1. Presentation and Analysis of a Multi-Dimensional Interpolation Function for Non-Uniform Data: Microsphere Projection Presented By: Will Dudziak

2. What We Will Cover… Introduce Interpolation Overview of Existing Techniques Introduce Microsphere Projection Analysis of Results –1D –2D –3D Conclusion and Final Remarks Q & A Hand with Reflecting Sphere M. C. Escher, 1935

3. Introduction – What is Interpolation? “the process of determining the value of a function between two points at which it has prescribed values.” The hill behind me is low, the mountain in front of me is high… therefore I must be somewhere in between low and high.

4. Introduction – Some Terminology “Control Points” or “Sample Points” Differentiable C 0, C 1, C 2, etc. Functional Interpolation Inverse-Distance p-value Monotonic Behavior

5. Introduction – Local vs. Global Interpolation Local Interpolation –Makes use of information from small set of ‘local’ points. Global Interpolation –Makes use of information from all points Key Difference: –Local interpolation applies different set of rules to close sample points than to far ones. vs.

6. Introduction – Exact vs. Inexact Interpolation Exact interpolations intersect ALL control points. Inexact interpolations follow general trends. Usages: –Exact: Largest problem with data is in scarcity of control points. –Inexact: Largest problem with data is in error margin of sampled value.

7. Popular Existing Interpolation Methods 1D –Polynomial Function Approximation –Piecewise Cubic Spline 2D –Thin-Plate Spline (TPS) 3D –Volume Spline –Multiquadric 1D, 2D, 3D –Nearest Neighbor –Shepard’s Method (inverse-distance weighting)

8. Popular Existing Interpolation Methods (1-D) Polynomial Function Interpolation Cubic Spline Interpolation

9. Radial Basis Function Premise: –Interpolation in N-dimensional space can be approximated by a function of form: A*f(d 1 ) + B*f(d 2 ) + C*f(d 3 ) + … + Z*f(d N ) ABC…Z are constants d i is the distance from interpolation location to control point i. Time Complexity: O(N 2 ) TPS, Volume Spline, and Multiquadric use RBF.

10. Shepard‘s Method Shepard’s Method, aka Inverse- distance Weighting. –Assign weights to each control point based on distance to interpolation location. –Based on weight distribution, interpolate value of point. Fundamental Problem with Shepard’s Method is the non-consideration of locality.

11. Microsphere Projection - Objectives Obeys Maximum Principle Differentiable (smooth) Non-consideration of redundant control points. (problem with Shepard’s method) Reasonable extrapolation ability

12. Microsphere Projection - Premise infinitesimally small sphere located at the point of interpolation. sphere is ‘illuminated’ by the surrounding sampled points. Based on the degree of illumination, a weight for all the control points assigned.

13. MS – Data Structure 1x Microsphere –2000x Sphere Segment 1x Index of control point with largest influence (int) 1x Projection influence (double) “Data Structure” not to be confused with “Data’s Structure”.

14. MS – Creating the Sphere Sphere needs lots of sections to be ‘spherical’. 2000 sections were used. Section unit vectors were created by: do x := rand(-1,1) y := rand(-1,1) z := rand(-1,1) vectorSize := sqrt (x*x + y*y + z*z) while ( vectorSize > 1 ) normalize ( Vector(x,y,z) )

15. MS – Control Point Projection Points are ‘projected’ on to the sphere. –Projection intensity varies: Inversely as the distance. Inversely as the angle of incidence. (governed by the cosine function) –Each section of the sphere only retains the information for the ONE POINT with largest projection value.

16. MS – Control Point Projection

17. Intensity Projection Function: for i := 0 to Number of Samples vector1 := sample[i].XYZLocation - interpolant.XYZLocation weight := pow(vector1.Size, -p) for j := 0 to Precision cosValue := CosValueBetweenVectors(vector1, S[j].Vector) if (cosValue * weight > S[j].Max_Illumination) S[j].Max_Illumination := cosValue * weight S[j].Brightest_Sample := i endif endfor

18. MS – Determining Final Interpolation Value Weight for each control point is assigned based on total sum of illumination on all the sections of the sphere. Weights are normalized such that Sum(weights) = 1. Final interpolation is: Σ(weight*control_point_value)

19. MS – Walkthrough 0.19 0.55 0.83 0.98 0.98 0.83 0.55 0.24 0.24 0.55 0.83 0.98 0.98 0.83 0.55 0.19 4.91 0.48 10.30 10dB 5dB 40dB Dist = 2 Dist = 1 [cos(θ) / dist 2 ] 0.48 0.04 1.00 0.48 * 10dB 0.48 * 5dB 0.04 * 40dB 8.8dB

20. 1-D Results – Case Study 1

21. 1-D Results – Case Study 2

22. 1-D Results – Source of Data

23. 1-D Results – Numeric Percentage of Data Sampled Method of Interpolation2.5%5%10%25%50%90% Microsphere Projection, p=20.17310.14290.11350.08280.06510.0541 Microsphere Projection, p=1 (piecewise linear)0.16880.13920.11050.08060.06350.0531 Piecewise Cubic Spline0.17440.14380.11370.08250.06440.0533 Shepard's Method, p=2 (inverse distance)0.16980.14100.11360.08480.06890.0601 Nearest Neighbor0.19060.15920.12810.09510.07690.0668 Average Value0.21860.21550.21340.21200.21170.2111 N (number of samples)3.642e6 1.821e67.284e53.642e5 RMS Error using Strict Interpolation on 1D Data

24. 1-D Results – Numeric Percentage of Data Sampled Method of Interpolation2.5%5%10%25%50%90% Microsphere Projection, p=20.19130.15030.11670.08380.06600.0545 Microsphere Projection, p=1 (piecewise linear)0.18820.14720.11390.08170.06450.0535 Piecewise Cubic Spline3733696.170.332.8300.34060.0584 Shepard's Method, p=2 (inverse distance)0.18270.14640.11610.08570.06970.0606 Nearest Neighbor0.20380.16400.12990.09570.07770.0675 Average Value0.22400.21750.21420.21240.21180.2114 N (number of samples)3.642e6 1.821e67.284e53.642e5 RMS Error using General Interpolation on 1D Data

25. 1-D Results – PCS Explanation In some cases, even asking for a small amount of extrapolation can be too much to ask.

26. 2-D Results – Case Study 1 Original Nearest Neighbor Shepard’s Method TPS MS, p=1 MS, p=2

27. 2-D Results – Case Study 1 TPS, Values truncated to 0-255 TPS, Areas interpolated beyond data range in red.

28. 2-D Results – Case Study 2 (Area of Interest) MS, p=1 MS, p=2 NN TPS

29. 2-D Results – Strict vs. General Interpolation

30. 2-D Results – Numeric Number of Points Sampled Method of Interpolation ( STRICT ) 1020501005001000 Microsphere Projection, p=20.2510.2310.2010.1780.1330.116 Microsphere Projection, p=1 (2d version of piecewise linear)0.2420.2220.1930.1720.1280.112 Thin-Plate Spline0.2690.2480.2140.1890.1380.121 Shepard's Method, p=2 (inverse distance)0.2410.2220.1970.1800.1500.140 Nearest Neighbor0.2890.2680.2340.2100.1590.140 Average Value0.2650.2590.2550.2510.2500.249 N (number of samples)1.0e5 Number of Points Sampled Method of Interpolation ( GENERAL ) 1020501005001000 Microsphere Projection, p=20.2580.2340.2040.1800.1330.116 Microsphere Projection, p=1 (2d version of piecewise linear)0.2470.2250.1960.1730.1290.112 Thin-Plate Spline0.3200.2730.2260.1950.1400.122 Shepard’s Method, p=2 (inverse distance)0.2420.2210.1970.1810.1510.139 Nearest Neighbor0.2920.2680.2370.2110.1600.141 Average Value0.2600.2540.250 0.2490.248 N (number of samples)1.0e5 Convex Hull Coverage43.6%62.3%80.0%88.2%96.6%98.1%

31. 3-D Results – Case Study Nearest Neighbor Shepard’s Method

32. 3-D Results – Case Study Volume Spline Multiquadric

33. 3-D Results – Case Study Microsphere Interpolation, p=2

34. 3-D Results – Numeric Method of InterpolationRelative RMS Error Microsphere Projection, p=20.080 Microsphere Projection, p=1 (piecewise linear)0.081 Volume Spline0.093 Multiquadric, r=10.077 Shepard's Method, p=2 (inverse distance)0.100 Nearest Neighbor0.110 Average Value0.168 N (Number of samples)145

35. Hyper-Dimensional Interpolation Microsphere Projection will work in any Cartesian coordinate system. Rotating Hypersphere. Oooooh, Aaaaah.

36. Hyper-Dimensional Interpolation As dimensionality increases, bounding box becomes much larger than convex hull. Greater need for algorithms which can both interpolate and extrapolate. Bounding BoxConvex Hull bb d is the d-dimensional bounding box ch d is the d-dimensional convex hull ║ x ║ is the volume of space enclosed by x

37. Conclusion Great interpolation ability Pretty good (and stable) extrapolation ability. Differentiable Preserves monotonic behavior Works well in any dimension Based on a simple physical model Buckminsterfullerene. Perhaps the smallest real-world microsphere possible, measuring around 0.000000001 meters.