Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

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Is a cubic Lagrange interpolation tensor product the same as bicubic interpolation?

I just implemented some interpolated texture sampling by sampling the 4x4 nearest pixels then doing Lagrange interpolation across the x axis to get four values to use Lagrange interpolation on across ...
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1answer
34 views

Alternatives / Extensions to the Thin Plate Splines method

Thin Plate Splines are a great method to find a smooth interpolating surface given scattered data. Essentially, the method involves calculating weights for a radial basis function centred around each ...
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Do Cubic Splines Minimize Error? [on hold]

Using cubic splines with first and second derivative conditions, will these serve to reduce error while interpolating (such as the Runge Phenomena)? Is there a proof of this?
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22 views

Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
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How can we calculate the tensor product of Lagrange basis polynomials?

Given data points $(x_i,y_i)$, the Lagrange basis polynomials are $$\mathcal l_j(x):=\sum_{i\ne j}\frac{x-x_i}{x_j-x_i}\;.$$ I'm reading a text targeting Smolyak's algorithm. In this text, they use ...
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2answers
55 views

function: bending the y=x line

My question has many relative questions but I didn't find anything exact to my needs. Let's take the function $f(x)=x$ with $x\in[0,100] $. I need to bend this and make it a curve. f will be a ...
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2answers
37 views

Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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31 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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53 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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22 views

Error estimate of polynomial quadratures missing some terms

Normally, for trapezoid rule and simpson's rule, etc, error analysis is done by using the error formula for interpolation. However, if the polynomial is restricted to some terms, for example, a ...
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1answer
40 views

Find function by 2 tangents and 2 points

I am looking for explicit function descriptions $F_1(s)$ and $F_2(s)$, following the line plotted. The line is just a description, but $F_1$ should never exceed $F_m$ and start at $s_0$ with a tangent ...
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Any one can give me solution of relationship between forward and backward interpolation operator? [closed]

Any one can give me solution of relationship between forward and backward interpolation operator ?
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1answer
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How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
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22 views

How does sinc interpolation work?

Every now and then I come across mention of sinc interpolation. Trying to read up on it, I have yet to get what it's about. I have done basic DSP work, have programmed stuff using FFT (using just a ...
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25 views

Function to turn results from a nearest-neighbour function into an inversely proportional version?

Short version: Given an input vector D of n values, what are the different methods that one can use to return a vector W such that each value in W is in inverse proportion to the magnitudes of the ...
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How to apply a time shift to a pulse-shape, spanned with spline functions?

I have a sampled pulse shape: $ h = [1, 0.5]$ and I do not know what is its real underlying continuous-time pulse. I want to compute the samples of $h(t-\Delta t)$. If I write the continuous pulse ...
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1answer
24 views

Curve fitting for 2D Data and Interpolation

I have polygon with $n$ Corner points where stresses are known to me. I have to fit a sutface $F(x,y)$, which can give the value of stress at anypoint inside the polygon. I fitted a curve using a ...
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1answer
32 views

Interpolating Random Points

I have a list of (x,y) co-ordinates that need to be interpolated. The co-ordinates are not necessarily part of a function. Therefore, polynomial interpolation will not work. Is there a way to use some ...
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24 views

Optimize to Find the Mahalanobis Distance to Minimize the Term

I have an optimization problem defined as following: Assuming we have a data set $ { \left\{ \left( {x}_{i}, {y}_{i} \right) \right\}}_{i = 1}^{N} $ where $ {x}_{i} \in {\mathbb{R}}^{d} $ and $ ...
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20 views

spline interpolation of 3 dimensional array

Iam trying to interpolate two files which are stored into array. here is the example of the files. File 1 : x y z 8 12 89 11 18 30 11 19 20 File2: x1 y1 z1 8 12 89 11 18 30 11 19 20 ...
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2answers
54 views

Numerical mathematics, Lagrange interpolation..

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
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1answer
37 views

What is the correct notation for curves?

What is the correct math notation to use is when referring to linear interpolation, curves, and points on curves? For instance, let's say we are talking about a quadratic Bezier curve. The control ...
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74 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
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1answer
28 views

Equivalance form for Slerp in quaternions interpolation

In all the books I have found that Slerp have two forms: A B I know that all the forms from A are equivalent but I don't know why the forms from A are equivalent with the form from B. Can ...
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49 views

Numerical mathematics, Lagrange interpolation

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
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2answers
31 views

Help understanding Product (capital PI) notation

On the wikipedia article for lagrange interpolation (https://en.wikipedia.org/wiki/Lagrange_polynomial), it shows the definition for the lagrange basis functions in a strange way - well strange to me ...
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1answer
20 views

A function for non-linear animation steps (large in the middle, small at the ends)

In a word game for Android I animate movement of letter tiles (for example when user selects "shuffle tiles" or "return tiles from game board" in menu) in a linear way (they have constant velocities) ...
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Find intersection between conotur point list and a line

Given: List of points representing a closed contour Task: Choose a random point on the contour and shoot a ray inside the contour and determine where the ray intersects the contour. This needs to be ...
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32 views

Interpolation at “extreme values.”

I am working with a meteorologist on a project. We are pulling data from METAR Observation stations on several variables (such as temperature, dew point, wind speed, etc.) throughout time. ...
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3answers
29 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
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18 views

How to use bilinear interpolation on 2d axis?

In my application i find 4 nearest points on the grid $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, $(x_4,y_4)$ for a signal that detected with unknown location using knn. Each of these points have a ...
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1answer
35 views

linear or bilinear interpolation

I want to know how to use linear and bilinear interpolation in 2D. Specifically the pairs $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ are given in a quadrilateral. In this case how to ...
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1answer
27 views

How to use bilinear interpolation?

I need an explanation about Bilinear Interpolation. I use KNN and find $4$ points which I need to use bilinear interpolation to find unknown position. I was unable to understand explanations in other ...
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2answers
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Formula for $N$-Dimensional linear interpolation

Linear interpolation between values $A$ and $B$ can be defined as: $f(x) = A(1-x)+Bx$ Bilinear interpolation between values $A,B,C,D$ is defined as: $f(x,y) = g(x)(1-y) + h(x)y$ where $g(x) = ...
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How to select the number of nodes in a spline interpolation?

I am writing a program to test the precision of different methods for imputing missing data in a time series. One of the methods I am going to test is a natural cubic spline interpolation. I'll be ...
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1answer
151 views

Integrating over a specific vector field

I am trying to show that the solution of the following integral is as follows: Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$ Where ...
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45 views

Number of points for local spline interpolation

I have a large scattered data set of (x,y = f(x)) points and I want to interpolate them to regularly spaced grid points. To do this I have chosen to use cubic splines as my interpolation method. The ...
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1answer
25 views

Interpolate between 3D plane and 3D hemisphere

I have a simple 3D plane whose points (different $x, y$ values, but all $z = 0$) need to be mapped to 3D Cartesian coordinates in order to form a hemisphere. However, I also would like to be able to ...
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1answer
36 views

Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
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When would a Fourier Product (made up term) exist for a finite sequence of the form $C_{\text{Max}}\prod _{i=1}^k A_i \cos \left( B_i n\right)$

Let us say that we are given a finite list of points of the form C = {i,$x_i$} where i goes from 0 to the card(C) that when plotted in the Euclidean plane has some vertical axis that splits the graph ...
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1answer
29 views

Cubic Uniform BSpline surface interpolation

I want to understand cubic BSpline surface( very hard for me to figure out). I prefer matrix form which presented here. Equation 4.12 in page 33, describes how data point should be presented ...
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4answers
66 views

Interpolation between 2 points on the perimeter of a circle?

I'm trying to produce movement on a unit circle from one point to another in equal increments, but I'm having trouble doing this without the use of angles (which isn't an option). Given 2 points on a ...
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36 views

How to proceed with this simple proof?

If $$\alpha_k = \sum_l a_l \ \ g((k-l)T-l\Delta T)$$ $$s_k = \sum_l \alpha_l \ \ q((k-l)T+k\Delta T)$$ where $a_l \in \pm1$ and $g(t) = \frac {\sin(\pi t/T)}{\pi t/T}$ and $q(t) = \frac {\sin(\pi ...
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44 views

What is the best way to interpolate over the 25th and 75th percentile of SAT scores?

The problem: I know the 25th and 75th percentiles of SAT scores for students admitted to a given university, and I want to interpolate over those two points in order to estimate all the percentiles ...
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1answer
20 views

Knowing two Vectors, and the distance to a 3rd, how to get the 3rd

If i know the two Vectors v1 and v2, which discripe points in a 2D space, and i also know that a vector v3 is on the line segment between v1 and v2, how can i get the x and y coodinates of v3 if the ...
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29 views

Interpolating a set of GPS points on ellipsoid earth model.

I have a set of GPS (latitude, longitude) co-ordinates along with the time at which the coordinates were collected. Additionally, I have the speed and the heading of the vehicle at those coordinates. ...
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periodic radial basis function

A have a point cloud ,described in spherical coordinates, which I need to fit with a smooth surface. I'm trying to do this with a bivariate radial basis function network, which operates on a spherical ...
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Interpolation of jacobians for point wise defined transformation

Here is my question: let's say that I have a transformation function T from the image A to the image B which is pointwise defined. That is, T(x) = J, where J is the jacobian representing the ...
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3answers
21 views

How can I characterise the error of an interpolated surface?

I am writing a program in which I can interpolate and display a surface by kernel interpolation. Lets say I interpolate a function $f(x)$ by the function $f^*(x)$. Clearly the error at any given point ...
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61 views

Positive linear combinations of intervals

Given two intervals at $i\in\{0,1\}$ $I_i=[-a_i,a_i]$ where $0<a_0<a_1=1-a_0<1$ and a third interval $I=[-a,a]$ where $0<a<\frac{1}2$, when is there an $\alpha,\beta\in\Bbb R$ such that ...