Ray Tracing II

1
CS148 Introduction to Computer Graphics and
Imaging Geometric Modeling
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Data Structures for Triangle Mesh
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Data Structure Separate Triangles

• Treat each triangle separately with its own
  vertices
• typedef float Point3
• struct Face
• Point v3
• mesh FacenFaces
• Storage 72 bytes per vertex
• No notion of neighbor triangles Individual
  triangles might not overlap with their vertices
  or edges

f.v0
f
f
f.v1
f.v2
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Data Structure Indexed Triangle Set

• Store each vertex only once each face contains
  indices to its three vertices
• typedef float Point3
• struct Face
• int vIndex3
• mesh.vertsPointnVerts
• mesh.facesFacenFaces
• Storage 12 (verts) 24 (faces) 36 bytes per
  vertex (approximate using f 2 v)
• By removing vertex redundancy we have a notion of
  neighbor. However, finding a specific neighbor
  requires a global search.

v3
v0
f
f
v1
v2
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Comparison

• Separate Triangles (Vertex Buffer only)
• Simple
• Redundant information
• Indexed Triangle Set (Vertex Buffer Index
  Buffer)
• Sharing vertices reduces memory usage
• Ensure integrity of the mesh (moving a vertex
  causes that vertex in all the polygons to
  be moved)
• Both formats are compact and directly accepted
  by GPUs
• Both can represent non-manifold meshes
• Neither is good at neighborhood
  access/modification

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Recall Calculating Normals at Vertices

• for f in mesh.faces()
• (v1, v2, v3) f.ccwVertices()
• f.N cross(v2-v1,v3-v1)
• if !areaWeighted f.N.normalize()
• for v in mesh.verts()
• v.N 0
• for f in v.faces()
• v.N f.N
• v.N.normalize()

For Indexed Triangle Set, this takes O(F) time!
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Triangle Neighbor Representation

• Starting from the Indexed Triangle Set, for each
  triangle we now store pointers to the three
  adjacent triangles, and for each vertex store a
  pointer to any one of its incident triangles
• struct Vertex
• Point pt
• Face f
• struct Face
• Vertex v3
• Face adjF3
• mesh.vertsPointnVerts
• mesh.facesFacenFaces
• How to find all the faces adjacent to a vertex
  efficiently?

v0
v0.f
f1
f2
F
v2.f
v1
v2
f0
v1.f
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Finding Next Face using Triangle Neighbor

• Find the next face clockwise around a vertex v
  from a face f
• Face fcwvf(Vert v, Face f)
• if( v f-gtv0 )
• return adjF 1
• if( v f-gtv1 )
• return adjF 2
• if( v f-gtv2 )
• return adjF 0
• Storage 36 (indexed triangles) 24 (face.adjF)
  4 (vert.f) 64 bytes per vertex (still less
  than separate triangles)
• Efficiently iterate through all triangles
  adjacent to a vertex

Edge-based representations are more general and
comprehensive, such as the winged-edge,
half-edge, and quad-edge data structures.
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Subdivision
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Recall Smooth Shading
The faceted silhouette cannot be smoothed by
shading.
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Subdivision Curves
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Subdivision Surfaces

• There exists smooth limit surfaces associated
  with a given input mesh
• The exact smooth limit surface depends on the
  subdivision algorithm used
• We define a refinement operation, that takes a
  coarse input mesh and generates a finer output
  mesh that is closer to the limit surface
• If we apply this refinement process infinitely,
  we will exactly achieve the target limit surface
• However, after just a few levels of refinement,
  any additional changes will be visually
  indistinguishable

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Refine a triangular mesh
Loop Subdivision Surface
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Loop Subdivision

• Properties
• Modifies existing vertices
• Applied to all triangulated surfaces
• Maintains higher order of continuity

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Subdivide Each Triangle into 4 Triangles
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Subdivide Each Triangle into 4 Triangles
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Second step Modify the vertices

• Where should the additional vertices be located?
• How should the original vertices be moved?

Additional/Odd (Black)
Original/Even (Gray)
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New Vertex Locations

• Compute positions of added vertices from
• the adjacent four original vertices using mask
• Without overwriting,
• update the original vertex positions from
• the six adjacent original vertices using mask
• Repeat until converged

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Semi-Regular Meshes

• Most of the mesh has vertices with degree 6
  (Regular point). If the mesh is topologically
  equivalent to a sphere, then not all the vertices
  can have degree 6
• Must have a few extraordinary points (degree ! 6)

Extraordinary point
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Weights at an Extraordinary Point

• Challenge find weights that generate a smooth
  surface (tangent plane continuous)
• Want the surface normal to be continuous
• This is a hard math problem!

Warren weights
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Example

• Example mesh

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Example

• Add vertices

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Example
Odd vertex refinement
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Example

• Odd vertex refinement

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Example

• Even vertex refinement

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Example

• Even vertex refinement

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Example

• Extraordinary vertex

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Example

• Extraordinary vertex

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Example

• Subdivided Surface

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Example

• Level 2

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Example

• Level 3

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Example

• Level 4

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Example

• Limit Surface

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Mesh Generation
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Marching Cubes

• Polygonization
• Convert implicit surface to polygonal mesh
• Render the polygons as before
• Two steps
• Partition space into cells
• Fit a polygon to the surface in each cell
• Gives a piecewise linear approximation of the
  surface in each cell

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Cell polygonization

• Need to find the cells that actually intersect
  the surface
• A simple method is exhaustive enumeration
• Divide all of space into a regular grid
• Traverse all cells, and polygonize the cells that
  contain the surface

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Surface vertices

• Determine where the surface intersects the cell
  edges
• 1st option Linearly interpolate function value
  from cell vertices to approximate (function might
  not be linear)
• 2nd option Numerically find the zero
• Either will work, but option 2 can be
  significantly more expensive

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Surface vertices

• Linearly interpolate function value from cell
  vertices to approximate the surface

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Surface polygons

• Given the vertices of our final polygonal
  surface, find the polygon faces of the surface
• Solution There are only a finite number of
  options, defined by the configurations of
  inside/outside of the cells vertices

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Surface polygons

• Consider 2D first
• 16 different possible configurations
• Using cell vertex inside/outside configuration to
  index into this table
• Note some of them are duplicates of others

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Surface polygons
Result
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Surface polygons

• In 3D, there are 256 configurations, which can be
  reduced to 15 cases.

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Ambiguity in Marching Cubes

• Some configurations are ambiguous
• Solution?
• Sample more
• Refine the cell
• Simple solution just sample once more in the
  middle of the cell

or
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Another Solution
Split a cube into six tetrahedra, and find
surface vertices and polygons in each tetrahedron
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Marching tetrahedra
Unambiguous Less cases More elements (more
edges)
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Voronoi Diagram

• Split the space into regions consisting of the
  set of points closest to a particular seed point
• Every edge is a part of the perpendicular
  bisector of two points

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Voronoi Diagram Generation

• A simple way to generate the Voronoi diagram
• Incrementally add points
• Calculate all the perpendicular bisectors between
  the existing points and the newly added point
• Remove each section of the perpendicular bisector
  lines that trespasses other Voronoi regions

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Voronoi Diagram Delaunay Mesh

• For every Voronoi edge (red), there exists a
  Delaunay edge (black) between the two
  corresponding points
• They are the dual graph to each other
• In 2D, the Delaunay mesh is a triangle mesh

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Delaunay Mesh

• In a 2D Delaunay triangulation, the circumcircle
  of every triangle contains no other points
• This property leads to high-quality elements
  (long, thin elements are avoided as much as
  possible)

• This is also a good property for a surface mesh
  in 3D

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Improving and Existing Mesh

• Edge-Flipping
• In-Circle Test - for a pair of adjacent
  triangles, test whether the circumcircle of one
  triangle contains the fourth point
• Flip the edge to make it locally-Delaunay

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Incremental Algorithm Sampling

• Sampling the boundary
• sampling density inversely proportional to the
  distance to the medial axis
• more samples for higher curvature regions
• Sampling the interior
• sampling density as continuous as possible,
  including the boundary and the interior

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Incremental Algorithm

• Consider adding sample points one at a time
• Find the triangle which the newly added point is
  inside (a giant bounding triangle can be used to
  guarantee that you can always find a triangle
  that contains the point)
• Split that triangle into three triangles
• For each of the three new triangles. do In-Circle
  test for it and its outer edge adjacent triangle
  flip the edge if the test fails
• For every flipped edge, its neighbors should also
  be tested recursively
• Delete triangles outside the object

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Generating a surface mesh
In 2d, we can extract the outer edges to form a
polygon
In 3d, we can extract the surface faces to form
an objects surface
The 3D volume needs sampling points on the
interior, and meshing in 3D is generally
significantly harder than in 2D. E.g. Delaunay
gives poor quality elements in 3D (unlike 2D)!
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3D object surface is technically 2D

• Meshing on the surface directly requires geodesic
  distances
• Instead, flatten a 3D mesh to 2D, use 2D Delaunay
  
• Hard to flatten exactly -- but could
  approximately flatten, then the result is
  approximately Delaunay
• Need to sew up boundary regions to make a
  continuous mesh leads to poor quality elements

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3D object surface is technically 2D

• Cut the object to flatten it, and add new sample
  points (blue) along the cut
• Generate the Delaunay with incremental algorithm
  connecting both original (black) and new (blue)
  sampling points
• sew up the boundary edges (reconnect blue points)
• Retest/remesh to improve the mesh, especially
  near the boundary edges

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3D Sampling

• Evenly sampling is not efficient and cannot
  capture details
• Usually need more points to represent higher
  curvature regions
• Similar to 2D

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3D In-Circle Test

• Once we get a 3D mesh, the In-Circle test can be
  used to improve it, especially near the
  boundaries where it was sewn together
• Need 3D version of the In-Circle test
• Use an ellipse to approximate the mapped circle
  in 2D plane
• Given three vertices on a curved surface,
  consider the infinite set of spheres through the
  three vertices. The centers of all the spheres
  lie on a single line. Choose the sphere whose
  center is on the surface and define In-Circle
  test as inside that sphere.

CS148 Lecture 6