Computing Shapes and Their Features from Point Samples

Tamal K. Dey The Ohio State University

Computing Shapes and Their Features from Point Samples

2/52Department of Computer and Information Science

Surface Reconstruction

`

Point Cloud

Surface Reconstruction


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)


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)


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|


Sampling


Medial Axis


f(x) is the distance to medial axis

Local Feature SizeAmenta-Bern-Eppstein 98

f(x)


Each x has a sample within f(x) distance

-samplingAmenta-Bern-Eppstein 98

x


-sampleε-sample is also ε'-sample for ε' > ε


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||


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)


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


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


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


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}


Curve samples and Voronoi


Crust Algorithm (2D)Amenta-Bern-Eppstein 98

Compute VP

Add Voronoi vertices

Compute Delaunay

Retain edges between

samples only


Nearest Neighbor AlgorithmDey-Kumar 99

Compute DP

For each p, compute nearest neighbor

For each p, compute its half-neighbor.


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


Normals and Voronoi Cells(3D) Amenta-Bern 98


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


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


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).


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)


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.


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)


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.


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.


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


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


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.


PolesP+

P-


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.


Crust in 3DAmenta-Bern 98

Introduce poles

Filter crust triangles from Delaunay

Filter by normals

Extract manifold


Manifold Extraction: Prunning

Remove Sharp edges with their triangles


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


Manifold Extraction: Walk

Walk inside or outside the possibly thickened surface


Cocone AlgorithmAmenta-Choi-Dey-Leekha 00

Simplified/improved the Crust

Only single Voronoi computation

Analysis is simpler

No normal filtering step

Proof of homeomorphism


Cocone

vp= p+ - p is the pole vector

Space spanned by vectors

within the Voronoi cell making

angle > 3/8 with vp or -vp


Cocone Algorithm


Candidate triangles computatione=(a,b); a=a-p; b= b-p


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


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.


No sharp edge

Lemma Sharp: No restricted Delaunay triangle has a sharp edge for ε < 0.06


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.


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.


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.


Undersampling Dey-Giesen 01

Boundaries

Small features

Non-smoothness


Boundaries


Small Features High curvature regions are often undersampled


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


Radius and HeightRadius r(p): radius of cocone

Height h(p): distance to the negative pole p-

cocone neighbors Np


Flatness Condition

Vertex p is flat if

1. Ratio condition: r(p) h(p)

2. Normal condition: vp,vq q with pNq


Boundary Detection(1st phase)

IsFlat(p,,) check ratio and normal condition for Vp;

if both are satisfied

return true

else

return false

end


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


ReconstructionCocone(P, ,)

Compute VP;

for each pP if pB compute T of triangles with duals intersecting Cp; endifenfor;

Extract manifold;end

B:= Boundary(P,,)


Data Set Sat


Data Set Engine


Nonsmoothness


Watertight Surfaces


Tight Cocone Dey-Goswami 02


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.


Result Sharp corners and edges of AutoPart can be reconstructed.


Dinosaur


Large Data

Octree subdivision


Cracks Cracks appear in surface computed from octree boxes


Padding Include a fraction from the neighbors to form the extended box


Surface Matching


Experimental DataPentium III,733Mhz,512Mb

Time

Memory


Lucy25

3.5 million points, 198 mints


David’s Head

2 mil points, 93 minutes


Noisy Data - Bunny

Front view Rear view


Noisy Data – Ram Head

Front view Rear view


Example movie file

Mannequin


• Female

Point data Tight Cocone Robust Cocone

Examples


• Mannequin

Point data Tight Cocone Robust Cocone

Examples


Cocone Software

Cocone: Reconstructs surfaces with boundaries.

Tight Cocone: Reconstructs watertight surfaces.

Available fromhttp://www.cis.ohio-state.edu/~tamaldey/cocone.html

Acknowledgement: CGAL


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]


Medial Axis• Medial Ball• Medial Axis -Sampling


Geometric Definitions

• Pole and Pole Vector

• Tangent Polygon • Umbrella Up


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..


Angle Condition• Angle Condition

[θ ]:

Max{σ in Up angle(nσ,(q-p)) }< /2- θ


‘Only Angle Condition’ Results

= 18 degrees

= 3 degrees = 32 degrees


‘Only Angle Condition’ Results

= 15 degrees

= 20 degrees

= 30 degrees


Ratio Condition

• Ratio Condition []:

Min{σ in Up ||p-q||/Rσ >


‘Only Ratio Condition’ Results

= 2

= 4

= 8


‘Only Ratio Condition’ Results

= 2

= 4

= 6


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


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


AnalysisLemma 1: If w lies in the segment mm’, 2 tan(arcsin2 ) .p pw w

p = O ()

S

m1

m’


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 )


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


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


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


Experimental Results


Medial Axis from a CAD model

CAD model

Point Sampling Medial Axis


Medial Axis


CAD model

Point Sampling


Example movie file

Anchor Medial


Segmentation and matchingDey-Giesen-Goswami 03

• Segment a shape into `features’• Match two shapes based on the

segmentation


Feature definitionFlow

Continuous

Discrete flow

Discretization


FlowShape :

dp xxpxh R allfor inf)( 2

xpxA p minarg)(• Anchor set:

• Height fuinction:


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


Features• F(x) = closure(S(x)) for a maximum x


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


Stable manifolds

• Gabriel edges are stable manifolds of saddles• Stable manifolds of maxima are shaded


Stable manifolds• Feature F(x) = closure(S(x)) for a maximum x


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 <


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.


Algorithm for )(~ xF


Merging• Small perturbations create insignificant features• Sampling artifacts introduce more segmentations

• Merge stable manifolds


Results (2D)


Results (3D)


Results (2D)


Results (3D)


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

http://www.cis.ohio-state.edu/~tamaldey/cocone.html

Computing Shapes and Their Features from Point Samples

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