Kinematics & Computational Geometry Ariel University Center

1. Computational Geometry Performance Problems with respect to GPGPU 2D 3D 4D +6D +10D Kinematics & Computational Geometry Ariel University Center 1

2. Agenda • Who are we: KCG lab AUC • Radio Frequency Motivation: • 2D Radio Map Compression • 3D Ray Tracing • Volumetric Compression: MRI, FMRI • Robot Configuration Space (6D-10D) • Open issue: Gaussian physical simulations: serial Fortran simulation challenge. KCG lab Ariel University Center 2

3. Who are we - (KCG) • KCG lab - Ariel University Center: • Founded 2008: Boaz Ben-Moshe & NirShvalb • 10 undergraduate students, 7 graduate, Postdoc • 6 founded projects: Magnet, Mafat • Simulation, Computational Geometry, Topology, Robotis, Optimization • http://www.ariel.ac.il/projects/kcg/ KCG lab Ariel University Center 3

4. RF planning Motivation • Radio elements: • Client: • Base Station: • Network: • Microwave - LOS • Satellite • Cable

5. Goal: design an ‘optimal’ network • Problems of interest: • Locating Base Stations • Frequency Assignment • Connectivity

6. Problems of interest: • Locating Base Stations: • Guarding like. • Complex objective function. • Frequency Assignment: • Connectivity:

7. Problems of interest: • Locating Base Stations: • Frequency Assignment: • Conflict free frequency • Connectivity:

8. Problems of interest: • Locating Base Stations: • Frequency Assignment: • Connectivity: • Smallest set of Relay Stations. • Back to the BS-locator.

9. Main Obstacles: • Huge inputs  simplify & approximation • Formalizing  objective function • NP hardness  efficient Heuristics

10. Approximating Visibility [BCK] • Given a terrain T and a view point p compute the set of points on the surface of T that are visible from p. • Alternatively: Paint T with two colors (red & blue) s.t. any blue (red) point is visible (invisible) from p.

11. left & right cross-sections  pizza slice.

12. Lets look at a specific pizza slice:

13. Radar-like generic algorithm Given Terrain (T), view point (vp), and fixed angle (a=A): while(int i=0;i<360) { S1=cross-section(i); S2=cross-section(i+a); if(close enough(S1, S2)) { extrapolate(S1, S2); a = A; i = min(360, i + a);} else a = a/2; }

14. Radar-like: Threshold Radar-like: 10 deg, low threshold | Radar-like: 10 deg, hi threshold

15. Error Measure exact radar approx xor Error value: xor-area / circle-area

16. Radar vs’ Naïve sampling Naïve sampling Radar visibility

17. Can we GPGPU the radar? • Sure, but: • Cross section distribute computation  memory management issues • Interpolation algorithm  2* over head

18. Generalizing radar-visibility to RF propagation model: • Discrete visibility (boolean) continues • Visibility a long a ray RF sampling

19. Approximating Radio-Maps General Frame work: Sampling Set (SP) Extrapolation DS

20. Compute two consecutive cross-sections.

21. Approximating Radio-Maps • Putting it all together: • Sensitive Radar algorithm • Sensitive 2D Simplification • Robust distance norm • Fine Tuning: • None grid sampling (2D)‏ • Parameters (terrain independent)‏

22. Approximating Radio-Maps • Grid Random TS F-Radar S-Radar • 5000 samples per radio-map

23. Radio-Maps: Interpolation • What is the best way to interpolate RM? • Triangulation? • Common way  slow, “heavy” • Numerical problems • Point location over head • Must be a better way  JPEG

24. DCT & DWT not just for DIP • Terrain Simplification [ICTS, BSS-07] • Radio Maps • Any “natural” 2D matrix • dip tools, ROI, • hardware, robustness • ‘fits’ GPU’s perfectly!

25. ICTS experimental results ICTS Original diff diff Qslim

26. Moving into 3D • Z-cam  2.5D • 3D Radio Path: ray tracing • MRI – Volumetric compression • FMRI – 3.5D

27. 3D Ray-Tracing • RF wireless communication in mind! • ISRC: The Israeli Short Range Consortium • Motivated by:

29. Problem of interests 29 29 • Given a geometric properties of a building B. and the position of both a transmitter T and a receiver R. • Find all 3D geometric paths between T and R with bounded length. CGK lab Ariel University Center

30. A building benchmark (complex): 30 30 * Two floors, 1600 m^2 * 400 walls, several types * Complex: doors, windows CGK lab Ariel University Center

31. Radio Paths Example‏ 31 31 • Radio Paths: • 2P, 0R • 3P, 1R • 0P, 1R • 2P, 4R – cut off • penetration point • reflection point 31

32. Now, all we got to do is to find the best path between all the pathes we found

35. 3D Ray-Tracing • Can it be implemented in using GPGPU? • Classic problem for distributed machines • Current tools are extremely slow! • Current research: looking into larger urban regions.

36. Volumetric Compression • MRI in mind • MRI – high resolution: say 1000^3 voxels • JP3D? http://www.jp3d.co.uk/main.html • Functional MRI: for research (no FDA issues) 3.5D • Current tools are very slow and limited: SPM05

37. Robot Configuration Space • Motion planning (minimal energy): • Very Simple Robot  6 deg of freedom (6 motors) • Each Motor ~20 possible positions  ~6*10^7 configurations. • Step Planning: shortest energy-path • Discussion!

38. Robot Configuration Space • Motion planning (minimal energy) • MRI – high resolution: say 1000^3 voxels • JP3D? http://www.jp3d.co.uk/main.html • Functional MRI: for research (no FDA issues) 3.5D • Current tools are very slow and limited: SPM05

39. physical simulations - serial FORTRAN Gaussian challenge • I know practically Nothing about this issue: • Fortran: large portion of the code is older then us! • Linda: A distributed version of Gaussian • SGI, Cray machines: strange speed-up behavior • Bar-Ilan has the sources – and seems to be intrested (yet scaptic). KCG lab Ariel University Center 39

40. Open Discussion • 2D: Radio Maps & terrains, • 3D: Z-cam, ray-tracing, MRI, FMRI • 6D-10D Robot Configuration (ant’s in mind) • Other: KCG lab Ariel University Center 40