matinf 4170/9170 - lecture 8 - 8/3-2017 · matinf 4170/9170 - lecture 8 - 8/3-2017 today: chapter 4...
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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
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Recap
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Proof: Insert more knots!
Problem of the week
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General formulas for knot insertion (4.2.2)Recall, for
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Computing discrete B-splines
Proof:
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Reccurence for discrete B-splines
B-splines
Discrete B-splines
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The Oslo-algorithms
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The Oslo-algorithms
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Knot insertion example: p=2
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In particular:
The B-spline coefficients are functions of the knots!
Observation
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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
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Symmetric affine functions
Multi-affine functions
In general p+1 terms
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The Blossom
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Blossoms of monomials
Example: g(x)=x2
(x1x2 + x1x3 + x2x3)/3
) = x1x2
Example: g(x)=x
In general:
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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
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Blossoms of B-splines
Proof:1. Each element of Rk(xi) is affine in xi
2. Symmetry by (3.7)3. Diagonal property holds