Simple Formulas for Quasiconformal Plane Deformations

Transcription

1 Simple Formulas for Quasiconformal Plane Deformations by Yaron Lipman, Vladimir Kim, and Thomas Funkhouser ACM TOG 212 Stephen Mann

2 Planar Shape Deformations Used in Mesh parameterization Animation shape interpolation others

3 Planar Shape Deformations Used in Mesh parameterization Animation shape interpolation others Issues Distort shape Bijective? Interpolation issues

4 Planar Shape Deformations Used in Mesh parameterization Animation shape interpolation others Issues Distort shape Bijective? Interpolation issues Conformal maps Preserve angles everywhere Bounded distortion: quasiconformal

5 Main Ideas Map four points to four points (interpolation) f

6 Main Ideas Map four points to four points (interpolation) Embed quads in circular grid f

7 Main Ideas Map four points to four points (interpolation) Embed quads in circular grid Map circular grid to parallelograms f m z m w

8 Main Ideas Map four points to four points (interpolation) Embed quads in circular grid Map circular grid to parallelograms Map parallelograms to each other f m z m 1 w A Map is f = m z A m 1 w

9 Circles Through Four Points Through 4 points, there exists unique 5th point z such that There exist four circles through consecutive points and z Opposite circles osculate z is outside z

10 Mobius Transformations Work in complex plane Construct a Mobius transformation to map quad to parallelogram m(z) = az + b, ad bc cz + d z m(z) (basically, map circles to lines, z to )

11 Details Look for Mobius transform that maps to parallelogram and linear transformation from square to same parallelogram: m z (z j ) = L(g j ), j = 1..4 where z j are four points, g j are corners of square

12 Details Look for Mobius transform that maps to parallelogram and linear transformation from square to same parallelogram: m z (z j ) = L(g j ), j = 1..4 where z j are four points, g j are corners of square Expand: az j + b cz j + d = g j + lg j, j = 1..4

13 Details Look for Mobius transform that maps to parallelogram and linear transformation from square to same parallelogram: m z (z j ) = L(g j ), j = 1..4 where z j are four points, g j are corners of square Expand: az j + b cz j + d = g j + lg j, j = 1..4 Multiply both sides by cz j + d gives 4 nonlinear equations in 5 unknowns (a, b, c, d, l) [ Z 1 ZG G ZG G ] (a, b, c, d, lc, ld) T =

14 Details Look for Mobius transform that maps to parallelogram and linear transformation from square to same parallelogram: m z (z j ) = L(g j ), j = 1..4 where z j are four points, g j are corners of square Expand: az j + b cz j + d = g j + lg j, j = 1..4 Multiply both sides by cz j + d gives 4 nonlinear equations in 5 unknowns (a, b, c, d, l) [ Z 1 ZG G ZG G ] (a, b, c, d, lc, ld) T = Solve using SVD and roots of degree 2 equation

15 Implementation Implemented in Octave (SVD, complex numbers, bad documentation)

16 Implementation Implemented in Octave (SVD, complex numbers, bad documentation) About 45 lines of code (plus testing, application code)

17 Implementation Implemented in Octave (SVD, complex numbers, bad documentation) About 45 lines of code (plus testing, application code) 4 8 hours work (errors, vagueness in paper, Octave struggles)

18 Implementation Implemented in Octave (SVD, complex numbers, bad documentation) About 45 lines of code (plus testing, application code) 4 8 hours work (errors, vagueness in paper, Octave struggles) Simple examples seem okay

19 Implementation Implemented in Octave (SVD, complex numbers, bad documentation) About 45 lines of code (plus testing, application code) 4 8 hours work (errors, vagueness in paper, Octave struggles) Simple examples seem okay...

20 Implementation Implemented in Octave (SVD, complex numbers, bad documentation) About 45 lines of code (plus testing, application code) 4 8 hours work (errors, vagueness in paper, Octave struggles) Simple examples seem okay...

21 Circle Examples Circle examples mixed (red: invalid data)

22 Restrictions on Deformation Paper mentions two restrictions: ad bc Counter-clockwise order Does not discuss boundary cases Does not discuss how restrictive conditions are

23 Restrictions on Deformation Paper mentions two restrictions: ad bc Counter-clockwise order Does not discuss boundary cases Does not discuss how restrictive conditions are Invalid circle examples give some feel

24 Restrictions on Deformation Paper mentions two restrictions: ad bc Counter-clockwise order Does not discuss boundary cases Does not discuss how restrictive conditions are Invalid circle examples give some feel Paper is big on circle not crossing using their method...

25 Restrictions on Deformation Paper mentions two restrictions: ad bc Counter-clockwise order Does not discuss boundary cases Does not discuss how restrictive conditions are Invalid circle examples give some feel Paper is big on circle not crossing using their method......but limits on where you can move points.

26 Restrictions on Deformation Paper mentions two restrictions: ad bc Counter-clockwise order Does not discuss boundary cases Does not discuss how restrictive conditions are Invalid circle examples give some feel Paper is big on circle not crossing using their method......but limits on where you can move points. And is not crossing good? Overly restrictive? Was circle a strawman example?

27 Local Version Deformation is global Very large distortion outside of quad

28 Local Version Deformation is global Very large distortion outside of quad Authors developed a local version Restricted operations Move opposite edges (i.e., to get a bend) Move one point (two edges left unchanged)

29 Local Version Deformation is global Very large distortion outside of quad Authors developed a local version Restricted operations Move opposite edges (i.e., to get a bend) Move one point (two edges left unchanged) Used variation on method that leaves solid edges unchanged (in f = m z A mw 1, use different A ; higher distortion) More engineering than math

30 Details Working with images of Mobius functions (parallelograms): Sample points on fixed edges Also sample δ inside fixed edges

31 Details Working with images of Mobius functions (parallelograms): Sample points on fixed edges Also sample δ inside fixed edges Map points to parallelograms m z m w c j d j

32 Details Working with images of Mobius functions (parallelograms): Sample points on fixed edges Also sample δ inside fixed edges Map points to parallelograms Use thin-plate splines to construct transformation to map z points onto w points m z m w c j d j ϕ(z) = j b j φ( z c j ) + A(z) ϕ(c j ) = d j

33 How To Solve ϕ(z) = j b jφ( z c j ) + A(z) Paper says done in a standard way and cites a book

34 How To Solve ϕ(z) = j b jφ( z c j ) + A(z) Paper says done in a standard way and cites a book I tried my own way (we have A, just solve part as linear system)

35 How To Solve ϕ(z) = j b jφ( z c j ) + A(z) Paper says done in a standard way and cites a book I tried my own way (we have A, just solve part as linear system)

36 How To Solve ϕ(z) = j b jφ( z c j ) + A(z) Paper says done in a standard way and cites a book I tried my own way (we have A, just solve part as linear system) Looked up map in book: it is standard. And brief: [ ] [ ] [ ] φ( ci c j ) P(c j ) bj dj P(c j ) T = A (Last page of paper 1/3 blank)

37 How To Solve ϕ(z) = j b jφ( z c j ) + A(z) Paper says done in a standard way and cites a book I tried my own way (we have A, just solve part as linear system) Looked up map in book: it is standard. And brief: [ ] [ ] [ ] φ( ci c j ) P(c j ) bj dj P(c j ) T = A (Last page of paper 1/3 blank)

38 Example

39 Example

40 Maps for Example (upside down, reverse order, inorder)

41 Another Example

42 Another Example

43 Map for Example

44 Other Radial Basis Functions? Thin plate splines: minimum solution of distance to affine map? Tried r 3 and 1/(r 2 + ɛ) Conformal Thin Plate r 2 log r r 3 Thin plate is less deformed in some places

45 Radial basis function: 1/(r 2 + ɛ) Thin Plate r 2 log r 1/(r 2 +.5) 1/(r 2 +.5) 1 r clearly bad; thin plate probably better than 1 r 2 +.5

46 Effect of n Method to reproduce boundaries only approximates ϕ(z) = n j=1 b jφ( z c j ) + A(z) They use n = 1. What is effect of n On caterpillar example: n = n = 5 n = 1

47 Normal Points Method uses normal points to interpolate cross boundary derivatives Ran caterpillar example with and without normal points: Normal points No normal points Both

48 Which Normal Points? z α p z α+1 z β+1 p z β Paper uses p = p + nδ [ z α z β+1 p z α+1 + z β z α+1 p z α ] (which is incorrect: need to normalized by by z α+1 z α )

49 Which Normal Points? z α p z α+1 z β+1 p z β Paper uses p = p + nδ [ z α z β+1 p z α+1 + z β z α+1 p z α ] (which is incorrect: need to normalized by by z α+1 z α ) What if we use simpler p = (1 δ)p + δ p

50 Perpendicular vs Affine 35 Perpendicular 35 Affine

51 Perpendicular vs Affine Maps 35 Perpendicular 35 Affine

52 Delta Paper chose δ =.1 δ =.1 δ =.1 δ =.1 Need to test with regular textures, etc., to decide what value best (but.1 looks bad)

53 Counter-clockwise Paper says ordered in counter-clockwise fashion (different order will lead to a different map). How different?

54 Counter-clockwise Paper says ordered in counter-clockwise fashion (different order will lead to a different map). How different? Counter-clockwise

55 Counter-clockwise Paper says ordered in counter-clockwise fashion (different order will lead to a different map). How different? Clockwise

56 What matters? Out of 3 stars: *** Perpendicular vs Affine normal points *** Counter-clockwise vs clockwise ** Value of n * Thin plate vs other radial basis * Normal points * Value of delta * Method to solve for thin plate coefficients : clearly needs more investigation

57 Code Line count Global method: 45 lines of code Local method: 142 (+25) lines of code (+32) Testing, Applications: 751 lines of code (but...)

58 Code Line count Global method: 45 lines of code Local method: 142 (+25) lines of code (+32) Testing, Applications: 751 lines of code (but...) Speed

59 Code Line count Global method: 45 lines of code Local method: 142 (+25) lines of code (+32) Testing, Applications: 751 lines of code (but...) Speed Really? It s Octave...

60 Extension: Scaling In their method, distance between white points connected with solids lines is fixed

61 Extension: Scaling For perpective, might want scaling No scaling

62 Extension: Scaling For perpective, might want scaling Scaling

63 Analysis Method seems to do reasonable job. Why? Low deformation Lines map to circles avoids kinks

64 Analysis Method seems to do reasonable job. Why? Low deformation Lines map to circles avoids kinks Comparison to other methods unfair? Doesn t map quads interiors to quad interiors

65 Can you break it? Yes, but...

66 Can you break it? Yes, but... (pixel dropout due to...?)

67 Can you break it? Yes, but... (pixel dropout due to...?)

68 Straight Sections Found several cases where map is straight and bend concentrated

69 Conclusions + Slick global method Nice math, easy to implement

70 Conclusions + Slick global method Nice math, easy to implement + Local method Doesn t have guarantees of global method Not so simple

71 Conclusions + Slick global method Nice math, easy to implement + Local method Doesn t have guarantees of global method Not so simple But seems reasonable (straight sections?)

72 Conclusions + Slick global method Nice math, easy to implement + Local method Doesn t have guarantees of global method Not so simple But seems reasonable (straight sections?) Restrictions on four points not discussed Circle editing seems oversold Cherry picked examples?

73 Conclusions + Slick global method Nice math, easy to implement + Local method Doesn t have guarantees of global method Not so simple But seems reasonable (straight sections?) Restrictions on four points not discussed Circle editing seems oversold Cherry picked examples? Geometric algebra reformulation?

74 Acknowledgements Several of the figures in this talk were based on figures in the Lipman, Kim, Funkhouser paper.