In MathsCell/bspline: B-Spline Interpolation and Regression

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B-spline recursive definition

Cox-de Boor formula:

$$ B_{i,0} := \left{ \begin{array}{ll} 1\ \ \text{if}\ t_i \leq x \lt t_{i+1} \ 0\ \ \text{otherwise.} \end{array} \right. $$ $$ B_{i,k} := \frac{x-t_i}{t_{i+k} - t_i} B_{i,k-1}(x) + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1,k-1}(x) $$

First derivative

$$ \frac{dB_{i,k}(x)}{dx} = k \left( \frac{B_{i,k-1}(x)}{t_{i+k} - t_i} - \frac{B_{i+1,k-1}(x)}{t_{i+k+1} - t_{i+1}} \right) $$

This implies that

$$ \frac{d}{dx}\sum_{i} \alpha_i B_{i,k} = \sum_{i=r-k+2}^{s-1} k \frac{\alpha_i - \alpha_{i-1}}{t_{i+k} - t_i} B_{i,k-1} \ \ \text{on}\ \ [t_r, t_s] $$

Integral

Let $$ \beta_i = k \frac{\alpha_i - \alpha_{i-1}}{t_{i+k} - t_i} $$ Then $$ \int_0^\zeta \frac{d}{dx}\sum_{i} \alpha_i B_{i,k} \ dx = \int_0^\zeta \sum_{i=r-k+2}^{s-1} \beta_i B_{i,k-1} \ dx $$ $$ \sum_{i} \alpha_i B_{i,k} \mid_0^\zeta = \int_0^\zeta \sum_{i=r-k+2}^{s-1} \beta_i B_{i,k-1} \ dx $$ If $\beta_i$ is considered as known and we have to find $\alpha_i$, we get $$ \alpha_i = \beta_i \frac{t_{i+k}-t_i}{k} + \alpha_{i-1} $$ here $\alpha_0$ being a freely defined constant. In term of cumulative sum, it gives $$ \alpha_j = \sum_{i=0}^j \beta_i \frac{t_{i+k}-t_i}{k} $$

MathsCell/bspline documentation built on Feb. 18, 2025, 12:40 p.m.