# Computational construction and visualization of non-uniform B-splines and their wavelets

Physics

## Submission Type

Oral Presentation

Mathematics

2022

CNS E 103

## Start Date

4-9-2022 11:15 AM

## End Date

4-9-2022 12:15 PM

## Abstract

Wavelet analysis is a mathematical process where a signal can be approximated by a linear combination of dilations and translations of a scaling function ϕ that can be further decomposed in multiresolution by use of a corresponding wavelet function. Being a multiresolution analysis, wavelet analysis is powerful for image and signal processing. An important class of scaling functions are B-splines, or basic splines. These are recursively defined piecewise polynomials with compact support [x0, xm], where a different polynomial is defined between each pair of m points, x0 < x1 < … < x­m, called the knots of ϕ. A B-spline is non-uniform if the knots are not equally spaced, which greatly complicates the visualization and definition of recursive B-splines. More precisely, for a B-spline with m knots, with maximum degree n of polynomial (in the group of piecewise defined polynomials), and the jth set of points within the knots satisfying j+n+1 ≤ m-1, we have the B-spline bj,n defined below. Note that the base case bj,0(x) is a function of magnitude 1 for x ∈ [x­j, x­j+1], and zero otherwise, and f(t) = (t-x)+n (which is the same as (t-x)n for t>x and zero otherwise).

bj,n(x) := (xj+n+1-xj) f[xj, xj+1, … , x­j+n+1]

The goal of this project is to create an automated method of constructing non-uniform B-splines and their corresponding wavelets. The geometry, especially for non-uniform B-splines, becomes increasingly complex to understand and is hard to individually calculate, making an automated method for constructing non-uniform B-splines very useful. It can then be used to investigate B-splines with different constraints on the knots.

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Apr 9th, 11:15 AM Apr 9th, 12:15 PM

Computational construction and visualization of non-uniform B-splines and their wavelets

CNS E 103

Wavelet analysis is a mathematical process where a signal can be approximated by a linear combination of dilations and translations of a scaling function ϕ that can be further decomposed in multiresolution by use of a corresponding wavelet function. Being a multiresolution analysis, wavelet analysis is powerful for image and signal processing. An important class of scaling functions are B-splines, or basic splines. These are recursively defined piecewise polynomials with compact support [x0, xm], where a different polynomial is defined between each pair of m points, x0 < x1 < … < x­m, called the knots of ϕ. A B-spline is non-uniform if the knots are not equally spaced, which greatly complicates the visualization and definition of recursive B-splines. More precisely, for a B-spline with m knots, with maximum degree n of polynomial (in the group of piecewise defined polynomials), and the jth set of points within the knots satisfying j+n+1 ≤ m-1, we have the B-spline bj,n defined below. Note that the base case bj,0(x) is a function of magnitude 1 for x ∈ [x­j, x­j+1], and zero otherwise, and f(t) = (t-x)+n (which is the same as (t-x)n for t>x and zero otherwise).

bj,n(x) := (xj+n+1-xj) f[xj, xj+1, … , x­j+n+1]

The goal of this project is to create an automated method of constructing non-uniform B-splines and their corresponding wavelets. The geometry, especially for non-uniform B-splines, becomes increasingly complex to understand and is hard to individually calculate, making an automated method for constructing non-uniform B-splines very useful. It can then be used to investigate B-splines with different constraints on the knots.