Isosurface Extraction and Spatial Filtering Using Persistent Octree (POT)

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

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Volumetric Data VisualizationVolumetric 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])

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Isosurface Extraction and RenderingIsosurface Extraction and Rendering

What is isosurface?

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].

Elevation Map: 2D-contours, eachwith a constant elevation

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Basic Steps in Isosurface ComputationBasic 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.

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Identifying Active CellsIdentifying 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 QueryActive 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 QuerySolutions 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).

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Spatial FilteringSpatial 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.

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View-dependent Isosurface RenderingView-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:Problem:How to How to examine the examine the active cells in active cells in a front-to-a front-to-back order?back order?

Problem:Problem:How to How to examine the examine the active cells in active cells in a front-to-a front-to-back order?back order?

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Value-based vs. Location-based PartitioningValue-based vs. Location-based Partitioning

Previous view-dependent algorithms often employ location-based 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 SolutionOur 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.

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Persistent Data StructuresPersistent 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.

A persistent data structure P

Version 2 of D

D(0) D(1) D(2) D(3)

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Handling Stabbing Queries Using Persistent Data StructuresHandling Stabbing Queries Using Persistent Data Structures

scalar value

sweeping line

V1V0 V5V4 V7V6 V9V8 V10V3V2

S1

S3

S2

S4

S5

isovalue

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A Standard OctreeA 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.

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The Compact OctreeThe 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.

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Update the Compact Octree – The InsertionUpdate the Compact Octree – The Insertion

insert

insert

insert

Case 1: Case 2:

Case 3:

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Update the Compact Octree – The DeletionUpdate the Compact Octree – The Deletion

Case 2:

Case 3:

An expensive delete operation. Fortunately, it does not happen

very often. Amortized update complexity:

still O(1). Not true for arbitrary

delete/insert operation.

Case 1:

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The Persistent Octree – An 1-D IllustrationThe Persistent Octree – An 1-D Illustration

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System SetupSystem 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 SetsData 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#Storage Cost#

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].

Data SetRichtmyer-Meshkov Instability

UNC

MR BrainHead Aneurysm

Standford Bunny

Storage Cost (byte)*

BONO+ 144M 1.5M 19M 20M

POT 230M 5.0M 76M 62M

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Performance Comparison of View-independent Isosurface ExtractionPerformance Comparison of View-independent Isosurface Extraction

Data Set Richtmyer-Meshkov Instability

Stanford Bunny

UNC

MR Brain

Head Aneurysm

Isovalue 190 1750 1750 55

BONO Index search time (ms)

710 64 64 22

POT Index search time (ms)

401 61 61 15

Speedup of POT index searching time relative to BONO

1.77 1.05 1.21 1.47

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Performance Comparison of View-dependent Isosurface ExtractionPerformance Comparison of View-dependent Isosurface Extraction

Data Set Richtmyer-Mashkov Instability

Stanford Bunny

UNC

MR Brain

Head Aneurysm

BONO # of nodes visited 212,992 19,897 7,318 17,260

Execution time (ms)

10,877 1,130 614 938

POT # of nodes visited 142,157 6,076 5,368 6,467

Execution time (ms)

9,511 1,040 602 806

Speedup of POT relative to BONO

# 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 POTOther 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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ConclusionsConclusions

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.

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Thank You!Thank You!

Questions?

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Isosurfacing By Ray CastingIsosurfacing 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 IsosurfacingAn Example of VD Isosurfacing

A front view of the MRI Head

The side view The final coverage mask

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More PicturesMore Pictures

A top view ofthe MRI-braindata

Stanfordbunny

Cutting plane: X=500 Cutting plane: Z=650

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Practical IssuesPractical 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).