Interpolation and Deformations A short...

Engineering Research Center for Computer Integrated Surgical Systems and Technology1 600.445 Fall 2000; Updated: 10 October 2002

Copyright © R. H. Taylor

Interpolation and DeformationsA short cookbook



Linear Interpolation

[ ]1

1

10 15 20

5

T

ρ

=

=

pr

[ ]2

2

40 30 20

20

T

ρ

=

=

pr

[ ]3

3

20 20 2

??

0

?

T

ρ

=

=

pr

10




[ ]1

1

10 15 20 T=

=

p

q a

rr r

[ ]2

2

40 30 20 T=

=

p

q b

rrr

[ ]3

3

20 20 20

???

T=

=

p

q

rr ( )1

3+ −a b arr r




2 2, Apr

1 1, Apr

( )3 1 2 1

3 ???A

λ= + −

=

p p p pr r r r

1

λ

( )1 2 1A A Aλ= + −

( ) 1 21 A Aλ λ= − +

( ) ( )( ) ( )

3 1 2 1

2 1 2 1

λ− −

=− −

p p p p

p p p p

r r r rir r r ri



Bilinear Interpolation

, 1i j+ur 1, 1i j+ +u

r

1,i j+ur

( )1 λ−λ

( )1 λ−λ

( )1 µ−

µ

,i jur




,i jur

, 1i j+ur 1, 1i j+ +u

r

1,i j+ur

( )1 λ−λ

( )1 µ−

µ

( )1 µ−

µ

( )( ) ( ) ( )( )( ) ( ) ( )1, 1 1, , 1 ,

, 1, , , 1 , 1, 1 ,

( , ) 1 1 1i j i j i j i j

i j i j i j i j i j i j i j

λ µ λ µ µ λ µ µ

λ µ λµ

+ + + +

+ + + +

= + − + − + −

= + − + − + −

u u u u u

u u u u u u u

r r r r rr r r r r r r




, ,,i j i jAur

, 1 , 1,i j i jA+ +ur

1, 1 1, 1,i j i jA+ + + +ur

1, 1,;i j i jA+ +ur

( ) { } { }, 1, 1, 1 , 1, interpolate( , , , , , )i j i j i j i jλ µ λ µ + + + +=u u u u ur r r r r

( ) { } { }, 1, 1, 1 , 1, interpolate( , , , , , )i j i j i j i jA A A A Aλ µ λ µ + + + +=



N-linear Interpolation

{ }

{ }

( )

1 2

, , 1

, ,

NlinearInterpolate( , )

1 Nli

1

Let

= , with 0

be a set of interpolation parameters, and let

be a set of constants. Then we define:

N

N N k

N

N

A A

Λ λ λ ≤ λ ≤

=

Λ =

− λ

A

A

…

…

{ }{ }

1

1

1 1 2

1 2 1 2

nearInterpolate( , , , )

NlinearInterpolate( , , , )

( ) ( , , )NlinearInterpolate( , )

1

NOTE: Sometimes in this situation we will use notation

N

N N

N

N N

N N

N

A A

A A

A A

−

−

−

− +

Λ

+ λ Λ

Λ = λ λ= Λ A

……

…



Barycentric Interpolation

1pr

2pr

2 1λ −p pr r

( ) 2 11− λ −p pr r

( ) ( ) 2 11λ = − λ + λp p pr r r( ) ( ) 2 1

1 2

1

1

λ = − λ + λ

= λ + µλ + µ =

p p p

p p

r r rr r




1pr 2p

r

3pr

( ), ????λ µ =pr ( ) ( )3 1 3 2 3

1 2 3

, ( )

(1 )

λ µ = + λ − + µ −

= λ + µ + − λ − µ

p p p p p p

p p p

r r r r r rr r r

( ) ( )

( )

3 1 3 2 3

1 2 3

1 2 3

, ( )

(1 )

, , 1 where

λ µ = + λ − + µ −

= λ + µ + − λ − µ

λ µ ν = λ + µ + ν λ + µ + ν =

p p p p p p

p p p

p p p p

r r r r r rr r r

r r r r

( ) ( )

( )( )

3 1 3 2 3

1 2 3

1 2 3

1 2 3

, ( )

(1 )

, , 1

, ,

where

A A A A

λ µ = + λ − + µ −

= λ + µ + − λ − µ

λ µ ν = λ + µ + ν λ + µ + ν =

λ µ ν = λ + µ + ν

p p p p p p

p p p

p p p p

r r r r r rr r r

r r r r

1A

2A

3A




1pr 2p

r

3pr

31 2

1 1 1 1

λ = µ ν

pp pprr rr

1A

2A

3A




{ }

{ }

1

1 2

, , 1 1

, ,

BarycentricInterpolate( , )

N

N

N k kk

A A

=

Λ λ λ ≤ λ ≤ λ =

=

Λ =

∑

A

A

r …

r …

rr

1

Let

= , with 0 and

be a set of interpolation parameters, and let

be a set of constants. Then we define:

1

( ) ( , , ) BarycentricInterpolate( , )

N

k kk

N N N

A=

Λ = λ

Λ = λ λ = Λ

∑A

A A A

rri

…1

NOTE: Sometimes in this situation we will use notation

NOTE: This is a special case of barycentric Bezier

polynomial interpo stlations (here, 1 degree)



Interpolation of functions

v

( )y v

01



Fitting of interpolation curves

• The discussion below follows (in part)

G. Farin, Curves and surfaces for computer-aided geometric design, a practical guide, Academic Press, Boston, 1990, chapter 10 and pp 281-284.



v

01

N,k

,

Note that many forms of polynomial may be usedfor the P ( ). One common (not very good) choice

is the power basis:

( )

Better choices are the Bernstein plynomials and the b-spline basis functi

= kN k

v

P v v

ons, which we will discuss ina moment

1-D Interpolation

{ }0 0

0

0 ,0

Given set of known values ( ),..., ( ) ,

find an approximating polynomial ( ,..., ; )

( ,..., ; ) ( )=

≅

= ∑

m m

N

N

N k N kk

y v y v

y P c c v

P c c v c P v



1-D Interpolation

{ }0 0

0

0 ,0

,0 0 , 0 0

,0 ,

Given set of known values ( ),..., ( ) ,

find an approximating polynomial ( ,..., ; )

( ,..., ; ) ( )

To do this, solve:

( ) ( )

( ) ( )

=

≅

=

∑

LM O M M

L

m m

N

N

N k N kk

N N N

N m N N m m

y v y v

y P c c v

P c c v c P v

P v P v c

P v P v c

0 ≅

Mm

y

y



Bezier and Bernstein Polynomials

• Excellent numerical stability for 0<v<1• There exist good ways to convert to more

conventional power basis

( ) ( )

( )

−

=

=

−

= −

=

= −

∑

∑

…00

,0

,

, , ; 1

( )

( ) 1

NN k N

N kk

N

k N kk

N k kN k

NP c c v c v v

k

c B v

Nwhere B v v v

k



Barycentric Bezier Polynomials

• Excellent numerical stability for c<0<1• There exist good ways to convert to more

conventional power basis

( )00

,0

,

, , ; ,

( , )

( , ) u+v=1

−

=

=

−

=

=

=

∑

∑

…N

N k NN k

k

N

k N kk

N k kN k

NP c c u v c u v

k

c B u v

Nwhere B u v u v

k



Bezier Curves

( )0 ,0

Suppose that the coefficients are multi-dimensional

vectors (e.g., 2D or 3D points). Then the polynomial

, , ; ( )

computed over the range 0 1 generates a Bezier

curve w

j

N

N k N kk

c

P c c v c B v

v=

=

≤ ≤

∑

uur

uur uur uur…

ith control vertices .jcuur

0c

1c

3c

2c



Bezier Curves: de Casteljau Algorithm

( )0 0

0

1 11

Given coefficients , Bezier curves can be generated

recursively by repeated linear interpolation:

, , ;

(1 )

j

NN

j jk k kj j j

c

P c c v b

where

b c

b v b vb− −+

=

== − +

uur

uur uur…

uur

0

rc

1

rc

3

rc

2

rc

( )1 v−v

v

v

( )1 v−( )1 v−



Iterative Form of deCasteljau Algorithm

1

0

Step 1: for 0

Step 2 : for 1 step 1 until do

for 0 step 1 until do(1 )

Step 3: return

j j

j j j

b c j N

k k N

j j N kb v b vb

b+

← ≤ ≤

← =← = −← − +



Advantages of Bezier Curves

• Numerically very robust• Many nice mathematical properties• Smooth

• “Global” (may be viewed as a disadvantage)



B-splines

0 1

0 2 2 1

1 1

Given

coefficient values { , , }

"knot points" { , , } with

D = "degree" of desired B-spline

Can define an interpolated curve ( , ; ) on

Then

(

+ −

+ − +

− + −

=

= ≤

≤ <

r rLL

L D

L D i i

D L D

c c

u u u u

P u u u u

P

C

u

C u

1

0

; ) ( )

where ( ) are B-spline basis polynomials (discussed

later)

+ −

=

= ∑ rL DD

j jj

Dj

u c N u

N u

C



deBoor Algorithm

1

1

0

Given , , as before, can evaluate ( , ; )recursively as follows: Step 1: Determine index such that

Step 2: Determine multplicity r such that

Step 3: Det for 1

i i

i r i r i

j j

D P u

i u u u

u u u

d c i D j

+

− − +

≤ <

= = =

= − + ≤

u c c u

L

1

11 1j 1

1 1

1

Step 4: Compute ( , ; ) recursively, where

d

D ri

j D k jk k kj j

j D k j j D k j

i

P u du u u u

d du u u u

−+

+ − −− −−

+ − − + − −

≤ +

=− −

= +− −

c u



B-spline basis functions

1

0

10

1 1 11

1 1

Given , , D as before

( , ; ) ( )

where

1( )

0 Otherwise

( ) ( ) ( ) for 0

+ −

=

−

− +− −+

+ − − +

=

≤ ≤=

− −

= + >− −

∑ rL DD

j jj

j jj

j j kk k kj j j

j k j j k j

P u c N u

u u uN u

u u u uN u N u N u k

u u u u

C u

C u



Some advantages of B-splines

• Efficient• Numerically stable • Smooth• Local



2D Interpolation

0 0

00 0 0

0

0

Consider the 2D polynomial

( , ) ( ) ( )

( )[ ( ), , ( )]

( )

where ( ) and ( ) can be arbitrary

functions (good choices Bernstein polynom

m n

ij i ji j

n

m

m mn n

i j

P u v c A u B v

c c B vA u A u

c c B v

A u B v

= =

=

=

∑∑L

L M O M ML

ials or B-Spline basis functions. Suppose that we have samples

( , ) for 0,...,

We want to find an approximating polynomial P. s s su v s N= =sy y



2D Interpolation: Finding the best fit

[ ][ ]

0

0

Given a set of sample values ( , ) corresponding to 2D coordinates

( , ), left hand side basis functions ( ), , ( ) and right hand side

basis functions ( ), , ( ) , the goal is to find the

s s s

s s m

n

u v

u v A u A u

B v B v

y

LL

00

01

0 0 0 1

matrix of

coefficients .

To do this, solve the least squares problem

( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

ij

s s s s s s s i s j s m s n sij

mn

u v A u B v A u B v A u B v A u B v

≈ •

C

c

cc

yc

c

M ML LM M




{ } { }{ }

0 0

0

A common special case arises when the ( , ) form a regular grid. In this

case we have , , and , , . For each value

, , solve the row least squares problem

( , )

u v

v

s s

s N s N

j N s

s s j

u v

u u u v v v

v v v N

u v

∈ ∈

∈

y

L LLM

M

0

0

00 01 0 0 0

10 11 1

0 1

( ) ( )

for the unknown m-vector . Then solve m n-variable least squares problems

( )

v v v

j

s m s

jm

m

m

N N N m

A u A u

B v

≈ •

≈

j

X

X

X

X X XX X X

X X X

M L ML M

M L M

LL

M M MM M O MM M M

L

1 0 0 00 10 0

0 1 1 1 1 01 11 1

0 1

0 1

0

( ) ( )( ) ( ) ( )

( ) ( ) ( )

for the vectors , , . Note that this latter step requires ov v v

n m

n m

n n mn

N N n N

j jn

B v B vB v B v B v

B v B v B v

•

c c cc c c

c c c

c c

L LL L

M M M M M O MM M O M LM M M

LL nly 1 SVD or

similar matrix computation.




• There are a number of caveats to the “grid” method on the previous slide. (E.g., you need enough data for each of the least squares problems). But where applicable the method can save computation time since it replaces a number of m and n variable least squares problems for one big m x n problem

• Note that there is a similar trick that you can play by grouping all the common ui elements together.

• Note that the y’s and the c’s do not have to be scalar numbers. They can be Vectors, Matrices, or other objects that have appropriate algebraic properties

Interpolation and Deformations A short...

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