Transcript Interactive View-Dependent Rendering with Conservative

Cache-Oblivious Mesh
Layouts
Sung-Eui Yoon,
2
Valerio Pascucci,
1
2
Peter Lindstrom
1
Dinesh Manocha
1: University of North Carolina - Chapel Hill
2: Lawrence Livermore National Laboratory
http://gamma.cs.unc.edu/COL
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Goal
• Compute cache-coherent layouts
of polygonal meshes
♦ For geometric processing and
visualization
♦ Handle any kinds of polygonal models
(e.g., irregular geometry)
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Motivation
• High growth rate of computational
power of CPUs and GPUs
Growth rate
during 1993 – 2004
50
45
40
35
30
25
20
15
10
5
0
Disk
access
speed
RAM
access
speed
CPU
speed
Courtesy:
http://www.hcibook.com/e3/online/moores-law/
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Memory Hierarchies and
Caches
Fast memory Slow memory
or cache
CPU or
GPU
Block
transfer
Disk
Access time: 100ns
102ns
106ns
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Cache-Coherent Layouts
• Cache-Aware
♦ Optimized for particular cache
parameters (e.g., block size)
• Cache-Oblivious
♦ Minimizes data access time without any
knowledge of cache parameters
♦ Directly applicable to various hardware
and memory hierarchies
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CAD Model –
Double Eagle Tanker Model
82 million triangles
Irregular distribution of geometry
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Isosurface and Scanned
Models
Isosurface
100M triangles
St. Matthew
372M triangles
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Main Contribution
• Algorithm to compute cacheoblivious layouts of polygonal
meshes
Cache-oblivious metric
Multilevel optimization
framework
Applicable to
hierarchical representations
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Live Demo – ViewDependent Rendering (VDR)
• Based on multiresolution hierarchy
♦ Dynamically computes simplification
♦ Cache-oblivious layout is used to minimize
GPU vertex cache misses
GeForce Go
6800 Ultra
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Related Work
• Cache-coherent algorithms
• Mesh layouts
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Cache-Coherent Algorithms
• Cache-aware [Coleman and
McKinley 95, Vitter 01, Sen et al.
02]
• Cache-oblivious [Frigo et al. 99,
Arge et al. 04]
Focus on specific problems such as
sorting and linear algebra computations
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Mesh Layouts
• Rendering sequences
♦ Triangle strips
♦ [Deering 95, Hoppe 99, Bogomjakov and
Gotsman 02]
• Processing sequences
♦ [Isenburg and Gumhold 03, Isenburg and
Lindstrom 04]
Assume that access pattern
globally follows the layout order!
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Mesh Layouts
• Space-filling curves
♦ [Sagan 94, Velho and Gomes 91, Pascucci
and Frank 01, Lindstrom and Pascucci 01,
Gopi and Eppstein 04]
Assume geometric regularity!
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Outline
• Overview
• Cache-oblivious metric
• Results
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Outline
• Overview
• Cache-oblivious metric
• Results
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Overview
va
vb
vd
Input graph
vc
Multilevel optimization
Cache-oblivious metric
Local permutations
Result 1D layout
va
vb vd
vc
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Graph-based
Representation
• Undirected graph, G = (V, E)
♦ Represents access patterns of
applications
• Vertex
va
vb
vd
vc
♦ Data element
♦ (e.g., mesh vertex or mesh triangle)
• Edge
♦ Connects two vertices if they are likely to
be accessed sequentially
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Problem Statement
• Vertex layout of G = (V, E)
♦ One-to-one mapping of vertices to indices
in the 1D layout
: V
 {1, ... , | V |}
va
vb
vd

1
2
3
4
va
vb vd
vc
vc
• Compute a  that minimizes the
expected number of cache misses
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Local Permutation
Vertex layout
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Terminology
• Edge span of (va, vb)
|  (va )   (vb ) |
 (va )  1
Layout mapping
|  (va )   (vc ) | 4
 (vc )  5
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Terminology
• Ei
♦ Set of edges having edge span i in the
layout
4
(va , vc )  E4
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Terminology
• Edge span distribution
♦ | E | where i is in [1, n]
i
1
4
3
1
2
1
| E1 | 4
| E2 | 1
| E3 | 1
| E4 | 1
4
Number
of edges
1
1 2 3 4
Edge span
1
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Cache Miss Ratio Function
(CMRF), pi
• Probability of a cache miss for a
given edge span i
Cache miss ratio =
Probability to have
a cache miss
pi
1
0
1
i
Edge span
n-1
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Number of Cache Misses at
Runtime
• Estimated by multiplying two
factors
♦ Runtime edge span distribution
♦ CMRF
(
p2
+
Edge span 2
p4
+
Edge span 4
p2)  (2,1)  ( p2, p4)
Edge span 2
1D Layout:
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Number of Cache Misses at
Runtime
Runtime edge span
distribution
CMRF
(
p2
+
Edge span 2
p4
+
Edge span 4
p2)  (2,1)  ( p2, p4)
Edge span 2
1D Layout:
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Expected Number of Cache
Misses
Edge span distribution of the layout
The number
of vertices
n 1
| E | p
i 1
i
i
♦ Approximate runtime edge span
distribution with one of the layout
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Outline
• Overview
• Cache-oblivious metric
• Results
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Cache-Oblivious Metric
• Decides if a local permutation
reduces number of cache misses
♦ Probabilistic formulation
♦ Reduces to geometric volume computation
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Does a Local Permutation
Decrease Cache Misses?
n 1
n 1
?
 | Ei | pi   (| Ei |  | Ei |) pi
i 1
i 1
| Ei |
| Ei |  | Ei |
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Does a Local Permutation
Decrease Cache Misses?
n 1
n 1
 | E | p   (| E |  | E |) p
i 1
i
i
i 1
n 1

 | E | p
i 1
i
i
i
i
i
0
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Monotonocity of CMRF,
pi
• Assume CMRF is a monotonically
increasing function of edge span
1
Cache miss
ratio
0
pi
1
i
Edge span
∞
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Exact Cache-Oblivious
Metric
n 1

 | E | p
i 1
i
i
0
Monotonicity of CMRF
where
0  p1  p2  ...  pn2  pn1  1
All the possible cache
configurations
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Geometric Formulation
Half hyperspace
p2
n 1
 | E | p
i 1
i
i
0
0
p1
where
0  p1  p2  ...  pn2  pn1  1 p2
Closed hyperspace

n 1
0
p1
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Geometric Volume
Computation
• Assume each CMRF to be equally
likely
n 1
 | E | p
i 1
i
i
p2
0
where
0  p1  p2  ...  pn2  pn1  1
0
p1
• Half hyperspace (blue area)
♦ Space of CMRFs that reduce cache misses
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Geometric Volume
Computation
Time complexity
n 1
♦ Exact: O(n
♦ Approximate:
) [Lasserre and Zeron 01]
5 [Kannan et al. 97]
O(n )
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Fast and Approximate
Volume Comparison
• Define a top polytope in closed
hyperspace
• Compute the centroid, C, of the
top polytope
Top polytope
Centroid, C
p2
0
p1
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Fast and Approximate
Volume Comparison
• Use the centroid for approximate
volume comparison
♦ The volume containing the centroid is likely
to be larger
Centroid, C
p2
0
p1
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Bound of Approximation
• 0.1% ~ 0.3% compared to the
exact metric
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Final Approximate Metric
Centroid
m
 | E
j 1
l( j)
Pack non-zero
| j 0
 | Ei | to 1,…, m
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Layout Optimization
• Find an optimal layout that
minimizes our metric
♦ Combinatorial optimization problem
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Multilevel Minimization
Step 1:
Coarsening
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Multilevel Minimization
Step 2:
Ordering of coarsest graph
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Multilevel Minimization
Step 3:
Refinement and
local optimization
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Outline
• Overview
• Cache-oblivious layouts
• Results
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Layout Computation Time
• Process 70 million vertices per
hour
♦ Takes 2.6 hours to lay out St. Matthew
model (372 million triangles)
♦ 2.4GHz of Pentium 4 PC with 1 GB main
memory
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Number of edges
Edge Span Distributions of
Different Layouts
Cache-oblivious layout
Original layout
Spectral layout
Edge span
>
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Applications
• View-dependent rendering
• Collision detection
• Isocontour extraction
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View-Dependent Rendering
• Layout vertices and triangles of
CHPM [Yoon et al. 04]
♦ Reduce misses of GPU vertex cache
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View-Dependent Rendering
Peak performance: 145 M tri / s on
GeForce 6800 Ultra
Models
# of Tri.
Our
layout
St.
Matthew
372M
106 M/s
Isosurface
100M
Double
Eagle
Tanker
82M
Simplification layout
[Yoon et al. 04]
23 M/s
4.5X
90 M/s
20 M/s
2.1X
22 M/s
47 M/s
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Realtime Captured Video – St.
Matthew Model
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Comparison with Other
Rendering Sequences
1.2
1.1
1
Cache
miss ratio 0.9
(misses 0.8
per
0.7
triangle) 0.6
0.5
0.4
0.3
Universal rendering sequences
[Bogomjakov and Gotsman 2002]
Our layout
8
32
16
Vertex cache size
64
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Comparison with Other
Rendering Sequences
1.2
[Hoppe 99]
1.1
Optimized for 16 vertex cache size
1
Cache 0.9
with FIFO replacement
miss ratio
0.8 Our layout
(misses
0.7
per
triangle) 0.6
0.5 Optimized for no particular cache size
0.4
0.3
64
32
16
8
Vertex cache size
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Performance during ViewDependent Rendering
0.9
[Hoppe 99]
0.85
0.8 Optimized for full resolution
Cache miss 0.75
ratio
0.7
(given cache
Our layout
0.65
size 32)
0.6
0.55
Optimized for various resolutions
0.5
100%
75%
50%
25%
10%
Resolution
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Comparison with Space Filling
Curve on Power Plant Model
2
1.9
1.8
1.7
1.6
Cache miss
1.5
ratio
1.4
1.3
1.2
1.1
1
Space filling curve (Z-curve)
Our layout
8
16
32
Vertex cache size
64
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Collision Detection
• Bounding volume hierarchies
♦ Widely used to accelerate the
performance of collision detection
♦ Traversed to find contacting area
♦ Uses pre-computed layouts of OBB trees
[Gottschalk et al. 96]
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Rigid Body Simulation
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Collision Detection Time
Depth-first layout
2X on average
Cache-oblivious layout
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Isocontour Extraction
• Contour tree [van
Kreveld et al. 97]
• Use mesh as the
input graph
• Extract an
isocontour that is
orthogonal to z-axis
Puget sound,
134 M triangles
Isocontour
z(x,y) = 500m
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Comparison – First
Extraction of Z(x,y) = 500m
Disk access time is bottleneck
25
Relative
Performance
over
Z-axis sorted
layout
20
21
15
13
10
5
0
2
Cacheoblivious
layout
1
Z-axis
sorted
Y-axis
sorted
Spectral
layout
Nearly optimized for particular isocontour
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Comparison – Second
Extraction of Z(x,y) = 500m
Relative
Performance
over
Z-axis sorted
layout
400
350
300
250
200
150
100
50
0
379
21
13
212
0.8
1
2
Cacheoblivious
layout
Z-axis
sorted
Y-axis
sorted
Spectral
layout
Memory and L1/L2 cache access times are bottleneck
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Limitations
• Assumptions on CMRF
♦ May not work well for all applications
• Does not compute global
optimum
♦ Greedy solution
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Advantages
• General
♦ Applicable to all kinds of polygonal models
♦ Works well for various applications
• Cache-oblivious
♦ Can have benefit from CPU/GPU cache to
memory and disk
• No modification of runtime
application
♦ Only layout computation
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OpenCCL: Cache-Coherent
Layouts of Graphs and Meshes
• Source codes for computing a
cache-coherent layout
• Easy to use
CLayoutGraph
Graph Oblivious
(NumVertex); Mesh Layout”
Google
“Cache
0
or
Graph.AddEdge (0, 1);
Graph.AddEdge
(0, 2);
Http://gamma.cs.unc.edu/COL
Graph.AddEdge (1, 2);
1
2
int Order [NumVertex];
Graph.ComputeOrdering (Order);
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Conclusion
• Novel algorithm for computing
cache-oblivious mesh layouts
♦ Cast the problem as an optimization
♦ Probabilistically compute the expected
number of caches misses
♦ Achieve significant improvements (2 to
20X) without modifying runtime
applications
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Ongoing and Future Work
• Apply to other applications
♦ Simplification and approximate collision
detection [Yoon et al. 04]
♦ Shortest path computation, etc.
• Investigate optimality
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Ongoing and Future Work
• Cache-Oblivious Layouts of
Bounding Volume Hierarchies
[Yoon and Manocha 05]
♦ Tech. Report, University of North Carolina
at Chapel Hill
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Acknowledgements
• Anonymous donor
♦ Power plant model
• Digital Michelangelo Project
♦ St. Matthew model at Stanford University
• LLNL ASCI VIEWS
♦ Isosurface model
• Newport news shipbuilding
♦ Double eagle tanker
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Acknowledgements
•
•
•
•
•
•
•
Army Research Office
DARPA
Intel Corporation
Lawrence Livermore Nat’l Lab.
National Science Foundation
Office of Naval Research
RDECOM
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Acknowledgements
•
•
•
•
•
•
Martin Isenburg
Dawoon Jung
Brandon Lloyd
Elise London
Brian Salomon
Avneesh Sud
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Questions?
Project URL
http://gamma.cs.unc.edu/COL
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL