how to Differentiate 3D points

Hi guys,

I have some 3D points of a 3D surface in x, y and z format. You can get the points here: http://textuploader.com/?p=6&id=EjEtc

Assuming 'M' is my 3D surface (i.e. made up of the 3D points), I would like to get the derivative of M with respect to the 'x' direction and the 'y' direction. That is, I want to differentiate M wrt to the x and y coordinates (i.e. gradient in x and y directions).

My data is not a uniform grid. Any ideas how to do this in Matlab ?

Answer 1

[dpb]

See

doc gradient

maybe? Look at alternate syntax with spacing inputs not just minimum case.

[Kurt]

dpb,

To get a better undetanding of my problem. Here is my code which does the desired differentiation, but on a uniform XYZ grid. I have no idea how to adapt this for the non-grid 3D data I posted in the 1st post.

function Run_Differentiate_3D()
clc;clear all;close all

   % Generate data
   [X2,Y2] = meshgrid(-2:.2:2, -2:.2:2);                                
  % surf(X,Y,Z)
  X2= X2(:);
  Y2 = Y2(:);
  Z2 =zeros(size(X2),1)  ;

% data here is a Grid
Surface_3D = [X2 Y2 Z2];

% Cx is derivative wrt to x coordinates, likewise Cy for y coords.
[Cx,Cy]=SurfaceGrad3D(Surface_3D);

end

function [Cu,Cv]=SurfaceGrad3D(x)

   r=size(x,1);
   c=size(x,2);
   Cu=spalloc(r*c,r*c,r*c*2);
   Cv=spalloc(r*c,r*c,r*c*2);

    for j=1:c-1
       for i=1:r
           Cu((c-1)*r+i,(c-1)*r+i)=1;
             Cu((c-1)*r+i,(c-2)*r+i)=-1;
            Cu((j-1)*r+i,(j-1)*r+i)=-1;
  %          
            Cu((j-1)*r+i,(j)*r+i)=1;
   %Cu((c-1)*r+i,(c-2)*r+i)=-1;
  % Cu((j-1)*r+i,(j-2)*r+i)=-1;
  %Cu((j-1)*r+i,(j)*r+i)=1;
       end;
   end;

for i=1:r-1
    for j=1:c
          Cv((j-1)*r+r,(j-1)*r+r)=1;
         Cv((j-1)*r+r,(j-1)*r+r-1)=-1;
         Cv((j-1)*r+i,(j-1)*r+i)=-1;

           Cv((j-1)*r+i,(j-1)*r+i+1)=1;

            %Cv((j-1)*r+i,(j-1)*r+i-1)=-1;

         % Cv((j-1)*r+i,(j-1)*r+i+1)=1;
      end;
  end;

end

[dpb]

Perhaps interp2 the data to a uniform grid and do the gradient there and then back to the sampled points?

What's the end result to be for?

[Kurt]

dpb,

Please see the following link for what I am trying to do and my end result:

http://tinypic.com/r/2925fk6/5

Essentially, I want to compute the Equation 1, Es(M). Where M is my 3D surface points. According to the code I posted above, I get dM/du and dM/dv, so the 3 differential terms of Equation 1 can easily be computed. But again, it assumes my 3D data is grid.

Assuming I given non-grid XYZ data, I want to compute Es(M). I dont really want to manipulate my original data by adding extra points through interpolation. Please let me know of any ideas.

thank you

[dpb]

If you can't generate a regular grid it's pretty difficult to envision how you'd do just a numerical difference. I guess I'd be for looking to fit a spline and do the gradients analytically, then.

[Kurt]

dpb,

can you tell me a bit more on fitting the spline an analytical grad? thanks

[Kurt]

dpb,

what do you think of this fex, can it do the job? http://www.mathworks.com/matlabcentral/fileexchange/29312-diffxy/content/diffxy/html/demo_diffxy.html

[dpb]

Dunno--give it a go and see...you'll have to see what its assumptions are on data ordering, etc., ...

On the spline, if the data are a (relatively) smooth surface, the idea is that a piecewise cubic poly should be a good representation of same. Since it's a poly of low order, one can analytically compute the derivatives from the coefficients.

For globally-smooth data, response surfaces are often used as well for the same purpose or to reduce high-complexity models to simply-evaluated RSMs for such purposes as MC simulation where the actual evaluation would be excessively costly.