數(shù)值分析15樣條插值

樣條插值多元插值《數(shù)值分析》14

http://www.tspline/先是雷諾和雪鐵龍工作的

PauldeCasteljau和PierreBézier,隨后美國通用汽車的其它人一起推動了現(xiàn)在稱為三次樣條和Bézier

樣條的建立。樣條是通過很少的控制點就能夠生成復(fù)雜平滑曲線的方法。Ref:/design/movies-before-after-green-screen-cgi/只給定離散數(shù)據(jù),不知道函數(shù),當然也不知道其導(dǎo)數(shù)(或?qū)?shù)值)。定義

1

給定區(qū)間[a,b]上的一個分劃:a=x0<x1<…<xn=b分段三次多項式滿足如下條件:

(1)Sk(xk-1)=yk-1,Sk(xk)=yk

(k=1,···,n)

(2)

S'k(xk)=S'k+1(xk)

(k=1,···,n-1)(3)S''k(xk)=S''k+1(xk)

(k=1,···,n-1)則稱

S(x)為三次樣條插值函數(shù)。當xk-1

≤x≤xk時,

Sk(x)=ak+bkx+ckx2+dkx3(k=1,…,n)條件:(1)Sk(xk-1)=yk-1,Sk(xk)=yk

(k=1,···,n)(2)S'k(xk)=S'k+1(xk)

(k=1,···,n-1)(3)S''k(xk)=S''k+1(xk)

(k=1,···,n-1)由樣條插值函數(shù)滿足條件,可以建立(4n-2)個方程！

n個三次多項式,共4n個待定系數(shù)。自然邊界條件:S''(x0)=0,S''(xn)=0例1已知f(–1)=1,f(0)=0,f(1)=1。構(gòu)造分段三次多項式是滿足自然邊界的樣條函數(shù)。解:如何構(gòu)造滿足條件的樣條插值函數(shù)？分段Hermite插值插值函數(shù)已經(jīng)滿足插值條件和一階導(dǎo)數(shù)連續(xù)。怎么保證二階導(dǎo)數(shù)連續(xù)

？

當xk-1

≤x≤xk時,

Sk(x)=ak+bkx+ckx2+dkx3(k=1,…,n)當

xk-1

≤x≤xk時,

Sk(x)由如下基函數(shù)組成當

xk≤x≤xk+1時,

Sk+1

(x)由如下基函數(shù)組成由(k=1,2,···,n-1)

待定未知數(shù)n+1個,建立n-1個方程。自然樣條的導(dǎo)數(shù)值滿足:設(shè)自然邊界條件成立即(k=1,2,······,n-1)自然樣條的導(dǎo)數(shù)值滿足:(k=1,2,······,n-1)回顧:嚴格主對角占優(yōu)矩陣一定是非奇異的。Demo

x=-5:5;y=1./(x.^2+1);u=-5:.01:5;v1=polyinterp(x,y,u);plot(x,y,'o',u,v1,'-')holdon,v2=piecelin(x,y,u);plot(u,v2,'r-')holdon,v3=spline(x,y,u);plot(u,v3,'b-')interpgui

樣條插值是插值函數(shù)的光滑性與局部單調(diào)性之間的折衷方案。rowcol二維插值a bc daa bbaa bbcc ddcc ddzero-orderfirst-ordera (a+b)/2 b(a+c)/2

(a+b+c+d)/4(b+d)/2c (c+d)/2 d雙線性插值

一維數(shù)據(jù)的插值interp1x=0:2:24;y=[22,21,19,18,20,24,27,32,31,28,26,23,22];xi=0:0.1:24;yi=interp1(x,y,xi,’spline’);plot(xi,yi,’-’,x,y,’o’);二維網(wǎng)格數(shù)據(jù)的插值interp2[x,y]=meshgrid(1:5,1:3);z=[8281808284;7963616581;8484828586];mesh(x,y,z)[xi,yi]=meshgrid(1:0.2:5,1:0.2:3);zi=interp2(x,y,z,xi,yi,'spline');mesh(xi,yi,zi);x=double(imread('fl_orig.pgm'));figure,subplot(221),imshow(x,[]);%helpinterp2(orimresize)%zero-orderinterpolation(replication)y0=interp2(x,'nearest');subplot(222),imshow(y0,[]);%first-orderinterpolation(bilinearinterpolation)y1=interp2(x,'linear');subplot(223),imshow(y1,[]);%bicubicinterpolationy2=interp2(x,'cubic');subplot(224),imshow(y2,[]);%%colorimagex=double(imread('fl_orig.ppm'));figure,subplot(221),imshow(x/255,[]);fori=1:3%zero-orderinterpolation(replication)y0(:,:,i)=interp2(x(:,:,i),'nearest');%first-orderinterpolation(bilinearinterpolation)y1(:,:,i)=interp2(x(:,:,i),'linear');%bicubicinterpolationy2(:,:,i)=interp2(x(:,:,i),'cubic');endsubplot(222),imshow(y0/255,[]);subplot(223),imshow(y1/255,[]);subplot(224),imshow(y2/255,[]);%Howtoimplementimageinterpolationbyyourself?figure,subplot(221),imshow(x,[]);%zero-orderinterpolation(replication)[M,N]=size(x);z0=zeros(2*M,2*N);z0(1:2:2*M,1:2:2*N)=x;h=[11;11];z0=filter2(h,z0);subplot(222),imshow(z0,[]);%first-orderinterpolation(bilinearinterpolation)z1=zeros(2*M,2*N);z1(1:2:2*M,1:2:2*N)=x;h=[121;242;121]/4;z1=filter2(h,z1);subplot(223),imshow(z1,[]);%cubicinterpolationz2=zeros(2*M,2*N);z2(1:2:2*M,1:2:2*N)=x;h1=[-1091690-1]/16;h=h1'*h1;z2=filter2(h,z2);subplot(224),imshow(z2,[]);Ref：Cubicconvolutioninterpolationfordigitalimageprocessing最近鄰插值low-resolutionimage