Bsplines seem slightly broken

[oscardssmith]

julia> using DataInterpolations, Plots
julia> using DataInterpolations: derivative
julia> u = [2.0, 1.0, 5.0, 4.0, 5.0, 4.0]
julia> t = [0.0, 2.0, 3.5, 4.0, 5.0, 6.5]
julia> bspline = BSplineInterpolation(u, t, 0, :Uniform, :Uniform; extrapolation_left=ExtrapolationType.Extension,  extrapolation_right=ExtrapolationType.Extension)
julia> plot(bspline)

Why isn't this matching ConstantInterpolation (e.g. having discontinuities at the data points)

For extra weirdness:

bspline = BSplineInterpolation(u, t, 2, :Uniform, :Uniform; extrapolation_left=ExtrapolationType.Extension,  extrapolation_right=ExtrapolationType.Extension)
plot(t->derivative(bspline, t))

[ChrisRackauckas]

@sathvikbhagavan I think you were looking at this last?

[sathvikbhagavan]

I will look into it

[SouthEndMusic]

Regarding the first plot: the discontinuities are at the knots. These knots are computed in the constructor, although I've never delved deeply into what actually happens there. In the BSpline interpolation I'm working on in NDInterpolations.jl, the knots will coincide with t.

[oscardssmith]

yeah, that feels like the right answer

[SouthEndMusic]

Now that I delved a bit deeper into B-Splines again for NDInteprolations.jl, I understand this a bit more. DataInterpolations.jl generally assumes that the input data consists of points (t,u). However, for B-Splines this is a bit different; B-Splines have knots and control points. Control points are not associated with points in the domain, but with basis functions which have support on a small number of intervals between knots.

Generally the number of distinct knots is not equal to the number of control points, which breaks the aforementioned assumption. The BSplineInterpolation somehow ties things together, in a way which is not explained in the doc(string)s but there is a reference: https://pages.mtu.edu/%7Eshene/COURSES/cs3621/NOTES/spline/B-spline/bspline-curve.html.

There's still a bug with the derivative though. I suggest adapting my implementation for NDInteprolations.jl (I'll create a PR there soon) which I think is quite clean and avoids some allocation/AD problems that are present in the current BSplineInterpolation in DataInterpolations.