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Searching for Structure in
Geometry Processing
Misha Kazhdan
What do we want to compute?
(What established theory can we apply?)
How do we compute it effectively?
(How do we implement efficiently/robustly?)
Geometry Processing in Detail
• Symmetry Detection
– Discrete
– Continuous
• Solid Reconstruction
– Computing the indicator function
– Iso-surface extraction
– Surface extension
Discrete Symmetry Detection
Identify the rotational (and reflective)
symmetries of a set of a shape
Reflective
4-Fold
Axial
What is a shape?
A shape is a finite set of points 𝑃 ⊂ ℝ2 .
What do we want to compute?
Given 𝑃, find 𝑐 ∈ ℝ2 and 𝜃 ∈ [0,2𝜋) such that a
rotation by angle 𝜃 about 𝑐 maps 𝑃 to itself.
1. Averaging commutes with affine maps:
⇒ 𝑐 is the center of mass.
2. The rotation is a bijection:
⇒ 𝜃 ∈ ∠𝑝𝑞 𝑞∈𝑃 .
Naïve Implementation
FindRotationalSymmetries( 𝑃 )
1. Fix 𝑝 ∈ 𝑃
2. For each 𝜃 ∈ ∠𝑝𝑞
3.
4.
5.
𝑞∈𝑃
𝑂 |𝑃|
Sym[𝜃] = true
For each 𝑟 ∈ 𝑃
If( 𝜃 𝑟 ∉ 𝑃 ): Sym[𝜃] = false
𝑂 |𝑃|
Naïve run-time complexity is 𝑂 𝑃
𝑂 |𝑃|
3 .
How can we compute it effectively?
Add structure to the problem:
Sort/order points by angle:
– Testing if 𝜃(𝑟) is in 𝑃 is sub-linear time.
– Rotations preserve circular order.
𝑝2
⇒ Rotational symmetries act
by circular shift.
𝑝1
𝜃: 𝑃 → 𝑃
𝑝𝑘 ↦ 𝑝𝑘+3
𝑝3
𝑝4
[Wolter et al., 1985] [Attalah, 1985]
𝑝5
𝑝0
Efficient Implementation
FindRotationalSymmetries( 𝑃 )
1.
2.
3.
SortPolarCoordinatesByAngle( 𝑃 )
𝑆 = 𝒓𝟎 ∠𝑝0 𝑝1 𝒓𝟏 ∠𝑝1 𝑝2 ⋯
FastSubstring( 𝑆 , 𝑆 ∘ 𝑆 )
𝑂 |𝑃| log |𝑃|
𝑂 |𝑃|
𝑂 |𝑃|
𝑆 ∘ 𝑆 = 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘
𝑆 = 𝟏 60∘ 𝟏 20∘ 𝟐 𝑆40=∘ 𝟏 60∘ 𝟏 20∘ 𝟐 𝑆40=∘ 𝟏 60∘ 𝟏 20∘ 𝟐 𝑆40=∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘ 𝟏 60∘ 𝟏 20∘ 𝟐 40∘
(𝟐, 100∘ )
(𝟏, 80∘ )
Complexity is 𝑂 𝑃 log 𝑃 .
(𝟏, 140∘ )
20∘
40∘
20∘ 40∘
(𝟐, 340∘ )
∘
60∘ (𝟏, 320 )
(𝟏, 260∘ )
[Wolter et al., 1985] [Attalah, 1985]
(𝟐, 220∘ )
(𝟏, 20∘ )
40∘
20∘
60∘
(𝟏, 200∘ )
60∘
[Knuth, Morris, and Pratt, 1977]
Given strings 𝐴 and 𝐵, find all matches of 𝐵 in 𝐴.
FindSubstring( 𝐴 , 𝐵 )
1. For 1 ≤ 𝑖 ≤ |𝐴|:
2.
For 1 ≤ 𝑗 ≤ |𝐵|:
3.
If( 𝑏𝑗 ≠ 𝑎𝑖+𝑗 ) break
4.
If( 𝑗 == 𝐵 + 1 ) output 𝑖
Naïve run-time complexity is 𝑂 𝐴 ⋅ 𝐵 .
𝑂 |𝐴|
𝑂 |𝐵|
[Knuth, Morris, and Pratt, 1977]
Example:
𝐵
𝐴=𝑎𝑏𝑎𝑏𝑎𝑏𝑎𝑐
𝑏𝑏
𝑎𝑎
𝑏𝑏
𝑎
𝑐𝑐
𝐵 𝐵=𝐵=𝑎=𝑎
𝑏𝑎
𝑐𝑎
– A shift of 𝐵 by 1 position could only match if the
first and last 4 letters of 𝐵 are the same.
– A shift of 𝐵 by 2 positions could only match if the
first and last 3 letters of 𝐵 are the same.
⋮
Pre-compute the prefix/suffix
structure of 𝐵:*
PMTof
𝑖 =
< 𝑖 𝑏|1𝐵|
⋯𝑏
⋯ 𝑏𝑖 only
– A shift
bymax
less𝑗than
positions
𝐵 𝐵
𝑗 = 𝑏𝑖−𝑗+1could
the “partial
match if a suffix of 𝐵 is also *aa.k.a.
prefix
of 𝐵.matching table”.
What do we want to compute?
Given 𝑃, find 𝑐 ∈ ℝ2 and 𝜃 ∈ [0,2𝜋) such that a
rotation by angle 𝜃 about 𝑐 maps 𝑃 to itself.
Only tells us about perfect matches.
⇓
Doesn’t work for noisy data.
Continuous Symmetry Detection
Measure the extent of rotational
(and reflective) symmetries of a shape
sym
1 2 3 4 56
[Zabrodsky et al., 1995]
𝑘-fold
What is a shape?
A shape is a function from the circle/disk
into the real values.*
Examples:
– Circular extent function
– Arc-length curvature
– Distance function
– Etc.
𝜃
This imbues “shape” with the structure of an inner-product space:
[Zahn et al., 1972] [Horn, 1984] [Ankerst et al., 1999] [Vranic and Saupe, 2001] …
⋅ 𝑆to
𝑝 𝑑𝑝 with the action of rotation.
∗𝑆
1 , 𝑆representation
2 = ∫ 𝑆1 𝑝
2 commute
The
needs
What is symmetry?
If 𝑔 is a symmetry of a shape, then 𝑔𝑛 is also a
symmetry of the shape, for all 𝑛 ∈ ℤ.
⇓
Symmetries are associated with groups.
What do we want to compute?
Given Φ: 𝑆1 → ℝ (or Φ: 𝐷 2 → ℝ) and given a
symmetry group 𝐺, compute the distance from
Φ to the nearest 𝐺-symmetric function.
1. Linear structure:
⇒ 𝐺-symmetric functions are a linear subspace.
2. Group + inner-product structure:
⇒ Projection is averaging
1
𝜋𝐺 Φ =
𝑔 Φ
𝐺
𝑔∈𝐺
What do we want to compute?
Given Φ: 𝑆1 → ℝ (or Φ: 𝐷 2 → ℝ) and given a
symmetry group 𝐺, compute the distance from
Φ to the nearest 𝐺-symmetric function.
Φ − 𝜋𝐺 Φ
2
1
=
2|𝐺|
Φ−𝑔 Φ
𝑔∈𝐺
2
Naïve Implementation
Naïve run-time complexity is:
1
3 −𝑔 Φ
2
Φ − 𝜋𝐺 Φ • =Circle: 𝑂 𝑁Φ
2|𝐺|
• Disk: 𝑂
𝑁4
𝑔∈𝐺
2
FindRotationalSymmetries( 𝑆 )
1.
2.
3.
4.
Φ = Function( 𝑆 )
For each 𝑘 ∈ [2, 𝐾]
For each 𝑗 ∈ [0, 𝑘 − 1]
sym 𝑘 +=
1
2𝑘
Φ−
𝑂 𝑁
𝑂 𝑘
2𝑗𝜋
𝑘
Φ
2
𝑂 𝑁 /𝑂 𝑁 2
How can we compute it effectively?
Computational Structure:
1. Memoization
– 𝐷 𝜃 = Φ − 𝜃 Φ 2 is evaluated 𝑂 𝑁 2 times
but the set of values only has complexity 𝑂 𝑁 .
⇒ Pre-compute the values 𝐷(𝜃) for every angle and
perform constant-time look up at run-time.
How can we compute it effectively?
Computational Structure:
2. Fast Correlation
– The distance can be expressed as:
Φ−𝜃 Φ 2 =2 𝐶 0 −𝐶 𝜃
with 𝐶 𝜃 = 〈Φ, 𝜃 Φ 〉 the auto-correlation of Φ.
⇒ Correlation in the spatial domain is multiplication
in the frequency domain – 𝑂(𝑁)/𝑂 𝑁 2 .
⇒ The change of basis to/from frequency space can
be done with an FFT – 𝑂(𝑁 log 𝑁) /𝑂(𝑁 2 log 𝑁)
[Gauss, 1805][Cooley and Tukey, 1965]
Efficient Implementation
FindRotationalSymmetries( 𝑆 )
1.
2.
3.
4.
Φ = Function( 𝑆 )
𝐶 = AutoCorrelation( Φ )
For each 𝑘 ∈ [2, 𝐾]
For each 𝑗 ∈ [0, 𝑘 − 1]
5.
sym 𝑘 +=
1
𝑘
𝐶 0 −𝐶
𝑂 𝑁 log 𝑁 /𝑂(𝑁 2 log 𝑁)
𝑂 𝑁
𝑂 𝑘
2𝑗𝜋
𝑘
Run-time complexity is:
• Circle: 𝑂 𝑁 2
• Disk: 𝑂 𝑁 2 log 𝑁
𝑂 1
Geometry Processing in Detail
• Symmetry Detection
– Discrete
– Continuous
• Solid Reconstruction
– Computing the indicator function
– Iso-surface extraction
– Surface extension
Solid Reconstruction
Fit a solid to a set of sample points
[Input] What is a shape?
A shape is a finite subset 𝑃 ⊂ ℝ3 × 𝑆 2 × ℝ>0 ,
corresponding to positions, normals, and area.
Shape is an object over which you
integrate 2-forms (i.e. a 2-current).
[Output] What is a shape?
A shape is a function telling us whether a
point is inside or outside the solid:
1
if 𝑝 ∈ 𝑆
𝜒𝑆 𝑝 =
0 otherwise
What do we want to compute?
Compute the (smoothed) indicator function:
1. Compute the gradients
2. Integrate the gradients
𝑃 = { 𝑝, 𝑛, 𝜎 }
[Kazhdan, 2005][Kazhdan et al., 2006]
𝛻𝜒
𝜒
Compute the Gradients
Since the indicator function is discontinuous, we
seek the gradient of the smoothed function:
𝜕 𝜕 𝜕
, ,
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜒𝑆 ∗ 𝐹
for some smoothing filter 𝐹: ℝ3 → ℝ.
Compute the Gradients
Convolution and differentiation commute:
𝜕
𝜕𝐹
𝜒𝑆 ∗ 𝐹 = 𝜒𝑆 ∗
.
𝜕𝑥
𝜕𝑥
Compute the Gradients
The convolution at point 𝑞 ∈ ℝ3 is the integral
of the signal times the shifted filter:
𝜕𝐹
𝜒𝑆 ∗
𝜕𝑥
𝑞
𝜕𝐹𝑞
=
𝜒𝑆 ⋅
𝑑𝑝 =
𝜕𝑥
ℝ3
with 𝐹𝑞 (𝑝) = 𝐹(𝑞 − 𝑝).
𝜕𝐹𝑞
𝑑𝑝
𝑆 𝜕𝑥
Compute the Gradients
The convolution at point 𝑞 ∈ ℝ3 is the integral
of the signal times the shifted filter, which is the
solid integral of the shifted filter:
𝜕𝐹
𝜒𝑆 ∗
𝜕𝑥
𝑞
𝜕𝐹𝑞
=
𝜒𝑆 ⋅
𝑑𝑝 =
𝜕𝑥
ℝ3
with 𝐹𝑞 𝑝 = 𝐹(𝑞 − 𝑝).
𝜕𝐹𝑞
𝑑𝑝
𝑆 𝜕𝑥
Compute the Gradients
The partial is the divergence of a vector field:
𝜕𝐹𝑞
𝑑𝑝 =
𝑆 𝜕𝑥
𝛻 ⋅ (𝐹𝑞 , 0,0) 𝑑𝑝 =
𝑆
〈 𝐹𝑞 , 0,0 , 𝑑𝑛〉 .
𝜕𝑆
Compute the Gradients
The partial is the divergence of a vector field, so
by Stokes’ Theorem:
𝜕𝐹𝑞
𝑑𝑝 =
𝑆 𝜕𝑥
𝛻 ⋅ (𝐹𝑞 , 0,0) 𝑑𝑝 =
𝑆
〈 𝐹𝑞 , 0,0 , 𝑑𝑛〉 .
𝜕𝑆
Compute the Gradients
Discretizing the integral, we get an estimate for
the gradient at 𝑞 as a sum over the input points:
𝛻 𝜒𝑆 ∗ 𝐹
≈
𝑞
=
𝐹𝑞 ⋅ 𝑑𝑛
𝜕𝑆
𝜎 ⋅𝐹 𝑞 −𝑝 ⋅𝑛.
𝑃 = { 𝑝, 𝑛, 𝜎 }
𝑝,𝑛,𝜎 ∈𝑃
𝛻𝜒
Integrate the Gradients
Given a target gradient field 𝑉, the best fit scalar
function is the solution to the Poisson Equation:
𝜒 = arg min 𝛻 𝜒 − 𝑉
⇕
2
∗
Δ𝜒 = 𝛻 ⋅ 𝑉.
𝛻𝜒
𝜒
∗ May
be subject to appropriate boundary conditions.
Naïve Implementation
ReconstructSolid( 𝑃 )
1. ComputeNormalsAndArea*( 𝑃 )
2. For 𝑞 ∈ 0,1 3 :
3.
𝑉 𝑞 =0
4.
∀ 𝑝, 𝑛 , 𝜎 ∈ 𝑃: 𝑉 𝑞 += 𝜎 ⋅ 𝐹 𝑞 − 𝑝 ⋅ 𝑛
𝑂 𝑁3
𝑂 |𝑃|
Note: 5. 𝑑𝑖𝑣 = 𝛻 ⋅ 𝑉
𝑂 𝑁3
Since6.thisSolvePoisson(
holds for any (smooth)
filter, we can choose 𝐹 ≥to𝑂be
𝑑𝑖𝑣 )
𝑁3
compactly supported so only samples close to 𝑞 contribute.
If we discretize over an 𝑁 × 𝑁 × 𝑁 grid and 𝑃 = 𝑂 𝑁 2 :
• Storage Complexity: 𝑂 𝑁 3
• Computational Complexity: 𝑂 𝑁 5
*[Rosenblatt,
1956] [Parzen, 1962] [Hoppe, 1994] [Mitra and Nguyen, 2003]
How can we compute it effectively?
Structure of the Input/Output:
• Points in 𝑃 should lie on a 2D manifold in ℝ3 .
• 𝜒 should be constant away from the samples.
⇒ Adapt a space-partition to 𝑃.
⇒ Use the same space partition to discretize 𝜒.
Choosing a Space Partition
• Supports discretization of the linear system:
– Octrees: [Losasso et al., 2004]
– Adapted Tetrahedralizations: [Alliez et al., 2007]
• Supports efficient (hierarchical) solver:
– Regular Grids: [Fedorenko, 1973] [Brandt, 1977]
– Octrees: [Kazhdan and Hoppe, 2013]
Efficient Implementation
ReconstructSolid( 𝑃 )
1.
2.
3.
4.
5.
6.
ComputeNormalsAndArea( 𝑃 )
𝑂 = ConstructOctree( 𝑃 )
For 𝑜 ∈ 𝑂:
𝑉 𝑜 =0
∀ 𝑝, 𝑛 ∈ 𝑃 s.t. 𝑝 − 𝑜 < Supp (𝐹):
𝑂 |𝑃| log |𝑃|
𝑂 |𝑃|
𝑂 1
𝑉 𝑜 += 𝜎 ⋅ 𝐹 𝑝 − 𝑜 ⋅ 𝑛
7.
∀𝑜 ∈ 𝑂: 𝑑𝑖𝑣 𝑜 = 𝛻 ⋅ 𝑉 (𝑜)
𝑂 |𝑃|
8.
HierarchicallySolvePoisson( 𝑑𝑖𝑣 )
𝑂 |𝑃|
• Storage Complexity: 𝑂 |𝑃|
• Computational Complexity: 𝑂 |𝑃| log |𝑃|
Iso-Surface Extraction
Extract the shape of the level-set
of a discretely sampled function
What is a shape?
A shape is a triangle mesh
What do we want to compute?
Find a shape separating points with negative
value from points with positive value.
1. Fit a function 𝑓: ℝ3 → ℝ to the samples.
2. Find the level-set: 𝑆 ≡ {𝑝 ∈ ℝ3 |𝑓 𝑝 = 0}.
Inverse Function Theorem:
In general, the level-set of a function
The level-set
will
be manifold/water-tight
cannot be
represented
by a triangle mesh.if zero
is not a singular value.
How can we compute it effectively?
1. Adapt the function:
For linear functions, the level-set is planar.
⇒ Given the samples:
I. Tetrahedralize the positions
II. Fit the linear interpolant
to each tet’s vertex values
III. Extract the level-set of the
Simplicial Structure:
global function.
[Doi and Koide, 1991]
Adjacent tets share triangular faces.
⇓
The global function is continuous.
How can we compute it effectively?
2. Adapt the geometry:
We need an approximate iso-surface that
separates the samples.
⇒ Given the samples:
I. Partition space into (convex) cells
II. Compute iso-edges on faces
that separate
Partition
Structure:vertices with
opposite
signs
Faces/edges
of the partition intersect trivially.
III. Extend the boundary⇓curve
surface
is water-tight/manifold.
to aSeparating
surface in
the interior
[Lorensen and Cline, 1987] [Schaefer and Warren, 2004]
Surface Extension
Find a surface whose boundary
interpolates a 3D curve
What is a shape?
A shape is a mapping from the unit disk into ℝ3 .
What do we want to compute?
Given a simple/closed curve 𝛾, find the smooth
parameterization Φ𝛾 that agrees with 𝛾:*
Φ𝛾 = arg min
𝛻Φ 2 𝑑𝜇
w/ Φ
𝐷2
𝑆1
=𝛾
– If 𝛾 lies on a convex domain,
the map Φ𝛾 is an embedding.
[Meeks and Yau, 1980]
– This is a minimal area surface,
which is a fixed point of MCF.
[PinkallMCF,
andwe
Polthier,
To discretize
need to 1993]
refine the mesh,
increasing the complexity of
the shape.
∗
Gradients, norms, and measures defined w.r.t. Φ
What do we want to compute?
Fixing the position of the vertices, we want a
triangulation that minimizes the area.
How can we compute it effectively?
Fixing the position of the vertices, we want a
triangulation that minimizes the area.
Leverage the structure of the solution:
– Sub-triangulations are minimal
– Sub-triangulations recur
⇒ Dynamic programming solution:
– 𝑂 𝑁 3 time
– 𝑂 𝑁 2 space
[Klincesk, 1980] [Barequet and Sharir, 1995]
Searching for Structure
Searching for Structure
What do we want to compute?
What is the structure of the class we are working with?
•
•
•
•
Ordered set
Inner-product space
Group representation
Currents
Searching for Structure
What do we want to compute?
What is the structure of the class we are working with?
How do we compute it effectively?
What is the structure of the computation we are performing?
•
•
•
•
Ordered set
• Memoization
Inner-product space • Divide-and-conquer
Group representation • Dynamic programming
Currents
Searching for Structure
What do we want to compute?
What is the structure of the class we are working with?
How do we compute it effectively?
What is the structure of the computation we are performing?
What is the structure of the instance we are considering?
•
•
•
•
• Partial matching table
Ordered set
• Memoization
Inner-product space • Divide-and-conquer • Adaptive octree
Group representation • Dynamic programming
Currents
Thank you
Questions
• Can iso-surface extraction be extended to more general space partitions?
• Can area minimizing triangulations be obtained through successive edge
flips? If so, how does the efficiency compare?
• Is there a Dirichlet minimizing triangulation [Rippa, 1990] for a closed
polygonal curve in 3D? Is it related to area minimizing triangulation?
• Does the optimality of the Delaunay triangulation translate into optimality
of the extracted level-set?
• Is the minimal area triangulation of a simple closed polygon on a convex
surface guaranteed to not self-intersect?
• Extend symmetry detection to circular/spherical parameterizations by
redefining the “action of 𝑆𝑂(2)/𝑆𝑂(3)” on functions.
• We don’t need the full power of linearity for symmetry detection. What is
the minimal structure needed to apply representation theory methods?