDS) method [6], [9] by considering non-linear relations instead of just a linear one in its original Gram matrix.Recently, DM was extended in several different ways to handle the orientation in local tangent spaces [10]- [13]. The relation between two patches is described by a matrix instead of a scalar value. The resulting kernel captures enriched similarities between local structures in the underlying manifold. These enriched similarities can be used to analyze local areas around data points instead of analyzing their specific locations. For example, this analysis can be beneficial in image processing (analyzing regions instead of individual pixels) and when the data points are perturbed so that their surrounding area is more important than their specific position. Since the constructions of these similarities are based on local tangent spaces, they provide methods to manipulate tangential vector fields (e.g., perform out-ofsample extensions). These manipulations are beneficial when the analyzed data consists of directional information in addition to positional information on the manifold. For example, the goal in [2] is ...

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Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Cited by 39 publications

References 37 publications

Learning from patches by efficient spectral decomposition of a structured kernel

Learning from patches by efficient spectral decomposition of a structured kernel

Topological Invariance of Biological Development

Dictionary Construction for Patch-to-Tensor Embedding

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