Surface Reconstruction and Distance Fields

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Surface Reconstruction and Distance Fields

• Gokul Varadhan

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Surface Representation

• Parametric
• Triangular mesh, Bezier surfaces, NURBS
• Implicit
• Represented as a function f(x,y,z) 0
• Volumetric representation
• One important implicit function
• is the distance function

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Distance Field

• Define an implicit function D(p) distance to
  the surface at point p

• D(x,y) distance to
• surface
• gt 0 outside the surface
• lt 0 inside the surface
• 0 on the surface

Use interpolation to compute distance at an
arbitrary point
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Distance Field

• Computing distance field
• For a triangular mesh minimum of distance to
  each of the triangles
• Fast marching methods can be used to compute
  distance fields quickly (Sethian99)
• Can compute quickly using graphics hardware (Hoff
  et al Hoff99)
• Could use an octree instead of a uniform grid
• Adaptively sampled distance fields (Frisken00)

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Graphics Hardware-based computation

• Consider the 2D case
• Define distance functions for primitives (point,
  line, polygons)

TENT
CONE
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Graphics Hardware-based computation

• Distance function for a polygon a collection of
  cones and tents
• Render the distance function using graphics
  hardware
• The Z-buffer holds the distance field
• For 3D distance field , refer to the paper
  Hoff99.

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Pros and Cons

• Advantages
• Fast operations
• Point classification does a point lie on,
  inside or outside surface ? Check the sign of the
  distance function at that point
• Union, Intersection, complement
• Independent of topology
• Topology can change during operations on distance
  fields
• Disadvantages
• Enumerating points on the surface
• Locating neighbours

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Boolean Operations
Min DistA, Dist B 0
Max DistA, Dist B 0
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Surface Reconstruction

• Objective obtain a triangular mesh
  representation
• To extract the surface
• Compute the zero-set p D(p) 0

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Distance-field based Surface reconstruction

• Marching Cubes Lorensen87
• Extended Marching Cubes Kobbelt01
• Dual Contouring Ju02

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Marching Cubes

• Given the distance field grid,
• Reconstruct the surface within each grid cell
• Once done with one cell (cube), march to the
  next.

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Marching Cubes

• Determine the topology of surface within the cube
  handle actual intersection later
• Each vertex of the cube is assigned a 1 if it
  lies outside the surface, 0 if inside
• 256 cases in all, can be reduced to 15 by
  symmetry
• Can enumerate different cases and store them in a
  table
• Use an index into the table

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Marching Cubes
For details, refer to the original paper
Lorensen87
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Marching Cubes
D2 gt 0
D1 lt 0
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Marching Cubes

• Handle each cell independently
• Because intersection points along grid edges are
  consistent between adjacent cells
• Reconstructed surface matches at cell boundaries
  and doesnt leave holes

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Extended Marching Cubes Kobbelt et al. 2001

• Marching cubes in general cannot reconstruct very
  sharp features and result in aliasing artifacts
• Problem normals dont converge
• Extended marching cubes (EMC) can reconstruct
  sharp features

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Extended Marching Cubes

• Detect if a cell contains a sharp feature
• If not use standard Marching Cubes
• Else use the method described next
• Instead of storing distance D at each grid point
• Store DX, DY, DZ, distances along X, Y Z axes
• Also store normals
• Axis-aligned distances provide exact intersection
  points
• Normals are used to reconstruct sharp features

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Extended Marching Cubes
Y
D2Y gt 0
X
Surface
D1Y lt 0
D3X gt 0
D1X lt 0
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Extended Marching Cubes
normal
normal
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Extended Marching Cubes

• This works only if there is atmost one sharp
  feature.
• The above method positions a vertex at the
  minimizer of the quadratic error function (QEF)
• Ex S (ni (x pi))2

Where pi and ni correspond to the intersections
and unit normals
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Dual Contouring Ju et al. 2002

• Use QEF criterion to position a vertex within
  each cell
• For each edge that exhibits a sign change,
  generate a quad connecting the minimizing
  vertices of the four cells sharing that edge

-
-


-

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Dual Contouring

• No need to explicitly test for features
• Mesh produced by dual contouring is dual to the
  mesh produced by marching cubes method
• Vertices correspond to faces and vice-versa
• Mesh have better aspect ratios
• Mesh vertices are allowed to move freely within
  the grid cells instead of being constrained to
  lie on the edges of the grid.

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Dual Contouring
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Applications CSG Modeling
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Point Cloud reconstruction

• Given a cloud of points, define a distance field
  (Hoppe et al Hoppe92)
• Estimate normal and hence the tangent plane at
  each point in the cloud
• Distance from a point in the cloud is defined to
  be the distance from its tangent plane
• Distance at each grid point is the minimum of
  distance from all the points in the cloud
• Once the distance field has been computed,
  extract the surface using any of the contouring
  methods we have discussed (MC, EMC or dual
  contouring)

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Point Cloud reconstruction
Cloud of 200 K points
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References

• Lorensen87 W. Lorensen, H.Cline, Marching
  Cubes a high resolution 3D surface
  reconstruction algorithm SIGGRAPH 87
• Kobbelt01 Leif P. Kobbelt, Mario Botsch, Ulrich
  Schwanecke, Hans-Peter Seidel, Feature Sensitive
  Surface Extraction from Volume Data, SIGGRAPH 01
  
• Ju02 Ju T., Losasso F., Schaefer S. and Warren
  J., Dual Contouring of Hermite Data, SIGGRAPH
  02
• Hoff99 K. Hoff, T. Culver, J. Keyser, M.Lin, D.
  Manocha, Fast Computation of generalized Voronoi
  diagrams using graphics hardware, SIGGRAPH 99

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References

• Frisken00 Perry and Frisken, "Adaptively
  Sampled Distance Fields (ADFs) Representing
  Shape for Computer Graphics", SIGGRAPH 2000
• Hoppe92 H. Hoppe, T. DeRose, T. Duchamp, J.
  McDonald, and W. Stuetzle. Surface reconstruction
  from unorganized points. SIGGRAPH '92
• Sethian99 J.A. Sethian, Level Set Methods and
  Fast Marching Methods Evolving Interfaces in
  Computational Geometry, Fluid Mechanics, Computer
  Vision, and Materials Science 1999