Interpolation - 2D mapping Tutorial 1: triangulationstockage.univ-brest.fr/~herbette/...interpolation... · Interpolation - 2D mapping 1 dimension Piecewise Cubic Hermite Interpolation

Interpolation - 2D mappingTutorial 1: triangulation

Measurements (Zk) at irregular points (x

k , y

k)

Ex: CTD stations, mooring, etc ...The “known Data”

X

Y

How to computesome values on the

regular spacedgrid points (+) ?

The” unknown data”Z

i

knowndata

estimateddata

linearcombination


Measurements (Zk) at irregular points (x

k , y

k)

Ex: CTD stations, mooring, etc ...The “known Data”

X

Y

How to computesome values on the

regular spacedgrid points (+) ?

The” unknown data”Z

i

knowndata

estimateddata

linearcombination

1

3

2


http://en.wikipedia.org/wiki/Barycentric_coordinate_system

Interpolation - 2D mapping1 dimension linear

interpolation

Data Points

Estimated Value

RT is the error asociated to the method.

We try to estimate the maximum possible error.

Estimation of the error:


interpolation


interpolation

http://fr.mathworks.com/moler/

The derivative is notcontinuous:

Piecewise linearinterpolation is notdifferentiable at the

xdata points

Matlab tutorial:Make a Piecewise linear interpolation porgram using matlabCompare results with the interp1 matlab command


Interpolation - 2D mapping1 dimension

Lagrange polynomial interpolation





VANDERMONDE MATRIX








Lagrange polynomial interpolationExemple 2:



Lagrange polynomial interpolationExemple 2:

Polynomial interpolation isdifferentiable at the xdata

points

Polynomial interpolationmay generate overshoots,

spikes


Interpolation - 2D mapping2 dimension bilinear

interpolation

Data Points

Estimated value

http://en.wikipedia.org/wiki/Bilinear_interpolation

Interpolation - 2D mapping2 dimension bilinear

interpolation

http://en.wikipedia.org/wiki/Bilinear_interpolation



Piecewise Cubic Hermite Interpolation

Hermite function :Functions that satisfy interpolation conditions derivatives

If we know P(xk),P'(x

k),P(x

k+1),P'(x

k+1) then piecewise cubic Hermite interpolation can reproduce

the data on the interval [xk x

k+1]

Problem: We usually do not know the values of derivatives


Piecewise cubic interpolation

Functions that satisfy interpolation conditions on derivatives: HERMITE interpolants

on the interval xk< x< x

k+1

s = x - xk

Conditions satisfiedby this function:

Knowing BOTH values of the functions and its FIRST derivatives at a discreteset of data points, then, we can reproduce the data on each interval x

k< x< x

k+1



Knowing BOTH values of the functions and its FIRST derivatives at a discreteset of data points, then, we can reproduce the data on each interval x

k< x< x

k+1

PROBLEMS:What happens if we do not know the derivatives ....???

There are different ways of estimating these derivatives.We look for a way that reduces the overshoots and spikes.

- splines- shape preserving cubics


Shape preserving cubic interpolation

We wish to determine the slope dk so that the function values do not overshoot

Piecewise linear interpolation(straight line)

Curve line (shape preserving)

JUMP in secondderivative !




Piecewise linear interpolation(straight line)

Curve line (shape preserving)

JUMP in secondderivative !




We still need to evaluate the slopes dk

Harmonic mean at interior points (it tends (compared to the arithmetic mean) to mitigate the impact of largeoutliers and aggravate the impact of small ones. Ex: speed of a car)

One-sided formula at end points ....

Arithmetic and Harmonic mean:A car travels a distance h1 and speed v1, and distance h2 at speed v2

Arithmetic mean:vbarith=(v1+V2)/2

Harmonic mean:t1=d1/v1t2=d2/v2

vbarharm=(d1+d2)/(d1/v1+d2/v2)




Cubic Spline interpolation

We add a constraint on thecontinuity of the second derivative


Cubic Spline interpolationWe add a constraint on the continuity of the second derivative

Valid on the interval:x

k< x< x

k+1


Cubic Spline interpolationWe add a constraint on the continuity of the second derivative:

P''(xk

+)=P''(xk

-)

This approach can be applied to the interior “knots”, k=2,...n-1=> (n-2) equations

We must add boundary conditions on the “first” and “end” intervals

Here written for hk=cte

“not a knot “ choice for x2 and x

n-1

We write the equations on the interval:x

1< x < x

3

xn-2

<x<xn



Functions that satisfy interpolation conditions on derivatives: HERMITE interpolants

on the interval xk< x< x

k+1

s = x - xk

Conditions satisfiedby this function:

Boundary Conditions for splines

● Natural Spline:

(N points, N-1 intervals)


● Not a Knot:


● Periodic boundary conditions:

x2

x3

x4

xN-3

xN-2

xN-1

xN

xN+1

x1

x0

GHOSTPOINT

GHOSTPOINT

L


● Periodic boundary conditions:

x2

x3

x4

xN-3

xN-2

xN-1

xN

xN+1

x1

x0

GHOSTPOINT

GHOSTPOINT

L

● These techniques can be adapted to 2D or 3D● If sparse data, and a smooth is what you are

looking for, then a spline surface is adapted● If very noisy data, or closely spaced data, then

a “smoothed spline”, may be better.● See Smith and Wessel (1990)

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Interpolation - 2D mapping Tutorial 1: triangulationstockage.univ-brest.fr/~herbette/...interpolation... · Interpolation - 2D mapping 1 dimension Piecewise Cubic Hermite Interpolation