Project Point cloud analysis based on papers in
MFCS 2020 
AIAA 2019 
PRL 2016 
CVPR 2014 
CTIC 2014
The mergegram of a dendrogram

@inproceedings{elkin2020mergegram, author = {Elkin, Y. and Kurlin, V.}, title = {The mergegram of a dendrogram and its stability}, booktitle = {Proceedings of MFCS 2020 (Mathematical Foundations of Computer Science)}, year = {2020} }
 DOI : https://doi.org/10.4230/LIPIcs.MFCS.2020.32
 Input : a dendrogram D of any hierarchical clustering on a point cloud.
 Output : the mergegram of the dendrogram D consisting of stable (birth,death) pairs of all intermediate clusters.
 Abstract. This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0dimensional case the distancebased persistence is determined by a singlelinkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edgelengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stableundernoise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.
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Vortex structures around tandem flapping wing objects

@inproceedings{ban2019development, title={Development of a Reconstruction Method for Major Vortex Structure around Tandem Flapping Wing Object via Vortex Trajectory Method}, author={Ban, Naohiko and Yamazaki, Wataru and Kurlin, Vitaliy}, booktitle={AIAA Scitech 2019 Forum}, pages={22242260}, year={2019} }
 Abstract. Flapping wing micro air vehicle (MAV) is expected to apply in unmanned operations under risky / ultimate conditions. The MAV has to fly in low Reynolds number conditions. Previous researches indicated that flights by flapping motions as insects had higher performance at low Reynolds numbers. In nature, some insects have four flapping wings and achieve excellent flight performance such as hovering and steep turn. This is considered to be the effect of aerodynamic interference between fore and aft wings. The fluid mechanics of the flapping wing are, however, more difficult than traditional fixed wing due to its complex unsteady fluid physics at low Reynolds numbers. Therefore, there are various flight mechanisms within the flapping wing flight and these have not yet been fully clarified. Particle Image Velocimetry (PIV) is an efficient flow measurement technique which can measure such complex flowfield at one time. The visualization of the vorticity distribution is important for understanding the flowfield of the flapping wing object. On the other hand, however, the vorticity distribution obtained through image processing such as PIV analysis is difficult to understand the detailed vortex flowfield due to the data error or noisiness in the flowfield. In this research, therefore, we developed a reconstruction method of vorticity distribution as postprocessing method of PIV measurement for the purpose to assist the physical understanding of the noisy vorticity distribution. The developed method could clearly reconstruct small vortex structures in the backward of the flapping wing object which was difficult to discriminate by conventional vorticity visualizations.
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Persistencebased segmentation of noisy 2D clouds

@article{kurlin2016fast, author = {Kurlin, V.}, title = {A fast persistencebased segmentation of noisy 2D clouds with provable guarantees}, journal = {Pattern Recognition Letters}, year = {2016}, volume = {83}, pages = {312} }
 DOI : 10.1016/j.patrec.2015.11.025
 Input : a sparse noisy cloud of boundary points near unknown contours.
 Output : a segmentation into most persistent regions bounded by contours.
 Run time : O(n log n) for any n points with real coordinates in the plane.
 Abstract. We design a new fast algorithm to automatically segment a 2D cloud of points into regions. The only input is a dotted image without any extra parameters, say a scanned blackandwhite map with almost closed curves or any image with detected edge points. The output is a hierarchy of segmentations into regions whose boundary contours have a long enough life span (persistence) in a sequence of nested neighborhoods of the input points. We give conditions on a noisy sample of a graph, when the boundaries of resulting regions are geometrically close to all original cycles in the unknown graph.
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Topologically persistent holes in noisy clouds

@inproceedings{kurlin2014fast, author = {Kurlin, V.}, title = {A fast and robust algorithm to count topologically persistent holes in noisy clouds}, booktitle = {Proceedings of CVPR 2014: Computer Vision and Pattern Recognition}, publisher = {IEEE}, year = {2014}, pages = {14581463} }
 DOI : 10.1109/CVPR.2014.189 Publisher : IEEE
 Input : a cloud of any points in the plane without extra parameters.
 Output : numbers of most persistent holes with associated probabilities.
 Run time : O(n log n) for any n points with real coordinates in the plane.
 Abstract. Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in unstructured clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We design the algorithm to count holes that are most persistent in the filtration of offsets (neighborhoods) around given points. The input is a cloud of n points in the plane without any userdefined parameters. The algorithm has the running time O(n log n) and space O(n). The output is the array (number of holes, relative persistence in the filtration). We prove theoretical guarantees when the algorithm finds the correct number of holes (components in the complement) of an unknown shape approximated by a cloud.
 C++ code : cloudanalysis.cpp (a betaversion, please email vitaliy.kurlin(at)gmail.com for support).
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Autocompletion of contours in sketches and 2D images

@inproceedings{kurlin2014auto, author = {Kurlin, V.}, title = {Autocompletion of contours in sketches, maps and sparse 2D Images based on topological persistence}, booktitle = {Proceedings of SYNASC 2014 workshop CTIC: Computational Topology in Image Context}, publisher = {IEEE}, year = {2014}, pages = {594601} }
 DOI : 10.1109/SYNASC.2014.85 Print ISBN : 9781479984473
 Input : a cloud of any points in the plane without extra parameters.
 Output : regions that are enclosed by most persistent closed contours.
 Run time : O(n log n) for any n points with real coordinates in the plane.
 Abstract. We design a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane. The only input can be a scanned map with almost closed curves, a handdrawn artistic sketch or any sparse dotted image in 2D without any extra parameters. The output is a hierarchy of closed contours that have a long enough life span (persistence) in a sequence of nested neighbourhoods of the input points. We prove theoretical guarantees when, for a given noisy sample of a graph in the plane, the output contours geometrically approximate the original contours in the unknown graph.
 C++ code : cloudanalysis.cpp (a betaversion, please email vitaliy.kurlin(at)gmail.com for support).
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