Geometry of sets and measures in euclidean spaces fractals and rectifiability

Naive Set of Theory.

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

Manifold-valued subdivision schemes based on geodesic inductive averaging

A method of determining a likelihood of an occurrence of a cardiac arrhythmia in a patient includes receiving three-dimensional imaging data of said patient's heart, constructing a whole-heart model for simulating at least one of electrophysiological activity or electromechanical activity of the patient's heart using the three-dimensional imaging data, simulating a response of the patient's heart to each of a plurality of stimulations to a corresponding plurality of different locations within the patient's heart using the whole-heart model, classifying each simulation outcome for each stimulation as one of a normal heart rhythm or a cardiac arrhythmia, calculating a likelihood index based on results of the classifying, and determining the likelihood of the occurrence of the cardiac arrhythmia in the patient based on the likelihood index. Software and data processing systems that implement the above methods are also provided.

System and method for personalized cardiac arrhythmia risk assessment by simulating arrhythmia inducibility

Short Communication to SMI 2011: Reconstruction of 3D objects from 2D cross-sections with the 4-point subdivision scheme adapted to sets

In this work we construct subdivision schemes refining general subsets of R^n and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). 
Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane-Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally we discuss the extension of the results obtained in the metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties.

Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric

We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive samples-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric. 
In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and obtain a set-valued analog of the Weierstrass approximation theorem. The rate of approximation of H\"older continuous SVFs by Bernstein operators is studied and shown to be asymptotically equal to that for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.

Bernstein-type approximation of set-valued functions in the symmetric difference metric

We study the approximation of univariate and multivariate set-valued
functions (SVFs) by the adaptation to SVFs of positive sample-based
approximation operators for real-valued functions. To this end, we
introduce a new weighted average of several sets and study its
properties. The approximation results are obtained in the space of
Lebesgue measurable sets with the symmetric difference metric.
 
In particular, we apply the new average of sets to adapt to SVFs the
classical Bernstein approximation operators, and show that these
operators approximate continuous SVFs. The rate of approximation of
Holder continuous SVFs by the adapted Bernstein operators is
studied and shown to be asymptotically equal to the one for
real-valued functions. Finally, the results obtained in the metric
space of sets are generalized to metric spaces endowed with an
average satisfying certain properties.

pdf/bernstein-type-approximation-of-set-valued-functions-in-the-5515jw7nsg.pdf

The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise linear boundaries.

https://www.mdpi.com/1999-4893/3/3/265/pdf

Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries

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Papers

Short Communication to SMI 2011: Reconstruction of 3D objects from 2D cross-sections with the 4-point subdivision scheme adapted to sets

Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric

Bernstein-type approximation of set-valued functions in the symmetric difference metric

Bernstein-type approximation of set-valued functions in the symmetric difference metric

Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries