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

Presentation and Analysis of a Multi-Dimensional Interpolation Function for Non-Uniform Data: Microsphere Projection

Presented By:Will Dudziak

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

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.

Introduction – Some Terminology

• “Control Points” or “Sample Points”

• Differentiable• C0, C1, C2, etc.• Functional Interpolation• Inverse-Distance p-value• Monotonic Behavior

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.

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.

InexactFunctional Approximation

ExactFunctional Approximation





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)

Popular Existing Interpolation Methods (1-D)

Polynomial FunctionInterpolation

1-Dimensional Polynomial Interpolationdegree=4

Interpolation

Control Points

First Derivative

1-Dimensional Cubic Spline Interpolation

Interpolation

Control Points

First Derivative

Cubic Spline Interpolation

Radial Basis Function

• Radial Basis Function• Premise:

– Interpolation in N-dimensional space can be approximated by a function of form:•A*f(d1) + B*f(d2) + C*f(d3) + … + Z*f(dN)•ABC…Z are constants•di is the distance from interpolation

location to control point i.•Time Complexity: O(N2)

• TPS, Volume Spline, and Multiquadric use RBF.

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.

Microsphere Projection - Objectives

• Obeys Maximum Principle• Differentiable (smooth)• Non-consideration of redundant

control points. (problem with Shepard’s method)

• Reasonable extrapolation ability

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.

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”.

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) )

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.



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 endforendfor

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)

MS – Walkthrough0.190.550.830.980.980.830.550.240.240.550.830.980.980.830.550.19

4.91

4.910.48

10.30

10dB

5dB

40dB

Dist = 2

Dist = 1

[cos(θ) / dist2]

0.48

0.480.04

1.00

0.48 * 10dB

0.48 * 5dB0.04 * 40dB

8.8dB

1-D Results – Case Study 1

1-Dimensional Nearest Neighbor Interpolation

Interpolation

Control Points

1-Dimensional Polynomial Interpolationdegree=4 Interpolation

Control Points

1-Dimensional Inverse-Distance Interpolationp=2 Interpolation

Control Points


Interpolation

Control Points

1-Dimensional MS Projectionp=1 Interpolation

Control Points


Control Points


1-Dimensional Nearest Neighbor Interpolation

Interpolation

Control Points

1-Dimensional Polynomial Interpolationdegree=4 Interpolation

Control Points

1-Dimensional Inverse-Distance Interpolationp=2 Interpolation

Control Points


Interpolation

Control Points


Control Points


Control Points

1-D Results – Source of Data

1-D Results – Numeric

Pixel 0 Pixel N

One Pixel Row

Strict Interpolation

General Interpolation

Randomly Sampled PixelTesting Region

Pixel 0 Pixel N

One Pixel Row




Percentage of Data Sampled

Method of Interpolation 2.5% 5% 10% 25% 50% 90%

Microsphere Projection, p=2 0.1731 0.1429 0.1135 0.0828 0.0651 0.0541

Microsphere Projection, p=1 (piecewise linear) 0.1688 0.1392 0.1105 0.0806 0.0635 0.0531

Piecewise Cubic Spline 0.1744 0.1438 0.1137 0.0825 0.0644 0.0533

Shepard's Method, p=2 (inverse distance) 0.1698 0.1410 0.1136 0.0848 0.0689 0.0601

Nearest Neighbor 0.1906 0.1592 0.1281 0.0951 0.0769 0.0668

Average Value 0.2186 0.2155 0.2134 0.2120 0.2117 0.2111

N (number of samples) 3.642e6 3.642e6 3.642e6 1.821e6 7.284e5 3.642e5

RMS Error using Strict Interpolation on 1D Data


Pixel 0 Pixel N

One Pixel Row




Pixel 0 Pixel N

One Pixel Row




Percentage of Data Sampled

Method of Interpolation 2.5% 5% 10% 25% 50% 90%


Microsphere Projection, p=1 (piecewise linear) 0.1882 0.1472 0.1139 0.0817 0.0645 0.0535

Piecewise Cubic Spline 3733 696.1 70.33 2.830 0.3406 0.0584



Average Value 0.2240 0.2175 0.2142 0.2124 0.2118 0.2114


RMS Error using General Interpolation on 1D Data

1-D Results – PCS Explanation

In some cases, even asking for a small amount of extrapolation can be too much to ask.


Original

NearestNeighbor

Shepard’s Method

TPS

MS, p=1

MS, p=2


TPS,Values

truncated to 0-255

TPS,Areas

interpolated beyond data

range in red.

2-D Results – Case Study 2 (Area of Interest)

MS, p=1

MS, p=2

NN

TPS

2-D Results – Strict vs. General Interpolation

Strict InterpolationLimited toConvex Hull

General InterpolationLimited toOriginal Data Range

Strict InterpolationLimited toConvex Hull

General InterpolationLimited toOriginal Data Range

2-D Results – NumericNumber of Points Sampled

Method of Interpolation (STRICT) 10 20 50 100 500 1000


Microsphere Projection, p=1 (2d version of piecewise linear) 0.242 0.222 0.193 0.172 0.128 0.112

Thin-Plate Spline 0.269 0.248 0.214 0.189 0.138 0.121



Average Value 0.265 0.259 0.255 0.251 0.250 0.249


Number of Points Sampled

Method of Interpolation (GENERAL) 10 20 50 100 500 1000


Microsphere Projection, p=1 (2d version of piecewise linear) 0.247 0.225 0.196 0.173 0.129 0.112

Thin-Plate Spline 0.320 0.273 0.226 0.195 0.140 0.122

Shepard’s Method, p=2 (inverse distance) 0.242 0.221 0.197 0.181 0.151 0.139


Average Value 0.260 0.254 0.250 0.250 0.249 0.248


Convex Hull Coverage 43.6% 62.3% 80.0% 88.2% 96.6% 98.1%

3-D Results – Case Study

100% below sample range. (very negative)

Within 0.01% of minimum value of sample range (zero)

Maximum value of sample range (very large)

100% beyond sample range (inconceivably large)

Range ofsampled data

Control Point






Control Point

Nearest Neighbor

Shepard’s Method







Control Point






Control Point

Volume Spline

Multiquadric







Control Point






Control Point

Microsphere Interpolation, p=2


Method of Interpolation Relative RMS Error

Microsphere Projection, p=2 0.080

Microsphere Projection, p=1 (piecewise linear) 0.081

Volume Spline 0.093

Multiquadric, r=1 0.077

Shepard's Method, p=2 (inverse distance) 0.100

Nearest Neighbor 0.110

Average Value 0.168

N (Number of samples) 145

Hyper-Dimensional Interpolation

• Microsphere Projection will work in any Cartesian coordinate system.

Rotating Hypersphere. Oooooh, Aaaaah.

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

d

d

ch

bbdas ,

bbd is the d-dimensional bounding boxchd is the d-dimensional convex hull

║x║ is the volume of space enclosed by x

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.

Documents

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