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Interpolation Dr. Mihail October 26, 2015 (Dr. Mihail) Interpolation October 26, 2015 1 / 11 Definitions 1 Interpolation: method of constructing new data points from sampling or experimentation. (Dr. Mihail) Interpolation October 26, 2015

SLIDE 1

Interpolation

Dr. Mihail

October 26, 2015

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SLIDE 2

Definitions

1 Interpolation: method of constructing new data points from sampling

r experimentation.

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SLIDE 3

Case study

Spoze we have

x f(x) 1 0.8415 2 0.9093 3 0.1411 4

0.7568

5

0.9589

6

0.2794

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SLIDE 4

Case study

Piecewise constant

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SLIDE 5

Case study

Piecewise linear

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Case study

Piecewise linear

Given two data points, say (xa, ya) and (xb, yb), the interpolant function is given by: y = ya + (yb − ya) x−xa

xb−xa

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Case study

Polynomial

f (x) = −0.0001521x6 −0.003130x5 +0.07321x4 −0.3577x3 +0.2255x2 +0.9038x

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Other methods

Interpolation Methods

Piecewise polynomial Spline Barycentric coordinates for interpolating on a triangle or tetrahedron Gaussian process And others...

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2D functions

Extend concept to multivariate functions

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Bilinear interpolation

Problem setting

Suppose you have a function f (x, y). You know the value of that function for a limited number of points (e.g., 4 points). Your goal is to approximate the function at arbitrary points (x, y).

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Bilinear interpolation

We first do linear interpolation along the X-axis: R1 = f (x, y1) ≈ x2 − x x2 − x1 f (Q11) + x − x1 x2 − x1 f (Q21) R2 = f (x, y2) ≈ x2 − x x2 − x1 f (Q12) + x − x1 x2 − x1 f (Q22)

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Bilinear interpolation

We then do linear interpolation along the Y-axis: f (x, y) ≈ y2 − y y2 − y1 f (x, y1) + y − y1 y2 − y1 f (x, y2)

f (x, y) ≈ 1 (x2 − x1)(y2 − y1) (f (Q11)(x2−x)(y2−y)+f (Q21)(x−x1)(y2−y)+f (Q12)(x2−x)(y−y1)+f (Q22)(x−x1)(y−y1)) (Dr. Mihail) Interpolation October 26, 2015 12 / 11