MATINF 4170/9170 - Lecture 8 - 8/3-2017

Today: Chapter 4• Weekly problem 3.2• Knot insertion• Blossoms

Last time: Chapter 3 and 4 • Recap: Differentiation and smoothness• B-splines as spline-basis• Knot insertion

Recap

Proof: Insert more knots!

Problem of the week

General formulas for knot insertion (4.2.2)Recall, for

Computing discrete B-splines

Proof:

Reccurence for discrete B-splines

B-splines

Discrete B-splines

The Oslo-algorithms

The Oslo-algorithms

Knot insertion example: p=2

In particular:

The B-spline coefficients are functions of the knots!

Observation

Affine functions in one variable

Affine functions in two variables

Characterized by

Blossoms (4.3)

Affine functions in three variables

In general 2p terms in affine functions of p variables

Characterized by

Symmetric affine functions

Multi-affine functions

In general p+1 terms

The Blossom

Blossoms of monomials

Example: g(x)=x2

(x1x2 + x1x3 + x2x3)/3

) = x1x2

Example: g(x)=x

In general:

Proof:

(4.24) Show that the RHS is the blossom

(4.23) Show that the RHS is the blossom for k=p. Differentiate p-k times wrt y

Blossoms of B-splines

Proof:1. Each element of Rk(xi) is affine in xi

2. Symmetry by (3.7)3. Diagonal property holds