Given a spline in B-form with control Points and a knot vector , how can we get the coefficients contained in the pp-form on . Is there an analytical equation ? Or are the 's determined via re-fitting? I just want to understand/reprodce what happends under the hood when I call fn2fm and convert from B-form to pp-form.
steps to convert spline from B-form to pp-form (fn2fm)
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x = [3.0,4.5,6.0,7.5,9.0,12.0,15.0];
y = [0 0.0343653 0.0694232 0.105143 0.141178 0.246013 0.630537];
f_bm = spapi(5, x, y);
f_pp = fn2fm(f_bm, 'pp');
Based on f_bm.knots, it is easy to compute f_pp.breaks. But how is f_pp.coefs calculated? Is there a linear system solved or is there an analytical equation for computing the coefficients based on the control points + knots + degree?
Thank you!
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Bruno Luong
le 13 Mar 2024
Modifié(e) : Bruno Luong
le 14 Mar 2024
The pp-form stores the polynomial of each each subiterval, the variable is x := (t-ti) where ti is the left knot of interval #i. https://www.mathworks.com/help/curvefit/the-ppform.html
In my BSFK FEX https://www.mathworks.com/matlabcentral/fileexchange/25872-free-knot-spline-approximation
there is a function private/Bspline2pp.m that just does the conversion to pp
It evalutes the spline in k points inside each subitervals then invert the Vandermond matrix to compute a row of pp.coefs. The last stage can be done with polyfit for coding convenience.
You can also compute from Taylor expansion of the polynomial wrt th the left knot as Torsen has suggested.
It requires to compute 0 to (k-1) order derivatives
Code to avoid using fn2fm (why?!) if this idea
WARNING this only works for non-duplicated knot vectors
x = [3.0,4.5,6.0,7.5,9.0,12.0,15.0];
y = [0 0.0343653 0.0694232 0.105143 0.141178 0.246013 0.630537];
f_bm = spapi(5, x, y);
f_pp = fn2fm(f_bm, 'pp'); % just used for checking correctness
k = f_bm.order;
breaks = f_bm.knots(k:end-k+1);
pieces = length(breaks)-1;
coefs = zeros(pieces,k);
Sd = f_bm;
xi = breaks(1:pieces);
for j=0:k-1
coefs(:,k-j) = fnval(Sd, xi)./factorial(j);
Sd = fnder(Sd);
end
pp = struct('form', 'pp', ...
'breaks', breaks, ...
'coefs', coefs,...
'piecs', pieces, ...
'order', k, ...
'dim', 1);
% Check the result
format long
f_pp.coefs
pp.coefs
% Note different last digit in coefs(3,4)
3 commentaires
Bruno Luong
le 13 Mar 2024
Modifié(e) : Bruno Luong
le 13 Mar 2024
I'm not sure, the De Boor algorithm and B-spline basis function evaluation are usually both based on Cox–de Boor recursion formula. It is just implement with different workfow. We never need to compute the explitly the basis functions if spline values are required to be calculated.
I suppose (not sure) when you call fnval with B-spline form input, it would use De Boor algorithm.
In my BSFK toolbox it is implement in function private/Berntein.m with vectorasation, if you want to dig in and see how it is implemented in MATLAB.
To me the most efficient evaluation is with pp-form. This is the whole point why this form is invented.
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