Databases
StackOverflow2013
Postsdbo
PObicubic interpolation of 3D surface
 

Note that there are some explanatory texts on larger screens.

plurals
  1. PObicubic interpolation of 3D surface
    primarykey
    data
    text
    <p>I am writing an algorithm in matlab to for bicubic interpolation of a surface <code>Psi(x,y)</code>. I have a bug in the code and cannot seem to track it down. I am trying a test case with <code>Psi=X^2-0.25</code> so that its easier to track down the bug. It seems as if my interpolation has an offset. My comments are included in my code. Any help would be appreciated.</p> <p>Plot of <code>Psi=X^2</code> in blue and interpolation in red <img src="https://i.stack.imgur.com/26UVE.jpg" alt="Plot of &lt;code&gt;Psi=X^2&lt;/code&gt; in blue and interpolation in red"></p> <p>Countour lines of <code>Psi</code> are plotted and the red dot is the point I am computing the interpolation about. The thick red line is the interpolation which is offset quite a bit from the red dot. <img src="https://i.stack.imgur.com/eXyhG.jpg" alt="Countour lines of &lt;code&gt;Psi&lt;/code&gt; are plotted and the red dot is the point I am computing the interpolation about. The thick red line is the interpolation which is offset quite a bit from the red dot."></p> <pre><code>function main() epsilon=0.000001; xMin=-1+epsilon; xMax= 1+epsilon; yMin=-1+epsilon; yMax= 1+epsilon; dx=0.1; Nx=ceil((xMax-xMin)/dx)+1; dy=0.1; Ny=ceil((yMax-yMin)/dy)+1; x=xMin:dx:xMax; x=x(1:Nx); y=yMin:dy:yMax; y=y(1:Ny); [XPolInX,XPolInY]=GetGhostMatricies(Nx,Ny); %Linear extrapolation matrix [D0x,D0y]=GetDiffMatricies(Nx,Ny,dx,dy); %derivative matricies: D0x is central differencing in x [X,Y]=meshgrid(x,y); Psi=X.^2-0.25; %Note that my algorithm is being written for a Psi that may not have a analytic representation. This Psi is only a test case. psi=zeros(Nx+2,Ny+2); %linearly extrapolate psi (for solving differential equation not shown here) psi(2:(Nx+1),2:(Ny+1))=Psi'; psi=(XPolInY*(XPolInX*psi)')'; %compute derivatives of psi psi_x =D0x*psi; psi_x =(XPolInY*(XPolInX*psi_x)')'; psi_y =(D0y*psi')'; psi_y =(XPolInY*(XPolInX*psi_y)')'; psi_xy=D0x*psi_y; psi_xy=(XPolInY*(XPolInX*psi_xy)')'; % i have verified that my derivatives are computed correctly biCubInv=GetBiCubicInverse(dx,dy); i=5; %lets compute the bicubic interpolation at this x(i), y(j) j=1; psiVoxel=[psi( i,j),psi( i+1,j),psi( i,j+1),psi( i+1,j+1),... psi_x( i,j),psi_x( i+1,j),psi_x( i,j+1),psi_x( i+1,j+1),... psi_y( i,j),psi_y( i+1,j),psi_y( i,j+1),psi_y( i+1,j+1),... psi_xy(i,j),psi_xy(i+1,j),psi_xy(i,j+1),psi_xy(i+1,j+1)]'; a=biCubInv*psiVoxel; %a=[a00 a01 ... a33]; polynomial coefficients; 1st index is power of (x-xi), 2nd index is power of (y-yj) xi=x(5); yj=y(1); clear x y x=(xi-.2):.01:(xi+.2); %this is a local region about the point we are interpolating y=(yj-.2):.01:(yj+.2); [dX,dY]=meshgrid(x,y); Psi=dX.^2-0.25; figure(2) %just plotting the 0 level contour of Psi here plot(xi,yj,'.r','MarkerSize',20) hold on contour(x,y,Psi,[0 0],'r','LineWidth',2) set(gca,'FontSize',14) axis([x(1) x(end) y(1) y(end)]) grid on set(gca,'xtick',(xi-.2):.1:(xi+.2)); set(gca,'ytick',(yj-.2):.1:(yj+.2)); xlabel('x') ylabel('y') [dX dY]=meshgrid(x-xi,y-yj); %P is my interpolating polynomial P = a(1) + a(5) *dY + a(9) *dY.^2 + a(13) *dY.^3 ... + a(2)*dX + a(6)*dX .*dY + a(10)*dX .*dY.^2 + a(14)*dX .*dY.^3 ... + a(3)*dX.^2 + a(7)*dX.^2.*dY + a(11)*dX.^2.*dY.^2 + a(15)*dX.^2.*dY.^3 ... + a(4)*dX.^3 + a(8)*dX.^3.*dY + a(12)*dX.^3.*dY.^2 + a(16)*dX.^3.*dY.^3 ; [c h]=contour(x,y,P) clabel(c,h) figure(3) plot(x,x.^2-.25) %this is the exact function hold on plot(x,P(1,:),'-r*') %See there is some offset here end %------------------------------------------------------------------------- function [XPolInX,XPolInY]=GetGhostMatricies(Nx,Ny) XPolInX=diag(ones(1,Nx+2),0); XPolInY=diag(ones(1,Ny+2),0); XPolInX(1,1) =0; XPolInX(1,2) =2; XPolInX(1,3) =-1; XPolInY(1,1) =0; XPolInY(1,2) =2; XPolInY(1,3) =-1; XPolInX(Nx+2,Nx+2)=0; XPolInX(Nx+2,Nx+1)=2; XPolInX(Nx+2,Nx)=-1; XPolInY(Ny+2,Ny+2)=0; XPolInY(Ny+2,Ny+1)=2; XPolInY(Ny+2,Ny)=-1; fprintf('Done GetGhostMatricies\n') end %------------------------------------------------------------------------- function [D0x,D0y]=GetDiffMatricies(Nx,Ny,dx,dy) D0x=diag(ones(1,Nx-1),1)-diag(ones(1,Nx-1),-1); D0y=diag(ones(1,Ny-1),1)-diag(ones(1,Ny-1),-1); D0x(1,1)=-3; D0x(1,2)=4; D0x(1,3)=-1; D0y(1,1)=-3; D0y(1,2)=4; D0y(1,3)=-1; D0x(Nx,Nx)=3; D0x(Nx,Nx-1)=-4; D0x(Nx,Nx-2)=1; D0y(Ny,Ny)=3; D0y(Ny,Ny-1)=-4; D0y(Ny,Ny-2)=1; %pad with ghost cells which are simply zeros tmp=D0x; D0x=zeros(Nx+2,Nx+2); D0x(2:(Nx+1),2:(Nx+1))=tmp; tmp=0; tmp=D0y; D0y=zeros(Ny+2,Ny+2); D0y(2:(Ny+1),2:(Ny+1))=tmp; tmp=0; %scale appropriatley by dx &amp; dy D0x=D0x/(2*dx); D0y=D0y/(2*dy); end %------------------------------------------------------------------------- function biCubInv=GetBiCubicInverse(dx,dy) %p(x,y)=a00+a01(x-xi)+a02(x-xi)^2+...+a33(x-xi)^3(y-yj)^3 %biCubic*a=[psi(i,j) psi(i+1,j) psi(i,j+1) psi(i+1,j+1) psi_x(i,j) ... psi_y(i,j) ... psi_xy(i,j) ... psi_xy(i+1,j+1)] %here, psi_x is the x derivative of psi %I verified that this matrix is correct by setting dx=dy=1 and comparing to the inverse here http://en.wikipedia.org/wiki/Bicubic_interpolation biCubic=[ %00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 1 dx dx^2 dx^3 0 0 0 0 0 0 0 0 0 0 0 0; 1 0 0 0 dy 0 0 0 dy^2 0 0 0 dy^3 0 0 0; 1 dx dx^2 dx^3 dy dx*dy dx^2*dy dx^3*dy dy^2 dx*dy^2 dx^2*dy^2 dx^3*dy^2 dy^3 dx*dy^3 dx^2*dy^3 dx^3*dy^3; 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 1 2*dx 3*dx^2 0 0 0 0 0 0 0 0 0 0 0 0; 0 1 0 0 0 dy 0 0 0 dy^2 0 0 0 dy^3 0 0; 0 1 2*dx 3*dx^2 0 dy 2*dx*dy 3*dx^2*dy 0 dy^2 2*dx*dy^2 3*dx^2*dy^2 0 dy^3 2*dx*dy^3 3*dx^2*dy^3; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 1 dx dx^2 dx^3 0 0 0 0 0 0 0 0; 0 0 0 0 1 0 0 0 2*dy 0 0 0 3*dy^2 0 0 0; 0 0 0 0 1 dx dx^2 dx^3 2*dy 2*dx*dy 2*dx^2*dy 2*dx^3*dy 3*dy^2 3*dx*dy^2 3*dx^2*dy^2 3*dx^3*dy^2; 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 1 2*dx 3*dx^2 0 0 0 0 0 0 0 0; 0 0 0 0 0 1 0 0 0 2*dy 0 0 0 3*dy^2 0 0; 0 0 0 0 0 1 2*dx 3*dx^2 0 2*dy 4*dx*dy 6*dx^2*dy 0 3*dy^2 6*dx*dy^2 9*dx^2*dy^2]; biCubInv=inv(biCubic); end %------------------------------------------------------------------------- </code></pre>
    singulars
    1. PO
      singulars
      1. PTAnswer
    1. This table or related slice is empty.
    1. PTQuestion
    1. USnamu
    1. USnamu
    plurals
    1. This table or related slice is empty.
    1. PL
      singulars
      1. LTLinked
    1. This table or related slice is empty.
    1. PO
      singulars
      1. PTAnswer
    1. This table or related slice is empty.
    1. This table or related slice is empty.
 

Querying!

 
Guidance
A row detail

Detail views are divided into sections. All the information in the data section comes from columns in the selected row. The other sections display data from other, related rows.

Related data can be related in a to-one or a to-many fashion. Captions of data related in a to-many fashion link to a list view showing a filtered view of the table.

Try moving around until you find a non-empty to-many entry and click on the label to get to one. You can move back to the root by clicking on the database name in the header.

SQuiL has stopped working due to an internal error.

If you are curious you may find further information in the browser console, which is accessible through the devtools (F12).

Reload