How can I 3d interpolate a function f: R^3 --> R^3 ?

Hello! I have a table (which exceeds the limits for me to create a meshgrid) which is of the kind:

x | y | z | fx | fy | fz

This 3d function (f) has repeated coordinates x, y, z (i.e. for fixed x0, y0, I have a set of z data corresponding to different values of fx, fy, fz). I tried to do interp3 having done previously meshgrid, but it does not work because of the size of the table.

I would like to find fx*, fy*, fz* such that fx* = fx(x*, y*, z*) and so on. Is there anything I could use? Thank you! Sorry if I have not explained myself properly, but I will leave the structure of my data (a sample) below:

-5.0000000000000003e-02 -5.0000000000000003e-02 4.1000000000000002e-02 -7.9951927903984449e-02 -7.9759897837000562e-02 -1.1193510633877023e-01

-5.0000000000000003e-02 -5.0000000000000003e-02 4.3000000000000003e-02 -7.5687538049114461e-02 -7.5592329497165670e-02 -8.9776172707900920e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 4.4999999999999998e-02 -7.0232531995898836e-02 -7.0632301003499667e-02 -7.3634053337554600e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 4.7000000000000000e-02 -6.6907808923732423e-02 -6.6544534197885738e-02 -6.1247548082081459e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 4.9000000000000002e-02 -6.2484890058519191e-02 -6.2255531287406893e-02 -4.9515426185261224e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 5.1000000000000004e-02 -5.8593779138299981e-02 -5.8438306650002582e-02 -4.0830627034238218e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 5.3000000000000005e-02 -5.5154062309008045e-02 -5.5049344468960537e-02 -3.3614960591879316e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 5.5000000000000000e-02 -5.2090952480478875e-02 -5.2296541426410242e-02 -2.7436886121766587e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 5.7000000000000002e-02 -4.8544831459857732e-02 -4.8816933529787172e-02 -2.1615647420514614e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 5.9000000000000004e-02 -4.5761096787988530e-02 -4.5943899781619980e-02 -1.7736320662827522e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 6.0999999999999999e-02 -4.3062395376749614e-02 -4.3205396827530287e-02 -1.4170468367842259e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 6.3000000000000000e-02 -4.0640523197885893e-02 -4.0627899289096873e-02 -1.0766430352291729e-02

-5.0000000000000003e-02 -5.0000000000000003e-02 6.5000000000000002e-02 -3.8189262345860293e-02 -3.8219490083574281e-02 -8.0298102353285952e-03

-5.0000000000000003e-02 -5.0000000000000003e-02 6.7000000000000004e-02 -3.5955144233611472e-02 -3.5970625678796879e-02 -5.6854763066810868e-03

-5.0000000000000003e-02 -5.0000000000000003e-02 6.9000000000000006e-02 -3.3853227037183693e-02 -3.3881101361149191e-02 -3.5386491816855065e-03

-5.0000000000000003e-02 -5.0000000000000003e-02 7.1000000000000008e-02 -3.1948568830853293e-02 -3.2187847593221519e-02 -1.8015823999897010e-03

-5.0000000000000003e-02 -5.0000000000000003e-02 7.3000000000000009e-02 -3.0064361772382288e-02 -3.0424370683854146e-02 -3.2209933750105250e-04

Answer 1

[John D'Errico]

This is a common problem, at least in the world of color modeling as I worked for many years. You will want to build 3 interpolant models, so essentially fx(x,y,z), fy(x,y,z), fz(x,y,z).

As long as the mapping is a 3d mapping, scatteredInterpolant is your best choice.

xyzuvw = [-5.0000000000000003e-02 -5.0000000000000003e-02 4.1000000000000002e-02  -7.9951927903984449e-02 -7.9759897837000562e-02 -1.1193510633877023e-01
-5.0000000000000003e-02 -5.0000000000000003e-02 4.3000000000000003e-02  -7.5687538049114461e-02 -7.5592329497165670e-02 -8.9776172707900920e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 4.4999999999999998e-02  -7.0232531995898836e-02 -7.0632301003499667e-02 -7.3634053337554600e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 4.7000000000000000e-02  -6.6907808923732423e-02 -6.6544534197885738e-02 -6.1247548082081459e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 4.9000000000000002e-02  -6.2484890058519191e-02 -6.2255531287406893e-02 -4.9515426185261224e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 5.1000000000000004e-02  -5.8593779138299981e-02 -5.8438306650002582e-02 -4.0830627034238218e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 5.3000000000000005e-02  -5.5154062309008045e-02 -5.5049344468960537e-02 -3.3614960591879316e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 5.5000000000000000e-02  -5.2090952480478875e-02 -5.2296541426410242e-02 -2.7436886121766587e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 5.7000000000000002e-02  -4.8544831459857732e-02 -4.8816933529787172e-02 -2.1615647420514614e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 5.9000000000000004e-02  -4.5761096787988530e-02 -4.5943899781619980e-02 -1.7736320662827522e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 6.0999999999999999e-02  -4.3062395376749614e-02 -4.3205396827530287e-02 -1.4170468367842259e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 6.3000000000000000e-02  -4.0640523197885893e-02 -4.0627899289096873e-02 -1.0766430352291729e-02
-5.0000000000000003e-02 -5.0000000000000003e-02 6.5000000000000002e-02  -3.8189262345860293e-02 -3.8219490083574281e-02 -8.0298102353285952e-03
-5.0000000000000003e-02 -5.0000000000000003e-02 6.7000000000000004e-02  -3.5955144233611472e-02 -3.5970625678796879e-02 -5.6854763066810868e-03
-5.0000000000000003e-02 -5.0000000000000003e-02 6.9000000000000006e-02  -3.3853227037183693e-02 -3.3881101361149191e-02 -3.5386491816855065e-03
-5.0000000000000003e-02 -5.0000000000000003e-02 7.1000000000000008e-02  -3.1948568830853293e-02 -3.2187847593221519e-02 -1.8015823999897010e-03
-5.0000000000000003e-02 -5.0000000000000003e-02 7.3000000000000009e-02  -3.0064361772382288e-02 -3.0424370683854146e-02 -3.2209933750105250e-04];

I would point out that your data is NOT amenable for a scattered interpolant. X and y are constant in this data, only z varies.

However, if I were to assume that x and y also vary, and that you have only posted the first 17 data points from your dataset, then you would do this:

umdl = scatteredInterpolant(xyzuvw(:,1),xyzuvw(:,2),xyzuvw(:,3),xyzuvw(:,4));

ans = 1×2

17 6

vmdl = scatteredInterpolant(xyzuvw(:,1),xyzuvw(:,2),xyzuvw(:,3),xyzuvw(:,5));
wmdl = scatteredInterpolant(xyzuvw(:,1),xyzuvw(:,2),xyzuvw(:,3),xyzuvw(:,6));

Now you can interpolate values for each of the outputs. I suppose you could batch them together, like this:

uvwpred = @(x,y,z) [umdl(x,y,z),vmdl(x,y,z),wmdl(x,y,z)];

[Juan Manuel Hussein Belda]

Thank you so much! That is a very good detailed option. It worked great, but I just ended up reshaping the table since it is ordered and then using interp3 because it worked faster :)

Answer 2

[KSSV]

Read about scatteredInterpolant and griddata

[Juan Manuel Hussein Belda]

Thank you! That option worked good, but I ended up working with reshape because it was faster, that is great!