CSL 859: Advanced Computer Graphics

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CSL 859 Advanced Computer Graphics

• Dept of Computer Sc. Engg.
• IIT Delhi

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Basic Depth Visibility

• Z-buffer
• Linear in pixel and geometry complexity
• Front to back
• Back to front
• Depth sorting
• Want to cull groups
• Of pixels
• Hiererchical Z-buffer
• Of primitives
• Occlusion Culling

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View-Frustum Visibility

• Clipping
• Scissoring
• Culling

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Clipping

• Clip with each boundary
• Out from In
• Output Intersection
• In from Out
• Output Intersection
• Output Vertex
• Out from Out
• Discard
• In from In
• Output Vertex

x 1
x-1
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Clipping

• Clip with each boundary
• Out from In
• Output Intersection
• In from Out
• Output Intersection
• Output Vertex
• Out from Out
• Discard
• In from In
• Output Vertex

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Clipping

• Clip with each boundary
• Out from In
• Output Intersection
• In from Out
• Output Intersection
• Output Vertex
• Out from Out
• Discard
• In from In
• Output Vertex

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Clipping

• Clip with each boundary
• Out from In
• Output Intersection
• In from Out
• Output Intersection
• Output Vertex
• Out from Out
• Discard
• In from In
• Output Vertex

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Intersection
v1t v0 (1-t)
x1t x0 (1-t) -1 t (1x0)/(x0-x1) y y1t
y0 (1-t)
v1
v0
x-1
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Homogeneous Space Clipping

• Clip coordinates projective coordinates
• -1 lt x/z lt 1 (after perspective divide)
• -1 lt x/w lt 1 (clip)
• -w lt x lt w
• -w lt y lt w
• 0 lt z lt w

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View-frustum Culling

• Project all vertices
• Then cull

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Group Culling

• Cull in World space

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Group Culling

• Cull in World space
• Group polygons
• Cull groups
• Plane test

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Groups

• Bounding volume hierarchies
• Bounding sphere
• Axis-aligned bounding boxes (AABB)
• Oriented bounding boxes (OBB)
• Discrete oriented planes (k-DOPs)
• Spatial Partitioning
• Octrees
• Binary space partition tree

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Hierarchies
Sphere AABB OBB kDOP
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Bounding Sphere Computation
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Bounding Sphere Computation

• Propose a center
• Find radius that includes all vertices
• Minimal volume obtained
• if 4 supporting vertices touch the sphere
• Test also degenerate case of 2 or 3 vertices
• Start with sphere through 2 vertices
• Test inclusion of other vertices in sequence
• If v is outside
• Compute new sphere also supported by v
• Restart if a previously included vertex is
  outside this new sphere

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AABB Computation

• Extrema along each primary axis

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OBB Computation

• One face and one edge of convex polyhedron are
  part of OBB
• OR
• Three edges of the convex polyhedron form part of
  the OBB

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Separating Axis Theorem

• Disjoint convex polyhedrons, A and B, are
  separated along at least one of
• An axis orthogonal to a face of A
• An axis orthogonal to a face of B
• An axis formed from the cross product of one edge
  from each of A and B

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SAT example Triangle/Box

• Box is axis-aligned
• 1) test the axes that are orthogonal to the faces
  of the box
• That is, x,y, and z

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Triangle/Box with SAT

• Assume that they overlapped on x,y,z
• Must continue testing
• 2) Axis orthogonal to face of triangle

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Triangle/Box with SAT

• If separating axis still not found
• 3) Test axis tebox x etriangle
• Example
• x-axis from box ebox(1,0,0)
• etrianglev1-v0
• Test all such combinations
• If there is at least one separating axis, then
  the objects do not intersect
• Otherwise they do

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Hierarchical Culling

• Test for a group
• If outside frustum
• Cull
• If inside frustum
• Display
• Otherwise,
• Subdivide group into smaller groups
• Recurse for each group

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BVH vs. Spatial Partitioning

• BVH SP
• - Object centric - Space centric
• - Spatial redundancy - Object redundancy

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BVH vs. Spatial Partitioning

• BVH SP
• - Object centric - Space centric
• - Spatial redundancy - Object redundancy

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BVH vs. Spatial Partitioning

• BVH SP
• - Object centric - Space centric
• - Spatial redundancy - Object redundancy

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BVH vs. Spatial Partitioning

• BVH SP
• - Object centric - Space centric
• - Spatial redundancy - Object redundancy

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Spatial Data Structures Subdivision

• Many others

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Uniform Spatial Subdivision

• Decompose the objects (the entire simulated
  environment) into identical cells arranged in a
  fixed, regular grids (equal size boxes or voxels)
• To represent an object, only need to decide which
  cells are occupied. To perform collision
  detection, check if any cell is occupied by two
  object
• Storage to represent an object at resolution of
  n voxels per dimension requires upto n3 cells
• Accuracy solids can only be approximated

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Octrees

• Quadtree is derived by subdividing a 2D-plane
  in both dimensions to form quadrants
• Octrees are a 3D-extension of quadtree
• Use divide-and-conquer
• Reduce storage requirements (in comparison to
  grids/voxels)

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Bounding Volume Hierarchies

• Model Hierarchy
• Each node has a simple volume that bounds a set
  of triangles
• Children contain volumes that each bound a
  different portion of the parents triangles
• A binary bounding volume hierarchy

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Designing BVH

• It should fit the original model as tightly as
  possible
• Testing two such volumes for overlap should be as
  fast as possible
• It should require the BV updates as infrequently
  as possible

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Observations

• Simple primitives (spheres, AABBs, etc.) do very
  well with respect to the second constraint. But
  they cannot fit some long skinny primitives
  tightly.
• More complex primitives (minimal ellipsoids,
  OBBs, etc.) provide tight fits, but checking for
  overlap between them is relatively expensive.
• Cost of BV updates needs to be considered.

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Trade-off in Choosing BVs

• increasing complexity tightness of fit
• decreasing cost of (overlap tests BV update)

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Fitting OBBs statistically

• Sample the convex hull of the geometry
• Find the mean and covariance matrix of the
  samples
• The mean will be the center of the box
• The eigenvectors of the covariance matrix are the
  principal directions axes
• The principle directions tend to align along the
  longest axis, then the next longest that is
  orthogonal, and then the other orthogonal axis

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Principal Components
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Back-face Culling
Sphere
Half the sphere is not visible
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Polygonal Backfacing?
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Polygon Groups
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Hierarchical Back-face Culling

• Cluster proximate polygons
• Keep orientations similar too
• For a group, find the half-space intersection
• If the eye lies in the common HS
• Cull group
• Coherence in traversal
• Subdivide half-space into partitions
• Query which partition eye lies in

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Back-Patch Culling
p.n gt e.n for all p
c-e.n gt rn
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Back-patch Visibility
p
n
e
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Back-Patch Culling

• Create bounding volume for object
• Compute planes tangent to volume
• passing through eye
• Compute half-space intersection of these planes
• Compute bounding cone of normals of the object
• If normal cone lies in common half-space
• Cull Object

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Silhouettes

• Edges between front and back faces
• Simple hack
• Render the front-facing polygons
• Render the back-facing polygons (in black)
• A common edge gets over-written
• Render them as wide lines?
• Offset backfaces closer?

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Results

• Wireframe Translation Fattening

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Exact Silhouettes

• For each edge check two adjacent faces
• Can compute hierarchically
• If an entire group front or back facing
• Discard
• Otherwise
• Subdivide
• But consider boundaries between groups

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Notion of Duality

• Dual of plane ax by cz 1 0 is
• point (a, b, c)
• And Dual of a point is a plane
• If point v is in ve half-space of plane P
• Dual(P) is in ve half-space of Dual (v)
• Dual of edge e between faces f1 and f2 is
• edge Dual(plane(f1))-Dual(plane(f2))

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Dual Approach

• Geometric duals
• silhouette-edge duals cross the view-point dual
  (plane)
• Coherence
• consecutive view-planes form a wedge
• Edge crossing the wedge is a silhouette update

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Duality
f1
p1
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Duality
f1
O
p1
p2
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Silhouette Update
View-plane
6
4
1
7
2
3
5
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Silhouette Algorithm Details

• Double-wedge point location query
• Partition space
• Octree, BAR-tree
• If region intersected by wedge
• recur

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Silhouette Algorithm Details
Lin
Lout
Adj Container
Tree Region
Double wedge
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Quiz 5

1. Explain how quadric error is used for mesh
   simplification.
2. Define Mean curvature and Gaussian curvature of a
   surface.

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Cells Portals

• Goal walk through architectural models
  (buildings, cities, catacombs)
• These divide naturally into cells
• Rooms, alcoves, corridors
• Transparent portals connect cells
• Doorways, entrances, windows
• Cells only see other cells through portals

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Cells Portals

• An example

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Cells Portals

• Idea
• Cells form the basic unit of PVS
• Create an adjacency graph of cells
• Starting with cell containing eyepoint, traverse
  graph, rendering visible cells
• A cell is only visible if it can be seen through
  a sequence of portals
• So cell visibility reduces to testing portal
  sequences for a line of sight

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Cells Portals
A
D
E
F
C
B
G
H
E
A
B
C
D
F
G
H
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Cells Portals
A
D
E
F
C
B
G
H
E
A
B
C
D
F
G
H
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Cells Portals
A
D
E
F
C
B
G
H
E
A
B
C
D
F
G
H
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Cells Portals
A
D
E
F
C
B
G
H
E
A
B
C
D
F
G
H
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Cells Portals
A
D
E
F
C
B
G
H
E
A
B
C
D
F
G
H
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Cells Portals
A
D
E
?
F
C
B
G
H
E
A
B
C
D
F
G
?
H
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Cells Portals
A
D
E
X
F
C
B
G
H
E
A
B
C
D
F
G
X
H
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Cells Portals

• View-independent solution find all cells a
  particular cell could possibly see
• C can only see A, D, E, and H

A
D
E
H
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Cells Portals

• View-independent solution find all cells a
  particular cell could possibly see
• H will never see F

A
D
E
C
B
G
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Cells and Portals

• Factor into View-independent view-dependent
• Portal-portal visibility line stabbing
• Linear program Teller 1993
• Cell-cell visibility stored in stab trees
• View-dependent eye-portal visibility stage
  further refines PVS at run time
• Slow pre-process
• Pointer following at render time

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Linear Programming

• Canonical form Find x that
• Maximizes cT x
• Objective function
• subject to Ax lt b
• -ve constraints
• where xi gt 0
• ve constraints

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Occlusion Query

• HW mechanism to determining visibility of a set
  of geometry
• After rendering geometry, query if any of the
  geometry could have or did modify the depth
  buffer.
• If Query answer is false, geometry could not
  have affected depth buffer
• If true, it could have or did modify depth
  buffer
• Ex Render bounding box, Query

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Occlusion Query

• Returns pixel count the no. of pixels that pass
• May issue multiple queries at once before asking
  for the result of any one
• Applications can overlap the time it takes for
  the queries to return with other CPU work

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Occlusion Query How to Use

• (Optional) Disable Depth/Color Buffers
• (Optional) Disable any other irrelevant state
• Generate occlusion queries
• Begin ith occlusion query
• Render ith (bounding) geometry
• End occlusion query
• Do other CPU computation now
• (Optional) Enable Depth/Color Buffers
• (Optional) Re-enable any other state
• Get pixel count of ith query
• If (count gt MAX_COUNT) render ith geometry

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Occlusion Query How to Use

• Generate occlusion queries
• Gluint queriesN
• GLuint pixelCount
• glGenOcclusionQueriesNV(N, queries)
• Loop over queries
• for (i 0 i lt N i)
• glBeginOcclusionQueryNV(queriesi)
• // render bounding box for ith
  geometry
• glEndOcclusionQueryNV()

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Occlusion Query How to Use

• Get pixel counts
• for (i 0 i lt N i)
• glGetOcclusionQueryuivNV(queriesi,
  GL_PIXEL_COUNT_NV,
• 
  pixelCount)
• if (pixelCount gt MAX_COUNT)
• // render ith geometry

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Incremental Object Culling

• How valuable is it?
• Bounding box still must be filled
• Must Query against something
• More intelligent queries possible
• Big win if you use query for object that you were
  going to render in any case
• Can amortize
• Skip query for visible objects for the next few
  frames
• Visible objects of last frame could occlude

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Incremental Object Culling

• Useful for multi-pass algorithms
• Render scene (roughly) from front to back
• Draw big occluders (walls) first
• Issue queries for other objects in the scene
• If query returns 0 in a pass, skip the object in
  subsequent passes
• Works well if the first pass sets up the depth
  buffer only

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From Region Visibility
Rest of the Scene
Equivalent center of projection
Occluding Segment
View Segment
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Visibility From Region in 3D
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Visibility From Region in 3D

• Find all supporting planes
• occluder and occludee in the same half-space
• Find a COP in that half-space of planes
• Clip occluders with planes parallel to supporting
  planes
• Render Scene
• Render clipped triangles
• Re-render triangles with occlusion query