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