Bounding Volume Hierarchies
CS 6965 Fall 2011
Fall 2011CS 6965 2
Fall 2011CS 6965
Heightfield3
Fall 2011CS 6965
C++ Operator?
4
#include <stdio.h>int main(){ int x = 10; while( x --> 0 ) // x goes to 0 { printf("%d ", x); }}
stackoverflow.com/questions/1642028/what-is-the-name-of-this-operator
Fall 2011CS 6965
C++ Operator?
5
That's not an operator -->. That's two separate operators, -- and >. Your post is decrementing x and then comparing x and 0 with the > operator
To better understand, the statement could be as follows:
while( (x--) > 0 )
Fall 2011CS 6965
Program 26
Fall 2011CS 6965
Backwards Lighting7
Fall 2011CS 6965
Light Direction
8
• costheta > 0
• Means N and V are on opposite sides
• Should flip the normal
• costheta < 0
• Means N and V are on the same side
• Keep the normal
Compute hit position (P= O+ tV
)Call primitive to get normal (N
) (normalized)
costheta = N⋅V
if (costheta < 0) normal =-normalColor light = scene.ambient*Kaforeach light source get CL and L
dist= L
,Ln
=L
L
cosphi = N⋅ Ln
if(cosphi > 0) if(!intersect with 0 < t < dist) light += CL * (Kd * cosphi)result=light*surface color
Fall 2011CS 6965
Library Calls
9
• Show up as something like this:
• This one was fixed by removing global variables
• Use #define or keep things locally
ERROR: undefined symbol: __dso_handlefailed to add instruction: ORIline 54assembler returned an error, exiting
Fall 2011CS 6965
Memset• We do have this one!
• Change this line:
• To this line:
10
@${LINKERDIR}/ln.py rt.s > ${TRAXOUTPUT}
@${LINKERDIR}/ln.py rt.s ${LIBDIR}/memset.s > ${TRAXOUTPUT}
Fall 2011CS 6965
Inheritance
• We can’t handle these in simhwrt yet
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_ZTV8gpSphere:# .long 0# .long _ZTI8gpSphere# .long _ZNK8gpSphere17intersectDetailedERK12RayPrimitiveR18IntersectionRecord# .long _ZNK8gpSphere14intersectQuickERK12RayPrimitive# .size _ZTV8gpSphere, 16
Fall 2011CS 6965
Calling Convention 12
Fall 2011CS 6965
Calling Convention13
Fall 2011CS 6965
Common Problems
14
void draw_line(const int& framebuffer, const int& width, const Vector& V1, const Vector& V2, const int& max_pixel){ int x0 = V1[0]; int x1 = V2[0]; int y0 = V1[1]; int y1 = V2[1];
int dx = x0 - x1; if (dx < 0) dx = -dx; int dy = y0 - y1 if (dy < 0) dy = -dy;
int sx, sy; if (x0 < x1) sx = 1; else sx = -1; if (y0 < y1) sy = 1; else sy = -1; int err = dx - dy; int e2; while (x0 != x1 || y0 != y1) {
Fall 2011CS 6965
Common Problems
15
int x0 = V1[0]; int x1 = V2[0]; int y0 = V1[1]; int y1 = V2[1];
trax_printi(x0); trax_printi(x1); trax_printi(y0); trax_printi(y1);
int dx = x0 - x1; if (dx < 0) dx = -dx; int dy = y0 - y1 if (dy < 0) dy = -dy;
int sx, sy; if (x0 < x1) sx = 1; else sx = -1; if (y0 < y1) sy = 1; else sy = -1; int err = dx - dy; int e2;
trax_printi(err);
Fall 2011CS 6965
Common Problems
16
int x0 = V1[0]; int x1 = V2[0]; int y0 = V1[1]; int y1 = V2[1];
trax_printi(x0); trax_printi(x1); trax_printi(y0); trax_printi(y1);
int dx = x0 - x1; if (dx < 0) dx = -dx; int dy = y0 - y1 if (dy < 0) dy = -dy;
int sx, sy; if (x0 < x1) sx = 1; else sx = -1; if (y0 < y1) sy = 1; else sy = -1;
trax_printi(dx); trax_printi(dy);
int err = dx - dy; int e2;
Fall 2011CS 6965
Common Problems
17
int x0 = V1[0]; int x1 = V2[0]; int y0 = V1[1]; int y1 = V2[1];
trax_printi(x0); trax_printi(x1); trax_printi(y0); trax_printi(y1);
int dx = x0 - x1; if (dx < 0) dx = -dx; int dy = y0 - y1 if (dy < 0) dy = -dy;
trax_printi(dx); trax_printi(dy);
int sx, sy; if (x0 < x1) sx = 1; else sx = -1; if (y0 < y1) sy = 1; else sy = -1;
int err = dx - dy; int e2;
Fall 2011CS 6965
Common Problems
18
trax_printi(x0); trax_printi(x1); trax_printi(y0); trax_printi(y1);
int dx = x0 - x1; if (dx < 0) dx = -dx; // dx = abs(dx) int dy = y0 - y1 if (dy < 0) dy = -dy; // dy = abs(dy)
trax_printi(dx); trax_printi(dy);
Fall 2011CS 6965
Common Problems
19
trax_printi(x0); trax_printi(x1); trax_printi(y0); trax_printi(y1);
int dx = x0 - x1; if (dx < 0) dx = -dx; // dx = abs(dx) int dy = y0 - y1 if (dy < 0) dy = -dy; // dy = abs(dy)
trax_printi(dx); trax_printi(dy);
// Write it a different way int dx = x0 > x1 ? x0 - x1 : x1 - x0; int dy = y0 > y1 ? y0 - y1 : y1 - y0;
Fall 2011CS 6965
Common Problems
20
void draw_line(const int& framebuffer, const int& width, const Vector& V1, const Vector& V2, const int& max_pixel){ int x0 = V1[0]; int x1 = V2[0]; int y0 = V1[1]; int y1 = V2[1];
int dx = x0 > x1 ? x0 - x1 : x1 - x0; int dy = y0 > y1 ? y0 - y1 : y1 - y0;
int sx, sy; if (x0 < x1) sx = 1; else sx = -1; if (y0 < y1) sy = 1; else sy = -1; int err = dx - dy; int e2;
while (x0 != x1 || y0 != y1) {
void draw_line(const int& framebuffer, const int& width, const Vector& V1, const Vector& V2, const int& max_pixel){ int x0 = V1[0]; int x1 = V2[0]; int y0 = V1[1]; int y1 = V2[1];
int dx = x0 - x1; if (dx < 0) dx = -dx; int dy = y0 - y1 if (dy < 0) dy = -dy;
int sx, sy; if (x0 < x1) sx = 1; else sx = -1; if (y0 < y1) sy = 1; else sy = -1; int err = dx - dy; int e2; while (x0 != x1 || y0 != y1) {
Before After
Fall 2011CS 6965
Bounding boxes• Box: easy
• bounds = Min(p1, p2), Max(p1, p2)
• Sphere:
• bounds = C-Vector(radius, radius,radius)
• C+Vector(radius, radius, radius)
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P1
P2
C
Fall 2011CS 6965
Bounding Boxes• Triangle:
• bounds = Min(p1,p2,p3), Max(p1,p2,p3)
• Plane
• infinite bounds
• You may want to consider removing planes from your renderer at this point
• Or keep them outside of your accel structures
22
P1
P2P3
Fall 2011CS 6965
Bounding boxes• Disc/Ring:
• Group:
• Union of object bounding boxes
• min of mins
• max of maxs
Cx ± rad Ny2 + Nz
2
N
= 1Similar for y/z
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Fall 2011CS 6965
Uses for bounding boxes
• Quick reject for expensive primitives
• To fill in grid cells
• Directly in Bounding Volume Hierarchy or similar
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Fall 2011CS 6965
Bounding Volume Hierarchy
• Observe that intersection is like a search
• We know that searching is O(n) for unsorted lists but O(log n) for sorted lists
• How do we sort objects from all directions simultaneously?
25
Fall 2011CS 6965
Bounding volume hierarchy
• Organize objects into a tree
• Group objects in the tree based on spatial relationships
• Each node in the tree contains a bounding box of all the objects below it
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Fall 2011CS 6965
BVH traversal
• At each level of the tree, intersect the ray with the bounding box
• miss: ray misses the entire subtree
• hit: recurse to both children
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Fall 2011CS 6965
BVH traversal
• At each level of the tree, intersect the ray with the bounding box
• miss: ray misses the entire subtree
• hit: recurse to both children
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Fall 2011CS 6965
BVH traversal
• At each level of the tree, intersect the ray with the bounding box
• miss: ray misses the entire subtree
• hit: recurse to both children
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Fall 2011CS 6965
BVH traversal
• At each level of the tree, intersect the ray with the bounding box
• miss: ray misses the entire subtree
• hit: recurse to both children
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Fall 2011CS 6965
BVH optimizations
• Stop if the current T value is closer than the BVH node
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Fall 2011CS 6965
BVH optimizations
• Stop if the current T value is closer than the BVH node
• Traverse down side of tree that is closer to origin of the ray first
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Fall 2011CS 6965
BVH optimizations
• Stop if the current T value is closer than the BVH node
• Traverse down side of tree that is closer to origin of the ray first
• Three or more way split
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Fall 2011CS 6965
Building a BVH• Determining optimal BVH structure is
NP-hard problem
• Heuristic approaches:
• Cost models (minimize volume or surface area)
• Spatial models
• Categories of approaches:
• Top down
• Bottom up
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okay okay
bad
Fall 2011CS 6965
Median cut BVH construction
• Top down approach:
• Sort objects by position on axis
• cycle through x,y,z
• use center of bounding box
• Insert tree node with half of objects on left and half on right
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Fall 2011CS 6965
Weghorst BVH construction
• Bottom up construction
• Add objects one at a time to tree
• Insert to subtree that would cause smallest increase to area
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Fall 2011CS 6965
Weghorst BVH construction
• Bottom up construction
• Add objects one at a time to tree
• Insert to subtree that would cause smallest increase to area
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Fall 2011CS 6965
Weghorst BVH construction
• Bottom up construction
• Add objects one at a time to tree
• Insert to subtree that would cause smallest increase to area
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Fall 2011CS 6965
Weghorst BVH construction
• Bottom up construction
• Add objects one at a time to tree
• Insert to subtree that would cause smallest increase to area
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Fall 2011CS 6965
k-d tree• Recursively divide space in half
• Alternate coordinate axes
• Cycle through axes or store axis split in each node
• What do you do with objects split by the plane?
• Hard part: where do you split in each dimension?
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Fall 2011CS 6965
BSP tree• Like a k-d tree where splitting planes
can be arbitrarily located
• Hard part: where do you split in each dimension?
• Harder part: how do you orient each plane?
• More storage/computation, tighter bounds
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Fall 2011CS 6965
BSP/k-d tree tradeoffs• Build is NP-hard problem
• Heuristic approaches are always used
• Traversal is quick
• Storage can be lower than BVH or grid
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Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• min: 0
• max: ∞
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
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Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
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Min: 0Max: ∞
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
45
Min: 0Max: ∞
Split: 1.0Stack: 1
New min: 0New max: 1.0
12
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
46
Min: 0Max: 1.0
Split: 1.3Stack: 1
New min: 0New max: 1.0
3 4
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
47
Min: 0Max: 1.0
Split: 0.4Stack: 1 5
New min: 0New max: 0.4
56
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
48
Min: 0.4Max: 1.0
Split: 0.7Stack: 1 8
New min: 0.4New max: 0.6
7 8
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
49
Min: 0.7Max: 1.0
Split:Stack: 1
New min: 0.4New max: 0.6
7 8
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
50
Min: 1.0Max: ∞
Split: 1.5Stack: 10
New min: 1.0New max: 1.5
9 10
Fall 2011CS 6965
K-d traversal• Keep track of intervals on ray:
• Compute t of split plane
• If t > max: goto near child
• If t < min: goto far child
• Otherwise: goto both children with updated intervals
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Min: 1.0Max: 1.5
Split: 1.4Stack: 10 11
New min: 1.0New max: 1.4
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Fall 2011CS 6965
K-d tree build• Optimal solution is NP-hard
• Spatial median split (a little like octree)
• Object median split (makes balanced trees)
• Cost model based (better)
• Cost of traversal
• Cost of intersection
• Probability of hit
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Fall 2011CS 6965
K-d tree build• Havran ‘01:
• Start with bounding box of scene
• Select split plane along each axis
• Start with spatial median
• Move toward bounding box of child nodes
• Recurse
• Stop if node contains 2-3 objects
• Stop if depth > max (20-30)
• Backtrack
• Combine children with identical content53
Fall 2011CS 6965
Acceleration Structure Summary
• Most common: Grid, Hierarchical Grid, k-d tree, BVH
• Grid based:
• P time deterministic build
• Grid resolution tradeoff
• Tree based:
• NP hard build
• Heuristic approaches
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Fall 2011CS 6965
Update SimHWRT• svn up
• make
• ./update.sh
• This one will take a while
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