Interpolation and Deformations A short cookbook

Transcription

1 Interpolation and Deformations A short cookbook Fall 2000; Updated: 29 September 205

2 Linear Interpolation p ρ 2 2 = [ ] = 20 T p ρ = [ ] = 5 p 3 = ρ3 =? 0?? T [ ] T Fall 2000; Updated: 29 September 205

3 Linear Interpolation p3 = q3 = a??? + 3 b a p = [ ] T q = a p q [ ] ( ) 2 2 T = = [ ] T b Fall 2000; Updated: 29 September 205

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 λ A 2 ( p3 p) ( p2 p) ( p p ) ( p p ) = Fall 2000; Updated: 29 September 205

5 Bilinear Interpolation u ( λ) u λ i+, j+ i, j+ ( µ ) u i, j λ µ ( λ) u i+, j Fall 2000; Updated: 29 September 205

6 Bilinear Interpolation u u i+, j+ i, j+ ( µ ) µ u i, j λ Fall 2000; Updated: 29 September 205 ( λ) ( µ ) u µ i+, j u( λ, µ ) = λ ( µ u ( µ ) ) u + + ( λ ) µ u + + ( µ ) u = u + λ u u + µ u u + λµ u u ( ) i, j i, j i, j i, j ( ) ( ) ( ) i, j i, j i, j i, j i, j i, j i, j

7 i, j + i, j + Bilinear Interpolation u, A u i+, j +, Ai +, j + u, A i, j i, j u ; A i+, j i+, j u u u u u ( λ, µ ) = interpolate( { λ, µ }{, } i, j, i+, j, i+, j +, i, j + ) ( λ, µ ) = interpolate( { λ, µ }{, } i, j, i+, j, i+, j +, i, j + ) A A A A A Fall 2000; Updated: 29 September 205

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: 29 September 205

9 Barycentric Interpolation p ( λ) p 2 p λ p p 2 p 2 p p p = λ p + µ p ( λ ) = ( λ ) 2 + λ λ + µ = Fall 2000; Updated: 29 September 205

10 Barycentric Interpolation p 3 A 3 p p 2 A p ( ( λ,, µ ) ) =???? p 3 + λ( ( p p ) 3 ) + µ ( ( p 2 p ) 3 ) = λ p + µ p2 2 + ( ( λ µ ) ) p3 3 p λ, µ, ν = λ p + µ p + ν p where λ + µ + ν = ( ) (,, ) A A λ µ ν = λ A + µ A + νa Fall 2000; Updated: 29 September 205

11 Barycentric Interpolation p 3 A 3 p p 2 A 2 A p p p2 p3 λ = µ ν Fall 2000; Updated: 29 September 205

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: 29 September 205 OTE: This is a special case of barycentric Bezier st polynomial interpolations (here, degree)

13 Interpolation of functions y( v) 0 v Fall 2000; Updated: 29 September 205

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: 29 September 205

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 P( c,..., c ; v ) P( c,..., c ; v ) = c P ( 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: 29 September 205

16 -D Interpolation 0 k, k k = 0 { y v y v } Given set of known values ( ),..., ( ), find an approximating polynomial y P( c,..., c ; v ) P( c,..., c ; v ) = c P ( v ) 0 0 m 0 m To do this, solve: P,0( v0 ) P, ( v0 ) c0 y0 " # " " " P,0 ( vm ) P, ( vm ) c ym Fall 2000; Updated: 29 September 205

17 Bezier and Bernstein Polynomials ( ) ( ) = k P c0,, c; v ck v v k = 0 k = k = 0, k ( v ) ( ) k where B, k ( v ) = v v k Excellent numerical stability for 0<v< There exist good ways to convert to more conventional power basis c B k k Fall 2000; Updated: 29 September 205

18 Barycentric Bezier Polynomials ( ) = c k P c 0,,c ;u,v where B,k (u,v) = k =0 k =0 Excellent numerical stability for 0<u,v< There exist good ways to convert to more conventional power basis k = c k B,k (u,v) k u k v k u k v k u+v= Fall 2000; Updated: 29 September 205

19 Bezier Curves " Suppose that the coefficients c are multi-dimensional vectors (e.g., 2D or 3D points). Then the polynomial " " " P c,, c ; v = c B ( v ) ( ) 0 k, k k = 0 computed over the range 0 v curve with control vertices c ". j j generates a Bezier c c 2 c Fall 2000; Updated: 29 September 205 c 3

20 Bezier Curves: de Casteljau Algorithm " Given coefficients c, Bezier curves can be generated recursively by repeated linear interpolation: " " P c c v = b where b (,, ; ) = 0 0 " c 0 j j k k ( ) k j = j + j + b v b vb j c0 v c ( v) v ( v) c2 v ( v) c Fall 2000; Updated: 29 September 205

21 Iterative Form of decasteljau Algorithm Step : b c for 0 j Step 2 : for k step until k = do for j 0 step until j = k do Step 3: return j j b ( v ) b + vb b j j j Fall 2000; Updated: 29 September 205

22 Advantages of Bezier Curves umerically very robust Many nice mathematical properties Smooth Global (may be viewed as a disadvantage) Fall 2000; Updated: 29 September 205

23 B-splines Given coefficient values C = { c 0,", c L+D } "knot points" u = {u 0,",u L+2D 2 } with u i u i+ D = "degree" of desired B-spline Can define an interpolated curve P(C,u; u) on u D u < u L+D Then P(C;u) = L+D c j j =0 D (u) j where j D (u) are B-spline basis polynomials (discussed later) Fall 2000; Updated: 29 September 205

24 B-Spline Polynomials Some useful references include L-09_BSplines_URBS.pdf Fall 2000; Updated: 29 September 205

25 B-spline polynomials & B-spline basis functions Given C, u,d as before where P(C,u;u) = j 0 (u) = L+D c j j =0 u j u u j 0 Otherwise j D (u) j k (u) = u u j k u j +k u j (u) + u u j +k k j u j +k u j + (u) for k > 0 j Fall 2000; Updated: 29 September 205

26 For a B-spline polynomial P(C,u;t) = L+D c j j=0 B-Spline Polynomials j D (u,t) the basis functions j D (u,t) are a function of the degree of the polynomial and the vector u = u 0,",u n of "knot points". The polynomial is "uniform" if the distance between knot points is evenly spaced and "non-uniform" otherwise Fall 2000; Updated: 29 September 205

27 deboor Algorithm (Farin) Given u, c, D as before, can evaluate P(c,u;u) recursively as follows: Step : Determine index i such that u i u < u i+ Step 2: Determine multplicity r such that u i r = u i r + = = u i Step 3: Set " d j 0 = c j for i D + j i + Step 4: Compute P(c,u;u) = d D r i+ recursively, where " d j k = u j+d k u " k dj + u j+d k u j u u j " k dj = α k " k j d j u j+d k u k j γ j + β j k " d j k γ j k Fall 2000; Updated: 29 September 205

28 deboor Algorithm: Example D=3, r=0 d j k = u j +D k u k + u j +D k u j dj 3 0 d i+ = p( { ", ci,"},3;u) 3 2 = α i+ d i + 3 βi+ 2 d i+ ( 2 ) 3 d i+ γ i+ ( ) 2 d i+ γ i+ 2 = α i+ d i + 2 βi+ k u u j α k k j d j dj = + β k k j d j u j +D k u k k j γ j γ j (( ) d 2 i + (u ui ) d 2 ) i+ (u i+ u i ) ( ) d i + (u ui ) d i+ = u i+ u ( ) = u i+2 u (u i+2 u i ) ( 2 + β ) 2 i d i γ i = ( (u i+ u) d i + (u u i ) d ) i (u i+ u i ) ( d 0 i + βi+ d 0 ) ( i+ γ i+ = ( u i+3 u) d 0 i + (u ui ) d 0 ) i+ (u i+3 u i ) ( 0 + β 0 ) i d i γ i = ( (u i+2 u) d 0 i + (u u i ) d 0 ) (u i i+2 u i ) ( d 0 i 2 + β d 0 ) ( i γ i i = (u i+ u) d 0 i 2 + (u u i 2 ) d 0 ) (u i i+ u i 2 ) d i 2 = αi 2 d i d i+ d i = α i+ d i = αi d i = α i Fall 2000; Updated: 29 September 205

29 deboor Algorithm (alternative formula) An alternative formulation from Wikipedia is given as: Given u, c, D as before, can evaluate P(c,u;u) recursively as follows: Step : Determine index i such that u i u < u i+ Step 2: Determine multplicity r such that u i r = u i r + = = u i Step 3: Set " d j 0 = c j for i D + j i + Step 4: Compute P(c,u;u) = d D r i+ recursively, where " d k j = ( α k,j ) d " k " j + α k k,j dj where α k,i = u u j u j+d+ k u j Source: Fall 2000; Updated: 29 September 205

30 Uniform B-Spline Polynomials Third degree uniform B-spline P(C,u;t) = c j 2 j (u,t) with t j = j j 3 j (u,t) = 6 (t j)2 if j t < j ( t j ) 3 + 3( t j ) + 3( t j ) + if j + t < j ( t j ) 3 6( t j ) + 4 if j + 2 t < j ( t j 3 ) if j + 3 t < j otherwise Fall 2000; Updated: 29 September 205

31 Some advantages of B-splines Efficient umerically stable Smooth Local Fall 2000; Updated: 29 September 205

32 2D Interpolation (tensor form) Consider the 2D polynomial m n P( u, v ) = c A ( u) B ( v ) i = 0 j = 0 ij i j where A ( u) and B ( v ) can be arbitrary i c00 c0n B0( v ) = [ A0 ( u),, Am ( u)] " # " " c c B ( v ) j 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: 29 September 205

33 2D Interpolation: Finding the best fit Given a set of sample values y s (u s,v s ) corresponding to 2D coordinates (u s,v s ), left hand side basis functions A 0 (u),,a m (u) and right hand side basis functions B 0 (v),,b n (v), the goal is to find the matrix C of coefficients c ij. To do this, solve the least squares problem " y s (u s,v s ) " " A 0 (u s )B 0 (v s ) " " A 0 (u s )B (v s ) " " A i (u s )B j (v s ) " " " A m (u s )B n (v s ) " " c 00 c 0 c ij c mn Fall 2000; Updated: 29 September 205

34 2D Interpolation: Sampling on a regular grid A common special case arises when the ( u, v ) form a regular grid. In this { 0 } { 0 } { 0 } case we have u u,, u and v v,, v. For each value v v,, v solve the row least squares problem j s v s s u " " " X j 0 ys( us, v j ) A0 ( us ) Am ( us ) " " " " X jm 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 j 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 c j 0,, c jn. ote that this latter step requires only SVD or similar matrix computation Fall 2000; Updated: 29 September 205

35 2D Interpolation: Sampling on a regular grid 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 objects that have appropriate algebraic properties Fall 2000; Updated: 29 September 205

36 -dimensional interpolation The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Define F " ( u) = A ( u ) A ( u ) i i i i Then solve the least squares problem c00 0 # # " " " c 0 0 " F00 0( us ) F0 0( us ) Fm m ( u ) n s y # s # # cm mn Fall 2000; Updated: 29 September 205

37 -dimensional interpolation The methods described earlier generalize naturally to dimensions. " P( u) = P( u,, u ) = c A ( u ) A ( u ) where A K i m m i in i i i = 0 i = 0 ( u) can be arbitrary functions (good choices are Bernstein polynomials or B-Spline basis functions). Suppose that we have samples " y = y ( u ) for s = 0,..., s s s We want to find coefficients of c approximating polynomial P. i in Fall 2000; Updated: 29 September 205

38 Example: 3D Calibration of Distortion Fall 2000; Updated: 29 September 205

39 Example: 3D Calibration of Distortion x y z c 000 c000 c000 " " x y z F 000( us ) # F555 ( us ) p s ps ps x y z c555 c555 c Fall 2000; Updated: 29 September 205

40 Example: 3D Calibration of Distortion The correction function will then look like this: p = CorrectDistortion( q) min max { u = ScaleToBox( q, q, q ) } return c B ( u ) B ( u ) B ( u ) i=0 j=0 k=0 i, j, k 5, i x 5, j y 5, k z Fall 2000; Updated: 29 September 205