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The Generalized Spherical Homeomorphism Theorem for Digital Images

L. Abrams, D.E. Fishkind, C.E. Priebe
2004 IEEE Transactions on Medical Imaging  
Using a canonical image-thickening technique and the authors' previously proven "spherical homeomorphism theorem for surfaces," we formulate and prove a spherical homeomorphism theorem which is valid for  ...  Index Terms-Digital topology, spherical homeomorphism.  ...  ACKNOWLEDGMENT The authors thank the referees for their helpful suggestions and comments.  ... 
doi:10.1109/tmi.2004.826049 pmid:15147017 fatcat:ljtvert66zaqlnauqkcg4ukr7q

A Genus Bound for Digital Image Boundaries

Lowell Abrams, Donniell E. Fishkind
2005 SIAM Journal on Discrete Mathematics  
Fishkind, and Priebe [1] , [2] proved-that the boundary of a digital image is topologically equivalent to a sphere if and only if certain related foreground and background graphs are both trees.  ...  In this manuscript we extend this result by proving upper and lower bounds on digital image boundary genus in terms of the foreground and background graphs, and we show that these bounds are best possible  ...  We thank the anonymous referees for their thoughtful suggestions.  ... 
doi:10.1137/s0895480104445691 fatcat:khvkjq4cabfkjb6bykmu4p5jha

Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

Alexander V. Evako
2017 Applied Mathematics and Physics  
A digital closed n-dimensional manifold with a locally spherical topology is a graph theoretic model for a continuous closed n-dimensional manifold.  ...  This paper defines and studies properties of a new class of digital n-dimensional spaces with a locally non-spherical topology. We prove that such spaces have the dimension n≥3.  ...  For the same reason as above, O(x) is a digital (n+m)-surface. This completes the proof. Figure 3 . Definition 4 Digital n-surfaces are called homeomorphic if they are homotopy equivalent.  ... 
doi:10.12691/amp-5-2-3 fatcat:7iduyoqtzbdvfnaxvgxrm7ctuy

Connectivity preserving digitization of blurred binary images in 2D and 3D

Peer Stelldinger, Ullrich Köthe
2006 Computers & graphics  
Since the input for any image analysis algorithm is a digital image, which does not need to have the same topological characteristics as the imaged real world, it is important to know, which shapes can  ...  Most existing approaches do not take into account the unavoidable blurring in real image acquisition systems or use extremely simplified and thus unrealistic models of digitization with blurring.  ...  Based on these definitions we are able to prove a sampling theorem for blurred binary images. 2 Sampling-Theorem for Blurred Binary Images In a previous paper we already proved a sampling theorem for  ... 
doi:10.1016/j.cag.2005.10.012 fatcat:zmurvv4qqra57hiuzurahww354

Self-affine Manifolds [article]

Gregory R. Conner, Jörg M. Thuswaldner
2015 arXiv   pre-print
Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established.  ...  Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions.  ...  We thank the referee for carefully reading the manuscript and valuable comments that helped to considerably improve the readability of this paper.  ... 
arXiv:1402.3000v3 fatcat:ojjh5lb5dzhubcta2pvgq2crjm

When Will a Sequence of Points in a Riemannian Submanifold Converge?

Tuyen Trung Truong
2020 Mathematics  
An answer to this question serves to understand the convergence behaviour for iterative algorithms for (constrained) optimisation problems, with many applications such as in Deep Learning.  ...  Assume that we know a priori some properties of the set A of cluster points of xn. The question is under what conditions that xn will converge.  ...  The author would like to thank the referees for helpful feedbacks. Conflicts of Interest: The author declares no conflict of interest.  ... 
doi:10.3390/math8111934 fatcat:3ueqdmsqgzdzfpp3njtzecxbgi

Generalized Polish spaces at regular uncountable cardinals [article]

Claudio Agostini, Luca Motto Ros, Philipp Schlicht
2021 arXiv   pre-print
In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard κ-Borel spaces for κ an uncountable (regular) cardinal satisfying  ...  As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces.  ...  Theorem 3.16. (1) Up to homeomorphism, the generalized Cantor space 2 κ is the unique nonempty (κ-)perfect (κ-Lindelöf, if κ is weakly compact) spherically complete G-Polish space. (2) Up to homeomorphism  ... 
arXiv:2107.02587v2 fatcat:lk4sm3rmkzgqdkbvaaf4cg4mg4

The Genus of a Digital Image Boundary Is Determined by Its Foreground, Background, and Reeb Graphs

Lowell Abrams, Donniell E. Fishkind
2007 Discrete & Computational Geometry  
of the natural height function on the boundary of the digital image to be a Morse function).  ...  information about the digital image and its complement, respectively, and the Reeb graph, relative to the natural height function, associated with the digital image's boundary.  ...  The following result, Theorem 1, was proved in [1] . Theorem 1. For any digital image I, max{r (G I ), r (G I C )} ≤ g(∂I) ≤ r (G I ) + r (G I C ).  ... 
doi:10.1007/s00454-007-1315-x fatcat:cap2f6nfojcnhlx3wihne67334

The Alexandroff Dimension of Quotients of R 2

Petra Wiederhold, Richard G. Wilson
2004 Discrete & Computational Geometry  
Theorem 1.1.  ...  For completeness, we recall the (recursive) definition of the small inductive dimension ind(X ) of a topological space X : For more details (and the definition of the other topological dimension functions  ...  Acknowledgment We thank the referees for their helpful comments.  ... 
doi:10.1007/s00454-004-0822-2 fatcat:t3b4g4yiijfztc4kvi5tltiy7u

A proof of the spherical homeomorphism conjecture for surfaces

L. Abrams, D.E. Fishkind, C.E. Priebe
2002 IEEE Transactions on Medical Imaging  
The basis for their correction strategy is a conjecture, which they call the spherical homeomorphism conjecture, stating that the boundary between the foreground region and the background region is topologically  ...  Due to noise and/or resolution issues, magnetic resonance imaging may see "handles" that need to be eliminated to reflect the true spherical topology.  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous referees for helpful comments.  ... 
doi:10.1109/tmi.2002.806590 pmid:12588040 fatcat:vdtwcj6shndyhcursndaxzzzj4

Integration of Topological Constraints in Medical Image Segmentation [chapter]

F. Ségonne, B. Fischl
2015 Handbook of Biomedical Imaging  
In this chapter we cover methods for integrating topological constraints into segmentation procedures in order to generate geometrically accurate and topologically correct models, which is critical for  ...  Conversely, topological "defects" or departures from the true topology of a structure due to segmentation errors can greatly reduce the utility of anatomical models.  ...  Their method is based on the theory of digital topology but is limited to 6-connectivity and has not been generalized for any other connectivity rule.  ... 
doi:10.1007/978-0-387-09749-7_13 fatcat:loum6vxikfdx7ja523hobm2zia

Optimal extension of Lipschitz embeddings in the plane [article]

Leonid V. Kovalev
2018 arXiv   pre-print
Furthermore, every Lipschitz embedding of the circle extends to a Lipschitz homeomorphism of the plane, again with a linear bound on the constant.  ...  We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved.  ...  This research was supported by the National Science Foundation grants DMS-1362453 and DMS-1764266.  ... 
arXiv:1808.04830v1 fatcat:3cwdljwiirevdbk7n2bsqtftxa

Developments in fractal geometry

Michael Barnsley, Andrew Vince
2013 Bulletin of Mathematical Sciences  
In engineering, fractal modelling has been successfully applied to image compression [14] . In multimedia, fractal modelling can be used for real-time image synthesis [97] .  ...  The rough objects are described in terms of the smooth systems that generate them. A key example is an attractor described in terms of the IFS that generates it.  ...  Acknowledgment We thank a referee for helpful suggestions.  ... 
doi:10.1007/s13373-013-0041-3 fatcat:jaiywnig3nhhzntflxfcbhuq2i

One Ring to Rule Them All: a simple solution to multi-view 3D-Reconstruction of shapes with unknown BRDF via a small Recurrent ResNet [article]

Ziang Cheng, Hongdong Li, Richard Hartley, Yinqiang Zheng, Imari Sato
2021 arXiv   pre-print
This paper proposes a simple method which solves an open problem of multi-view 3D-Reconstruction for objects with unknown and generic surface materials, imaged by a freely moving camera and a freely moving  ...  We encourage the reader to view our demo video for better visualizations.  ...  We wish to thank Prof Boxin Shi for sharing the DiLiGent-MV dataset. We thank Dr Art Subpa-asa for drawing the artistic diagrams for the paper.  ... 
arXiv:2104.05014v1 fatcat:mlvm2cveqbdphkxx2hhgq6yomq

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization [article]

Greg Kuperberg
2017 arXiv   pre-print
The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated  ...  The new result is that the homeomorphism problem is elementary recursive, i.e., that the computational complexity is bounded by a bounded tower of exponentials.  ...  Finally in the spherical case, we want to pass from the unoriented to the oriented homeomorphism type of N when N is spherical but not lens. (Note that every such N is chiral.)  ... 
arXiv:1508.06720v4 fatcat:txzxycnltzdrllwawlllm6m5zu
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