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174 results on '"cubical complex"'

2. A topological framework for identifying phenomenological bifurcations in stochastic dynamical systems.

Author

Tanweer, Sunia, A. Khasawneh, Firas, Munch, Elizabeth, and R. Tempelman, Joseph

Abstract

Changes in the parameters of dynamical systems can cause the state of the system to shift between different qualitative regimes. These shifts, known as bifurcations, are critical to study as they can indicate when the system is about to undergo harmful changes in its behavior. In stochastic dynamical systems, there is particular interest in P-type (phenomenological) bifurcations, which can include transitions from a monostable state to multi-stable states, the appearance of stochastic limit cycles and other features in the probability density function (PDF) of the system's state. Current practices are limited to systems with small state spaces, cannot detect all possible behaviors of the PDFs and mandate human intervention for visually identifying the change in the PDF. In contrast, this study presents a new approach based on Topological Data Analysis that uses superlevel persistence to mathematically quantify P-type bifurcations in stochastic systems through a "homological bifurcation plot"—which shows the changing ranks of 0th and 1st homology groups, through Betti vectors. Using these plots, we demonstrate the successful detection of P-bifurcations on the stochastic Duffing, Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and elaborate on how to generate an estimated homological bifurcation plot given a kernel density estimate (KDE) of these systems by employing a tool for finding topological consistency between PDFs and KDEs. [ABSTRACT FROM AUTHOR]

Published
2024
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3. Capturing Shape Information with Multi-scale Topological Loss Terms for 3D Reconstruction

Author

Waibel, Dominik J. E., Atwell, Scott, Meier, Matthias, Marr, Carsten, Rieck, Bastian, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Wang, Linwei, editor, Dou, Qi, editor, Fletcher, P. Thomas, editor, Speidel, Stefanie, editor, and Li, Shuo, editor

Published
2022
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5. ON MULTI-DETERMINANT FUNCTORS FOR TRIANGULATED CATEGORIES.

Author

ALDROVANDI, ETTORE and LESTER, CYNTHIA

Subjects
*TRIANGULATED categories
Abstract

We extend Deligne's notion of determinant functor to tensor triangulated categories. Specifically, to account for the multiexact structure of the tensor, we define a determinant functor on the 2-multicategory of triangulated categories and we provide a multicategorical version of the universal determinant functor for triangulated categories whose multiexactness properties are conveniently captured by a certain complex modeled by cubical shapes, which we introduce along the way. We then show that for a tensor triangulated category whose tensor admits a Verdier structure the resulting determinant functor takes values in a categorical ring. [ABSTRACT FROM AUTHOR]

Published
2023

6. All About Cells

Author

Kolb, Lisa, Krömker, Susanne, Müller, Stefan C., editor, Plath, Peter J., editor, Radons, Günter, editor, and Fuchs, Armin, editor

Published
2018
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9. Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images

Author

Heiss, Teresa, Wagner, Hubert, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Felsberg, Michael, editor, Heyden, Anders, editor, and Krüger, Norbert, editor

Published
2017
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10. A Polynomial Time Algorithm to Compute Geodesics in CAT(0) Cubical Complexes.

Author

Hayashi, Koyo

Subjects
*POLYNOMIAL time algorithms, *GEODESICS, *CATS
Abstract

This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Owen and Provan's algorithm (IEEE/ACM Trans Comput Biol Bioinform 8(1):2–13, 2011) as a subroutine. Our algorithm is applicable to any path in any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes. [ABSTRACT FROM AUTHOR]

Published
2021
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11. Cubical Homology-Based Machine Learning: An Application in Image Classification

Author

Seungho Choe and Sheela Ramanna

Subjects
cubical complex, cubical homology, image classification, deep learning, persistent homology, Mathematics, QA1-939
Abstract

Persistent homology is a powerful tool in topological data analysis (TDA) to compute, study, and encode efficiently multi-scale topological features and is being increasingly used in digital image classification. The topological features represent a number of connected components, cycles, and voids that describe the shape of data. Persistent homology extracts the birth and death of these topological features through a filtration process. The lifespan of these features can be represented using persistent diagrams (topological signatures). Cubical homology is a more efficient method for extracting topological features from a 2D image and uses a collection of cubes to compute the homology, which fits the digital image structure of grids. In this research, we propose a cubical homology-based algorithm for extracting topological features from 2D images to generate their topological signatures. Additionally, we propose a novel score measure, which measures the significance of each of the sub-simplices in terms of persistence. In addition, gray-level co-occurrence matrix (GLCM) and contrast limited adapting histogram equalization (CLAHE) are used as supplementary methods for extracting features. Supervised machine learning models are trained on selected image datasets to study the efficacy of the extracted topological features. Among the eight tested models with six published image datasets of varying pixel sizes, classes, and distributions, our experiments demonstrate that cubical homology-based machine learning with the deep residual network (ResNet 1D) and Light Gradient Boosting Machine (lightGBM) shows promise with the extracted topological features.

Published
2022
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12. Random walks and Brownian motion on cubical complexes.

Author

Nye, Tom M.W.

Subjects
*BROWNIAN motion, *RANDOM walks, *METRIC spaces, *WIENER processes, *CUBES
Abstract

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on a single cube, combined with a distribution on walks between cubes. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and applications in this context. [ABSTRACT FROM AUTHOR]

Published
2020
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15. Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Author

Gonzalez-Lorenzo, Aldo, Juda, Mateusz, Bac, Alexandra, Mari, Jean-Luc, Real, Pedro, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Bac, Alexandra, editor, and Mari, Jean-Luc, editor

Published
2016
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16. Computation of Cubical Steenrod Squares

Author

Krčál, Marek, Pilarczyk, Paweł, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Bac, Alexandra, editor, and Mari, Jean-Luc, editor

Published
2016
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17. Topological Comparisons of Fluvial Reservoir Rock Volumes Using Betti Numbers: Application to CO Storage Uncertainty Analysis

Author

Dahrabou, Asmae, Viseur, Sophie, Gonzalez-Lorenzo, Aldo, Rohmer, Jérémy, Bac, Alexandra, Real, Pedro, Mari, Jean-Luc, Audigane, Pascal, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Bac, Alexandra, editor, and Mari, Jean-Luc, editor

Published
2016
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18. Homotopic Thinning in 2D and 3D Cubical Complexes Based on Critical Kernels

Author

Couprie, Michel, Bertrand, Gilles, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Normand, Nicolas, editor, Guédon, Jeanpierre, editor, and Autrusseau, Florent, editor

Published
2016
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19. Two Measures for the Homology Groups of Binary Volumes

Author

Gonzalez-Lorenzo, Aldo, Bac, Alexandra, Mari, Jean-Luc, Real, Pedro, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Normand, Nicolas, editor, Guédon, Jeanpierre, editor, and Autrusseau, Florent, editor

Published
2016
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20. Cellular Skeletons: A New Approach to Topological Skeletons with Geometric Features

Author

Gonzalez-Lorenzo, Aldo, Bac, Alexandra, Mari, Jean-Luc, Real, Pedro, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Azzopardi, George, editor, and Petkov, Nicolai, editor

Published
2015
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21. Perspectives

Author

Fajstrup, Lisbeth, Goubault, Eric, Haucourt, Emmanuel, Mimram, Samuel, Raussen, Martin, Fajstrup, Lisbeth, Goubault, Eric, Haucourt, Emmanuel, Mimram, Samuel, and Raussen, Martin

Published
2016
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23. Clear and Compress: Computing Persistent Homology in Chunks

Author

Bauer, Ulrich, Kerber, Michael, Reininghaus, Jan, Farin, Gerald, Series editor, Hege, Hans-Christian, Series editor, Hoffman, David, Series editor, Johnson, Christopher R., Series editor, Polthier, Konrad, Series editor, Rumpf, Martin, Series editor, Bremer, Peer-Timo, editor, Hotz, Ingrid, editor, Pascucci, Valerio, editor, and Peikert, Ronald, editor

Published
2014
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24. Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes.

Author

Rungtanapirom, Nithi, Stix, Jakob, and Vdovina, Alina

Subjects
*INFINITE series (Mathematics), *FINITE groups, *CUBES, *ALGEBRA, *VERTEX operator algebras, *QUATERNIONS
Abstract

We construct vertex transitive lattices on products of trees of arbitrary dimension d ≥ 1 based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher-dimensional cubical version of Ramanujan graphs (optimal expanders). [ABSTRACT FROM AUTHOR]

Published
2019
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25. SurfCut: Surfaces of Minimal Paths from Topological Structures.

Author

Algarni, Marei and Sundaramoorthi, Ganesh

Subjects
*TOPOLOGY, *EUCLIDEAN algorithm, *EIKONAL equation, *NUMERICAL analysis, *MATHEMATICAL analysis
Abstract

We present SurfCut, an algorithm for extracting a smooth, simple surface with an unknown 3D curve boundary from a noisy 3D image and a seed point. Our method is built on the novel observation that ridge curves of the Euclidean length of minimal paths ending on a level set of the solution of the eikonal equation lie on the surface. Our method extracts these ridges and cuts them to form the surface boundary. Our surface extraction algorithm is built on the novel observation that the surface lies in a valley of the eikonal equation solution. The resulting surface is a collection of minimal paths. Using the framework of cubical complexes and Morse theory, we design algorithms to extract ridges and valleys robustly. Experiments on three 3D datasets show the robustness of our method, and that it achieves higher accuracy with lower computational cost than state-of-the-art. [ABSTRACT FROM AUTHOR]

Published
2019
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26. The Hirsch Lemma

Author

Griffiths, Phillip, Morgan, John, Bass, Hyman, Series editor, Oesterlé, Joseph, Series editor, Tschinkel, Yuri, Series editor, Weinstein, Alan, Series editor, Griffiths, Phillip, and Morgan, John

Published
2013
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27. A ∞ -Structures and C ∞ -Structures

Author

Griffiths, Phillip, Morgan, John, Bass, Hyman, Series editor, Oesterlé, Joseph, Series editor, Tschinkel, Yuri, Series editor, Weinstein, Alan, Series editor, Griffiths, Phillip, and Morgan, John

Published
2013
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29. A Quasi-linear Algorithm to Compute the Tree of Shapes of nD Images

Author

Géraud, Thierry, Carlinet, Edwin, Crozet, Sébastien, Najman, Laurent, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Hendriks, Cris L. Luengo, editor, Borgefors, Gunilla, editor, and Strand, Robin, editor

Published
2013
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30. A Parallel Thinning Algorithm for Grayscale Images

Author

Couprie, Michel, Bezerra, Nivando, Bertrand, Gilles, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Gonzalez-Diaz, Rocio, editor, Jimenez, Maria-Jose, editor, and Medrano, Belen, editor

Published
2013
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33. Writing Reusable Digital Topology Algorithms in a Generic Image Processing Framework

Author

Levillain, Roland, Géraud, Thierry, Najman, Laurent, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Köthe, Ullrich, editor, Montanvert, Annick, editor, and Soille, Pierre, editor

Published
2012
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34. Some Morphological Operators on Simplicial Complex Spaces

Author

Dias, Fábio, Cousty, Jean, Najman, Laurent, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Debled-Rennesson, Isabelle, editor, Domenjoud, Eric, editor, Kerautret, Bertrand, editor, and Even, Philippe, editor

Published
2011
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35. Well-Composed Cell Complexes

Author

Gonzalez-Diaz, Rocio, Jimenez, Maria-Jose, Medrano, Belen, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Debled-Rennesson, Isabelle, editor, Domenjoud, Eric, editor, Kerautret, Bertrand, editor, and Even, Philippe, editor

Published
2011
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36. COMBINATORIAL APPROACH TO DETECTION OF FIXED POINTS, PERIODIC ORBITS, AND SYMBOLIC DYNAMICS.

Author

Gidea, Marian and Shmalo, Yitzchak

Subjects
ALGORITHMS, DYNAMICAL systems, MATHEMATICAL analysis, GEOMETRY, MANIFOLDS (Mathematics)
Abstract

We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of 'correctly aligned windows'. We subdivide 'windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way, we encode the information on the dynamics into combinatorial structure. We use a version of Sperner's Lemma to infer that, if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. The method developed here does not require the computation of algebraic topology-type invariants, as only combinatorial information is needed; our arguments are elementary. [ABSTRACT FROM AUTHOR]

Published
2018
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37. Limit Theorems for Random Cubical Homology.

Author

Hiraoka, Yasuaki and Tsunoda, Kenkichi

Subjects
*BETTI numbers, *LIMIT theorems, *IMAGE processing, *DIGITAL images, *TOPOLOGY
Abstract

This paper studies random cubical sets in Rd. Given a cubical set X⊂Rd, a random variable ωQ∈[0,1] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ωQ≤t for each t∈[0,1]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown. [ABSTRACT FROM AUTHOR]

Published
2018
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38. Surface Thinning in 3D Cubical Complexes

Author

Chaussard, John, Couprie, Michel, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Wiederhold, Petra, editor, and Barneva, Reneta P., editor

Published
2009
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40. Cell Complexes

Author

Cohen, Arjeh M., editor, Cohen, Henri, editor, Eisenbud, David, editor, Singer, Michael F., editor, Sturmfels, Bernd, editor, and Kozlov, Dmitry

Published
2008
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43. Minimal Non-simple and Minimal Non-cosimple Sets in Binary Images on Cell Complexes

Author

Kong, T. Yung, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Kuba, Attila, editor, Nyúl, László G., editor, and Palágyi, Kálmán, editor

Published
2006
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46. Efficiently Storing Well-Composed Polyhedral Complexes Computed Over 3D Binary Images.

Author

Gonzalez-Diaz, Rocio, Jimenez, Maria-Jose, and Medrano, Belen

Abstract

A 3D binary image I can be naturally represented by a combinatorial-algebraic structure called cubical complex and denoted by Q( I), whose basic building blocks are vertices, edges, square faces and cubes. In Gonzalez-Diaz et al. (Discret Appl Math 183:59-77, 2015), we presented a method to 'locally repair' Q( I) to obtain a polyhedral complex P( I) (whose basic building blocks are vertices, edges, specific polygons and polyhedra), homotopy equivalent to Q( I), satisfying that its boundary surface is a 2D manifold. P( I) is called a well-composed polyhedral complex over the picture I. Besides, we developed a new codification system for P( I), encoding geometric information of the cells of P( I) under the form of a 3D grayscale image, and the boundary face relations of the cells of P( I) under the form of a set of structuring elements. In this paper, we build upon (Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological and geometric information of P( I), it is enough to store just one 3D point per polyhedron and hence neither grayscale image nor set of structuring elements are needed. From this 'minimal' codification of P( I), we finally present a method to compute the 2-cells in the boundary surface of P( I). [ABSTRACT FROM AUTHOR]

Published
2017
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48. M к -Polyhedral Complexes of Bounded Curvature

Author

Bridson, Martin R., Haefliger, André, Chern, S. S., editor, Eckmann, B., editor, de la Harpe, P., editor, Hironaka, H., editor, Hirzebruch, F., editor, Hitchin, N., editor, Hörmander, L., editor, Knus, M.-A., editor, Kupiainen, A., editor, Lannes, J., editor, Lebeau, G., editor, Ratner, M., editor, Serre, D., editor, Sinai, Ya. G., editor, Sloane, N. J. A., editor, Tits, J., editor, Waldschmidt, M., editor, Watanabe, S., editor, Berger, M., editor, Coates, J., editor, Varadhan, S. R. S., editor, Bridson, Martin R., and Haefliger, André

Published
1999
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49. M κ —Polyhedral Complexes

Author

Bridson, Martin R., Haefliger, André, Chern, S. S., editor, Eckmann, B., editor, de la Harpe, P., editor, Hironaka, H., editor, Hirzebruch, F., editor, Hitchin, N., editor, Hörmander, L., editor, Knus, M.-A., editor, Kupiainen, A., editor, Lannes, J., editor, Lebeau, G., editor, Ratner, M., editor, Serre, D., editor, Sinai, Ya. G., editor, Sloane, N. J. A., editor, Tits, J., editor, Waldschmidt, M., editor, Watanabe, S., editor, Berger, M., editor, Coates, J., editor, Varadhan, S. R. S., editor, Bridson, Martin R., and Haefliger, André

Published
1999
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50. Capturing shape information with multi-scale topological loss terms for 3D reconstruction

Author

Waibel, D.J.E., Atwell, S., Meier, M., Marr, C., and Rieck, B.

Subjects
3d Shape Prediction, Cubical Complex, Topological Loss
Abstract

Reconstructing 3D objects from 2D images is both challenging for our brains and machine learning algorithms. To support this spatial reasoning task, contextual information about the overall shape of an object is critical. However, such information is not captured by established loss terms (e.g. Dice loss). We propose to complement geometrical shape information by including multi-scale topological features, such as connected components, cycles, and voids, in the reconstruction loss. Our method uses cubical complexes to calculate topological features of 3D volume data and employs an optimal transport distance to guide the reconstruction process. This topology-aware loss is fully differentiable, computationally efficient, and can be added to any neural network. We demonstrate the utility of our loss by incorporating it into SHAPR, a model for predicting the 3D cell shape of individual cells based on 2D microscopy images. Using a hybrid loss that leverages both geometrical and topological information of single objects to assess their shape, we find that topological information substantially improves the quality of reconstructions, thus highlighting its ability to extract more relevant features from image datasets.

Published
2022

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