Physics of the Earth and Planetary Interiors 163 (2007) 292–298

Visualization of wavelet compressed mantle convection data

Benjamin J. Kadlec a,∗, Daniel E. Goldstein b,David A. Yuen c, Alexei Vezolainen d

a Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309, United Statesb NorthWest Research Associates, Inc., Colorado Research Associates Division, Boulder, CO 80301, United States

c Department of Geology and Geophysics and Minnesota Supercomputing Institute,University of Minnesota, Minneapolis, MN 55455, United States

d Department of Mechanical Engineering, University of Colorado at Boulder, Boulder, CO 80309, United States

Received 5 February 2007; accepted 20 March 2007

Abstract

Wavelet compression methodologies promise significant computational and storage cost savings through the benefit of producingresults on adapted grids that significantly reduce storage and data manipulation costs. Yet, with these storage methodologies comenew challenges in the visualization of temporally adaptive datasets. Wavelet compressed three-dimensional mantle convectionsimulations are visualized as a challenging case study. We employ adaptive 3D visualization using cache friendly octal tree volumevisualization as an effective visualization approach. To explore the localized multi-scale structures in the datasets, the wavelet

coefficients are also visualized allowing visualization of energy contained in local physical regions as well as in local wavenumberspace. Our technique allows for exploring localized multi-scale structures using multi-resolution visualization from the compressedstate of large geophysical simulations.© 2007 Elsevier B.V. All rights reserved.

isualiza

Keywords: 3D visualization; Wavelets; Mantle convection; Volume v

1. Introduction

The race towards petascale computing has createdan explosion in data from large-scale computationsof nonlinear phenomena in many fields, includingthe geo- and environmental sciences. Today, the geo-sciences are facing an onslaught of information coming

from terabyte-scale numerical simulations of nonlinearphenomena and higher resolution data from satellitemissions and precise laboratory experiments. In addi-

∗ Corresponding author.E-mail addresses: kadlec@msi.umn.edu (B.J. Kadlec),

dan@cora.nwra.com (D.E. Goldstein), daveyuen@gmail.com(D.A. Yuen), alexei.vezolainen@colorado.edu (A. Vezolainen).

0031-9201/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.pepi.2007.03.009

tion

tion, great torrents of data are being produced by newearthquake models coupled with sensing information,which can easily overwhelm researchers due to theirhigh dynamic range and multi-scale character in spaceand time. As scientists in the geo and environmentaldisciplines continue their march towards petascale com-puting, the cost of large-scale data analysis will be a direproblem requiring novel tools to process and representdata more efficiently. The efficient storage and subse-quent visualization of these large datasets is a trade offin storage costs versus data quality.

The last decade and a half has witnessed the devel-opment of wavelets or wavelet analysis. Wavelets have awide range of application in areas such as feature extrac-tion, data compression, numerical simulation (Goldstein

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t al., 2005; Goldstein and Vasilyev, 2004; Schneider etl., 2005), and visualization (Mallat, 1999). Waveletsre an effective tool for efficiently representing dataith localized multi-scale structures due to the inher-

nt localized support of the wavelet basis functions inoth physical and wavenumber space. This dual-spaceocalization property of wavelets allows for waveletompressed datasets to represent localized multi-scaletructures with a high level of compression and minimaloss of important features.

The ability of wavelets to efficiently represent dataontaining localized multi-scale structure makes them andeal tool for the simulation of geophysical systems andhe subsequent storage and visualization of simulationnd experimental results (Yuen et al., 2004; Vecsey et al.,003; Vezolainen et al., 2005) In this work our emphasisill be on the issues and possible dramatic advantages toe gained by representing existing geo-physical datasetsn a wavelet compressed form for storage and visualiza-ion.

This paper is organized as follows: we will firstescribe the basic properties of second generationavelets. Then we will discuss the de-noising properties

hrough wavelet thresholding, which allow decomposi-ion of localized multi-scale coherent structures fromn incoherent background. Next we will motivate theeed for visualization in our work and describe the visu-lization techniques we derive. Following this we willresent, as an example case, the visualization of a man-

le convection simulation dataset at different levels ofavelet compression. We will conclude with some dis-

ussion of future improvements that are being studiedor the visualization of wavelet compressed datasets.

Fig. 1. Lifted interpolating wavelet ψ, of ord

anetary Interiors 163 (2007) 292–298 293

2. Wavelets

Wavelets are basis functions which are localized inboth physical space (due to their finite support) andwavenumber space (due to their vanishing moments),e.g. Fig. 1. For comparison, the classical Fourier trans-form is based on functions (sines and cosines) that arewell localized in wavenumber space, but do not providelocalization in physical space due to their global sup-port. Because of this space/scale localization, the wavelettransform provides both spatial and scale (frequency)information while the Fourier transform on the otherhand only provides frequency information.

A scalar field f(x) can be represented in terms ofwavelet basis functions as

f (x) =∑

l∈L0

c0l φ

0l (x) +

+∞∑

j=0

2n−1∑

μ=1

∑

k ∈ κμ,jdμ,jk ψ

μ,jk (x), (1)

where φ0l (x) and ψμ,jk are respectively n-dimensional

scaling functions and wavelets of different families (μ)and levels of resolution (j). One may think of a waveletdecomposition as a multi-level or multi-resolution rep-resentation of a function, where each level of resolutionj (except the coarsest one) consists of wavelets ψjk or

family of wavelets ψμ,jk having the same scale butlocated at different positions. Scaling function coeffi-cients represent the averaged values of the field, whilethe wavelet coefficients represent the details of the field

at different scales. The wavelet functions have a zeromean, while the scaling functions do not. Note that inn-dimensions there are 2n − 1 distinctive n-dimensionalwavelets (Daubechies, 1992). Also note that due to the

er 6 (a) and its Fourier transform (b).

and Pl

294 B.J. Kadlec et al. / Physics of the Earth

local support of both scaling functions and wavelets,there is a one-to-one correspondence between the loca-tion of each scaling function or wavelet with a grid point.As a result each scaling function coefficient c0

l and each

wavelet coefficient dμ,jk is uniquely associated with asingle grid point with the indices l and k, respectively.

Traditionally, one-dimensional first generationwavelets ψjk are defined as translates and dilates of

one basic wavelet ψ, i.e. ψjk(x) = ψ(2jx− k). Secondgeneration wavelets (Sweldens, 1996, 1998) are ageneralization of first generation wavelets that suppliesadditional freedom to deal with arbitrary boundaryconditions, and irregular sampling intervals. Secondgeneration wavelets form a Riesz basis for L2 space, withthe wavelets being local in both space and frequencyand often having many vanishing polynomial moments,but without the translation and dilation invarianceof their first generation cousins. Despite the loss ofthese two fundamental properties of wavelet bases,second generation wavelets retain many of the usefulfeatures of first generation wavelets, including a fastO(N) transform. The construction of second generationwavelets is based on the lifting scheme that is discussedin detail by Sweldens (1996, 1998).

For this study we use a set of second generationwavelets known in the literature as lifted interpolat-ing wavelets (Vasilyev and Bowman, 2000; Sweldens,1996). In particular a lifted interpolating wavelet oforder 6, which is shown in Fig. 1 along with its Fouriertransform is used in this work. For a more in-depth dis-cussion on the construction of these wavelets the readeris referred to the papers by Sweldens (1996, 1998), andVasilyev and Bowman (2000). For a more general dis-cussion on wavelets we refer the reader to the booksof Daubechies (1992), Mallat (1999), and Foufoula-Georgiou and Kumar (1994).

2.1. Wavelet filters

Wavelet filtering is performed in wavelet space usingwavelet coefficient thresholding, which can be consid-ered as a nonlinear filter that depends on each flowrealization. The wavelet thresholding filter is definedby

f̄ >ε(x) =∑

l∈L0

c0l φ

0l (x)

++∞∑

j=0

2n−1∑

μ=1

∑

k ∈ κμ,j|dμ,jk

|>∈ ||f ||WTF

dμ,jk ψ

μ,jk (x), (2)

anetary Interiors 163 (2007) 292–298

where f(x) is a scalar field, ε> 0 stands for the non-dimensional (relative) threshold parameter, and ||·||WTFbeing the Wavelet Threshold Filtering (WTF) norm thatprovides the (absolute) dimensional scaling for filteredvariable f. For instance, in the case of velocity, the (abso-lute) dimensional scaling can be specified as the L2 norm(||ui||WTF = ||ui||2) or the L∞ norm (||ui||WTF = ||ui||∞).Note that once the WTF-norm ||·||WTF is specified, thewavelet thresholding filter (2) is uniquely defined by thenon-dimensional threshold parameter, ε.

The reconstruction error due to wavelet filtering withnon-dimensional threshold parameter ε can be shown tobe (Vasilyev, 2003; Donoho, 1992):

||f (x) − f̄ >ε(x)||2 ≤ Cε||f ||WTF, (3)

for a sufficiently smooth function f(x), where C is oforder unity.

2.2. Wavelet compression and wavelet de-noising

The major strength of wavelet filtering decomposi-tion (2), is the ability to compress signals. For functionsthat contain isolated small scales on a large-scale back-ground, most wavelet coefficients are small, thus, we canretain good approximation even after discarding a largenumber of wavelets with small coefficients. Intuitively,the coefficient dμ,jk will be small unless u(x) has varia-tion on the scale of j in the immediate vicinity of waveletψμ,jk (x).Another important property of wavelet analysis used

in this work is the ability of wavelets to de-noise signals.The wavelet de-noising procedure, also called wavelet-shrinkage, was introduced by Donoho (1993, 1994)based on orthogonal wavelet decompositions. It can bedescribed as follows: given a function that consists ofa smooth function with superimposed noise, one per-forms a forward wavelet transform and sets to zero“noisy” wavelet coefficients (i.e. those wavelet coeffi-cients whose modulus squared is less than the noisevariance σ2), otherwise the wavelet coefficient is kept.This procedure is known as hard thresholding. Donoho(1993) demonstrated that hard thresholding is optimalfor de-noising signals in the presence of Gaussian whitenoise. In the case of the plume data discussed in thiswork the “noise” is actually the areas of low grade,less coherent, temperature between the coherent plumeregions.

3. Need for localized multi-scale visualization

The multitude of data being generated from high-resolution simulations of multi-scale phenomena, such

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s convection in the Earth’s mantle, requires novelechniques for efficiently analyzing numerical results.nteractive visualization is the most feasible and nat-ral tool for allowing human observation to extractcientific insight from such complex data. Advances inigh-performance graphics processing and large high-esolution LCD displays have allowed interactive 3Disualization to be a realizable dream that allows giga-ized datasets to be investigated on commodity graphicsorkstations.As the range of scales and depth of resolution used

ithin simulations become more vast and precise, newnnovations must be developed to maintain and increasehe interaction available with many of today’s simula-ions. Harnessing the multi-scale capabilities of waveletnalysis and coupling it with direct visualization is aovel solution to this problem. Wavelet analysis allowsor the compression of datasets while retaining theulti-scale physical features of the dataset. In order

or this to be visually realized, proper choice of visu-lization techniques and parameters must be made. Inhe case of our mantle convection data, we employ vol-me visualization to capture the structure and behaviorf plume upwellings and downwellings. Representinghese plumes as they were intended to be modeledequires data that includes connectivity between pointsnd appropriate opacity-values that enhance plumeeatures. In addition, three-dimensional interaction isecessary in order to determine the best visual represen-ation of a dataset by looking at different perspectives,hile recognizing key physical features Fig. 2.The powerful result of our work is being able to

isualize large three-dimensional datasets with only aompressed subset of the full volume, allowing for larger

atasets to be interactively explored. As the size ofatasets continue to increase faster than the rate of mem-ry and storage technology, this research will be timelyo everyone computing volumes of data on the order of

ig. 2. Volume rendered mantle convection simulation with Rayleighumber 10ˆ8 shows the localized multi-scale plume structures to beavelet compressed.

anetary Interiors 163 (2007) 292–298 295

terabytes and larger. The end goal is allowing researchersto conduct scientific visualization with a highly reducednumber of bytes that are still able to represent the keyphysical information in a dataset.

4. Visualization technique

In this work we use direct wavelet space visualiza-tion and unstructured volume visualization in real spaceto study mantle convection data. This unit problem rep-resents the complex visualization and data storage andmanagement issues seen with larger more complex geo-physical datasets from experiment and simulation. Directwavelet space visualization is a technique for low costqualitative data analysis. The unstructured volume visu-alization technique used falls between direct waveletspace visualization and structured volume visualizationtechniques in computation and memory cost, yet thequality of this technique is on par with full structuredvolume visualization techniques.

The direct visualization of wavelet coefficient mag-nitudes allows for a quick low cost view of localizedfeatures in a dataset. By interactively thresholding therange of wavelet coefficient magnitudes from below, asmall subset of the data can be used to highlight thelocation and general physical character of data features.This can be done easily by representing point centereddata as spherical glyphs scaled and colored by waveletcoefficient magnitude. The cost of rendering glyphs issignificantly lower than volume visualization allowingfor exploration of even the largest datasets at interactivespeeds for physically significant features. This techniquecan be used to determine areas of interest that can thenbe explored further with more costly visualization tech-niques.

Volume visualization is an ideal technique for cap-turing structure and behavior of physical features in adataset. Direct volume visualization of wavelet com-pressed datasets is a way to significantly reduce thecost of volume visualization in comparison with vol-ume visualization on a full regular mesh. We note that,although state of the art regular mesh volume visualiza-tion algorithms are faster on a per cell basis, significantcomputational saving, in terms of the full data domain,can be realized with the unstructured volume visualiza-tion applied to the wavelet compressed data due to thesignificant levels of compression realized.

In this work we use an octal tree-based ray-tracing

volume rendering technique developed at the Multi-Scale Modeling and Simulation Laboratory at theUniversity of Colorado-Boulder. First an octal tree-basedcell structure must be created from wavelet compressed

and Planetary Interiors 163 (2007) 292–298

Table 12563 Mantle convection wavelet compression

Compression (%) Points Epsilon

0.00 16,777,216 0.0000016.90 13,939,031 0.0005066.20 5,667,187 0.0050094.96 845,846 0.05000

296 B.J. Kadlec et al. / Physics of the Earth

points that lie on a subset of the original data’s regu-lar mesh. This cell creation is done recursively basedon wavelet level. The multi-level nature of the waveletcoefficients allows for a multi-resolution algorithm thatis initiated on the scaling function points that are allretained in the wavelet thresholding algorithm. Thepoints associated with increasing wavelet level are theniterated on at increasing level to create an efficientmesh representation. The technique is implemented in acache-friendly fashion such that requests are minimizedfor information stored in main memory outside of thecache.

5. Results

Mantle convection simulation data at 2563 with con-stant viscosity and Rayleigh number of 108 has beenwavelet filtered at increasing threshold (ε) values result-ing in dataset wavelet compression rates from 0% to99.27%. Table 1 displays statistics on the range of epsilonvalues in terms of significant wavelet coefficients, num-ber of points used in rendering, and fraction of data used.We consider the significant wavelet coefficients those

that were above the threshold value thus were retained.

Fig. 3 shows the wavelet space coefficients of thewavelet transformed mantle convection simulation data.The spherical glyphs are scaled and colored by coef-

Fig. 3. Wavelet coefficients from the mantle convection simulation plotted acoefficient magnitude.

98.30 284,642 0.1000099.27 121,955 0.20000

ficient magnitude. In this figure the scaling functioncoefficients, that represent the mean of the dataset, arenot shown. Only wavelet coefficients with magnitudegreater than 0.05 are shown to highlight the significantplume structures. This is also a clear visual representa-tion of the collocation of the retained data points afterwavelet threshold filtering and areas of localized struc-tures, in this case plumes, in the dataset.

Fig. 4 shows three-dimensional volume visualiza-tion directly on unstructured meshes generated from thewavelet compressed datasets. Three-dimensional visual-ization was completed using ray-tracing with 5002 rays

and an average of 170 points per ray. We show renderingresults at six compressions (0%, 16.9%, 66.2%, 94.96%,98.3%, and 99.27%). Threshold values of approximately0.05 (94.96% compression) and lower generate render-

s spherical glyphs that are scaled and colored according to wavelet

B.J. Kadlec et al. / Physics of the Earth and Planetary Interiors 163 (2007) 292–298 297

%, 98.3

i(hm

Fig. 4. Mantle convection simulation at 0%, 16.9%, 66.2%, 94.96

ng results that appear unchanged from the raw datatop four images). The lower two images show that atigher wavelet compression rates critical feature infor-ation is lost. The middle right plot corresponds to the

%, and 99.27% wavelet compression (left to right, top to bottom).

wavelet coefficients shown in Fig. 3. We can see from thisthat visualizing the wavelet coefficients themselves high-lights the regions of plume activity accurately. Clearlythe direct volume visualization does a better job of cap-

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298 B.J. Kadlec et al. / Physics of the Earth

turing the structure and behavior of the plume upwellingsand downwellings.

6. Conclusion

Mantle convection data has been visualized bothin wavelet space and real space in a significantlywavelet compressed form and is compared to regularuncompressed visualization. Wavelet coefficient visu-alizations have been shown that highlight the jointspace/wavenumber representation of the wavelet coef-ficients. The significant plume structures in the datasetare clearly seen in the low overhead wavelet coefficientplots (Fig. 3). Direct volume rendering on the waveletcompressed datasets has been shown to compare wellwith the full dataset volume visualization up to a com-pression of 94.96% (Fig. 4). The techniques developedin this and ongoing work along with dynamically adap-tive wavelet simulation techniques being developed inrelated work, will allow for simulation on a supercom-puter, transfer of results to a visualization workstationand subsequent visualization to be all done at a fractionof the cost of using un-adapted Cartesian or zonal meshrepresentations.

Future work is to develop more specialized algorithmsfor cell generation from adaptive wavelet collocationdatasets and volume rendering techniques based onhigher order interpolations using pre-calculated deriva-tives. These will allow for increased fidelity volumerendering on the wavelet collocation mesh. This is ofhighest importance in regions where the wavelet collo-cation points are extremely sparse causing a ghostingeffect when cell-based linear interpolation methods areused.

Acknowledgements

We would like to thank Fabien W. Dubuffet for hismantle convection solution and Oleg Vasilyev and Gor-don Erlebacher for stimulating discussions on waveletsand visualization.

anetary Interiors 163 (2007) 292–298

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