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Questions tagged [spline]

A smooth piecewise-defined curve formed by joining segments together, end-to-end. The segments are usually described by polynomial or rational functions. Splines are typically used for approximation or data fitting.

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How to vector-derivate this matrix-vector multiplication?

Edit: Possible error in the book? See bottom I'm reading "Functional Data Analysis" by Ramsay & Silverman. The text contains the following (p. 87), regarding how smoothing spline coefficients are ...
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Get approximation of integral using spline and simpson rule

I would like some help figuring out how to do this: I was given a function - $f$ and $f'$ With 7 consecutive derivatives in [a,b] ,samples: $x_k = a+kh$ $h=\frac{b-a}{n}$ , a spline function S:[a,b]$...
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Spline interpolation - why cube with 2nd derivative?

Background For spline interpolation, it looks the degree 3 cubic spline is accepted as the better way and in my understanding it requires 1st and 2nd derivatives at the joints to be the same. ...
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A question on the existence and uniqueness of a cubic Hermite interpolant

I have been trying to solve a particular problem that establishes both the existence and uniqueness of a cubic hermite interpolant on some generic interval $[a,b]$. Briefly, for a function $f$ we ...
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finding the moments of a spline defined by Kronecker delta

having the Spline function of equidistant knots $x_i$ defined by $S_j(x_k)=\delta_{jk},$ where $j,k=0,1,2,...,n$ and $S_j^{(2)}(x_0)=S_j^{(2)}(x_n)=0$ , how can I find the moments $M_1,...,M_{n-1}$ ? ...
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Determine the values of the following function is a natural cubic spline.

$$s(x)=\begin{cases} 3+ax-9x^2+bx^3,\quad &1\le x\le 2\\ 1+c(x-2)+d(x-2)^2+t(x-2)^3, &2\le x\le 3 \end{cases} $$ This is what I have: $$\begin{cases} s_1= bx^3-9x^2+ax+3 &s_2= t(x-2)^3+d(...
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Intuition Sobolev spaces and smoothing splines

With inputs $X_1, \dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $\hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of ...
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Insert new control points over an image

I'm trying to make an automatic non-rigid image registration using B-Splines as a deformation model, but I'm having some problems: Initially, I put a few control points (5x5), and after a long time, ...
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Cubic runout spline (cubic spline)

I was studying cubic spline interpolation and then I stumbled upon "Cubic runout spline". The idea behind it is that you set the boundary conditions for second derivatives to be: $f_1''(x_0) = 2f_1''(...
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Not-a-knot cubic spline interpolation using tridiagonal solver

I am trying to write my own cubic spline interpolator. Given the formula for the cubic spline $$S_n(x) = a_n+b_n(x-x_n)+c_n(x-x_n)^2+d_n(x-x_n)^3$$ my interpolator works perfectly for the natural ...
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Question about linear splines.

I have a question about linear splines. A computer package I am using has the option to use linear splines, but there is no user manual I can find. I am having problems with the kind of inputs the ...
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Convergence Rate of Smoothing Splines

Consider the regression model with data $(Y_i, x_i)_{i=1}^{n}$ (i.e. real-valued target and univariate predictor $x_i$) $$ Y_i = f(x_i) + \epsilon $$ with $\epsilon$ iid, $E(\epsilon) = 0$ and $\...
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Spline interpolation base on average of function

Let's choose the domain as the unit square and divide it into N^2 subdomain. Now I only know the average of function on each subdomain. I know i can regard the average as the value of the function at ...
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Is it just repeat control points and knots if we want to redraw the curve repeatedly

Suppose I have a cubic B-spline curve, it has $57$ knots vectors and $53$ control points. The knot vector is like $(0,0,0,0,1,2,...,50,50,50,50)$ The curve is like this If we want to generate the ...
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Is $f$ in the vector space of cubic spline functions?

Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the ...
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eigenfunction representation with spline: show that coefficients fall faster than order of eigenvalues

I'm trying to understand the Proof of Theorem 3 in Bühlmann & Yu 2003 (Boosting with the $L_2$-Loss). The paper considers some projection matrix $S$ corresponding to a smoothing spline of degree $...
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Approximation of piecewise linear functions by constant function

Let $f(x) = \begin{cases} A_1 x + B_1 &\mbox{if } x \in [a, x_0] \\ A_2 x + B_2 & \mbox{if } x \in [x_0, b] \end{cases}$ and $f \in C[a, b]$ i.e. f has the "angle" form. Denote $E_t(f) = ||...
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Estimating Spline curve by OLS. Is a good idea to fix the knots at Chebyshev sites?

I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or ...
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Relation between a Bezier curve and B-Spline curve

While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out. I ...
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Least squares and Gram matrix of B-spline derivatives

The Gram matrix of a B-spline basis is defined as $$ G_{ij} = \int_S B_i(x) B_j(x) dx $$ where the integral is taken over the full support $S$ of the B-splines. This matrix is positive definite and so ...
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Derivation of moments of normalized B splines

I am reading Ramsey's paper on Monotone splines https://projecteuclid.org/euclid.ss/1177012761 . On page 3 he specifies that each $M_{i}$ has a properties of a probability density function over the ...
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B-Spline how to create control points for a curve to pass through knot values

I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?
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Fitting splines to famous curves programmatically

this may be a bit of a naive question. I am looking for a way to input a Cartesian description of a famous curve and map that to some spline say NURBS to make spline paths. Is this at all possible? ...
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“Standard methods of the calculus of variations” or “Do you read German?”

I'm trying to understand an article of Reinsch (1967), on smoothing spline functions. The author uses some rules that unfortunately I couldn't find. The minimized functional is: $$ \int_{x_0}^{x_n} g'...
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NURBS circle without all the double knots?

I've been looking at various examples of a circle parametrized as a degree-2 NURBS curve, e.g.: NURBS circle example on wikipedia Philip Schneider's "NURB Curves: A Guide for the Uninitiated" David ...
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Drawing De Boor's algorithm

Assume we have 4 control points $[c_0, c_1, c_2, c_3]$ and uniform knot sequence $[0,1,2,3]$ If we were to draw an quadratic bezier we would be forced to use only 3 of the control points and then we ...
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Would the use of cubic splines increase the number of data points to interpolate from result in smaller error and avoid Runge's phenomena?

I am implementing the cubic spline method to interpolate the function: $$f(x)=\sin(x);\ -π≤0≤π$$ I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the ...
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Confusing myself with Cox de boor's algorithm

I am using the recursive definition to understand the algorithm, mainly: $$ B_{i,0}(t) = \begin{cases} 1 & \text{if} & t_i \leq t < t_{i+1} \\ 0 & \text{otherwise} \end{...
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Polynomial representation: Marsden's Identity.

Marsden's Identity states that: For every $\tau$ in $\mathbb{R }$: $(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}B_{j,k,t}$ with: $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$ Following de Boor'...
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Representing rectangular function using divided differences.

I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the ...
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Finding B-spline for space spanned by Multi-dimensional Spline.

It's known that a B-spline of degree $p$ , $B_j^p(x)$ is completely determined by a knot vector $(u_j,u_{j+1},...,u_{j+p+1})$. One could define it as: $B_j^p(x)=[u_j,u_{j+1},...,u_{j+p+1}](\cdot - x)^...
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Approximate a ($y²-x²=1$) hyperbola with line segments and elliptic ($a(x-x_0)²+b(y-y_0)²=1$) arcs

On some IT graphic systems, you have tools which draw line segments and circle or elipse arcs, but which do not draw parabolas or hyperbolas, and in many cases, those system keep track of graphical ...
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Adaptive knot selection for B-spline fitting.

When fitting a B-spline for regression purposes I've seen a lot of cases where knots are fixed uniformly ,but in some situations this could lead to poor estimations because the behaviour of the curve ...
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Aggregate and interpolate overlapping time-series data

I'm trying to aggregate counter data from two different types of measurements. The first type of measure gives an exact value of the counter on a given day. ...
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Interval Spanned by $(n+1)$ points in $\mathbb{R}$

Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$. If you are familiar with B-splines, this function $f$ can be interpreted as a ...
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Fitting a spline: find coefficients using Fourier Transform?

I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way. Given that $$s(x)=\sum_kc(k)\...
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What does this notation mean? Double arrow with $z$ above

I am reading a paper about digital filtering (for the very first time) and I found this notation (double arrow with $z$ above) which I do not quite understand. Could you please give me some hint? ...
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“fastest” curve through n points

I'm programming an AI for a race game, where my car has to drive through some checkpoints. If I let it drive straight in direction of the checkpoints, it has to slow down and make a huge turn after ...
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Order of convergence for spline interpolation in a Sobolev norm

We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{\text{loc}}(\mathbb{R})$. Let us be more precise: Let $h\in \mathbb{R}_{>...
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$[t_j,t_{j+1},…,t_{j+k}]f$ Divided Difference on B-splines.

While reading Moments and Fourier Transform of B-splines (Neuman,E.) I found a different notation for the B-splines. The author define $M_{j,k}(x)$ with knots $t_j<t_{j+1}<...<t_{j+k}$ as: $$...
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Conditions for two B-Splines to represent the same curve

What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space? Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($...
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B splines recursion

Given that by definition the i-th B-spline of order k is: $$B_{i,k}=w_{i,k}B_{i,k-1}+(1-w_{i+1,k})B_{i+1,k-1}$$ where $w_{j,k}=\frac{x-t_j}{t_{j+k-1}-t_j}$ We can define the spline space as $$S_{k,...
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Monotonic and smooth interpolation between three points

The problem I have is the following: Given three $x,y$ points, $(-1, -y_{0})$, $(0,0)$ and $(1,y_{2})$, where $y_{0} \geq 0$ and $y_2 \geq 0$, I want to interpolate smoothly between them with a ...
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Discrete norm approximation of the $L^p$ norm for spline functions

In Theorem 5.2 in Lynche (1988) "A data reduction strategy for splines with applications to the approximation of functions and data", a bound for the difference between the $(l_2,t)$ and $L^2$ norms ...
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How to refine NURBS mesh in isogeometric analysis?

Let we have a coarse description of 2D domain using NURBS. That is, we have two sets of knot vectors $\{\xi_1, \dots, \xi_n\}$, $\{\eta_1, \dots, \eta_m\}$, the set of control vectors $\{B_{ij}\}$, ...
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B and P splines bibliography recommendation.

I am following a course in Statistical Learning and we are using Elements of Statistical Learning(Hastie et. al.) for a very first introduction to smoothing splines, B-spines and P-splines, but ...
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Fitting a cubic spline model with two knots in $R$

We are given data from a set of data from an example given by: battery voltage drop in a guided missile motor observed over the time of missile flight. The set of data has three columns of data: ...
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How to calculate smoothing spline coefficients

I am attempting to calculate smoothing spline coefficients based on the description in Reinsch's 1967 paper, but I'm having some trouble. The first derivative is not continuous. Here are the ...
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103 views

Strong convex hull property of B-spline

B-spline curve is contained in the convex hull of its control polygon. Here is one example, the degree of the curve is 5, The first segment is $u\in [u_5,u_6)$ . The control polygon is from $P_0 ... ...
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Convert continuous Bezier curve to B-Spline

Is there an algorithm/process for converting a sequence of bezier curves into a b-spline? I've found much discussion of the reverse, but nothing for this. I'm attempting to make a spline editor in ...