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I have read that in contrast of the thin-plate splines, B-splines are locally controlled, which makes them computationally efficient even for large number of control points.

I didn't understand what does it mean by saying that B-splines are locally controlled. Can anyone help with this?

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It means that value of B-spline at a point depends just on few control points localized nearby. And vice versa, if you modify a control point, or coefficient of one basisfunction it will affect just some local nighborhood.

In case of cubic B-spline (which is most common e.g. in computer graphics ) in 1D value $f(x)$ at any point depends on value of 4 control points (resp. basisfunctions ) as

$ f(x) = \sum_{i=1..3} \alpha_i \phi ( x - i ) $

It is nice visible here http://www.brnt.eu/phd/ucbs-basis.png or here http://www.cs.berkeley.edu/~sequin/CS284/IMGS/bsplinebasics.gif

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  • $\begingroup$ Thank you very much. Your answer was very helpful. I would like to clarify one more thing. In case of cubic B-splines in 1D any point depends on value of 4 control points regardless how coarse or fine is the grid? It's always affect 4 control points? $\endgroup$
    – Gina
    Commented Mar 24, 2016 at 10:31
  • $\begingroup$ Also here: brnt.eu/phd/ucbs-basis.png the ks are the points affected? Because for each colorful function correspond 5 k values. $\endgroup$
    – Gina
    Commented Mar 24, 2016 at 10:36
  • $\begingroup$ yes, if you sample an interval of $x$ by finer grid than basis function $\phi(x)$ has smaller extend (are more contracted, perhabs I should write $\phi_i( (x/L) - i)$ where $L$ is the sampling step size ) so at any point $x$ just 4 of them are non-zero. But because they are more contracted, they are sharper, so you can reproduce finer details of your function which you want to approximate. $\endgroup$ Commented Mar 24, 2016 at 13:00

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