Computing Shapes and Their Features from Point Samples
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Transcript of Computing Shapes and Their Features from Point Samples
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Tamal K. Dey The Ohio State University
Computing Shapes and Their Features from Point Samples
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Surface Reconstruction
`
Point Cloud
Surface Reconstruction
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Algorithms
1. Alpha-shapes (Edelsbrunner, Mücke 94)
2. Crust (Amenta, Bern 98)
3. Natural Neighbors (Boissonnat, Cazals 00)
4. Cocone (Amenta, Choi, Dey, Leekha, 00)
5. Tight Cocone (Dey, Goswami, 02)
6. Power Crust (Amenta, Choi, Kolluri 01)
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Basic Topology
d-ball Bd {x in Rd | ||x|| ≤ 1} d-sphere Sd {x in Rd+1 | ||x||=1}
Homeomorphism h: T1 → T2 where
h is continuous, bijective and has continuous inverse
K-Manifold : neighborhoods homeomorphic to open k-ball2-sphere, torus, double torus are 2-manifolds
K-manifold with boundary: interior points, boundary pointsBd is a d-manifold with boundary where bd(Bd)=S(d-1)
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Basic Topology Smooth Manifolds Triangulation
k-simplex Simplicial complex K: (i) t in K if t is a face of t' in
K
(ii) t1, t2 in K => t1∩ t2 face of both
K is a triangulation of a topological space T if
T ≈ |K|
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Sampling
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Medial Axis
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f(x) is the distance to medial axis
Local Feature SizeAmenta-Bern-Eppstein 98
f(x)
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Each x has a sample within f(x) distance
-samplingAmenta-Bern-Eppstein 98
x
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-sampleε-sample is also ε'-sample for ε' > ε
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Lipschitz Property of f()Lemma (Lipschitz Continuity):
f(x) ≤ f(y) + ||x-y||
Proof: Let m be a point on M with f(y)=||y-m||
By triangular inequality
||x-m|| ≤ ||y-m|| + ||x-y||
f(x) ≤ ||x-m|| ≤ f(y)+||x-y||
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FTL: Feature translation lemma
Lemma (Feature Translation): If ||x-y|| ≤ ε f(x) then
1/(1+ ε)f(y) ≤ f(x) ≤ 1/(1- ε)f(y)
Exercise 1: Prove it. Also prove ||x-y|| ≤ ε /(1-
ε)f(y)
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FBL: Feature Ball LemmaLemma (Feature Ball): If a d-ball B intersects a k-
manifold Σ at more than two points with either (i) B∩Σ is not a k-ball, or (ii) bd(B)∩Σ is not a (k-1)-sphere, then B contains a medial axis point. Exercise 2: Prove
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Voronoi/Delaunay Diagrams
Voronoi diagram VP: collection of Voronoi cells {Vp}
Vp={x in R3 | ||x-p|| ≤ ||x-q|| for all q in P}
Voronoi facet, Voronoi edge, Voronoi vertex
Delaunay triangulation DP: Dual of VP, a simplicial complex
Delaunay edge, triangle, tetrahedra
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Delaunay propertiesEmptiness : A simplex t is in DP if and only if there is
a circumscribing ball of t that does not contain any point of P inside.
Proof: If a k-simplex t, 0 ≤ k ≤ 3, is in a DP, its dual (3-k)-dimensional Voronoi element has a point x that is equidistant from the (k+1) vertices of t. Also these vertices are closest to x among all points of P. This only means the ball centered at x with the vertices of t on the bounding sphere is empty.
Exercise 3: Show that if t has an empty circumball, t is in DP
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Restricted Voronoi/Delaunay
Restricted Voronoi: VP,Σ = {Vp Σ =Vp∩ Σ | p in P}
Restricted Delaunay: Dp, Σ ={A k-simplex is Conv R where
∩Vp, Σ ≠ Ø for p in R}
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Curve samples and Voronoi
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Crust Algorithm (2D)Amenta-Bern-Eppstein 98
Compute VP
Add Voronoi vertices
Compute Delaunay
Retain edges between
samples only
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Nearest Neighbor AlgorithmDey-Kumar 99
Compute DP
For each p, compute nearest neighbor
For each p, compute its half-neighbor.
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Difficulties in 3DVoronoi vertices can come
close to the surface… slivers are nasty.
There is no unique `correct’ surface for reference
or
or……
Voronoi vertex
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Normals and Voronoi Cells(3D) Amenta-Bern 98
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Long Voronoi cellsLemma (Medial): Let m1 and m2 be the centers of two medial balls at p. Vp contains m1, m2.Exercise 4: Prove it
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NL : Normal LemmaLemma (Normal) : Let v be a point in Vp with ||v-p||>μf(p). Then,
angle((v-p),np)≤ arcsin ε/μ(1- ε) + arcsin ε /(1- ε). Exercise 5: Prove NL
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NVL: Normal Variation lemma
Lemma (Normal Variation) : Let x and y be two points with ||x-y||≤r f(x) for r< 1/3. Then, angle(nx,ny) ≤ r/(1-3r).
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ENL: Edge Normal LemmaLemma (Edge Normal): angle((p-q),np) > /2 – arcsin ||p-q||/2f(p).
Proof: sin θ = ||p-q||/2||m-p|| ≤ ||p-q||/2f(p)
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TNL: Triangle Normal Lemma
Lemma (Triangle Normal) : angle(npqr,np) ≤ α + arcsin((2/√3) sin 2α)where α ≤ arcsin d/f(p) and d, the circumradius, is sufficiently small.
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TopologyClosed Ball property (Edelsbrunner, Shah 94): If restricted Voronoi cellis a closed ball in each dimension, then DP, Σ is homeomorphic to Σ.
Assume P is an e-sample of Σ where e is sufficiently small. It can be shown that (P, Σ) satisfies the closed ball property.(proof from Cheng-Dey-Edelsbrunner-Sullivan 02)
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SDL: Short Distance Lemma
Lemma Short Distance : x, y two points in Vp,Σ. (i) ||x-p||< ε/(1- ε)f(p)(ii) ||x-y|| < 2ε/(1- ε)f(x).
Exercise 6: Prove it.
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LDL: Long Distance Lemma
Lemma (Long Distance) : Suppose L intersects S in two points x, y andmakes angle less than ξ with nx. Then ||x-y||>2f(x)cos ξ.
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VEL: Voronoi Edge LemmaLemma (Voronoi Edge) : A Voronoi edge intersects Σ in a single point.
Proof: x ≤ angle(npqr,np) + angle(np,nx) ≤ O(ε) + O(ε) by TNL and NVL.
2f(x)cos O(ε) ≤ ||x-y|| ≤ O(ε) f(x)by SDL and LDL
Contradiction when ε is sufficiently smallExercise 7: Prove it
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VFL: Voronoi Facet LemmaLemma (Voronoi Facet): A Voronoi facet intersects Σ in a 1-ball.
Proof: angle(Lnx)≤ angle(Lnp) + angle(np,nx)
≤ O(ε) + O(ε) by ENL and NVL. 2f(x)cos O(ε) ≤ ||x-y|| ≤ O(ε) f(x)
by SDL and LDL Contradiction when ε is sufficiently small
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VCL: Voronoi Cell LemmaLemma (Voronoi Cell) : A Voronoi cell intersects Σ in a 2-ball.
Proof: show that handles and connected components of Σ cannot be in the cell. Then show that if the cell intersects Σ in multiple disks, we reach a contradiction with SDL and LDL.
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PolesP+
P-
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PVL: Pole Vector LemmaP+
P-
npvp
Lemma (Pole Vector) : angle((p+-p),np)=2arcsin ε /(1- ε).Proof: ||p+-p||> f(p) since Vp contains a medial axis point (Medial Lemma). Plug this in Normal Lemma.
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Crust in 3DAmenta-Bern 98
Introduce poles
Filter crust triangles from Delaunay
Filter by normals
Extract manifold
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Manifold Extraction: Prunning
Remove Sharp edges with their triangles
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Why Prunning Works?Crust triangles include restricted Delaunay triangles
The underlying space of the restricted Delaunay
triangles is homeomorphic to the sampled surface
No edge of the restricted triangles is sharp
After prunning, at least the surface made by the
restricted Delaunay triangles remains
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Manifold Extraction: Walk
Walk inside or outside the possibly thickened surface
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Cocone AlgorithmAmenta-Choi-Dey-Leekha 00
Simplified/improved the Crust
Only single Voronoi computation
Analysis is simpler
No normal filtering step
Proof of homeomorphism
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Cocone
vp= p+ - p is the pole vector
Space spanned by vectors
within the Voronoi cell making
angle > 3/8 with vp or -vp
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Cocone Algorithm
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Candidate triangles computatione=(a,b); a=a-p; b= b-p
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Candidate Triangle Properties
Candidate triangles include the restricted
Delaunay triangles
Their circumradii are small O()f(p)
Their normals make only O() angle with the
surface normals at the vertices
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Restricted Delaunay property
Claim: Let y in Vp∩ Σ . Then, angle(np,(y-p)) > /2- εExercise 8: Prove it.
Lemma (Restricted Delaunay): All restricted triangles are in T for ε <0.1.Proof: Let y in e∩Σ where e is the dual edge for a triangle.
angle((y-p),vp)>angle((y-p),np)-angle(np,vp)
> /2- ε -angle(np,vp)
> 3/8 by PVL for ε < 0.1.
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No sharp edge
Lemma Sharp: No restricted Delaunay triangle has a sharp edge for ε < 0.06
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Small radius and flatnessLemma (Small Triangle): The circumradius r of any candidate triangle is O(ε)f(p) where p is any of its vertex and ε < 0.06.
Proof: There is y in dual edge so that angle(vp,(y-p))>3/8.By PVL angle(np,(y-p)) > 3/8-2arcsin ε /(1- ε).Use contrapositive of NL to conclude ||y-p||=O(ε)f(p) for ε <0.06.
Lemma (Flat Triangle): For each candidate triangle pqrangle(npqr,np)=O(ε)
Proof: Follows from STL and TNL.
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Homeomorphism
Let M be the triangulated surface obtained after the manifold extraction.
Define h: R3 -> Σ where h(q) is the closest point on Σ. h is welldefined except at the medial axis points.
Lemma Homeomorphism: The restriction of h to M, h: M -> Σ,is a homeomorphism.
Proof: Use STL, FTL for the proof, see ACDL00.
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Cocone GuaranteesTheorem:
Any point x is within O(f(x) distance from a point in the output. Conversely, any point of the output surface has a point x within O()f(x) distance for ε <0.06.
Theorem:
The output surface computed by Cocone from an -sample is homeomorphic to the sampled surface for ε < 0.06.
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Undersampling Dey-Giesen 01
Boundaries
Small features
Non-smoothness
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Boundaries
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Small Features High curvature regions are often undersampled
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Well Sampled Patch and Boundary Vertices
is well sampled if ε-sampling
holds for
Restricted Voronoi on defines boundary vertices
p is interior if restricted cell has no boundary point otherwise p is boundary vertex
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Radius and HeightRadius r(p): radius of cocone
Height h(p): distance to the negative pole p-
cocone neighbors Np
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Flatness Condition
Vertex p is flat if
1. Ratio condition: r(p) h(p)
2. Normal condition: vp,vq q with pNq
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Boundary Detection(1st phase)
IsFlat(p,,) check ratio and normal condition for Vp;
if both are satisfied
return true
else
return false
end
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Boundary Detection(2nd phase)
Boundary(P,,) Compute the set R of flat vertices; while pR and pNq with qR and r(p)h(p) and vp,vq R:=Rp; endwhile return P\Rend
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ReconstructionCocone(P, ,)
Compute VP;
for each pP if pB compute T of triangles with duals intersecting Cp; endifenfor;
Extract manifold;end
B:= Boundary(P,,)
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Data Set Sat
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Data Set Engine
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Nonsmoothness
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Watertight Surfaces
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Tight Cocone Dey-Goswami 02
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Tight COCONE Principle Compute the Delaunay triangulation of the input point set.
Use COCONE along with detection of undersampling to get an initial
surface with undersampled regions identified.
Stitch the holes from the existing Delaunay triangles without inserting any
new point.
Effectively, the output surface bounds one or more solids.
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Result Sharp corners and edges of AutoPart can be reconstructed.
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Dinosaur
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Large Data
Octree subdivision
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Cracks Cracks appear in surface computed from octree boxes
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Padding Include a fraction from the neighbors to form the extended box
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Surface Matching
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Experimental DataPentium III,733Mhz,512Mb
Time
Memory
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Lucy25
3.5 million points, 198 mints
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David’s Head
2 mil points, 93 minutes
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Noisy Data - Bunny
Front view Rear view
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Noisy Data – Ram Head
Front view Rear view
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Example movie file
Mannequin
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• Female
Point data Tight Cocone Robust Cocone
Examples
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• Mannequin
Point data Tight Cocone Robust Cocone
Examples
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Cocone Software
Cocone: Reconstructs surfaces with boundaries.
Tight Cocone: Reconstructs watertight surfaces.
Available fromhttp://www.cis.ohio-state.edu/~tamaldey/cocone.html
Acknowledgement: CGAL
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Medial axis from point sample
• Earlier work did not have guarantees[Attali-Montanvert-Lachaud 01]
• Power shape : guarantees topology, uses power diagram[Amenta-Choi-Kolluri 01]
• Medial : Approximates the medial axis as a Voronoi subcomplex and has converegence guarantee.[Dey-Zhao 02]
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Medial Axis• Medial Ball• Medial Axis -Sampling
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Geometric Definitions
• Pole and Pole Vector
• Tangent Polygon • Umbrella Up
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Filtering conditions
• Medial axis point m• Medial angle θ• Angle and Ratio
Conditions
Our goal: : approximate the medial axis as a approximate the medial axis as a subset of Voronoi facetssubset of Voronoi facets..
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Angle Condition• Angle Condition
[θ ]:
Max{σ in Up angle(nσ,(q-p)) }< /2- θ
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‘Only Angle Condition’ Results
= 18 degrees
= 3 degrees = 32 degrees
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‘Only Angle Condition’ Results
= 15 degrees
= 20 degrees
= 30 degrees
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Ratio Condition
• Ratio Condition []:
Min{σ in Up ||p-q||/Rσ >
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‘Only Ratio Condition’ Results
= 2
= 4
= 8
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‘Only Ratio Condition’ Results
= 2
= 4
= 6
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Algorithm• Each of Angle and Ratio conditions
individually is not sufficient. • Combination of both conditions
• First Angle, then Ratio• The Angle condition captures the Delaunay edges
which lie away from the surface.• The Ratio condition captures the the Delaunay edges
which make small angles with the umbrella triangles but are comparatively larger than their circumradii.
• Allows fixed values of θ and
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Algorithm
)cloure(Output 11endfor10
endfor9endif8
Dual78 Condtion or Ratio Condition Angle satisfies if6
edgeDelaunayeachfor5; Compute4
eachfor3;2
;and Compute1
8
F
pqF:Fpq
UpqUPp
FDV
P
p
p
PP
)(M EDIAL
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AnalysisLemma 1: If w lies in the segment mm’, 2 tan(arcsin2 ) .p pw w
p = O ()
S
m1
m’
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Analysis (continued)
Lemma 2: Let F = Dual pq be a Voronoi facet
where pq satisfies the angle condition [ ] with
2p + p. Any point w in F with
is at a distance
from m where m and are the center and radius
of the medial ball at p with .
w p 1 (2 tan(arcsin 2 ))
sin( ) p pp
0pm pwt t
p, p = O ()
>( - p )
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Analysis (continued)Lemma 3: Let w be any point in the Voronoi facet F=Dual pq with
where is the radius of the medial ball at p with center m so that .
If pq satisfies the Ratio condition then either
0pm pwt t
0[ 8]
1/8( )p pw m O or
1/8( ) .pw p
w p
Proof
If , Lemma 2 gives 7 /8p 1/8( )p pO
If , ……7 /8p 1/8( )p
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Analysis (continued)
Lemma 4: Let w Vp be a point such that , where is the
radius of a medial ball at p with the center m and , and .
Then, for sufficiently small >0, if the medial angle of p
is larger than 1/3.
w p
0pm pwt t 1/ 4
3/ 4( )w m O
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TheoremsTheorem:
if by Angle: (i) , apply Lemma 4
Otherwise Lemma 2
if by Ratio: (i) , apply Lemma 4
Otherwise Lemma 3
Theorem:
0lim .L M
0lim .M L
w p
w p
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Experimental Results
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Experimental Results
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Experimental Results
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Experimental Results
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Medial Axis from a CAD model
CAD model
Point Sampling Medial Axis
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Medial Axis
Medial Axis from a CAD model
CAD model
Point Sampling
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Medial Axis
Medial Axis from a CAD model
CAD model
Point Sampling
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Example movie file
Anchor Medial
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Segmentation and matchingDey-Giesen-Goswami 03
• Segment a shape into `features’• Match two shapes based on the
segmentation
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Feature definitionFlow
Continuous
Discrete flow
Discretization
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FlowShape :
dp xxpxh R allfor inf)( 2
xpxA p minarg)(• Anchor set:
• Height fuinction:
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Flow
• Vector field v :
)()()(
xdxxdxxv
if x is regular and 0 otherwise
• Flow induced by v Fix points of are the critical points of h
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Features• F(x) = closure(S(x)) for a maximum x
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Flow by discrete set• Driver d(x): closest point on dual to the Voronoi
object containing x•
Vector field:
• This also induces a flow )()()(
xdxxdxxv
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Stable manifolds
• Gabriel edges are stable manifolds of saddles• Stable manifolds of maxima are shaded
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Stable manifolds• Feature F(x) = closure(S(x)) for a maximum x
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Flow relation• t < t’ if the circumcenters of t and t’ lie on the same side of
the edge shared by them.• Collect all triangles related by the transitive closure of <
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Flow relation in 3D• In 2D there is at most one t’ so that t< t’• Exercise 9: Show an example in 3D where a tetrahedron t <
t1 and t < t2. Show that there cannot be any third tetrahedron t3 so that t< t3.
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Algorithm for )(~ xF
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Merging• Small perturbations create insignificant features• Sampling artifacts introduce more segmentations
• Merge stable manifolds
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Results (2D)
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Results (3D)
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Results (2D)
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Results (3D)
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Open problems• Design an algorithm to reconstruct non-smooth surface
• Design an algorithm for medial axis approximaionwith topological guarantee
• Prove an approximation result for feature segmentation
Software: Cocone, Medial : http://www.cis.ohio-state.edu/~tamaldey/cocone.htmlSegmatch: http://www.cis.ohio-state.edu/~tamaldey/segmatch.html