pixel voxel representations of graphs
play

Pixel & Voxel Representations of Graphs Md. Jawaherul Alam - PowerPoint PPT Presentation

Nov 09, 2023 •599 likes •1.49k views

Northridge, Los Angeles September 26, 2015 Graph Drawing Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Blsius Ignaz Rutuer Torsuen Ueckerdt Alexander Wolff . Motivation Build contact representation of graphs

  1. Northridge, Los Angeles – September 26, 2015 Graph Drawing Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Bläsius Ignaz Rutuer Torsuen Ueckerdt Alexander Wolff

  2. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  3. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  4. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  5. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  6. . Contact Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize polygonal complexity GD 2015 9 9 8 6 5 8 7 7 4 6 2 5 1 9 4 3 2 3 1 8 2 7 3 5 9 9 4 6 1 8 8 4 5 7 7 6 6 6 5 5 3 4 7 4 1 2 3 3 2 8 2 1 1 9 ■ Vertices ⇒ Geometric objects (polygons, arcs, polyhedra) ■ Edges ⇒ Contacts

  7. . Contact Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize polygonal complexity GD 2015 9 9 8 6 5 8 7 7 4 6 2 5 1 9 4 3 2 3 1 8 2 7 3 5 9 9 4 6 1 8 8 4 5 7 7 6 6 6 5 5 3 4 7 4 1 2 3 3 2 8 2 1 1 9 ■ Vertices ⇒ Geometric objects (polygons, arcs, polyhedra) ■ Edges ⇒ Contacts

  8. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  9. . Motivation Build contact representation of graphs from unit blocks How many unit blocks are required? Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  10. . Motivation Build contact representation of graphs from unit blocks How many unit blocks are required? Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  11. . Pixel and Voxel Representations Pixel in 2D, Voxel in 3D Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Building contact representation from unit blocks

  12. . Pixel and Voxel Representations Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Building contact representation from unit blocks ■ Pixel in 2D, Voxel in 3D

  13. . Pixel and Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels/voxels GD 2015 1 1 5 3 2 4 3 5 5 4 4 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of pixels/voxels) ■ Edges ⇒ Adjacent (face-to-face) pixels/voxels in two blobs

  14. . Pixel and Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels/voxels GD 2015 1 1 5 3 2 4 3 5 5 4 4 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of pixels/voxels) ■ Edges ⇒ Adjacent (face-to-face) pixels/voxels in two blobs

  15. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 5 2 4 3 5 4 2 3 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  16. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 5 3 5 2 4 3 4 5 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  17. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 5 3 5 2 4 3 4 5 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  18. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 5 3 5 33 23 2 4 3 4 5 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  19. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 1 5 5 3 12 3 5 33 23 4 2 4 3 4 5 2 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  20. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 1 5 5 3 12 3 5 33 23 4 2 4 3 4 5 2 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  21. . Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of voxels GD 2015 1 46 3 8 4 5 3 1 5 5 4 4 2 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of voxel) ■ Edges ⇒ Adjacent (face-to-face) voxels in two blobs

  22. . Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of voxels GD 2015 1 46 3 8 4 5 3 1 5 5 4 4 2 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of voxel) ■ Edges ⇒ Adjacent (face-to-face) voxels in two blobs

  23. . Related Work Contact Representations Point-contact with circles [Koebe, 1936] Point-contact with triangles [De Fraysseix et al., 1994] Side-contact with hexagons [Gansner et al., 2010], [Bonichon et al., 2010] Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  24. . Related Work Contact Representations [Gansner et al., 2010], [Bonichon et al., 2010] Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Point-contact with circles [Koebe, 1936] ■ Point-contact with triangles [De Fraysseix et al., 1994] ■ Side-contact with hexagons

  25. . Related Work Contact Representations with Rectilinear Polygons [Yeap and Sarrafzadeh, 1993], [He, 1999], [Liao et al., 2003] Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact with 8-sided rectilinear polygons:

  26. . Contact Representation of nonplanar graphs Md. Jawaherul Alam Pixel & Voxel Representations Related Work GD 2015 [Felsner and Francis, 2011] [Thomassen, 1986], [Bremner et al., 2012] Contact Representations in 3D ■ Contact representation of planar graphs with cuboids ■ Improper contact representation of planar graphs with cubes 2 3 4 1 5 6 7 8 9

  27. . [Felsner and Francis, 2011] Md. Jawaherul Alam Pixel & Voxel Representations Related Work GD 2015 Contact Representations in 3D [Thomassen, 1986], [Bremner et al., 2012] ■ Contact representation of planar graphs with cuboids ■ Improper contact representation of planar graphs with cubes ■ Contact Representation of nonplanar graphs 2 3 4 1 5 6 7 8 9

  28. . Related Work Vertex Contact Graphs of Paths on a Grid (VCPG) Mosaic Drawing Contact of square or hexagonal tilies [Cano et al., 2015] Same representation, different objective! Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact graphs of grid paths [Aerts and Felsner, 2014]

  29. . Related Work Vertex Contact Graphs of Paths on a Grid (VCPG) Mosaic Drawing Same representation, different objective! Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact graphs of grid paths [Aerts and Felsner, 2014] ■ Contact of square or hexagonal tilies [Cano et al., 2015]

  30. . Related Work Vertex Contact Graphs of Paths on a Grid (VCPG) Mosaic Drawing Same representation, different objective! Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact graphs of grid paths [Aerts and Felsner, 2014] ■ Contact of square or hexagonal tilies [Cano et al., 2015]

  31. . Min-Pixel-Representation Md. Jawaherul Alam Pixel & Voxel Representations Our Result Min-Voxel-Representation GD 2015 Find a grid drawing with unit edge lengths Input: a planar max-degree-4 graph Reduction from: NP-complete in both 2D and 3D Computational Complexity ■ Finding minimum-size representation is 1 5 2 4 3 1 2 3 4 5 6 7 1 2 3 4 5 6 7

  32. . Min-Pixel-Representation Md. Jawaherul Alam Pixel & Voxel Representations Our Result Min-Voxel-Representation GD 2015 Find a grid drawing with unit edge lengths Computational Complexity NP-complete in both 2D and 3D ■ Finding minimum-size representation is 1 5 Reduction from: P 2 4 3 Input: a planar max-degree-4 graph G P ⇓ 1 2 3 4 ⇓ 5 6 7 1 2 3 4 5 6 7

  33. . , Md. Jawaherul Alam Pixel & Voxel Representations Our Result voxels are sufficient , For a graph with genus voxels are necessary and sufficient GD 2015 For a graph with treewidth voxels are sufficient Voxel Representation Computational Complexity Pixel Representation NP-complete in both 2D and 3D ■ Finding minimum-size representation is 1 5 2 3 4 ■ For a k -outerplanar graph, Θ( kn ) pixels are necessary and sufficient 1 2 3 4 5 6 7 1 2 3 4 5 6 7

  34. . , Md. Jawaherul Alam Pixel & Voxel Representations Our Result voxels are sufficient , For a graph with genus voxels are necessary and sufficient GD 2015 For a graph with treewidth Voxel Representation Computational Complexity Pixel Representation NP-complete in both 2D and 3D ■ Finding minimum-size representation is 1 5 2 3 4 ■ For a k -outerplanar graph, Θ( kn ) pixels are necessary and sufficient ■ O ( n 2 ) voxels are sufficient 1 2 3 4 5 6 7 1 2 3 4 5 6 7

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Blsius Ignaz Rutuer

Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Blsius Ignaz Rutuer

Northridge, Los Angeles September 26, 2015 Graph Drawing Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Blsius Ignaz Rutuer Torsuen Ueckerdt Alexander Wolff . Motivation Build contact representation of graphs

1.65k views • 88 slides
1 Splatting Splatting Algorithm: Process from closest voxel to furthest voxel

1 Splatting Splatting Algorithm: Process from closest voxel to furthest voxel

Object Order Splatting Render image one voxel at a time Volume Rendering has three key steps Reconstruction Integration or projection for each voxel ... - transfer function Material attributes and classification - determine

279 views • 17 slides
Graphs and their representations After this lesson, you should be able to define the

Graphs and their representations After this lesson, you should be able to define the

Review Re A E C F B D Graphs and their representations After this lesson, you should be able to define the major terminology relating to graphs implement a graph in code, using various conventions https :// www.google.com /

567 views • 22 slides
Graphs and their representations After this lesson, you should be able to define the

Graphs and their representations After this lesson, you should be able to define the

A E C F B D Graphs and their representations After this lesson, you should be able to define the major terminology relating to graphs implement a graph in code, using various conventions

233 views • 21 slides
Pixel Presentation What is Pixel Pixel is an education and training institution with a specific

Pixel Presentation What is Pixel Pixel is an education and training institution with a specific

Pixel Presentation What is Pixel Pixel is an education and training institution with a specific expertise in International cooperation and project management Where is Pixel Pixel Quality and Accreditations Pixel has received the following

1.37k views • 67 slides
Graphs and their representations After this lesson, you should be able to explain what

Graphs and their representations After this lesson, you should be able to explain what

Graphs and their representations After this lesson, you should be able to explain what makes a graph different than a tree implement simple graph algorithms https://www.google.com/maps/preview#!data=!1m4!1m3!1d989355!2d-

505 views • 15 slides
Graphs Introduction Types Classes Slides by Christopher M. Bourke Representations Instructor:

Graphs Introduction Types Classes Slides by Christopher M. Bourke Representations Instructor:

Graphs CSE235 Graphs Introduction Types Classes Slides by Christopher M. Bourke Representations Instructor: Berthe Y. Choueiry Isomorphism Connectivity Euler & Hamiltonian Spring 2006 Computer Science & Engineering 235

783 views • 56 slides
Pixel Art What is pixel art? Pixel art is a digital art form that is created in raster in its

Pixel Art What is pixel art? Pixel art is a digital art form that is created in raster in its

VA 402 class presentation by Berrin Sun Supervised by Ekmel Ertan Pixel Art What is pixel art? Pixel art is a digital art form that is created in raster in its original size, and that is edited in pixel level. Types isometric non-isometric

590 views • 40 slides
Contact Representations of Planar Graphs Graduiertenkolleg MDS TU Berlin April 20., 2015 Stefan

Contact Representations of Planar Graphs Graduiertenkolleg MDS TU Berlin April 20., 2015 Stefan

Contact Representations of Planar Graphs Graduiertenkolleg MDS TU Berlin April 20., 2015 Stefan Felsner felsner@math.tu-berlin.de Overview A Survey of Results and Problems Straight Line Triangle Representations Discs a a b b c c

528 views • 49 slides
Old and New Challenges in Coloring Graphs with Geometric Representations Bartosz Walczak

Old and New Challenges in Coloring Graphs with Geometric Representations Bartosz Walczak

Old and New Challenges in Coloring Graphs with Geometric Representations Bartosz Walczak Jagiellonian University Krakw, Poland The 27th International Symposium on Graph Drawing and Network Visualization Chromatic number, denoted :

978 views • 72 slides
New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs

New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs

New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs Archontia Giannopoulou George B. Mertzios School of Engineering and Computing Sciences, Durham University, UK Algorithmic Graph Theory on the Adriatic

1.04k views • 83 slides
PRACTICAL REAL-TIME VOXEL-BASED GLOBAL ILLUMINATION FOR CURRENT GPUS Alexey Panteleev NVIDIA

PRACTICAL REAL-TIME VOXEL-BASED GLOBAL ILLUMINATION FOR CURRENT GPUS Alexey Panteleev NVIDIA

PRACTICAL REAL-TIME VOXEL-BASED GLOBAL ILLUMINATION FOR CURRENT GPUS Alexey Panteleev NVIDIA OUTLINE Introduction: what is Global Illumination? Screenshots Overview of Voxel Cone Tracing Implementation details Voxel clipmaps

585 views • 54 slides
Graphical representations of a corpus, and clustering on graphs University of Birmingham, 11

Graphical representations of a corpus, and clustering on graphs University of Birmingham, 11

Graphical representations of a corpus, and clustering on graphs University of Birmingham, 11 February 2016 Yves van Gennip (University of Nottingham) Joint work with R. Carrington, A. Hennessey, M. Mahlberg, S. Preston, K. Severn, V. Wiegand

398 views • 12 slides
A Massively Scalable Architecture for Learning Representations from Heterogeneous Graphs NVIDIA

A Massively Scalable Architecture for Learning Representations from Heterogeneous Graphs NVIDIA

A Massively Scalable Architecture for Learning Representations from Heterogeneous Graphs NVIDIA GPU Technology Conference 2019 - San Jose, CA C. Bayan Bruss Anish Khazane 1. Overview & Background TODAYS TALK 2. Our Approach How to

680 views • 54 slides
Weak Unit Disk Contact Representations for Graphs without Embedding Jonas Cleve, Freie

Weak Unit Disk Contact Representations for Graphs without Embedding Jonas Cleve, Freie

Weak Unit Disk Contact Representations for Graphs without Embedding Jonas Cleve, Freie Universitt Berlin EuroCG 2020 March 1618 Wrzburg, Germany Unit Disk Contact Representations Unit disk contact representation (UDCR) of G = ( V ,

823 views • 71 slides
Geometric representations of planar graphs and maps Eric Fusy (CNRS/LIX) Summer school on

Geometric representations of planar graphs and maps Eric Fusy (CNRS/LIX) Summer school on

Geometric representations of planar graphs and maps Eric Fusy (CNRS/LIX) Summer school on random geometry, Bogota, may 2016 Overview of the course Planar graphs and planar maps - structural aspects 1 1 2 2 2 2 3 3 -

1.74k views • 160 slides
Audio Decomposition Meinard Mller International Audio Laboratories Erlangen

Audio Decomposition Meinard Mller International Audio Laboratories Erlangen

Lecture Music Processing Audio Decomposition Meinard Mller International Audio Laboratories Erlangen meinard.mueller@audiolabs-erlangen.de Book: Fundamentals of Music Processing Meinard Mller Fundamentals of Music Processing Audio,

666 views • 39 slides
Design Patterns Credits: Xiaochuan Yi, U of Georgia Nenad Medvidovi Sommerville, Chapter 18

Design Patterns Credits: Xiaochuan Yi, U of Georgia Nenad Medvidovi Sommerville, Chapter 18

Design Patterns Credits: Xiaochuan Yi, U of Georgia Nenad Medvidovi Sommerville, Chapter 18 dofactory.com Instructor: Peter Baumann email: p.baumann@jacobs-university.de tel: -3178 office: room 88, Research 1 CONGRESS.SYS Corrupted:

403 views • 25 slides
R. Avakian R.H. Avakian JLAB, November 1, 2004 R.H. Avakian JLAB, November 1, 2004 Coherent

R. Avakian R.H. Avakian JLAB, November 1, 2004 R.H. Avakian JLAB, November 1, 2004 Coherent

JLab Nov. 1 R. Avakian R.H. Avakian JLAB, November 1, 2004 R.H. Avakian JLAB, November 1, 2004 Coherent radiation in Crystalls Crystalls Coherent radiation in 1. Coherent Bremsstrahlung (CBS) 2. Parametric X-ray Radiation 3. Channeling

492 views • 34 slides
Slides: alturl.com/pxo7e Epistemic Network Analysis of students longer written assignments as

Slides: alturl.com/pxo7e Epistemic Network Analysis of students longer written assignments as

Slides: alturl.com/pxo7e Epistemic Network Analysis of students longer written assignments as formative/summative evaluation With Simon Skov Fougt Associate professor, ph.d. Department for School and Learning Metropolitan University

658 views • 16 slides
Development of immersion gratings and grisms at LLNL Dispersing elements for astronomy: New

Development of immersion gratings and grisms at LLNL Dispersing elements for astronomy: New

Development of immersion gratings and grisms at LLNL Dispersing elements for astronomy: New trends and possibilities Milan, Italy Paul J. Kuzmenko, LLNL October 10, 2017 LLNL-PRES-740702 This work was performed under the auspices of the U.S.

466 views • 21 slides
Ali Kamandi Spring 2007 kamandi@sharif.edu Sharif University of Technology Internet History

Ali Kamandi Spring 2007 kamandi@sharif.edu Sharif University of Technology Internet History

Ali Kamandi Spring 2007 kamandi@sharif.edu Sharif University of Technology Internet History Internet Evolution Internet Pioneers Internet Growth Conclusion 1836 Telegraph invented by Cooke and Wheatstone Morse code to

591 views • 37 slides
Communication Optimal and Tiled Algorithm for 2D Linear Algebra Fr. Feb 20 2009 NSF

Communication Optimal and Tiled Algorithm for 2D Linear Algebra Fr. Feb 20 2009 NSF

Communication Optimal and Tiled Algorithm for 2D Linear Algebra Fr. Feb 20 2009 NSF Workshop: Future Directions in Tensor- Based Computation and Modeling Software and Language: How Do We Build an Infrastructure that Supports

1.06k views • 63 slides
4.4 Shortest Paths in a Graph shortest path from Princeton CS department to Einstein's house

4.4 Shortest Paths in a Graph shortest path from Princeton CS department to Einstein's house

4.4 Shortest Paths in a Graph shortest path from Princeton CS department to Einstein's house Shortest Path Problem Shortest path network. Directed graph G = (V, E). Source s, destination t. Length e = length of edge e. Shortest

226 views • 6 slides

More recommend