Wavelet-based Multiresolutional Representation of Computational Field Simulation Datasets

Zhifan Zhu3v4, Raghu Machiraju1*3, Bryan Fry213, Robert Moorhead

‘Department of Computer Science

2Department of Electrical and Computer Engineering Mississippi State University, Mississippi

Abstract ‘Ibis paper addresses multiresolutional representation of

datasets arising from a computational field simulation. The approach determines the regions of interest, breaks the volume into variable size blocks to localize the information, and then codes each block using a wavelet transform. The blocks are then ranked by visual information content so that the most informative wavelet coefficients can be embedded in a bitstream for progressive trans- mission or access. The technique is demonstrated on a widely-used Computational Field Simulation dataset.

CR Categories and Subject Descriptors: 1.3.2 [Computer Graph- ics]: Graphics Systems; I.3.8 [Computer Graphics]: Applications; 1.4.2 [Image Processing]: Compression (Coding).

Additional Keywords: wavelet transform, structure detection, human visual system, progressive transmission

1 INTRODUCTION Given the advances in hardware and tbe proliferation of new

numerical and data analysis techniques, the solution grids from computational simulations are continuously becoming larger. For instance, even 100 time steps using a grid size of 69x43~15 from a computational field simulation (CFS) requires the storage of the computational grid, a vector field (velocity) and two scalar fields (density, energy) all in floating point format making the storage requirement 136 Mbytes. Therefore, it is useful for the datasets to be stored in compressed format. Both lossy and lossless schemes have been used to store datasets. The efficiency of any of these compression schemes is measured in terms of the number of bits assigned per element of the dataset. For computational field simu- lations it is important that any representation be-inherently lossless. However, it is also imperative that the compression scheme allow for selective display of important structures in a dataset.

Similarly, the size of the datasets becomes a burden when they are transmitted over usually a slow network. It is not uncommon that a domain for a simulation is spatially divided into several zones and executed on a multicomputer or a cluster of networked processors, Each zone has its own geometric grid and a designated processor, or solver. Visualizing the solution data from the solvers in real-time or near-real-time provides instantaneous feedback and therefore allows tracking of the simulation. Such systems are becoming increasingly available [8][18]. In this case, visualization is done in a distributed manner, i.e., the solution data are trans- ferred from the solvers to a graphics workstation over the network and selectively visualized. The advantages of progressive transmis- sion schemes especially those afforded by multiresolutional meth-

3 NSF Engineering Research Center for Computational Field Simulation, I?O.Box 9621, Mississippi State University, MS 39762, (zhu,mghu,bfry,jm) @erc.msstate.edu

O-61 66-8262-0197 $10.00 Copyright 1997 IEEE.

3NSF Eng. Res. Center for Computational Field Simulation 4Center F0r.Ai.r Sea Technology

Mississippi State University

ods are well-known. A visualization is first created at a low resolution and progressively updated. Thus, the solvers constitute a continuous multi-channel, multi-resolution data source which feeds the visualizer.

A compression and progressive transmission scheme for visu- alization should maximize the information presented while mini- mizing the data rate. This problem can be explained by using an information-rate curve. If the information in a data volume is uni- formly distributed over the data domain and the bit budget is uni- formly allocated for each point, this curve will be linear. In most cases, however, the information contained in a dataset is non-uni- formly distributed, resulting in a non-linear curve. As a result some regions of the volume face bit starvation and are not adequately represented: The desired information-rate curve should have a sharp rise as close to the zero-rate as possible, so that most of the data information could be visualized as early as possible. Also, to achieve an effective progressive transmission scheme it is impor- tant that an embedded scheme be used, where the information is ranked by significance and transmitted accordingly over the chan- nel [28]. Moreover, CFS datasets are generally very sparse, since tbe computational experiment attempts to study and focus on very specific phenomenon and sometimes at very specific locations in the domain. Most of the information is packed in a very small por- tion of the entire volume. Thus, an embedded scheme wouId be ideal for visualizing CFS datasets unlike medical data sets which are more dense.

In this paper, we present a wavelet-based muhiresolutional data representation scheme that has the following salient features:

l allows for lossless compression of floating point data

l facilitates seIective and embedded display or transmission, allowing the user to stop transmission at any desired time.

l does not require that the resolution of the source volumes be dyadic (power-of- two).

The third feature is essential for CFS datasets which come in a variety of shapes and sizes, often dictated by the underlying geom- etry and the desired local resolution of structures. Many multireso- lutional schemes impose power-of-2 restrictions on the resolution. A commonly used solution is padding the dataset until power-of- fwo resolutions are obtained. This increases the quantity of data that must be transmitted, sometimes significantly.

Our compression scheme is two-pass in nature. In the first pass, a spatial subdivision is obtained which delineates regions of high spatial frequency. The subdivision is obtained by creating a correla on mask from the dyadic wavelet transform. This step may require the padding of the of the volume. An octree embedding of this mask provides the required subdivision. In the second phase each block of the partition is coded separately using an appropriate function from a wavelet library. It is noteworthy that this step does not require any padding of the original volume. Later the wavelet

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coefficients in each of the blocks are sorted and packed based on the presence of significant structures. The visual content associated with coefficients within each block is estimated by employing a computational model of the Human visual System (HVS). The final result is an embedded encoding of the volume. Thus, our method draws from results in information theory, image coding/ analysis, and low-level vision models. Our focus is not on the development of full-fledged compression scheme including quanti- zation of coefficients. Rather, we develop a multiresolution repre sentation scheme which can be very easily extended to lossy or lossless coding. Other papers [S][30] present the details of a plausi- ble complete coding scheme.

In Section 2 we motivate the need for a variable block sized multiresolution representation and review past reported methods in volume compression. In Section 3 we present our scheme for mul- tiresolution volume representation and in Section 4 we present our embedded progressive transmission scheme. Results are presented in Section 5 and in Section 6 we draw conclusions and discuss future work

2 WAVELET-BASEDMULTIRESOLUTION REPRESEN-TATION

In this section we first describe transform coding methods especially those based on wavelets. Specifically, we describe the first and second generation coding methods and draw adequate dis- tinctions between them. Later, reported work including methods besides wavelets will be described briefly. In this paper we use the words signal, image, and volume synonymously.

*artsform Coding - Wavelet based compression methods belong to the class of Transform Coding (TC) methods. In these methods the image or volume is projected to a transformed space spanned by adequate basis functions. Thus, the original underlying

signal can be written as f = ~ciBi(n), where the BiS are the

basis functions and the CiS are the coefficients obtained as a result of the inner product of the signal with the basis functions. The most popular TC method is the discrete cosine transform (DCI). In the case of the DCT the basis functions are cosines of infinite extent. The goal of any TC method including DCT is to decorre- late the signal in the projected space so that a few coefficients can adequately represent the signal. Compression can be obtained by retaining only a portion of the coefficients. The size of the retained portion is dictated by the desired quality of tbe resulting image. Although efficient, the DCT does not succinctly represent percep- tually important structures like comers and edges. Many non-zero coefficients are needed to represent edges with high fidelity. There exist multiresolution methods (Gaussian, Laplacian, etc.) which are implemented as pyramid transforms. These methods again do not fare well in compressing edges. However, wavelet transforms have the potential to perform this task well.

Wavelet Pansform - The wavelet transform has gained much popularity due to some very attractive properties including compact support of the basis functions and the ability to represent functions of arbitrary smoothness [5]. A wavelet transform [21] projects the signal into a ortbogonal/biorthogonal basis space spanned by functions called wavelets. Wavelets offer useful time- frequency localization and efficient multiresolution hierarchies. The original image is split into several subimages of smaller reso- lutions guided by a dyadic subdivision of the frequency spectrum (see Figure 3a). Thus at each level of the transform four subimages (LL, HL, LH, HH; d-low, H=high)-amobtained, each spanning either the high or low portions of the 2D spectrum. Similarly, for

3D volumes eight subvolumes result at each level of the transform, In addition to signal de-correlation, the wavelet transform provides a natural multiresolutional data representation. The original signal can be reconstructed from coarse to tine resolutions by using incre- mentally more wavelet coefficients.

In [13][22][23][35] techniques in compressed rendering in the wavelet domain are reported. The essential coefficients based on magnitude are retained. It has been shown in 171 that an image can be suitably represented by the first N coefficients within a certain error tolerance measured in spaces spanned by smooth functions, These results although significant do not easily adapt to practical use. Methods which retain only N coefficients are often classiflcd as first generation methods. Although, much progress has been made in the field of coding, general purpose techniques are still lacking [9].

Much information in an image is concentrated in a few regions. These are dominated by structures characterized by dis- continuities in intensity and manifest as edges (images) and bound- aries (volumes). As pointed out in [9], second generation methods are driven by structures in an image. Moreover, it is essential that a human observer or a model of tbe HVS be included in the coding scheme. Edge detection and representation with edges have been used in [22]. Maximas are searched for in a multi-dimensional scale space created by a wavelet transform resulting in the genera- tion of multiscale edges. However, it is often the case that texture is not preserved. Also, search-based approaches for identifying edges are cumbersome especially for 3D volumes. Finally, edge based methods suffer from the disadvantage of being error prone in the presence of noise. General structure-extraction coding methods exist and they attempt to segment an image into edge and texture regions. The edge image is coded with high-accuracy and the tcx- mre is coded with very low bit rate. Although satisfactory, the com- pression ratios obtained are not significantly large. Higher order methods which attempt to capture the correlation of the signal across scales are still needed. A significant method, wavelet prob- ing, was presented in [4] to segment the image into smooth regions which are coded very efficiently. The identified edge regions arc then coded separately by employing the same technique. Although the method may yield high coding efficiency, it is not suitable for embedded coding, since the coefficients are not assigned edgc- based significance.

Since ,the wavelet transform is performed using a single mother wavelet, the transform is essentially signal independent. To adapt to the statistical properties of the data, wavelet packet tech- niques [3] decompose the signal by choosing from a library of wavelet packets the wavelets that best represent the signal. Esscn- tially a KL-transform is performed on the image which is inhcr- ently expensive. Several extensions and improvements have been proposed to the basic algorithm which attempt to expedite the pro- cedure of finding the best basis [25]. However, the computational expense is still staggering. An aspect of the wavelet packet trans- form however is amenable for use. Spatial subdivision of the image is conducted and normally a quad-/octree is used in conjunction with the subdivision of the frequency bands. Each block ensuing from the spatial subdivision can be coded with a different wavelet basis, the choice being dictated by a variety of reasons including measures of function smoothness.

Spatial subdivision has been used in compression techniques other than TC methods. For instance, in [31] a quadtree was used to first segment the image. Later each block was subjected to an application of vector quantization (VQ) algorithm. The spatial sub- division can also be conducted in the wavelet domain with the

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Volume

Dyadic WT Structure Detection

Octree ---) Subdivision

w-r of Each Block Wavelet

Coefl icients

Multiresolution Pyramid

Faglation go$k;le-Sized

Figure 1. Schematic Diagram of Coding Algorithm. The volume is subjected to a wavelet transform (WT). A correlation mask is found which is then subdivided by an octree. The ensuing blocks are coded again.

I

intent of locating significant regions. In [26] a zerotree wavelet structure is created from the wavelet space. This allows for a grouping of wavelet coefficients of similar significance in a hierar- chical fashion and facilitates an embedded representation that is well suited for progressive transmission. In [ 161 this technique was expanded to include a computational model of the HVS.

Our method uses a spatial subdivision method in the wavelet domain. We also allow embedded coding and employ a method developed in [20][36] to locate the significant structures of a vol- ume. Then, the spatial domain is decomposed into blocks, with each block being coded independently. Finally, an ordering is done in a linear fashion among all the blocks. We describe our method in greater detail in Section 3 and Section 4. In the following section we brielly review previous work in volume coding.

ing methods the DCT, wavelet, Gaussian and Laplacian image pyr- amids have found common use. However. the work reuorted in both [I21 and [37] emphasizes less the compression a&t and more the rendering of images from the compressed domain. There exist other methods which have been borrowed from image coding and are suitable for hardware implementation [17]. Other wavelet methods reported in [8][28][30] belong to the first generation class; the compression schemes used therein do not adapt to structures in the datasets. Rather, they apply standard and computationally via- ble techniques for compression and transmission.

2.1 Related Work

Previous volume compression methods include predictive methods [IO], fractal methods [2], vector quantization [24]. DCT (371, wavelets ([13][23][35]). and gaussian pyramids (121. The predictive method (followed by Huffman coding) reported in [IO] provided sufficient improvement over the standard UNIX utilities like gzip, but did not provide a high level of compression. Also, it was not amenable to embedded coding for progressive transmis- sion. Ning and Hesselink [24] used VQ to compress volume data and achieved a high level of compression. However, the method is not again very amenable to progressive transmission given the asymmetric nature of the coder and decoder. The coding effort is generally orders of magnitude higher than expended in the decod- ing component. The fractal technique of [2] is expensive and is not amenable to progressive transmission. Among the transform cod-

3 A VARIABLE-BLOCKSIZE VOLUME COMPRESSION ALGORITHM

The volume representation scheme reported herein has the four components shown in Figure 1. The block representation allows for the ranking and measurement of important information in a signal. The progressive transmission scheme requires another component to order the blocks (Section 4). The first three compo- nents facilitate the detection of significant structures while the fourth component achieves the actual coding. We now describe each component in detail.

Wavelet ‘ikansforrns - We use the orthogonal wavelet trans- form to obtain the mask and the biorthogonal wavelet transform to code the image in each block. The Daubechies filter of length 8 was used for the first task while the second task employed second order spline wavelets. The chosen length of 8 for the Daubechies filter suffices since it can represent fourth degree polynomial func- tions accurately which can suitably describe all practical images and volumes. In [32] a study of different biorthogonal wavelets for coding efficiency is conducted. Our choice of the second wavelet

’

Figure 2. Subdivision of a slice of CFS data. The significant regions-of-interest are marked with arrows. (a) Mask driven quadtree decomposition of original slice. (b) Mask. The mask has the highest value in the marked regions, which are regions of significant structures. The mask is shown as a grayscale image. with brighter values indicating higher mask values.

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ias dictated by that study. We chose the biorthogonal filter of sec- ond order since it provides adequate results.

The reported method should work for all images with any res- olution. However, its functioning -depends on the partition of the 1 image into regions which are then coded individually. The parti- . tioning is achieved through a dyadic wavelet transform. Padding is required for resolutions which are not powers-of-two. Once the partition is obtained the additional regions are no longer consid- ered and only the original volume is considered. In either case, the dyadic wavelet transform is conducted through the pyramid scheme of Mallat [21] by the repeated application of a pair of suit- able low- and bandpass filters. More details on the wavelet trans- form can be found in much reported work including [.5][21]. It noti remains to identify the coefficients which contribute to structures in an image,

Structure Detection - We use a wavelet based analysis algo- rithm that locates significant structures which persist a&oss die scales of the wavelet decomposition. As a result of the induced subdivision of the frequency spectrum of the image, the image is now decomposed into a pyramid consisting of bandpass images. A denoising stage follows which allows for thresholding of the coef- ficients at separate levels based on a noise model, e.g., Gaussian [6]. The subband images are then combined through a process bf multiplication to yield a correlation mask of the same size as the image. If a redundant transform is used then no interpolation of the subband images is needed, however, for the orthogonal wavelet an interpolation step is required. If the mask at any location is set to zero, it implies that no scale coherent structures persist at that point. Wherever the mask is set to zero, the corresponding coeffi- cients in the pyramid are set to zero. An image and the associated mask are shown in Figure 2. As a result of this structure-based thresholding operation only the coefficients contributing to coher- ent features remain. The mask generation can be iterative allowing for more coefficients to be neglected. The 3D subband combining algorithm is similar to the ones presented in [20][27][36]. The algorithm in [27] uses the sum of binary thresholded wavelet coef- ficients. However, the algorithms in both [20] and [36] attempt to detect correlated structures across the scales by employing the product of the wavelet transform. It can be easily shown that the product is more effective since it arises naturally from the shape of the wavelet transform which is in the form of a cone [14]. In [36] the emphasis is on denoising and detecting edges, while in [20] the algorithm identifies structures of arbitrary singularity &d assigns pixel-wise saliency values in regions populated by singular struc- tures. The correlation mask has been used to enhance structures in images [20] and towards the development of image comparison metrics [ 111. We are exploring its use in volume compression in this paper.

Octree Subdivision of the Mask - The spatial subdivision allows the identification of coherent and similar regions in the image. Several options exist including region growing and recur- sive spatial decomposition methods. Region growing algorithms although useful are not very amenable to user control and can result in blocks of odd shape and size and hence we resort to recur- sive spatial subdivision methods based on quad-/octrees. The level of the tree is controlled by the user thus producing square/cubic blocks. Otherwise, a crude homogeneity measure can be employed to control the subdivision process. We show the spatial subdivision obtained using a slice of the data in Figure 2a. There will be some blocks which do not carry any information; they arise in the prox- imity of the discontinuities. In the following section we describe how the wavelet coefficients in thb blocks can be packed for pro- gressive transmission.

4 PROGRESSIVE TRANSMISSION

In this section we describe the packing of data for progrcssivo transmission. The blocks obtained from the spatial subdivision of the mask are sorted based on the information content within each block. A single bitstream is created from the wavelet transform coefficients of the partitioned blocks for transmission or storage, The rationale behind this scheme is to insert the most vital infor- mation at the beginning of the bitstream. Such a scheme then allows for transmission until the desired target rate is achieved. The information content can be measured in a variety of ways including the mean-square error. However, the MSE is not a viable measure of saliency in an image [29]. Rather, a weighted MSE based on perceptual considerations is gaining much popularity and is being used in image coding extensively [34]. The resulting bitstream Is constructed from coefficients obtained from blocks in an order determined the presence of significant structures.

We now describe the Contrast Sensitivity Function (CSF) and show how it can be used to determine the perceptual saliency,

Human Visual System - The human cortex is often modeled as a linear system and its response to a visual excitation in the receptive field is expressed as a convolution of the impulse response of the visual cortex with the input stimulus. Equivalently, the response to an input can be obtained by multiplying the Con- trast Sensitivity Function (CSF) with the Fourier transform of tho input. The CSF measures the response of the visual system to dif- ferent frequencies. It is important to note that the response is lower for higher frequencies. Attempts have been made to measure the CSF of the human visual system. For instance, Mannos and Sakrl-

.son [19], after conducting a series of psychophysical experiments on human subjects, found that the CSF can be modeled by the function in Equation 1 (Figure 3a). Here,& is spatial frcqucncy in cycles per degree.

-(o.114Ql’1 C&f)= [0.0492 + 0.2964fJe

A 2D (or a 3D) version can be easily obtained and is shown in (Figure3b). The discrete orthogonal wavelet transform does not seme as a computational model for the HVS. To include the CSP into our methods, a frequency decomposition induced by a dyadic wavelet transform is imposed on the CSF (Figure 3a). Thus, for each band a weight can be computed which is then applied to the wavelet transform. The coefficients at each level and location ara modulated using a perceptual weight obtained by integrating the CSF over the area occupied by the band in a 3D frequency space. Figure 3a shows the CSF function mapped to image frequency and the superimposed wavelet transform subdivision in 1D. Thus, after m iterations of the wavelet transform the weight for level m is:

where FB, is the frequency subband in question. This weight mca- sures the area of the CSF curve in each octave band (Figure 3a), The weights are precomputed and stored in a table. In Figure 4b wo show the normalized weights for a 2D subdivision [16], Tho weights can be used to provide importance to wavelet coefllcients within each block. We employ a 3D version of the CSF to code a

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Figure 3.(a) I-D Contrast Sensitivity Function. The dyadic subdivision (of the wavelet transform) is imposed on the image frequency. (b) 2D Version of the CSF.

given volume. In reality, there is no physical justification for such a function. However, it is not uncommon to employ modulation transfer functions (MTF) to alter the significance of.coefficients in a transformed space. Thus, we can conceivably construct a device which accentuates frequencies in the central portion of a 3D spec- trum. To avoid necessary confusion we, however, use the terms CSF and MTF synonymously. We now show how the perceptual weight can be incorporated into a packing scheme.

Perceptual Weighted Embedded Pacldng Scheme - The following scheme is designed to select the next significant coefli- cient based on the structure information estimated in each block. As a result a wavelet coefficient bitstream is constructed with the coefficients carrying the most information being placed in the stream as early as possible. After data partitioning, the data volume is represented by N variable-size blocks, Bi, 0 ,< i 5 N . Each block B, has the following entities:

l The starting spatial address (i&k) and the block size.

l The information quantity contained in the block, Hi, which is defined as the sum of all correlation mask values contained in the block.

l The wavelet coefficients computed for the data block.

The measurement of the information in a block does not have to be exact, AS long as Hi is a monotonic function it will suffh.

Since the mask is created from the product and non-linearly enhanced with a logarithm function, Hi will be monotonic. Blocks of the same size will have the same CSF weight but different His. Thus the visual significance of a wavelet coefficient is a relative quantity and is determined by two factors:

l the quantity of information in the block to which the wavelet coefficient belongs, namely Hi

l the position of the wavelet coefficient inside the block. ,

During embedding, the relative significance of the blocks and coefficients has to be constantly updated. To dp so, we calculate for each block RH,, the remaining information, using a CSF which is pre-computed for each possible block size. This function should have the following properties

where NI is the number of coefficients in a block.

Thus, a 2k x 2k x 2k block will have Sk3 coefficients, each measuring the information available in a subband. In the case of a Zlevel, 2D wavelet transform, the dyadic decomposition results in the subdivision shown in Figure 4. Thus each band in the dyadic decomposition is pre-computed and assigned a value, i.e., each position in the block (ij,k) is .assigned a normalized, weight NCSF(i,j,k) (prefix N indicating normalization). A possible candi- date for the function RH is

RHi(mj = Hi (4)

The function RH essentially computes the perceptual informa- tion that can still be assigned to the block after m coefficients have been embedded. When a coefficient is embeddedin the bitstream the corresponding normalized weight is subtracted from value of RH. The coefli&ents inside a block are selected in a zig-zag order similar to the selection scheme in JPEG [33J. Once a coeffibient is embedded, the corresponding perceptual weight is subtracted. This scheme distinguishes between blocks that have the same spatial dimension but different information content. Given our &tree spa- tial subdivision of the original domain, several such similar sized blocks can exist, some with little information. The packing algo- rithm can be summarized as follows:

Step 1: Sort the blocks in descending order, using Hi as the key.

Step 2: Insert into the bitstream the lowest subband, coefficient for each block and update RHi accordingly. Re-sort the blocks based on RHp

Step 3: Insert-into the bitstream coefficients (determined by the 2 patterns) from the first block in the sequence and update RHi until the first block is no longer the most significant block.

Step 4: Insert-sort the first, block into the block sequence based on RHi; go back to Step 3. In 1261 a variant of this embedded packing scheme is described.

The block addresses and the ordering information must be coded and stored precisely first in the bitstream. This can be done either by:

l coding the block addresses in the ordered sequence, or

l creating a ‘bit-volume’ and putting the block’s sequence num- ber at the block’s reference address in the volume, and then coding the bit-volume.

We now provide the results of applying our coding scheme to a dataset from a simulation.

0

7c/4

7d2 0.0516 0.0516 0.0219 0.0219

0.0516 0.0516 0.0219 0.0219

04 Figure 4. Embedding of Dyadic Subbands. (a) The subband LL] has been split into 4 more subbands. Each of the 4 subbands of the Zlevel transform has an associated coefficient. The selection of the bands and within bands is shown in the form of Z. Thus the order among the subbands is LLI, LJYl, HLI and HHl. In lieu of LLI the subbands L&, LHz, H& and HH2 are embedded. (b) The normalized weights for Zlevel decomposition of the CSF. The weights are used in the computation of the remaining information function.

5 RESULTS We tested our methods on a dataset obtained froth a simula-

tion of the non-steady flow of fluid between turbine blades. The flow of fluid is axi-symmetric. ColorPlate shows a slice of the density field juxtaposed with the body of the turbine [15]. The regions of interest are the leading and trailing edges of the blade and the tip of the propeller. The grid is curvilinear and is of size 69x43x15. The solution of the simulation yielded two scalar fields (density, p; energy, e) and a vector field (u, v, w). We do not com- press the grid in this exercise, rather we restrict ourselves to the scalar fields and components of the velocity field. Thus, the grid file and the compressed solution comprise the data that needs to be transmitted or stored. To determine the correlation mask we create a larger grid of size 128x64~16 through zero-padding. An octree subdivision of the mask provides the required blocks. Starting with the original data in each block, we then perform a wavelet trans- form. Finally the coefficients and the blocks are all embedded into a bitstream. We now discuss the rate-distortion and rate-informa- tion curves for the dataset in question.

Rate-Distortion Curve: The rate-distortion curve is plotted for the components of the dataset (Figure Sa). We define the rate to be the percent of the coefficients retained. Thus a rate of one (1) would imply that all coefficients are retained, while a rate of zero implies that none are’ retained. We plot the distortion for all the fields. The distortion at a given rate is measured by finding the dif- ference between the actual and the partially reconstructed dataset in the mean-square-error (MSE) sense. It is evident that the degra- dation is non-uniform over the various fields. The v and the w-com- ponent of the velocity yield similar behavior; they require that 80% of the signal be retained to obtain a very desirable error of 60dB. Given the axi-symmetric flow there exists more disturbances in directions other than the flow direction. In reality, this is a very encouraging statistic. The other fields fare even better and the same distortion is obtained at the even lower rate of 20%. Essentially, the datasets are very sparse and are amenable to extreme compression.

Information-Rate Curve: We also plot the information that is embedded in the bitstream at different rates (Figure 5b). The information embedded is given by the quantity E, defined as:

‘I c RHi

E=l-i CHi ‘ i

(5)

Both the quantities Hi, RHi are defined in Section 4. The denominator in the ratio measures the information content and is computed a. the sum of the mask values over all the blocks, In essence it measures the information in terms of significant struc- tures..The numerator on the other hand, measures the information retained.

Although the rate-distortion curve is different for at least two of the components the information-rate curve is similar for all of them. This is the case since we measure information in terms of structures as captured by the mask. The masks of all five compo- nents are very similar signifying that the regions of discontinuitics are similar. The behavior of the information-rate curve is borne out by the visual examination of the components. It is noteworthy that the octree s- jtial subdivision of the correlation masks for all the componen&. yielded almost the same number of blocks. This leads one to conclude that the discontinuities occur in the same areas of the grid for all components. Since, the quantity E is a gross mea- sure of perceptually significant structures in a dataset, the quantity evaluates to the same value for all components. However, the num- ber of significant blocks is much less in the case of v, w compo- nents indicating that significant structures in these datasets arc much sharper or discontinuous. Thus, the rate-distortion curve is less steep and requires more coefficients to reconstruct.

Progressive Iso-surface Rendering: To show the efficiency of our approach we display the iso-surfaces for a derived quantity, namely pressure. We use FAST [l], a tool often used by CFS cngl- neers to visualiie their’grids (Note: The checker-board pattern In the background iif in’ColorPlate(b)-(f) is due to a well-known artl- fact of the screen door transparency rendering process used in FAST). Foi incompressible flow, pressure at any point can be com- puted as follows:

p = (7-l) [ e-ip(u2+v2+w2)]

where y is a gas constant. ColorPlate shows the iso-surface of value 0.80 (the range of pressure is [0.44,1.02]) for a rate of 100% (full-resolution). The two significant surfaces exist near the tip of

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normalized rate (4 (b)

Figure 5. Performance of our coding scheme. (a) Rate-Distortion curve. The distortion is measured in decibels and is given by -2Ologtu(Error). It is plotted along the Y-axis and the rate along the X-axis (b) Rate-Information curve. The information in each block is measured by the sum of the mask values.

the propeller and along the blade. It is important to note that in ColorPlate even at a rate of 20% the iso-surface along the blade is captured at very high fidelity. This is a testimony to the fact that our packing scheme was effective in embedding significant struc- tures very early in the bitstream. Another important observation is that the behavior of our scheme was not affected even though we applied our methods on a non-linear derived quantity, namely pres- sure.

To further show the effectiveness of our methods, we render isosurfaces at a lower pressure of 0.75. ColorPlate shows the iso-surfaces at the full normalized rate. We see a profusion of sur- faces of varying smoothness and curvature. Once again we show the iso-surfaces at rates of 20% (ColorPlate( and 50% (Color- Plate(f)). It is again worthwhile noting that the highly curved sur- face in the middle is captured with high fidelity even at 20%. The smoother surface at the extreme right is captured accurately at the rate of 30% (not shown). The surface at extreme left is depicted with significantly high fidelity. This once again demonstrates the ranking induced among structures through our methods. The results contained in [S] do not show the same amount of structure sensitivity; instead the degradation at rates less than full is more uniform. In [S] CFS datasets are compressed using a non-adaptive blocking scheme.

The time expended in various stages of the algorithm is not significant. For the given dataset the time expended is less than 10 seconds. Compressing floating point numbers to achieve high com- pression rates is not a trivial task. Usually an entropy scheme like Huffman coding or even vector quantization can be employed to compress the wavelet coefficients [8]. It suffices to say that any fur- ther compression and quantization schemes in the waveIet domain will only enhance the efficiency of the reported scheme. The decoding cost is also small given the regular pattern of embedding the coefficients and the fact that biorthogonal wavelets were cho- sen to actually encode the blocks.

6 CONCLUSIONS AND FUTURE WORK We presented a multiresolution representation of a volume

that is amenable to high compression ratios and progressive trans-

mission. The results obtained bear testimony to the use of the method. The method is computationally cheap and can include aspects of the Human Visual System easily. Improvements can come in the form 0E

l better wavelet transforms especially redundant ones

l better methods to determine the correlation mask

l better packing schemes

l adaptive choice of the wavelet function from a library

New work can include:

l further quantlzation of the coefficients

l detecting coherency between successive frames in a time vary- ing simulation

l datasets from other domains (medical) and modalities

The embedded methods developed herein can be used in many other applications besides progressive transmission. For instance, the embedded method would be very useful in storage and database access of datasets. The first magnitude-significant coefficients resulting from a wavelet transform could be used to search for images in a database. It is conceivable that more effec- tive and robust search performance is obtained when the coeffi- cients are ordered by the structures to which they contribute. Also, by including a computational model of the HVS the search can result in perceptually closer matches.

7 ACKNOWLEDGEMENTS We acknowledge the contributions of Prof. Roni Yagel, The

Ohio State University and Ajeet Gaddipatti of the Biomedical Engineering Program. The Ohio State University, towards the development of the wavelet combining schemes. One of us (Raghu Machiraju) developed the schemes in collaboration with them. This work was supported by the NSF Engineering Research Center for Computational Field Simulation, Mississippi State University.

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