CASC VP 1

Global Static Indexing for Real-time Exploration of Very Large Regular Grids

Valerio Pascucci and Randall J. Frankpascucci,rjfrank@llnl.gov

Lawrence Livermore National Laboratory

CASC VP 2

Outline• Motivation

• Previous work

• Data layout

• 2n tree indexing

• Performance for slicing large grids

• Conclusions and future work

k

I

i j

(bla bla[vis`01], bla bla bla [sc`00] ……)

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We must achieve real-time interaction with large datasets on a wide variety of platforms.

The problem

• Extremely large datasets 0.5TB/timestep (8k,8k,8k,(time)).

• Interactive rendering for real-time data exploration.

• Target platforms: desktop,parallel server, cluster.

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Previous Work• Out-of-core geometric algorithms

[Goodrich, Tsay, Vengroff, Vitter ‘93]

[Vitter ‘00][Matias, Segal, Vitter ‘00]

[Asano, Ranjan, Roos, Welzl ’95][Arge Miltersen ’99]

• Out-of-core visualization[Chiang, Silva ‘97][Sutton, Hansen ‘99]

[Livnat,Shen,Johnson ’96][El-Sana, Chiang’00]

[Bajaj, Pascucci, Thompson, Zhang ‘99]

• Space filling curves(image processing, multidimensional databse, geomtric datastructure …)

[Bandou, Kamata.’99][Balmelli, Kovacevic, Vetterli ’99]

[Parashar, Browne, Edwards, Klimkowski ’97]

[Niedermeier, Reinhardt, sanders ‘97][wise’00]

[Hans Sagan ’94] [Lawder king ’00][Griebel Zumbusch ‘99]

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We apply three fundamental techniques to the visualization of large simulation data.

Our approach

• Multi-resolution geometric representation:– adaptive view-dependent refinement;– minimal geometric output for selected error tolerance.

• Cache oblivious external memory data layouts:– exploit spatial and resolution coherency;– no need for complicated paging techniques.

• Progressive processing: – continuously improved rendering;– scalability with the resources without budgeting.

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We focus on the progressive computation of slices (any orientation) of large 3D rectilinear grids.

• Rectilinear grid• Sub-sampling octtree

• 1D order = hierarchical3D Z-order curve

• Coarse to fine slice refinement

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General Data Layout

Grouping the data by level of resolution

Data coherent Progressive refinement of a hierarchical

geometric data-structure

Grouping the data by geometric proximity

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General Data Layout

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General Data Layout

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General Data Layout

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General Data Layout

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General Data Layout

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General Data Layout

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General Data Layout

nD to 1D mapping:

I → I*

I → l find the level of resolution lCl (pre)compute the number of elements in the levels coarser than l

I → I’ index of the element within its level of resolution

I* = Cl + I’

C0 C1 C2 C3

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We exploit the correlation of bin/quad/oct-trees with the Lebesgue space-filling curves.

The Lebesgue curve is also known as Z-order, Morton, …. Curve.Special case of the general definition introduced by Guiseppe Peano in 1890.

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We turn the recursive definition of the Z-order curve into a hierarchical subsampling scheme.

(a) (b)

(c) (d)

coarse data new level data

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We obtain a multi-resolution hierarchical representation which is not exactly a 2n-tree.

• Not exactly a quad-tree ……

Level 0

Level 1

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We obtain a multi-resolution hierarchical representation which is not exactly a 2n-tree.

• Not exactly a quad-tree ……

Level 0

Level 1

Level 2

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Level 3

We obtain a multi-resolution hierarchical representation which is not exactly a 2n-tree.

• Not exactly a quad-tree ……

Level 0

Level 1

Level 2

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The 1D index I* can be computed in a simple and efficient way in any dimension.

I*

I

1514131211109876543210

1514131110976532112840Z-order 4

3210Z-order 2

0Z-order 0

210

Cl = 22(l-1) _I_22lI’ =I* = Cl + I’

_I_22(l+1)- - 1

I

i j

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The 1D index I* can be computed in a simple and efficient way in any dimension.

I*

I

ki j

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Overall the hierarchical Z-order yields a cache oblivious hierarchical data layout.

Distribution in the grid of each constant size block of data

0 1 2 3 4 5 6 7 8 9

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Theoretical analysis shows a gain of orders of magnitude independently of the block size.

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32*32*32

3-bits shift

1-bit shift

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1-bit shift

32K blocks 64K blocks

Cache oblivious !!!

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Real speedup matches theoretical expectations:more than 10x improvement, platform scalable.

• 2048x2048x1920 dataset (we have run up to 8192x8192x7680)

• 20MB memory cache

• Translation and rotation tests (average over 3 primary axis)250Mhz SGI Onyx (1024x1024)

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500Mhz PIII Laptop (512x512)

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Conclusions and Future Work

• Results: an implicit scheme for coherent spatial, multi-resolution regular grid data access

• Simple address remapping

• Read/write access

• No additional data overhead

• Supports progressive access

• Near term applications

• Volume rendering

• Time critical iso-contouring

• Future work

• “View dependent” parameterization

• Unstructured/temporal hierarchies

• Improved interpolation

• Distributed implementation

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UCRL-VG-143542

This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48.