Multiresolution topological simplification

Persistent homology has been devised as a promising tool for the topological simplification of complex data. However, it is computationally intractable for large data sets. In this work, we introduce multiresolution persistent homology for tackling large data sets. Our basic idea is to match the resolution with the scale of interest so as to create a topological microscopy for the underlying data. We utilize flexibility-rigidity index (FRI) to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution, we are able to focus the topological lens on a desirable scale. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA and RNA molecules. In particular, the topological persistence of a virus capsid with 240 protein monomers is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks and graphs.Comment: 22 pages and 14 figure

Multiresolution Tree Networks for 3D Point Cloud Processing

We present multiresolution tree-structured networks to process point clouds for 3D shape understanding and generation tasks. Our network represents a 3D shape as a set of locality-preserving 1D ordered list of points at multiple resolutions. This allows efficient feed-forward processing through 1D convolutions, coarse-to-fine analysis through a multi-grid architecture, and it leads to faster convergence and small memory footprint during training. The proposed tree-structured encoders can be used to classify shapes and outperform existing point-based architectures on shape classification benchmarks, while tree-structured decoders can be used for generating point clouds directly and they outperform existing approaches for image-to-shape inference tasks learned using the ShapeNet dataset. Our model also allows unsupervised learning of point-cloud based shapes by using a variational autoencoder, leading to higher-quality generated shapes.Comment: Accepted to ECCV 2018. 23 pages, including supplemental materia

Molecular multiresolution surfaces

The surface of a molecule determines much of its chemical and physical property, and is of great interest and importance. In this Letter, we introduce the concept of molecular multiresolution surfaces as a new paradigm of multiscale biological modeling. Molecular multiresolution surfaces contain not only a family of molecular surfaces, corresponding to different probe radii, but also the solvent accessible surface and van der Waals surface as limiting cases. All the proposed surfaces are generated by a novel approach, the diffusion map of continuum solvent over the van der Waals surface of a molecule. A new local spectral evolution kernel is introduced for the numerical integration of the diffusion equation in a single time step.Comment: 11 pages, 4 figure

A Coarse-to-Fine Multiscale Mesh Representation and its Applications

We present a novel coarse-to-fine framework that derives a semi-regular multiscale mesh representation of an original input mesh via remeshing. Our approach differs from the conventional mesh wavelet transform strategy in two ways. First, based on a lazy wavelet framework, it can convert an input mesh into a multiresolution representation through a single remeshing procedure. By contrast, the conventional strategy requires two steps: remeshing and mesh wavelet transform. Second, the proposed method can conditionally convert input mesh models into ones sharing the same adjacency matrix, so it is able be invariant against the triangular tilings of the inputs. Our experiment results show that the proposed multiresolution representation method is efficient in various applications, such as 3D shape property analysis, mesh scalable coding and mesh morphing

Shape optimisation with multiresolution subdivision surfaces and immersed finite elements

We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors. Based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness. During shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information. By virtue of the multiresolution editing semantics, updating the coarse control mesh vertex coordinates leads to large-scale geometry changes and, conversely, updating the fine control mesh coordinates leads to small-scale geometry changes. In our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum. This approach effectively prevents the appearance of non-physical boundary geometry oscillations and control mesh pathologies, like inverted elements. Independent of the fineness of the control mesh used for optimisation, on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution. With the immersed finite element method there is no need to maintain an analysis suitable domain mesh. In some of the presented two- and three-dimensional elasticity examples the topology derivative is used for creating new holes inside the domain

Multiresolution analysis on compact Riemannian manifolds

In the chapter "Multiresolution Analysis on Compact Riemannian Manifolds" Isaac Pesenson describes multiscale analysis, sampling, interpolation and approximation of functions defined on manifolds. His main achievements are: construction on manifolds of bandlimited and space-localized frames which have Parseval property and construction of variational splines on manifolds. Such frames and splines enable multiscale analysis on arbitrary compact manifolds, and they already found a number of important applications (statistics, CMB, crystallography) related to such manifolds as two-dimensional sphere and group of its rotations.Comment: published in "Multiscale Analysis and Nonlinear Dynamics: From Genes to the Brain", First Edition. Edited by Misha Meyer Pesenson. 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGa

Quantum points/patterns, Part 1. From geometrical points to quantum points in a sheaf framework

We consider some generalization of the theory of quantum states, which is based on the analysis of long standing problems and unsatisfactory situation with the possible interpretations of quantum mechanics. We demonstrate that the consideration of quantum states as sheaves can provide, in principle, more deep understanding of some phenomena. The key ingredients of the proposed construction are the families of sections of sheaves with values in the category of the functional realizations of infinite-dimensional Hilbert spaces with special (multiscale) filtration. Three different symmetries, kinematical (on space-time), hidden/dynamical (on sections of sheaves), unified (on filtration of the full scale of spaces) are generic objects generating the full zoo of quantum phenomena (e.g., faster than light propagation).Comment: 10 pages, LaTeX, spie.cls, Submitted to Proc. of SPIE Meeting, The Nature of Light: What are Photons? IV, San Diego, CA, August, 201

Multiresolution Representations for Piecewise-Smooth Signals on Graphs

What is a mathematically rigorous way to describe the taxi-pickup distribution in Manhattan, or the profile information in online social networks? A deep understanding of representing those data not only provides insights to the data properties, but also benefits to many subsequent processing procedures, such as denoising, sampling, recovery and localization. In this paper, we model those complex and irregular data as piecewise-smooth graph signals and propose a graph dictionary to effectively represent those graph signals. We first propose the graph multiresolution analysis, which provides a principle to design good representations. We then propose a coarse-to-fine approach, which iteratively partitions a graph into two subgraphs until we reach individual nodes. This approach efficiently implements the graph multiresolution analysis and the induced graph dictionary promotes sparse representations piecewise-smooth graph signals. Finally, we validate the proposed graph dictionary on two tasks: approximation and localization. The empirical results show that the proposed graph dictionary outperforms eight other representation methods on six datasets, including traffic networks, social networks and point cloud meshes

A quantitative structure comparison with persistent similarity

Biomolecular structure comparison not only reveals evolutionary relationships, but also sheds light on biological functional properties. However, traditional definitions of structure or sequence similarity always involve superposition or alignment and are computationally inefficient. In this paper, I propose a new method called persistent similarity, which is based on a newly-invented method in algebraic topology, known as persistent homology. Different from all previous topological methods, persistent homology is able to embed a geometric measurement into topological invariants, thus provides a bridge between geometry and topology. Further, with the proposed persistent Betti function (PBF), topological information derived from the persistent homology analysis can be uniquely represented by a series of continuous one-dimensional (1D) functions. In this way, any complicated biomolecular structure can be reduced to several simple 1D PBFs for comparison. Persistent similarity is then defined as the quotient of sizes of intersect areas and union areas between two correspondingly PBFs. If structures have no significant topological properties, a pseudo-barcode is introduced to insure a better comparison. Moreover, a multiscale biomolecular representation is introduced through the multiscale rigidity function. It naturally induces a multiscale persistent similarity. The multiscale persistent similarity enables an objective-oriented comparison. State differently, it facilitates the comparison of structures in any particular scale of interest. Finally, the proposed method is validated by four different cases. It is found that the persistent similarity can be used to describe the intrinsic similarities and differences between the structures very well.Comment: 20 PAGES, 13 PICTURE

High order implicit time integration schemes on multiresolution adaptive grids for stiff PDEs

We consider high order, implicit Runge-Kutta schemes to solve time-dependent stiff PDEs on dynamically adapted grids generated by multiresolution analysis for unsteady problems disclosing localized fronts. The multiresolution finite volume scheme yields highly compressed representations within a user-defined accuracy tolerance, hence strong reductions of computational requirements to solve large, coupled nonlinear systems of equations. SDIRK and RadauIIA Runge-Kutta schemes are implemented with particular interest in those with L-stability properties and accuracy-based time-stepping capabilities. Numerical evidence is provided of the computational efficiency of the numerical strategy to cope with highly unsteady problems modeling various physical scenarios with a broad spectrum of time and space scales