數值方法 2008, Applied Mathematics NDHU 1 Spline interpolation

數值方法 2008, Applied Mathematics NDHU 1

Spline interpolation

Spline A flexible piece of wood, hard rubber, or

metal used in drawing curves Spline

(mathematics) - Wikipedia, the free encyclopedia


http://en.wikipedia.org/wiki/Spline_(mathematics)



Before computers were used, numerical calculations were done by hand. Functions such as the step function were used but polynomials were generally preferred. With the advent of computers, splines first replaced polynomials in interpolation, and then served in construction of smooth and flexible shapes in computer graphics.[5]數值方法 2008, Applied Mathematics NDHU 3

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

Control Polygon


Spline

Spline Pictures


https://www.google.com.tw/search?num=10&hl=zh-TW&site=imghp&tbm=isch&source=hp&biw=1093&bih=450&q=spline&oq=spline&gs_l=img.3..0l2j0i24l8.1108.2113.0.2504.6.6.0.0.0.0.64.343.6.6.0...0.0...1ac.1.Ropc8SnKxYI

https://www.google.com.tw/search?num=10&hl=zh-TW&site=imghp&tbm=isch&source=hp&biw=1093&bih=450&q=spline&oq=spline&gs_l=img.3..0l2j0i24l8.1108.2113.0.2504.6.6.0.0.0.0.64.343.6.6.0...0.0...1ac.1.Ropc8SnKxYI

Spline approximation

Matlab Spline toolbox Input

paired data (x,y) x_in : a set of horizontal coordinates

Output: y_out, spline Interpolation at x_in



Example I: three input arguments

Input: x = 0:10; y = sin(x); x_in = 0:.25:10;Output: y_out = spline(x,y,x_in);Figure: plot(x,y,'o',x_in,y_out)


Example II: Two input arguments


x = -4:4; y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];

cs = spline(x,y);

xx = linspace(-4,4,101); plot(x,y,'o');

plot(xx,ppval(cs,xx),'-‘);

INPUT

OUTPUT

FIGURE

Polynomial output


Example III

2 D splines



2-D spline

demo_2d_spline.m


Polygon

figure;hold onpoints = [0 0; 1 0; 1 1; 0 2; -1 1; -1 0; 0 -1; 0 -2].';plot(points(1,:),points(2,:),'k'), axis([-2 2 -2.1 2.2]), grid offtitle('Control polygon')


2D spline

>> plot(points(1,:),points(2,:),'ok')>> fnplt( cscvn(points), 'g',1.5 )

CSCVN `Natural' or periodic interpolating cubic spline curve CS = CSCVN(POINTS) returns a parametric `natural' cubic spline that interpolates

to the given points POINTS(:,i) at parameter values t(i) , i=1,2,..., with t(i) chosen by Eugene Lee's centripetal

scheme, i.e., as accumulated square root of chord-length. When first and last point coincide and there are no double

points, then a parametric *periodic* cubic spline is constructed

instead. However, double points result in corners.數值方法 2008, Applied Mathematics NDHU 15


For example, x1=[1 0 -1 0 1];x2=[0 1 0 -1 0]; plot( x1,x2, ' ok ');hold on curve = cscvn([x1;x2]) ; fnplt( curve )

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


shows a (circular) curve through the four vertices of the standard diamond (because of the periodic boundary conditions enforced), while

x1=[1 0 -1 -1 0 1];x2=[0 1 0 0 -1 0]; plot( x1,x2, ' ok ');hold on curve = cscvn([x1;x2]) ; fnplt( curve )

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

shows a corner at the double point as well as at the curve endpoint.

Spline

A spline is composed of a set of piecewise polynomials

Linear spline Quadratic spline Cubic spline


Spline approximation

Linear spline First order piecewise polynomials

Quadratic spline Second order piecewise polynomials

Cubic spline Cubic piecewise polynomials



L=length(x_in)n=length(x)

for j=1:L

for k=1:n

if x(k) >= x_in(j)breakend

EXIT

function y_out=my_spline1(x,y,x_in)

i=k-1;a=(y(i+1)-y(i))/(x(i+1)-x(i));y_out(j)=y(i)+a*(x_in(j)-x(i));

Procedure: my_spline1

Function y_out=my_spline(x,y,x_in) L=length(x_in); for j=1:N

A. Find i that satisfies x(i) <= xx(i) <x(i+1) B. Set g to the slop of a line that connects

(x(i),y(i)) and (x(i+1),y(i+1)). C. y_out(j) = y(i)+g*(x_in(j)-x(i))


Line interpolating


(x(i),y(i))

(x(i+1),y(i+1))

(x_in(j),y_out(j)=?)

)()1(

)()1(*)]()(_[)()(_

ixix

iyiyixjinxiyjouty

Linear spline


x = 0:10; y = sin(x);x_in = 0:.25:10;y_out = my_spline1(x,y,x_in);plot(x,y,'o',x_in,y_out)

my_spline1.m



Quadratic Spline

polynomial quadratic a is

if )()( 1

i

iii

f

where

xxxxfxS


Quadratic spline

),( 11 ii yx

),( 11 ii yx

iiii cxbxaxf 2)(

112

11 )( iiii cxbxaxf

),( ii yx


Quadratic splines

Quadratic spline interpolation Determine all fi

i all )(' forzxSAssume ii


quadratic polynomial

iiiiii

iii yxxzxx

xx

zzxf

)()()(2

)( 2

1

1

iiii

iii zxx

xx

zzxf

)()(

)(1

1'

iii zxf )('

11' )( iii zxf

Smoothness criteria

The slopes of fi and fi+1 are identical at point Pi


)()( ''1 iiii xfxf

),( 11 ii yx

),( ii yx

),( 11 ii yx

)(xfi

)(1 xfi


Smoothness Criteria Checking Check iiiii zxfxf )()( ''

1

iiii

iii zxx

xx

zzxf

)()(

)(1

1'

111

1'1 )(

)()(

iiii

iii zxx

xx

zzxf

iiiiii

iiii zzxx

xx

zzxf

11

1

1'1 )(

)()(

Fitting criteria

Pi denotes point (xi yi)

fi passes points Pi and Pi+1


),( 11 ii yx

),( ii yx

),( 11 ii yx

)(xfi

)(1 xfi

11)( iii yxf

iii yxf )(


Fitting Criteria Checking - I Check iii yxf )(

iiiiii

iii yxxzxx

xx

zzxf

)()()(2

)( 2

1

1

i

iiiiiiii

iiii

y

yxxzxxxx

zzxf

)()()(2

)( 2

1

1


Fitting Criteria Checking - II Check

iiiiii

iii yxxzxx

xx

zzxf

)()()(2

)( 2

1

1

111

12

11

11

)(2

)()()(2

)(

iiiiii

iiiiiiii

iiii

yyxxzz

yxxzxxxx

zzxf

11)( iii yxf


Recurrence relations

111 )(2

iiiiii yyxxzz

1,...,0,21

11

niz

xx

yyz i

ii

iii


)1( 1,...,0,21

11 eqniz

xx

yyz i

ii

iii

determined becan ,...,

0Set

1

0

nzz

z

Assume

determined are )( All xfi


Equations for Implementation

1-n0,...,i

eq(2) )()()(2

)( 2

1

1

iiii

ii

iii yxxzxx

xx

zzxf

)1( 1,...,0,2

,...,0),,( : data Paired

1

11 eqniz

xx

yyz

niyx

iii

iii

ii


Re-indexing

n1,...,i

eq(2) )()()(2

)( 2

1

1

iiii

ii

iii yxxzxx

xx

zzxf

)1( ,...,1,2

1,...,1),,( : data Paired

1

11 eqniz

xx

yyz

niyx

iii

iii

ii


L=length(x_in)K=length(x)n=K-1;z(1)=0

for j=1:L

for k=1:K

if x(k) >= x_in(j)breakend

EXIT

function y_out=my_spline1(x,y,x_in)

i=k-1a=(z(i+1)-z(i))/(x(i+1)-x(i));b=z(i);c=y(i)y_out(j)=a*(x_in(j)-x(i))^2+b*(x_in(j)-x(i)) +c;

for i=1:n

a=(y(i+1)-y(i))/(x(i+1)-x(i))z(i+1)=2*a-z(i)

Procedure: my_spline2

Function y_out=my_spline2(x,y,x_in) n=length(x)-1;L=length(x_in); for i=1:n

Determine zi+1 by eq(1)

for j=1:L A. Find j that satisfies x(i) <= x_in(i) <x(i+1) B. substitute x_in(j) to eq(2) C. Set y_out(j) to the result of step B



Cubic spline

),( 11 ii yx

),( ii yx

),( 11 ii yx

iiiii dxcxbxaxf 23)(

112

13

11 )( iiiii dxcxbxaxf

Criteria of cubic spline

Fitting criteria First order smoothness criteria Second order smoothness criteria



Fitting criteria

),( 11 ii yx

),( ii yx

),( 11 ii yx

)(xfi

)(1 xfi

11)( iii yxf

iii yxf )(

Smoothness criteria I

The slopes of fi and fi+1 are identical at point Pi


)()( ''1 iiii xfxf

),( 11 ii yx

),( ii yx

),( 11 ii yx

)(xfi

)(1 xfi


Smoothness criteria II

The curvatures of fi and fi+1 are identical at point Pi

)()( ''1 iiii xfxf

),( 11 ii yx

),( ii yx

),( 11 ii yx

)(xfi

)(1 xfi )()( ''''1 iiii xfxf

Pi

Exercise 6 due to 10/29

Problem 3 of Ex 5


Problem 2

Draw a flow chart to illustrate linear spline interpolation

Implement the flow chart by matlab codes

Give examples to test your matlab codes


Problem 3

Draw a flow chart to illustrate quadratic spline interpolation

Implement the flow chart by matlab codes

Give examples to test your matlab codes


Documents

數值方法 2008, Applied Mathematics NDHU 1 Spline interpolation