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Optimization of Inverse Snyder Polyhedral Projection

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Optimization of Inverse Snyder Polyhedral Projection

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  1. Optimization of Inverse Snyder Polyhedral Projection October 5, 2011

  2. Inverse Snyder Optimization • What is Snyder? • Inversion Issues • Optimizations: • Operation Reduction • Iteration Reduction • Iteration Elimination • Results • Summary (NASA, 2000)

  3. What is Snyder? • Projecting Spherical Earth to Planar Map

  4. What is Snyder? • Projecting Spherical Earth to Planar Map F(p) F-1(p) (NASA, 2000) (Google Maps, 2010 – Mercator Projection)

  5. What is Snyder? • Projecting Spherical Earth to Planar Map Mercator Projection

  6. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area

  7. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area Mollweide (NASA, 2000-2006)

  8. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area Mollweide (NASA, 2000-2006) Lambert Azimuthal Equal-Area

  9. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area Werner Mollweide (NASA, 2000-2006) Lambert Azimuthal Equal-Area

  10. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area P P-1

  11. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area (Snyder, 1992)

  12. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area • Industrial applications • in virtual worlds • Truncated Icosahedron: • Angular Deformation: < 3.75o • Scale Variation: < 3.3% • Icosahedron: • Angular Deformation: < 17.27o • Scale Variation: < 16.3% PYXIS Innovation

  13. What is Snyder? • Projecting Spherical Earth to Planar Map • Preserve Area • Industrial applications • in virtual worlds • Truncated Icosahedron: • Angular Deformation: < 3.75o • Scale Variation: < 3.3% • Icosahedron: • Angular Deformation: < 17.27o • Scale Variation: < 16.3% Snyder’s Icosahedron Face

  14. Constructing the Projection • Identify Symmetric Region

  15. Constructing the Projection B’ G g A’ C’

  16. Constructing the Projection B’ B’ G g G g H D’ Az A’ A’ C’ C’

  17. Constructing the Projection B’ B’ B G g G g g H D D’ Az’ Az A’ A’ C A C’ C’

  18. Constructing the Projection • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth B’ B’ B G g G g g H D D’ Az’ Az A’ A’ C A C’ C’

  19. Constructing the Projection • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth B’ B’ B G g G g g H D D’ Az’ Az A’ A’ C A C’ C’

  20. Constructing the Projection • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth linear non-linear, trigonometric functions B’ B’ B G g G g g H D D’ Az’ Az A’ A’ C A C’ C’

  21. Constructing the Projection • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth B’ B’ B G g G g g H D D’ Az’ Az A’ A’ C A C’ C’

  22. Constructing the Projection non-linear with inverse trig. funcs • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth B’ B’ B G g G g g H D D’ Az’ Az A’ A’ C A C’ C’

  23. Constructing the Projection non-linear with inverse trig. funcs • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth B’ B’ B G g G g g H D D’ Az’ Az q d’ A’ A’ C A C’ C’

  24. Constructing the Projection non-linear with inverse trig. funcs • Find planar azimuth: Az’ • Position P’ based on d’ from q • Unwrap azimuth B’ B’ B G g G g g H D D’ Az’ Az q d’ A’ A’ C A C’ C’

  25. Inverse Projection B g C A

  26. Inverse Projection B B g g D Az’ d’ C A C A

  27. Inverse Projection B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  28. Inversion Issues • Find spherical azimuth: Az B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  29. Inversion Issues • Find spherical azimuth: Az linear non-linear, trigonometric functions B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  30. Inversion Issues • Find spherical azimuth: Az linear non-linear, trigonometric functions non-linear, inverse trig. functions B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  31. Inversion Issues • Find spherical azimuth: Az linear Use Iterative Method to Solve For Az non-linear, trigonometric functions non-linear, inverse trig. functions B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  32. Inversion Issues • Frequently called! (PYXIS, 2011)

  33. Optimizations • Operation Reduction B G g H D P Az q A C B’ D’ P’ Az’ d’ C’ A’

  34. Optimizations • Operation Reduction • Iteration Reduction B G g H D P Az q A C B’ D’ P’ Az’ d’ C’ A’

  35. Optimizations • Operation Reduction • Iteration Reduction • Iteration Avoidance B G g H D P Az q A C B’ D’ P’ Az’ d’ C’ A’

  36. Operation Reduction • Reduce repetitive calls • 2π, H • cos and sin calls • Pre-computation of values • Trigonometric calls (eg. sincos)

  37. Operation Reduction • Reduce repetitive calls • 2π, H • cos and sin calls • Pre-computation of values • Trigonometric calls (eg. sincos) • Nominal speed up • Note: No look-up table for cos and sin (would increase error)

  38. Iteration Reduction • Recall finding spherical azimuth: Az linear non-linear, trigonometric functions non-linear, inverse trig. functions B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  39. Iteration Reduction • Recall finding spherical azimuth: Az linear non-linear, trigonometric functions non-linear, inverse trig. functions B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  40. Iteration Reduction • Recall finding spherical azimuth: Az linear non-linear, trigonometric functions non-linear, inverse trig. functions B’ B B G g g g H D D’ Az’ Az q d’ A’ C A C A C’

  41. Iteration Reduction Newton Raphson: Iterative Solution Finding

  42. Iteration Reduction Newton Raphson: Iterative Solution Finding • Idea: Consider treating the iterative solution as a one-dimensional function

  43. Iteration Reduction • Idea: Consider treating the iterative solution as a one-dimensional function

  44. Iteration Reduction Polynomial Approximating Azimuthal Shift

  45. Iteration Reduction • Use polynomial for improved initial estimate of Newton-Raphson

  46. Iteration Elimination • Idea: Skip the iteration entirely, using this approximating function! • Note: Will need to evaluate error

  47. Results Need to: • Determine Runtime Improvements • Contrast Original with Iteration Reduction  especially regarding iteration drop • Establish Error for Elimination Approach

  48. Results: Runtime Improvements • Approach: • Profile inverse Snyder method using gprof • Ran 100 times, against four (4) quality levels • Quality: Quality 10 - Flat & Projected

  49. Results: Runtime Improvements

  50. Results: Iteration Reduction • Original vs Reduced