1. Mappings Representation and Distortion
Roi Poranne and Shahar Kovalsky

2. Mappings Representation and Distortion
Modeling
[Sorkine & Alexa 07]
[Ju et al. 05]
[Poranne & Lipman 14]
[Lipman et al. 07]
[Weber et al. 09]
[Jacobson 07]

3. Mappings Representation and Distortion
Parameterization
[Lévy et al. 02]
[Schuler et al. 13]
[Fu et al. 15]
[Weber et al. 12]
[Mullen et al. 08]

4. Mappings Representation and Distortion
Surface mappings
[Ovsjanikov et al. 12]
[Panozzo et al. 13]
[Kim et al. 11]
[Schreiner et al. 04]
[Kraevoy and Sheffer 04]
[Jin et al. 08]

5. “Real” Applications
Brain/colon mapping
[Gu et al.]

6. Biological Morphology
“Real” Applications
Biological Morphology
[Boyer et al. 2012]

7. Medical segmentation/registration
“Real” Applications
Medical segmentation/registration
[Levi and Gotsman 12]

8. Definition Examples Mapping / Map : A smooth function between shapes /
spaces
Examples
𝑓:ℝ→ℝ
𝑓 𝑥 = 𝑥 2

9. 𝑓 𝑧 =𝑧+ 1 𝑧
𝑓: ℝ 2 → ℝ 2

10. 𝑓:𝕄→ ℝ 2

11. 𝑓:𝕄→ 𝕊 2
We can also parameterize to other canonical domains like a sphere which you can see here

12. 𝑓:𝕄→ 𝕄 ′

13. 𝑓: ℝ 3 → ℝ 3

14. 𝑓: ℝ 3 → ℝ 3
All
cases
are
based on
the
same
concepts
and
use
similar
techniques
Our
focus : 2D
mappings

15. Outline Roi Shahar Representation Distortion of mappings
Optimization of mappings

16. 2D Maps : Representation
𝐟 𝐱 = 𝑢 𝐱 𝑣 𝐱

17. 2D Maps : Representation𝐟
𝐟 𝐱 = 𝑢 𝐱 𝑣 𝐱
Solution: Represent maps as linear combination of basis functions
The set of all maps
𝑓 𝑥 : ℝ 2 → ℝ 2
is too big to handle!

18. 2D Maps : Representation
𝑓 1 , 𝑓 2 , 𝑓 3 ,…, 𝑓 𝑛

19. 2D Maps : Representation
𝑓 1 , 𝑓 2 , 𝑓 3 ,…, 𝑓 𝑛
𝐟 𝐱 = 𝑢 𝐱 𝑣 𝐱

20. 2D Maps : Representation
𝑓 1 , 𝑓 2 , 𝑓 3 ,…, 𝑓 𝑛
𝐟 𝐱 = 𝑢 𝐱 𝑣 𝐱
= ∑ 𝑎 𝑖 𝑓 𝑖 𝐱 ∑ 𝑏 𝑖 𝑓 𝑖 𝐱

21. 2D Maps : Representation
𝑓 1 𝑥,𝑦 =𝑥
𝑓 2 𝑥,𝑦 =𝑦
𝑥,𝑦 ⇒ 2 𝑓 1 , 𝑓 2

22. 2D Maps : Representation
𝑓 1 𝑥,𝑦 =𝑥
𝑓 2 𝑥,𝑦 =𝑦
𝑥,𝑦 ⇒ 𝑓 1 + 𝑓 2 , 𝑓 2

23. 2D Maps : Representation
𝑓 1 𝑥,𝑦 =𝑥
𝑓 2 𝑥,𝑦 = 𝑦
𝑓 3 𝑥,𝑦
𝑥,𝑦 ⇒ 𝑓 1 + 𝑓 3 , 𝑓 2 + 𝑓 3

24. Mappings for deformations
𝐩 𝑖

25. Mappings for deformations
𝐪 𝑖

26. Deformation as an interpolation problem
𝐟 𝐩 𝑖 = 𝐜 𝑖 𝑓 𝑖 𝐱
𝒒 𝑖
𝐜 𝑖 =?
𝐩 𝑖
𝐟 𝐩 𝑖 = 𝐪 𝑖 , ∀𝑖
𝐜 𝑖 𝑓 𝑖 𝐩 𝑖 = 𝐪 𝑖 , ∀𝑖

27. Example: Thin Plate Spline
Solve the problem
min 𝐸 TPS (𝐟) = 𝜕 2 𝐟 𝜕 𝑥 𝜕 2 𝐟 𝜕𝑥𝜕𝑦 𝜕 2 𝐟 𝜕 𝑦
Bending energy
s.t. 𝐟 𝐩 𝑖 = 𝐪 𝑖 , ∀𝑖
General solution
𝐟 𝐩 𝑖 = 𝐜 0 + 𝐜 𝑥 𝒙+ 𝐜 𝑦 𝒚+ 𝐜 𝑖 ϕ ‖𝐱− 𝐩 𝑖 ‖
ϕ 𝑟 = 𝑟 2 log 𝑟

28. Hermite interpolation
Interpolate derivatives

29. Hermite interpolation
Interpolate derivatives

30. Hermite interpolation
Interpolate derivatives
𝐟 𝐩 𝑖 = 𝐪 𝑖
𝐃𝐟 𝐩 𝑖 = 𝐃 𝑖

31. Hermite interpolation
Interpolate derivatives
𝐜 𝑖 𝑓 𝑖 𝐩 𝑖 = 𝐪 𝑖
𝐜 𝑖 𝛻 𝑓 𝑖 𝐩 𝑖 = 𝐃 𝑖

32. Example: Linear Blend Skinning
𝐟 𝐱 = 𝑤 𝑖 𝐱 𝐓 𝐢 𝒙+ 𝐪 𝐢
Weights
Affine
transformations

33. Example: Linear Blend Skinning
𝐟 𝐱 = 𝑤 𝑖 𝐱 𝐓 𝑖 𝒙+ 𝐪 𝑖

34. Example: Linear Blend Skinning
𝐟 𝐱 = 𝑤 𝑖 𝐱 𝐓 𝑖 𝒙+ 𝐪 𝑖
𝑤 𝑖 𝐩 𝑗 = 𝛿 𝑖𝑗
Lagrange property
Hermite (derivative) property
𝛻 𝑤 𝑖 𝐩 𝑗 =𝟎

35. Deformation as an boundary interpolation
Target
𝐟 𝐱
Source

36. Example: Barycentric Coordinates
[Li et al.]
[Lipman et al]
[Schneider et al.]
[Li et al.]
[Weber et al.]
[Weber et. al]

37. Example: Barycentric Coordinates
𝛼 𝑖 (𝑥)
Stages:
Source shape
Polygonal cage
Coordinates
𝑓 𝐱 = 𝑖=1 𝑛 𝛼 𝑖 𝐱 𝐪 𝑖

38. Example: Barycentric Coordinates
𝛼 𝑖 (𝑥)
Stages:
Source shape
Polygonal cage
Coordinates
𝑓 𝐱 = 𝑖=1 𝑛 𝛼 𝑖 𝐱 𝐪 𝑖

39. Example: Barycentric Coordinates
𝐪 𝑖
𝛼 𝑖 (𝑥)
Stages:
Source shape
Polygonal cage
Coordinates
Manipulate cage
Apply deformation
𝑓 𝐱 = 𝑖=1 𝑛 𝛼 𝑖 𝐱 𝐪 𝑖

40. Mean-value coordinates
Cauchy coordinates
Harmonic coordinates

41. Example: Hermite Bary’ Coordinates
𝑓 𝑖
𝑓 𝑥 = 𝑖=1 𝑛 𝛼 𝑖 𝑥 𝑓 𝑖

42. Example: Hermite Bary’ Coordinates
𝑓 𝑖
𝑓 𝑥 = 𝑖=1 𝑛 𝛼 𝑖 𝑥 𝑓 𝑖

43. 𝑓 𝑖
𝑓 𝑖
𝑓 𝑥 = 𝑖=1 𝑛 𝛼 𝑖 𝑥 𝑓 𝑖
𝑑 𝑖
+ 𝑖=1 𝑛 𝛽 𝑖 𝑥 𝑑 𝑖

46. Basis functions

47. What are good maps? Bijectivity Low distortion Local Not Bijective
Lower
distortion
Bijective

48. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

49. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

50. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

51. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

52. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

53. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

54. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective
Still Bijective!

55. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

56. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

57. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

58. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective
Not
Bijective!

59. Globally Bijective VS. Locally Bijective
Two
Pre-images
Not
Bijective!

60. Globally Bijective VS. Locally Bijective
And the other is there.
Not
Bijective!

61. Globally Bijective VS. Locally Bijective𝐔
This map is however locally bijective. We know this is true because every neighborhood we look at, for example this one, is bijectivly mapped to its image. Of course, if we want to prove this using the definition we will have to work harder, but there actually a simpler sufficient condition, which I’ll show in a minute
Only Locally
Bijective!
𝐟(𝐔)

62. Locally Bijection – Non-example
Ok, now not all mappings are locally bijective. Here’s a non-example. Let start with this square and begin deforming it.

63. Locally Bijection – Non-example

64. Locally Bijection – Non-example

65. Locally Bijection – Non-example

66. Locally Bijection – Non-example
Ok so far so good

67. Locally Bijection – Non-example

68. Locally Bijection – Non-example

69. Locally Bijection – Non-example

70. Locally Bijection – Non-example

71. Locally Bijection – Sufficient condition𝐟
𝐟 𝐱 = 𝑢 𝐱 𝑣 𝐱
The Jacobian:
𝒥𝐟 𝐱 = 𝜕 𝑥 𝑢 𝐱 𝜕 𝑥 𝑣 𝐱 𝜕 𝑦 𝑢 𝐱 𝜕 𝑦 𝑣 𝐱

72. Locally Bijection – Sufficient condition𝐟
𝐟 𝐱 = 𝑢 𝐱 𝑣 𝐱
The Jacobian:
𝒥𝐟 𝐱 = 𝜕 𝑥 𝑢 𝐱 𝜕 𝑥 𝑣 𝐱 𝜕 𝑦 𝑢 𝐱 𝜕 𝑦 𝑣 𝐱 = 𝛻𝑢 𝐱 𝛻𝑣 𝐱
The Condition:
det 𝒥𝐟 𝐱 >0, ∀𝑥

73. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

74. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective

75. Globally Bijective VS. Locally Bijective
𝐟 is bijective
𝐟:𝐔→𝐟(𝐔) is bijective
Google: “Global inversion theorems”

76. What are good maps? Bijectivity Low distortion Local Not Bijective
Lower
distortion
Bijective

77. Distortion - Types
Conformal distortion
Isometric distortion

78. Distortion - Types Conformal distortion Isometric distortion
The distortion is a function
of the Jacobian at a point

79. Distortion - LSCM LSCM – Least Squares Conformal Map We want the
[Lévy et al. 2002]
Distortion - LSCM
LSCM – Least Squares Conformal Map We
want
the
Jacobian to be a
similarity
matrix
𝜕 𝑥 𝑢 𝜕 𝑥 𝑣 𝜕 𝑦 𝑢 𝜕 𝑦 𝑣
𝛼 𝛽 −𝛽 𝛼
𝜕 𝑥 𝑢 =
𝜕 𝑦 𝑣
Cauchy-Riemann
Equations
𝜕 𝑦 𝑢 =
− 𝜕 𝑥 𝑣

80. Distortion - LSCM LSCM – Least Squares Conformal Map We want the
Jacobian to be a
similarity
matrix
𝜕 𝑥 𝑢 𝜕 𝑥 𝑣 𝜕 𝑦 𝑢 𝜕 𝑦 𝑣
𝛼 𝛽 −𝛽 𝛼
𝒟 LSCM = 2
𝜕 𝑥 𝑢 −
𝜕 𝑦 𝑣 + 2
𝜕 𝑦 𝑢
𝜕 𝑥 𝑣 +

81. Quick Notation Change 𝒥𝐟 𝐀 𝜕 𝑥 𝑢 𝜕 𝑥 𝑣 𝜕 𝑦 𝑢 𝜕 𝑦 𝑣 𝑎 𝑐 𝑏 𝑑 𝒟 LSCM = 2
𝜕 𝑥 𝑢 𝜕 𝑥 𝑣 𝜕 𝑦 𝑢 𝜕 𝑦 𝑣
𝑎 𝑐 𝑏 𝑑
𝒟 LSCM = 2
𝜕 𝑥 𝑢 −
𝜕 𝑦 𝑣 + 2
𝜕 𝑦 𝑢
𝜕 𝑥 𝑣 + 2 𝑎 − 𝑑 + 2 𝑏 𝑐 +

82. Distortion – ASAP ASAP– As Similar As Possible 𝒟 ASAP = 𝐀− 𝒮 𝐀 𝐹 2
[Liu et al. 2008]
Distortion – ASAP
ASAP– As Similar As Possible
𝒟 ASAP =
𝐀− 𝒮 𝐀 𝐹 2
Closest
Similarity
Jacobian
How to compute closest similarity?

83. Distortion – ASAP ASAP– As Similar As Possible
How to compute closest similarity?

84. Distortion – ASAP ASAP– As Similar As Possible
How to compute closest similarity?
In 2D:
min 𝒮 𝐀−𝒮 𝐹 2
s.t. 𝒮 is similarity
min 𝛼,𝛽 𝑎 𝑐 𝑏 𝑑 − 𝛼 𝛽 −𝛽 𝛼 𝐹 2

85. Distortion – ASAP ASAP– As Similar As Possible
How to compute closest similarity?
In 2D:
min 𝒮 𝐀−𝒮 𝐹 2
s.t. 𝒮 is similarity
1 2 𝑎+𝑑 𝑐−𝑏 𝑏−𝑐 𝑎+𝑑 𝒮=

86. Distortion – ASAP ASAP– As Similar As Possible
How to compute closest similarity?
In 2D:
1 2 𝑎+𝑑 𝑐−𝑏 𝑏−𝑐 𝑎+𝑑
1 2 𝑎−𝑑 𝑐+𝑏 𝑏+𝑐 𝑑−𝑎 𝐀= +

87. Distortion – ASAP 𝒮 𝐴 𝒮 𝐴 ⊥ ASAP– As Similar As Possible
How to compute closest similarity?
In 2D:
𝒮 𝐴
𝒮 𝐴 ⊥
1 2 𝑎+𝑑 𝑐−𝑏 𝑏−𝑐 𝑎+𝑑
1 2 𝑎−𝑑 𝑐+𝑏 𝑏+𝑐 𝑑−𝑎 𝐀= +
Similarity
Anti-Similarity

88. Distortion – ASAP 𝑎−𝑑 2 + 𝑏+𝑐 2 ASAP– As Similar As Possible
𝐀− 𝒮 𝐀 𝐹 2
𝒮 𝐴 ⊥
Jacobian
Closest
Similarity
Measure of antisimilarity
𝑎−𝑑 𝑐+𝑏 𝑏+𝑐 𝑑−𝑎 𝐹 2
𝑎−𝑑 𝑏+𝑐 2

89. Distortion – ASAP LSCM = ASAP ASAP– As Similar As Possible 𝐀− 𝒮 𝐀 𝐹 2
𝐀− 𝒮 𝐀 𝐹 2
𝒮 𝐴 ⊥
Jacobian
Closest
Similarity
LSCM = ASAP
Measure of antisimilarity
𝑎−𝑑 𝑐+𝑏 𝑏+𝑐 𝑑−𝑎 𝐹 2

90. Distortion - ARAP ARAP– As Rigid As Possible 𝒟 ARAP = 𝐀− ℛ 𝐀 𝐹 2
𝐀− ℛ 𝐀 𝐹 2
Closest
Rotation
Jacobian
How to compute closest Rotation?

91. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆
𝑉 𝑇
𝜎 𝜎 2
𝜎 1 > 𝜎 2

92. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆
𝑉 𝑇

93. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆
𝑉 𝑇

94. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆
𝑉 𝑇

95. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆
𝑉 𝑇

96. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆
𝑉 𝑇
𝑈 and 𝑉 are not rotations!

97. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇
1 0 0 −1

98. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇

99. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇

100. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇

101. Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇

102. Signed Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇

103. Signed Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇
𝜎 − 𝜎 2
𝜎 1 > 𝜎 2
Now 𝑈 and 𝑉 are rotations!

104. Signed Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇
𝜎 − 𝜎 2
𝜎 1 > 𝜎 2
Now 𝑈 and 𝑉 are rotations!
What if 𝑈 and 𝑉 both had reflections?

105. Signed Singular Value Decomposition
Every Matrix 𝑀 has a factorization of the form 𝑀 = 𝑈 𝑆 𝑅
𝑉 𝑇
𝜎 − 𝜎 2
𝜎 1 > 𝜎 2
Now 𝑈 and 𝑉 are rotations!
What if 𝑈 and 𝑉 both had reflections?
sign det 𝑀 =sign 𝜎 2

106. Singular values in 2D Closed form expression 𝛼= 𝛻𝑢+ 𝑣 2 𝛻 𝛽= 𝛻𝑢+ 𝑣 2 𝛻
𝒥𝐟= 𝜕 𝑥 𝑢 𝜕 𝑥 𝑣 𝜕 𝑦 𝑢 𝜕 𝑦 𝑣 = 𝛻𝑢 𝛻𝑣
𝛼= 𝛻𝑢+ 𝑣 2 𝛻
𝛽= 𝛻𝑢+ 𝑣 2 𝛻
𝜎 1 = 𝛼 + 𝛽
𝜎 2 = 𝛼 − 𝛽

107. Distortion - ARAP ARAP– As Rigid As Possible 𝒟 ARAP = 𝐀− ℛ 𝐀 𝐹 2
𝐀− ℛ 𝐀 𝐹 2
Closest
Rotation
Jacobian
How to compute closest Rotation?

108. Distortion - ARAP ARAP– As Rigid As Possible 𝒟 ARAP = 𝐀− ℛ 𝐀 𝐹 2
𝐀− ℛ 𝐀 𝐹 2
Closest
Rotation
Jacobian
𝐀=𝑈𝑆 𝑉 𝑇
ℛ 𝐀 =𝑈 𝑉 𝑇
Proof: Using Lagrange multipliers

109. Distortion – ASAP ASAP– As Similar As Possible 𝒟 ASAP = 𝐀− 𝒮 𝐀 𝐹 2
𝐀− 𝒮 𝐀 𝐹 2
Closest
Similarity
Jacobian
𝐀=𝑈𝑆 𝑉 𝑇
𝒮 𝐀 = 𝜎 𝑈 𝑉 𝑇
𝜎 = 𝜎 1 + 𝜎 2 2

110. Distortion - ARAP ARAP– As Rigid As Possible 𝒟 ARAP = 𝐀− ℛ 𝐀 𝐹 2
𝐀− ℛ 𝐀 𝐹 2
Closest
Rotation
Jacobian
𝐀=𝑈𝑆 𝑉 𝑇
ℛ 𝐀 =𝑈 𝑉 𝑇
Proof: Using Lagrange multipliers

111. Distortion - ARAP ARAP– As Rigid As Possible 𝒟 ARAP = 𝐀− ℛ 𝐀 𝐹 2
𝐀− ℛ 𝐀 𝐹 2
= 𝐀−𝑈 𝑉 𝑇 𝐹 2
= 𝑈𝑆 𝑉 𝑇 −𝑈 𝑉 𝑇 𝐹 2
= 𝑈(𝑆−𝐼) 𝑉 𝑇 𝐹 2
= (𝑆−𝐼) 𝐹 2
= 𝜎 1 − 𝜎 2 −1 2

112. Distortion - ASAP ASAP– As Similar As Possible 𝒟 ASAP = 𝐀− 𝒮 𝐀 𝐹 2
𝐀− 𝒮 𝐀 𝐹 2
= 𝑈𝑆 𝑉 𝑇 − 𝜎 𝑈 𝑉 𝑇 𝐹 2
= 𝑈(𝑆− 𝜎 𝐼) 𝑉 𝑇 𝐹 2
= 𝜎 1 − 𝜎 𝜎 2 − 𝜎 2
= 𝜎 1 − 𝜎 2 2

113. Distortion bias
𝒟 ASAP (𝐴)
<
𝒟 ASAP (2𝐴)
2 𝜎 1 −2 𝜎 2 2
𝜎 1 − 𝜎 2 2

114. Conformal distortion
𝜎 1 𝜎 2