Download Isosurface Extraction and Spatial Filtering Using Persistent Octree

Isosurface Extraction and
Spatial Filtering Using
Persistent Octree (POT)
Qingmin Shi and Joseph JaJa
Institute for Advanced Computer Studies and
Department of Electrical and Computer Engineering
University of Maryland, College Park
Volumetric Data Visualization
NLM Visible Human
([Lorensen Vis’95])
LLNL Richtmyer-Meshkov Instability
([Shi, JaJa Vis’06])
Tetra Meshes of Ribosome 30S
([Zhang, Xu, and Bajaj,
Computer Aided Geometric
Design 2006])
2
Isosurface Extraction and Rendering
 What is isosurface?
Elevation Map: 2D-contours, each
with a constant elevation
Isosurface: A 3D-contour -- A 2D-surface
embedded in a 3D space with constant
density (isovalue) C.
Image created from the UNC CT Head data set
using VolPack [Lacroute and Levoy, SIGGRAPH
’94].
3
Basic Steps in Isosurface Computation
 Finding the active cells that are cut by the isosurface.
 Piece-wise linear approximation (triangulation) of the
isosurface patches in each active cell.
• The marching cubes algorithm was used in this paper.
 Rendering triangles using either hardware or software.
4
Identifying Active Cells
 Why is this step important?
• Volume size can be very large.
• The size of the whole Richtmyer-Meshkov instability data set is
about 2.1 TB!
• The isosurface in general only occupies a small
subspace.
• Examining all cells is very expensive and unnecessary.
 Solution: value based partitioning.
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Active Cell Identification As A Stabbing Query
 Let mini and maxi be the
minimum and maximum
scalar value of all the
vertices of a cell i.
 A cell is active if and only if
the isovalue C is within the
range [mini, maxi].
 Finding active cells is
equivalent to stabbing a
set of horizontal segments
with an infinite vertical line.
C
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Solutions to the Stabbing Query
 Optimal solutions:
• Priority tree (McCreight SIAM J. Comp.’85).
• Interval tree (Cignoni et al. Volvis’96, Chiang and Silva Vis’97).
• Space: O(N), query time: O(log N + K)
N : number of cells; K : number of active cells.
 Good practical solutions
• Span space (Livnat, Shen, Johnson Vis’96).
• Span triangles (von Rymon-Lipinski et al. Vis’04).
• Fixed-sized buckets (Waters, Co, and Joy EuroVis’05).
7
Spatial Filtering
 Only a subset of the active cells need to be processed:
• View-dependent rendering – cells that are visible.
• Ray casting – the first active cell that intersects each ray.
• 4D isosurface visualization – cells that intersect the cutting
hyperplane.
 Benefits:
• Reduced indexing structure search time.
• Reduced triangulation time.
• Reduced rendering time.
8
View-dependent Isosurface Rendering
 Render the isosurface patches in a front-to-back order.
 Keep a coverage mask to tell which part of the screen has been
covered.
 Do not triangulate and render active cells whose projection to the
screen has already been covered.
Problem:
How to examine
the active cells
in a front-toback order?
9
Value-based vs. Location-based Partitioning
 Previous view-dependent algorithms often employ locationbased partitioning.
• Example: a min-max octree equipped with min, max
values for each internal node.
• Problem: identifying active cells is inefficient.
 Active cells reported by a value-based partitioning structure
do not obey any spatial ordering.
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Our Solution
 Build one octree to organize the active cells corresponding to EACH
possible isovalue.
 Compress the sequence of octrees into a persistent octree (POT).
 The benefits of both approaches:
• The size of the POT is linear in the input size O(N), even for
continuous data types!
• Identifying visible active cells is done by a front-to-back traversal of
an octree that consists of ONLY the active cells – O(K)!
• Spatial filtering is efficient using the octree.
11
Persistent Data Structures
 Consider a dynamic data
structure D.
 Each insertion/deletion produces
a new version of D.
 The entire evolution history of D
is recorded as P.
 Any particular version of D can
be searched by knowing its
version number.
 If the cost of each update is
small, the entire persistent data
structure is small.
Version 2 of D
D(0)
D(1) D(2) D(3)
A persistent data structure P
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Handling Stabbing Queries Using Persistent
Data Structures
isovalue
sweeping line
S1
S2
S3
S4
S5
scalar value
V0
V1
V2
V3
V4 V5 V6 V7 V8 V9
V10
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A Standard Octree




Orange nodes represent active cells (regions).
Gray nodes represent mixed regions.
White (non-active) nodes can be omitted.
Not suitable to be made persistent.
• Why? A single update operation requires significant change of the data
structure.
14
The Compact Octree
 What’s good about the compact octree?
On average, only a CONSTANT number of changes need to be made to the
tree.
 So?
The corresponding persistent octree requires LINEAR space.
15
Update the Compact Octree – The Insertion
Case 2:
Case 1:
insert
insert
Case 3:
insert
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Update the Compact Octree – The Deletion
Case 1:
Case 3:
Case 2:
 An expensive delete operation.
 Fortunately, it does not happen
very often.
 Amortized update complexity:
still O(1).
 Not true for arbitrary
delete/insert operation.
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The Persistent Octree – An 1-D Illustration
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System Setup
 A Linux PC with dual Xeon 3.0 GHz processors (only one
was used).
 8 GB memory.
 150 GB local disk.
 NVidia6800 Ultra GPU card.
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Data Sets
 Downsampled LLNL Richtmyer-Meshkov Instability Data
Set (byte, 1024 x 1024 x 960 x 35 time steps).
 Stanford Bunny (short, 512 x 512 x 360).
 UNC MR Brain (short, 256 x 256 x 109).
 Head Aneurysm (byte 512 x 512 x 512) (Michael Meißner,
Viatronix Inc., USA).
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Storage Cost#
RichtmyerMeshkov
Instability
UNC
MR Brain
Head
Aneurysm
Standford
Bunny
BONO+
144M
1.5M
19M
20M
POT
230M
5.0M
76M
62M
Data Set
Storage Cost
(byte)*
The POT is about 1.6 to 4 times bigger.
# Both BONO and POT are built on 4x4x4 cell blocks.
* The above table only lists the sizes of the indexing structures. The sizes of the
raw data sets are the same for both BONO and POT and thus are omitted here.
+ [Wilhelms and Van Gelder, ACM Trans. on Graphics, 92].
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Performance Comparison of View-independent
Isosurface Extraction
Data Set
RichtmyerMeshkov
Instability
Stanford
Bunny
UNC
MR Brain
Head
Aneurysm
Isovalue
190
1750
1750
55
BONO Index search time 710
(ms)
64
64
22
POT
61
61
15
1.05
1.21
1.47
Index search time 401
(ms)
Speedup of POT index
searching time relative to
BONO
1.77
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Performance Comparison of View-dependent
Isosurface Extraction
Data Set
BONO
POT
Speedup
of POT
relative
to BONO
RichtmyerMashkov
Instability
Stanford
Bunny
UNC
MR Brain
Head
Aneurysm
# of nodes visited
212,992
19,897
7,318
17,260
Execution time
(ms)
10,877
1,130
614
938
# of nodes visited
142,157
6,076
5,368
6,467
Execution time
(ms)
9,511
1,040
602
806
# of nodes visited
1.50
3.27
1.36
2.67
Execution time
1.14
1.09
1.02
1.16
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Other Applications of POT
 Isosurface rendering by Ray-casting.
• Go straight to the intersecting active cell.
 Rendering time-varying data (4-D isosurface)
• A 4-D isosurface can only be visualized by slicing it using a 4-D
hyperplane.
• POT can be used to quickly identify cells that are both active and
relevant (cut by the hyperplane).
 Multi-resolution isosurface rendering
• Triangulate supercells at different resolutions.
• Internal nodes in a POT naturally represent active supercells.
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Conclusions
 Good things about POT.
• Simultaneous pruning in both value and viewing space.
• Fast search time and reasonable storage cost.
• Hierarchical organization of active cells facilitates spatial
search.
 Further improvements.
• Need tests on continuous data types.
• Improve the search time and storage cost.
• How to incorporate this technique into parallel/distributed
isosurface rendering systems.
25
Thank You!
Questions?
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Isosurfacing By Ray Casting
 The basic scheme:
• Shoot a ray from each pixel towards the volume.
• Identify the foremost active cell intersected by the ray.
• Analytically compute the position of the intersection point.
• Shade the intersection point.
 The bottle neck: traversal of non-active cells.
 Using POT:
• No active cells in the octree.
• No need to examine non-active regions.
 An example:
• Standard ray-casting: 9 cells to examine.
• POT: 3 nodes to visit.
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An Example of VD Isosurfacing
A front view of the
MRI Head
The side view
The final coverage mask
28
More Pictures
A top view of
the MRI-brain
data
Cutting plane: X=500
Stanford
bunny
Cutting plane: Z=650
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Practical Issues
 Build POT on 4x4x4 cubes
rather than individual cells.
 Sequentialize the POT (an
acyclic graph) on disk.
 Use hierarchical coverage
mask to speedup visibility
test (Greene
SIGGRAPH’96, Livnat and
Hansen Vis’98).
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