How interp2 deal with edges on bicubic interpolation?

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Jérémy Talbot-Pâquet
评论： Bruno Luong 2021-7-9
Hi, I'm trying to make my own algorithm for 2d interpolation.
The upper left figure shows the data used for the interpolation. The upper right is the data interpolated with a linear method. The lower left is the result of matlab interp 2 cubic method and the lower right is an algorithm I made for cubic interpolation in 2d.
I wanted to know what interp2 does when you have data on the edge (for example the point where the X is). You need 2 points both sides to interpolate with the cubic method so what I did is interpolate linearly on the edge of the map. I'm on MATLAB 2016.
At the moment, my method uses separable convolution so I interpolate on rows first and then I interpolate on columns.
From what I understand from the code, interp2 extrapolates to be able to interpolate near edges, but I don't know which method is used to do so? Is it simply linear extrapolation?

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Bruno Luong
Bruno Luong 2021-7-7
编辑：Bruno Luong 2021-7-7
The boundary handling is descriibed in the section Boundary Condition of this Cubic interpolation reference
Especially the equation (19) gives the value of the left outside pixel from the inside pixels
c(-1) = 3*f(x0) - 3*f(x1) + f(x2)
Likewise eqt (25) is for the right-side
c(N+1) = 3*f(x_N) - 3*f(x_{N-1}) + f(x_{N-2})
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Josh Meyer
Josh Meyer 2021-7-9
编辑：Josh Meyer 2021-7-9
It's important to note that the reference paper mentioned in the 2013 thread applied to interp2(..,'cubic') and interp1(..,'v5cubic') when it was originally posted. However, at that time interp1(..,'cubic') had different behavior and was the same as interp1(..,'pchip'), so the reference paper didn't apply there. In other words, interp2 and interp1 had similar options named 'cubic' that actually were different methods, and I believe this is the source of your confusion.
As of R2020b the behavior of interp1(..,'cubic') is the same as interp1(..,'v5cubic'), so the reference paper now applies for interp1's cubic option as well. The details of this change are at the bottom of the interp1 page.
Bruno Luong
Bruno Luong 2021-7-9
interp(...'pchip') is NOT the same as interp1(..., 'v5cubic'). The later uses convolution formula on the paper. And 'pchip' preserves data monotony and it is a non-linear method.
So 'cubic' in R2013 cannot match both IMO.
Anyway it is not big deal, I'm only interested in what happens now.

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Kelly Kearney
Kelly Kearney 2021-7-7
Ran in:
interp2 does not extrapolate. By default, it sets any points outside the data bounds to NaN; you can alternatively choose a different scalar value to assign to all extrapolated points. Alternatively, you could use griddedInterpolant, which allows you to explicitly assign an algorithm to the extrapolated points:
[x,y,z] = peaks(5);
[xi,yi] = meshgrid(linspace(-4,4,50));
zi1 = interp2(x,y,z,xi,yi,'cubic');
F = griddedInterpolant(x',y',z,'cubic','linear');
zi2 = F(xi',yi');
subplot(2,1,1);
hi = imagesc(xi(1,:),yi(:,1),zi1, 'AlphaData', ~isnan(zi1), 'alphadatamapping', 'scaled');
hold on;
scatter(x(:), y(:), [], z(:), 'filled', 'markeredgecolor', 'w');
title('interp2: cubic')
subplot(2,1,2);
hi = imagesc(xi(1,:),yi(:,1),zi2, 'AlphaData', ~isnan(zi2), 'alphadatamapping', 'scaled');
hold on;
scatter(x(:), y(:), [], z(:), 'filled', 'markeredgecolor', 'w');
title('griddedInterpolant: cubic, linear')
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Jérémy Talbot-Pâquet
Jérémy Talbot-Pâquet 2021-7-7
编辑：Jérémy Talbot-Pâquet 2021-7-7
Thank you for your answer. Sorry if I wasn't clear, I meant that somewhere in interp2 for the cubic method, the array is expanded outside the domain to be able to interpolate on the edges since you need 2 points both sides. But you're right, interp2 does not return an extrapolated array unless you give an extrapval argument.
I haven't thought of GriddedInterpolant it could help me to expand the array to more than 1 point. Thank you

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Matt J
Matt J 2021-7-7
编辑：Matt J 2021-7-7
Ran in:
Based on a little reverse engineering, I believe I have discovered that the 'cubic' method of interp1 uses quadratic extrapolation to pad 1 element at the beginning and end of the array. I think we can be pretty confident that the obvious generalization to the 2D case is used for interp2.
To verify this, the test below checks that the extrapolation rule leads to the same interpolated values regardless of whether the signal starts at the beginning of the array or in the middle:
v1=[rand(1,5) 0 0 0 0 0],
v1 = 1×10
0.1348 0.1215 0.0821 0.6977 0.1237 0 0 0 0 0
t1=0:numel(v1)-1;
F1=@(x) interp1(t1,v1,x,'cubic');
p=polyfit(0:2, v1(1:3),2);
extrap=polyval(p,-1), %quadratic extrapolation
extrap = 0.1222
v2=circshift(v1,5); %shift v1 and insert extrapolated value
v2(5)=extrap,
v2 = 1×10
0 0 0 0 0.1222 0.1348 0.1215 0.0821 0.6977 0.1237
t2=t1-5;
F2=@(x) interp1(t2,v2,x,'cubic');
t=linspace(0,1,30);
plot(t,F1(t),'-',t,F2(t),'o')
xlabel t
ylabel 'Interpolated Values'
legend('v1(t1)','v2(t2)')

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