Interpolation and Deformations A short cookbook

Transcription

1 Interpolation and Deformations A short cookbook Fall 2000; Updated: 0 October 2002

2 Linear Interpolation p ρ 2 2 = [ ] = 20 T p ρ = [ ] = 5 p 3 = ρ3 =? 0?? T [ ] T Fall 2000; Updated: 0 October 2002

3 Linear Interpolation p q = = p q [ ] T a p q [ ] = = a??? + b a 3 ( ) 2 2 T = = [ ] T b Fall 2000; Updated: 0 October 2002

4 Linear Interpolation p 2, A2 λ p, A p = p + λ p p ( ) ( ) 3 2 A 3 =??? A + λ A A λ = 2 ( λ) A λa 2 ( p3 p) i( p2 p) ( p p ) i( p p ) = Fall 2000; Updated: 0 October 2002

5 Bilinear Interpolation u ( λ ) u λ i+, + i, + ( µ ) u i, λ µ ( λ) u i+, Fall 2000; Updated: 0 October 2002

6 Bilinear Interpolation u u i+, + i, + ( µ ) µ u i, λ Fall 2000; Updated: 0 October 2002 ( λ ) ( µ ) u µ i+, u( λµ, ) = λ ( µ u ( µ ) ) u + + ( λ ) µ u + + ( µ ) u = u + λ u u + µ u u + λµ u u ( ) i, i, i, i, ( ) ( ) ( ) i, i, i, i, i, i, i,

7 i, + i, + Bilinear Interpolation u, A u i+, +, Ai+, + u, A i, i, u ; A i+, i+, u, interpolate(,, u, u, u, u ) ( λµ ) = { λµ } { } i, i+, i+, + i, + ( λµ ) = { λµ } { } i, i+, i+, + i, + A, interpolate(,, A, A, A, A ) Fall 2000; Updated: 0 October 2002

8 Let { } { A,, A } 2 -linear Interpolation Λ = λ,, λ, with 0 λ k be a set of interpolation parameters, and let A = be a set of constants. Then we define: linearinterpolate( Λ, A) = ( λ ) A( Λ ) = A( λ,, λ ) linearinterpolate( Λ, A,, A ) { } 2 { A } A +λ linearinterpolate( Λ,,, ) = linearinterpolate( Λ, A) OTE: Sometimes in this situation we will use notation Fall 2000; Updated: 0 October 2002

9 Barycentric Interpolation p ( λ) p2 p λ p p 2 p 2 p p p =λ p +µ p ( λ ) = ( λ ) 2 +λ λ+µ= Fall 2000; Updated: 0 October 2002

10 Barycentric Interpolation p 3 A 3 p p 2 A p ( ( λµ,, ) ) =???? p 3 +λ( ( p p) 3 ) +µ ( ( p 2 p ) 3 ) =λ p +µ p2 2 + ( ( λ µ ) ) p3 3 p λµν,, =λ p +µ p +ν p where λ+µ+ν= ( ) (,, ) A A λµν =λ A +µ A +νa Fall 2000; Updated: 0 October 2002

11 Barycentric Interpolation p 3 A 3 p p 2 A 2 A p p p2 p3 λ = µ ν Fall 2000; Updated: 0 October 2002

12 Let Barycentric Interpolation Λ = λ,, λ, with 0 λ and λ = { } { A,, A } 2 k k k = be a set of interpolation parameters, and let A = be a set of constants. Then we define: BarycentricInterpolate( Λ, A) = Λ i A = λ k= k A k OTE: Sometimes in this situation we will use notation A( Λ ) = A( λ,, λ ) = BarycentricInterpolate( Λ, A) Fall 2000; Updated: 0 October 2002 OTE: This is a special case of barycentric Bezier st polynomial interpolations (here, degree)

13 Interpolation of functions yv () 0 v Fall 2000; Updated: 0 October 2002

14 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, 990, chapter 0 and pp Fall 2000; Updated: 0 October 2002

15 ote that many forms of polynomial may be used for the P k,,k is the power basis: P ( v) v a moment 0 ( v). One common (not very good) choice = k -D Interpolation 0 k k, k = 0 { y v y v } Given set of known values ( ),..., ( ), 0 0 find an approximating polynomial y Pc (,..., c ; v) Pc (,..., c ; v) = cp ( v) Better choices are the Bernstein plynomials and the b-spline basis functions, which we will discuss in m 0 m v Fall 2000; Updated: 0 October 2002

16 -D Interpolation 0 k k, k = 0 { y v y v } Given set of known values ( ),..., ( ), 0 0 find an approximating polynomial y Pc (,..., c ; v) Pc (,..., c ; v) = cp ( v) m 0 m To do this, solve: P,0( v0 ) P, ( v0) c0 y0 P,0 ( vm) P, ( vm) cm y m Fall 2000; Updated: 0 October 2002

17 Bezier and Bernstein Polynomials ( ) ( ) = k P c0,, c; v ck v v k = 0 k = k = 0 k, ( v) ( ) k where Bk, ( v) = v v k Excellent numerical stability for 0<v< There exist good ways to convert to more conventional power basis cb k k Fall 2000; Updated: 0 October 2002

18 Barycentric Bezier Polynomials ( ) 0,, ;, = k P c c uv ck u v k = 0 k k = 0 k, (, uv) k k where Bk, ( uv, ) = u v u+v= k Excellent numerical stability for c<0< = There exist good ways to convert to more conventional power basis cb k Fall 2000; Updated: 0 October 2002

19 Bezier Curves Suppose that the coefficients c are multi-dimensional vectors (e.g., 2D or 3D points). Then the polynomial Pc,, c ; v = cb ( v) ( ) 0 k k, k = 0 computed over the range 0 v curve with control vertices c. generates a Bezier c c 2 c Fall 2000; Updated: 0 October 2002 c 3

20 Bezier Curves: de Castelau Algorithm Given coefficients c, Bezier curves can be generated recursively by repeated linear interpolation: Pc c v = b (,, ; ) where b = c k k ( ) k = + + b vb vb c0 v c ( v) v ( v) c 2 v ( v) c Fall 2000; Updated: 0 October 2002

21 Iterative Form of decastelau Algorithm Step : b c for 0 Step 2: for k step until k = do for 0 step until = k do Step 3: return b ( vb ) + vb b Fall 2000; Updated: 0 October 2002

22 Advantages of Bezier Curves umerically very robust Many nice mathematical properties Smooth Global (may be viewed as a disadvantage) Fall 2000; Updated: 0 October 2002

23 B-splines Given coefficient values C = { c,, c } 0 L+ D "knot points" u = { u,, u } with u u 0 L+ 2D 2 i i+ D = "degree" of desired B-spline Can define an interpolated curve P( Cu, ; u) on u u < u D L+ D Then L+ D P( ; ) = D C u c ( u) where later) = 0 D () u are B-spline basis polynomials (discussed Fall 2000; Updated: 0 October 2002

24 deboor Algorithm Given u, c, D as before, can evaluate P(,; cuu) recursively as follows: Step : Step 2: Determine index i such that u u < u Determine multplicity r such that u = u = = u i r i r+ i 0 Step 3: Det for d = c i D+ D r i + i + Step 4: Compute P(,; cuu) = d recursively, where d u u u u = d + i i+ k + D k k k u+ D k u u+ D k u d Fall 2000; Updated: 0 October 2002

25 B-spline basis functions Given Cu,,D as before where L+ D (, ; ) = D P Cuu c ( u) = 0 0 u u u ( u) = 0 Otherwise u u u u ( u) = ( u) + ( u) for k > 0 k k + k k + u+ k u u+ k u Fall 2000; Updated: 0 October 2002

26 Some advantages of B-splines Efficient umerically stable Smooth Local Fall 2000; Updated: 0 October 2002

27 Consider the 2D polynomial i= 0 = 0 2D Interpolation Puv (, ) = ca( ub ) ( v) where m n i i A( u) and B ( v) can be arbitrary i c00 c0n B0( v) = [ A0 ( u),, Am ( u)] c c B ( v) m0 mn n functions (good choices Bernstein polynomials or B-Spline basis functions. Suppose that we have samples ys = y( u, v ) for s = 0,..., s s s We want to find an approximating polynomial P Fall 2000; Updated: 0 October 2002

28 2D Interpolation: Finding the best fit Given a set of sample values y ( u, v ) corresponding to 2D coordinates [ B v B v ] [ ] ( u, v ), left hand side basis functions A ( u),, A () u and right hand side s s 0 m basis functions ( ),, ( ), the goal is to find the coefficients c. i 0 n s s s matrix C of To do this, solve the least squares problem c00 c 0 s( us, v ) y s A0( us) B0( vs) A0( us) B( vs) Ai( us) B( vs) Am( us) Bn( vs) ci cmn Fall 2000; Updated: 0 October 2002

29 { 0 } 2D Interpolation: Finding the best fit A common special case arises when the ( u, v ) form a regular grid. In this { 0 } { 0 } case we have u u,, u and v v,, v. For each value v v,, v solve the row least squares problem s v s s u X 0 ys( us, v ) A0( us) Am( us) X m for the unknown m-vector X. Then solve m n-variable least squares problems X X X X X X X X X m m 0 m v v v s s v B ( v ) B( v0) Bn( v0) c00 c0 cm0 B0( v) B( v) Bn( v) c0 c c m c0n c n cmn B0( v ) B( v ) ( ) v B v n v v for the vectors c0,, cn. ote that this latter step requires only SVD or similar matrix computation Fall 2000; Updated: 0 October 2002

30 2D Interpolation: Finding the best fit 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 ote that there is a similar trick that you can play by grouping all the common u i elements together. ote that the y s and the c s do not have to be scalar numbers. They can be Vectors, Matrices, or other obects that have appropriate algebraic properties Fall 2000; Updated: 0 October 2002