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

## Major

Physics

## Submission Type

Oral Presentation

## Area of Study or Work

Mathematics

## Expected Graduation Date

2022

## Location

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 [x_{0}, x_{m}], where a different polynomial is defined between each pair of m points, x_{0 }< x_{1 }< … < x_{m}, 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 j^{th} set of points within the knots satisfying j+n+1 ≤ m-1, we have the B-spline b_{j,n }defined below. Note that the base case b_{j,0}(x) is a function of magnitude 1 for x ∈ [x_{j}, x_{j+1}], and zero otherwise, and f(t) = (t-x)_{+}^{n} (which is the same as (t-x)^{n} for t>x and zero otherwise).

b_{j,n}(x) := (x_{j+n+1}-x_{j}) f[x_{j}, x_{j+1}, … , x_{j+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.

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 [x_{0}, x_{m}], where a different polynomial is defined between each pair of m points, x_{0 }< x_{1 }< … < x_{m}, 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 j^{th} set of points within the knots satisfying j+n+1 ≤ m-1, we have the B-spline b_{j,n }defined below. Note that the base case b_{j,0}(x) is a function of magnitude 1 for x ∈ [x_{j}, x_{j+1}], and zero otherwise, and f(t) = (t-x)_{+}^{n} (which is the same as (t-x)^{n} for t>x and zero otherwise).

b_{j,n}(x) := (x_{j+n+1}-x_{j}) f[x_{j}, x_{j+1}, … , x_{j+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.