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Detecting Undersampling in Surface Reconstruction

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Title: Detecting Undersampling in Surface Reconstruction


1
Detecting Undersampling in Surface Reconstruction
  • Tamal K. Dey and Joachim Giesen
  • Ohio State University

2
Surface Reconstruction
  • A sample and PL approximation

3
Previous and recent works
  • Functional approach
  • Tangent plane HDeDDMS92
  • Natural Neighbors BC00
  • Alpha shapes
  • EM94
  • BBCS97
  • Voronoi filtering
  • Crust AB98
  • Cocone ACDL00

4
Local feature size and sampling
  • Medial axis
  • Local feature size f(p)
  • ?-sampling
  • ? ? d(p)/f(p)

5
Crust Algorithm
  • Compute VP
  • Add Voronoi vertices
  • Compute Delaunay
  • Retain edges between samples only

6
Crust in 3D
  • Introduce poles
  • Filter crust triangles
  • Filter by normals
  • Extract manifold

7
Cocones
Space spanned by vectors making angle ? ?/8 with
horizontal
  • Compute cocones
  • Filter triangles whose duals intersect cocones
  • Extract manifold

8
Boundaries
  • Only part of a surface is well sampled

9
High curvature
  • High curvature regions are often undersampled

10
Non-smoothness
  • Impossible to sample densely
  • ? ? 0

11
Well sampled patch and boundary vertices
  • S ? F is well sampled if e-sampling holds for S
  • Restricted Voronoi on S defines boundary vertices
  • p is interior if restricted cell has no boundary
    point otherwise p is boundary vertex

12
Cocones, radius and height
  • cocones space spanned by vectors making ? ?/8
    with the horizontal
  • radius r(p) radius of cocone
  • height h(p) min distance to the poles
  • cocone neighbors Np

13
Flatness condition
  • Vertex p is flat if

1. Ratio condition r(p) ? ? h(p)
2. Normal condition ?v(p),v(q) ? ? ?q with p?Nq
14
Boundary Detection(1st phase)
IsFlat(p,?,?) check ratio and normal condition
for Vp if both are satisfied
return true else return
false end
15
Boundary detection(2nd phase)
Boundary(P,?,?) Compute the set R of flat
vertices while ?p?R and p?Nq with q?R
and r(p)??h(p) and ?v(p),v(q) ??
RR?p endwhile return P\R end
16
Reconstruction
Cocone(P, ?,?)
Compute VP
B Boundary(P,?,?)
for each p?P if p?B compute T of triangles with
duals intersecting Cp
endif enfor
Extract manifold end
17
Correctness
Definition An interior vertex is deep if no
boundary vertex contains it as cocone neighbor.
Theorem 1 All deep interior vertices are flat.
18
Correctness
Assumptions (i) S?S with points ?f(p) away
from sample points p defines same set of boundary
vertices as S does, where ???. (ii) Each boundary
vertex has an interior vertex as neighbor. (iii)
Each interior vertex is connected to a deep
interior vertex only through interior vertex
neighborhoods.
Theorem 2 Boundary vertices cannot be flat.
Theorem 3 Boundary() detects all and only
boundary vertices.
19
Implementation
  • Co-cone is implemented in CGAL
  • Floating point arithmetic is faster, but
    produces numerical errors
  • Exact arithmetic with integers is slow
  • Use floating point filters
  • Difficulty in manifold extraction step with
    false boundary vertices due to noise and
    numerical error

20
Umbrella check
  • Check if a vertex has an umbrella
  • Declare a vertex without umbrella as a boundary
    vertex
  • Prune triangles with sharp edges only if they
    are not incident with boundary vertices

21
Parameters
  • theory ? 0.01, higher ? in practice
  • theory ? 1.3?, ? 0.66-0.99 produce good
    result. Smaller ? detects more boundaries
  • theory ? 0.14 radians, ?/6 produces good
    result
  • cocone angle ?/8 in theory and ?/8 in practice

? ? 0.23
? ? 0.99
22
Data set foot
23
Data set Mannequin
24
Data set cactus
25
Data Set Sat
26
Data set Engine
27
Data set Oilpump
28
Nonsmoothness Repaired
29
Noise Detection (Outliers)
Cleaned
Outliers
30
Boundary Detection Helps Modeling by Parts
31
Arithmetic Precision
  • Floating point is fast, but causes numerical
    error
  • Exact arithmetic is slow, but produces robust
    results

32
Precision
Floating point
Exact arithmetic
33
Timings (Exact-Double)
Name Delaunay
Boundary
Reconstruction Cactus
130- 40-
23- Cat
120-50 108-58
68-51 Engine
831-405 108-58
79-59 foot
233-99 224-119
138-103 Mannequin
135-57 137-73
85-64 Oilpump
409-153 314-166
197-141 Club
171-74 179-95
112-84
34
Timings
Name points
triangles
Reconstruction(sec.) Halfsphere
245 486
0.58 Mannequin 12772
25339 74 Foot
20021 39995
122 Oilpump
30931 61548
194 Monkeysaddle 10000
19596 345
PIII, 933Mhz, 512MB
35
Conclusions
  • Introduced a measure radius/height ratio for
    skininess of Voronoi cells
  • Helps in detecting boundaries and sharp features
  • Recently we have used the radius/height ratio for
    sample decimation (CCCG01)
  • Used it for supersize data (PVG01)
  • Can we use it to eliminate noise?
  • More applications

543,652 points 143 -gt 28 min
3.5 million points Unfin-gt 198 min
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