Interpolation - 2D mapping Tutorial 1: triangulation

Transcription

1 Tutorial 1: triangulation Measurements (Zk) at irregular points (xk, yk) Ex: CTD stations, mooring, etc... The known Data How to compute some values on the regular spaced grid points (+)? The unknown data Zi Y estimated data known data linear combination X

2 Tutorial 1: triangulation Measurements (Zk) at irregular points (xk, yk) Ex: CTD stations, mooring, etc... The known Data How to compute some values on the regular spaced grid points (+)? The unknown data Zi Y estimated data known data linear combination X

3 Tutorial 1: triangulation 3 2 1

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7 1 dimension linear interpolation Data Points Estimated Value

8 RT is the error asociated to the method. We try to estimate the maximum possible error. Estimation of the error:

9 1 dimension linear interpolation

10 1 dimension linear interpolation The derivative is not continuous: Piecewise linear interpolation is not differentiable at the xdata points

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

12 1 dimension Lagrange polynomial interpolation

13 1 dimension Lagrange polynomial interpolation

14 1 dimension Lagrange polynomial interpolation

15 1 dimension Lagrange polynomial interpolation

16 1 dimension Lagrange polynomial interpolation

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18 1 dimension Lagrange polynomial interpolation VANDERMONDE MATRIX

19 1 dimension Lagrange polynomial interpolation

20 1 dimension Lagrange polynomial interpolation Exemple 2:

21 1 dimension Lagrange polynomial interpolation Exemple 2: Polynomial interpolation is differentiable at the xdata points Polynomial interpolation may generate overshoots, spikes

22 2 dimension bilinear interpolation Estimated value Data Points

23 2 dimension bilinear interpolation

24 1 dimension Piecewise Cubic Hermite Interpolation Hermite function :Functions that satisfy interpolation conditions derivatives If we know P(xk),P'(xk),P(xk+1),P'(xk+1) then piecewise cubic Hermite interpolation can reproduce the data on the interval [xk xk+1] Problem: We usually do not know the values of derivatives

25 1 dimension Piecewise cubic interpolation Functions that satisfy interpolation conditions on derivatives: HERMITE interpolants on the interval xk< x< xk+1 s = x - xk Conditions satisfied by this function: Knowing BOTH values of the functions and its FIRST derivatives at a discrete set of data points, then, we can reproduce the data on each interval xk< x< xk+1

26 1 dimension Piecewise cubic interpolation Knowing BOTH values of the functions and its FIRST derivatives at a discrete set of data points, then, we can reproduce the data on each interval xk< x< xk+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

27 1 dimension Shape preserving cubic interpolation We wish to determine the slope dk so that the function values do not overshoot Curve line (shape preserving) JUMP in second derivative! Piecewise linear interpolation (straight line)

28 1 dimension Shape preserving cubic interpolation We wish to determine the slope dk so that the function values do not overshoot Curve line (shape preserving) JUMP in second derivative! Piecewise linear interpolation (straight line)

29 1 dimension Shape preserving cubic interpolation We wish to determine the slope dk so that the function values do not overshoot

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32 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 large outliers and aggravate the impact of small ones. Ex: speed of a car)

33 One-sided formula at end points...

34 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/v1 t2=d2/v2 vbarharm=(d1+d2)/(d1/v1+d2/v2)

35 1 dimension Shape preserving cubic interpolation

36 1 dimension Cubic Spline interpolation We add a constraint on the continuity of the second derivative

37 1 dimension Cubic Spline interpolation We add a constraint on the continuity of the second derivative Valid on the interval: xk< x< xk+1

38 1 dimension Cubic Spline interpolation We 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 xn-1 We write the equations on the interval: x1< x < x3 xn-2<x<xn

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40 1 dimension Piecewise cubic interpolation Functions that satisfy interpolation conditions on derivatives: HERMITE interpolants on the interval xk< x< xk+1 s = x - xk Conditions satisfied by this function:

41 Boundary Conditions for splines Natural Spline: (N points, N-1 intervals)

42 Boundary Conditions for splines Not a Knot:

43 Boundary Conditions for splines Periodic boundary conditions: GHOST POINT x0 x1 x2 x3 x4 xn-3 L GHOST POINT xn-2 xn-1 xn xn+1

44 Boundary Conditions for splines Periodic boundary conditions: GHOST POINT x0 x1 x2 x3 x4 xn-3 L GHOST POINT xn-2 xn-1 xn xn+1

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47 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)