Roerdink, Jos B.T.M.

Westenberg, Michel A.

English

This paper is concerned with X-ray volume visualization by means of wavelet splatting, a wavelet-based extension to splatting. Wavelet splatting allows multiresolution visualization of volume data. While a user is interacting with the data, only low resolution images are computed. When interaction ceases, the image is refined incrementally. We discuss a particular implementation of wavelet splatting which was proposed previously, and show that certain rendering artefacts appear in the low resolution images, due to the particular ordering of the wavelet coefficients. We propose a new variant that uses a different ordering, and computes low resolution images based on the wavelet approximation coefficients only. This variant does not suffer from artefacts, and is faster by a factor of two to three

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   University of GroningenX-ray volume rendering by hierarchical wavelet splattingWestenberg, Michel A.; Roerdink, Jos B.T.M.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Westenberg, M. A., & Roerdink, J. B. T. M. (2000). X-ray volume rendering by hierarchical waveletsplatting. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra (Eds.), 15THINTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS (pp. 159-162).(INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). IEEE (The Institute of Electrical andElectronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 06-06-2022X-Ray Volume Rendering by Hierarchical Wavelet SplattingMichel A. Westenberg and Jos B.T.M. RoerdinkInstitute for Mathematics and Computing ScienceUniversity of GroningenP.O. Box 800, 9700 AV Groningen, The Netherlandsmichel@cs.rug.nl, roe@cs.rug.nlAbstractThis paper is concerned with X-ray volume visualizationby means of wavelet splatting, a wavelet-based extension tosplatting. Wavelet splatting allows multiresolution visual-ization of volume data. While a user is interacting with thedata, only low resolution images are computed. When in-teraction ceases, the image is refined incrementally. We dis-cuss a particular implementation of wavelet splatting whichwas proposed previously, and show that certain renderingartefacts appear in the low resolution images, due to theparticular ordering of the wavelet coefficients. We proposea new variant that uses a different ordering, and computeslow resolution images based on the wavelet approximationcoefficients only. This variant does not suffer from artefacts,and is faster by a factor of two to three.1 IntroductionX-ray volume rendering remains an interesting tech-nique for medical applications, because physicians arewell-trained in interpreting X-ray like images for diagno-sis. X-ray volume rendering is a direct volume renderingmethod [2], based upon integrating the 3-D data along theline of sight. The underlying mathematical principle is theX-ray transform, well-known from computerized tomogra-phy [3]. The X-ray transformPθf of f(x), x = (x, y, z) ∈R3, is defined byPθf(u, v) =∫Rf(uu + vv + tθ) dt, (1)whereu andv are two mutually orthogonal vectors in theview plane perpendicular to a direction vectorθ (cf. Fig.1).Usual implementations are based either on ray-casting,Fourier rendering, or splatting. Ray-casting is an imagebased rendering method in which a ‘virtual’ light ray is castfrom the view plane into the volume data. For X-ray render-ing, the voxel values are simply added along this light ray.P (u,v)yθxθ uvzFigure 1. Parameters u, v, and θ of the X-raytransform.Fourier rendering [6] is a fast method to compute the X-raytransform by using frequency domain techniques. Splat-ting [9] is an object order rendering method in which thevoxels are projected to the view plane. In the discrete case,we have a collection of samplesck, k = (k, l,m) ∈ Z3,of the functionf . The splatting algorithm reconstructsffrom its samples by convolution with a reconstruction fil-ter φ, and then computes a mapping to the screen. This isexpressed byPθf(u, v) =∑kck∫Rφk(uu + vv + tθ) dt. (2)The integral overφ results in a two-dimensional function,which is called afootprint. For orthographic projection, allvoxels have the same footprint for a fixed viewing directionθ. A mapping to the view plane by accumulation of thefootprints weighted by the voxels forms the image.Previously, Lippert et al. [4, 5] proposed a wavelet-basedextension to splatting, calledwavelet splatting. Waveletsplatting modifies the splatting algorithm by using waveletsas 3-D reconstruction filters. One of the reasons to usewavelets is that rendering the data sets is time consumingdue to their size. This produces a need for compressionmethods and mechanisms to visualize the data incremen-1tally (‘progressive refinement’). For this purpose, wavelet-based multiresolution models have been developed, whichallow systematic decomposition of the data into versions atdifferent levels of resolution. Wavelet-based methods alsohave proved to perform well in compression of images [1]and compression of volume data [7].In this paper, we compare two different implementationsof wavelet splatting, and come up with a new variant, calledhierarchical wavelet splatting, which takes advantage of thestrengths of both methods. The first method uses the hi-erarchical ordering provided by the wavelet transform [8]during rendering. The second method imposes a global or-dering on the wavelet coefficients [5]. It will turn out thatthe global ordering of the coefficients introduces artefacts inlow resolution images. The hierarchical ordering does notsuffer from these artefacts. The advantage of the global or-dering is that images of high resolution are obtained usinga small number of wavelet coefficients.The remainder of this paper is organized as follows. Sec-tion 2 reviews the fundamentals of wavelet splatting, dis-cusses the two different implementations, and gives a com-parison. In Section3, we introduce hierarchical waveletsplatting. Finally, some experimental results are presentedin Section4, and discussed in Section5.2 Wavelet splattingWe summarize basic wavelet concepts, and describewavelet splatting. This is followed by a description of twodifferent implementations and a comparison between them.2.1 Basic wavelet splattingA 1-D biorthogonal wavelet basis can be constructedfrom a scaling functionφ with associated waveletψ, anddual scaling functioñφ and dual wavelet̃ψ. The corre-sponding basis functions are{φj,k} and {ψj,k}, j, k ∈Z, whereφj,k(x) = 2−j/2φ(2−jx − k) andψj,k(x) =2−j/2ψ(2−jx − k). The dual basis functions are definedsimilarly. The parametersj andk denote scale and trans-lation, respectively. From the 1-D basis, one constructs a3-D separable wavelet basis (of the so-called nonstandardtype) with eight basis functions, i.e. one scaling functionΦ0j,k(x, y) and seven wavelet basis functionsΨτj,k(x, y),τ ∈ T = {1, 2, . . . , 7}, by taking tensor products of the1-D scaling functionsφj,k and waveletsψj,k.Wavelet splatting [4, 5] modifies the basic splatting al-gorithm in two ways: (i) it uses wavelets as reconstructionfilters, and (ii) it provides a mechanism to visualize data atdifferent levels of detail. First, the algorithm performs a 3-D wavelet decomposition of the volume data. Substitutionof the expansion of on the 3-D wavelet basis in the X-raytransform (1) results in:Pθf(u, v) =∑kcMk∫RΦ0M,k(uu + vv + tθ) dt+M∑j=1∑τ∈T∑kdj,τk∫RΨτj,k(uu + vv + tθ) dt. (3)The approximationcoefficients arecMk = 〈f, Φ̃0M,k〉, andthedetail coefficients aredj,τk = 〈f, Ψ̃τj,k〉, where〈·, ·〉 de-notes the inner product in the spaceL2(R3) of square inte-grable functions onR3.This equation expressesPθf(u, v) as a weighted sum-mation of integrals along the line of sight, resulting in 2-Dfunctions on the view plane: the footprints. The integralshave to be evaluated only once for a given viewing directionat the coarsest scalej = M and translationk = (0, 0, 0),yielding eightprototypefootprints. The footprints for otherscales and translations can be computed by rescaling andshifting. Prototype footprints can be computed efficientlyby slicing their 3-D Fourier transforms [4]. When analyticalexpressions exist for the Fourier transforms of the scalingfunction and wavelet, as is the case of the Haar and cardinalB-spline wavelets, no interpolation from discrete samples isnecessary.2.2 Hierarchical orderingA straightforward implementation of (3) uses thehier-archical ordering of the decomposition levels. Renderingstarts by using the approximation coefficientscMk to com-pute a low resolution image. This image is then refinedby using the detail coefficientsdM,τk , followed by the de-tail coefficientsdM−1,τk , . . . , d0,τk . We recently proposedtwo-stage wavelet splatting[8], which provides a fast ren-dering algorithm for the hierarchical ordering. Two-stagewavelet splatting splits the splatting process in two stages:(i) coefficient projection and accumulation, and (ii) a finalconvolution with the footprint. This is more efficient thana direct implementation of (3), which blends the footprintinto the view plane for each voxel. This requires interpo-lation of all elements of the footprint, whereas two-stagesplatting only interpolates a single voxel. The final convo-lution requires no interpolation. We incorporated two-stagesplatting into wavelet splatting, and found speedups in ren-dering time up to a factor of three for Haar and B-splinewavelets [8].2.3 Global orderingLippert et al. [5] proposed an implementation of waveletsplatting which is targeted at networked applications. In or-der to transmit the volume data efficiently, they developed2a compression technique that allows progressive transmis-sion. The data compression pipeline converts the wavelettransformed data set into a sequential bitstream. First, a sep-arate sequence of coefficients for each wavelet type and de-composition level is generated, and the coefficients in eachsequence are sorted in descending order. Next, lossy com-pression involving deltacoding, normalization, and quan-tization is performed on each sequence. Finally, all se-quences are merged and sorted in descending order. Thisstep imposes aglobal ordering on the wavelet coefficients,in which the hierarchical order defined by the wavelet de-composition levels is lost. The client decompresses thestream and performs the actual splatting. The frame ratecan be controlled by the user, who can define the amountof coefficients that is used as a first coarse representation ofthe volume.2.4 ComparisonWe compare image quality of two-stage wavelet splat-ting (TWS), which uses a hierarchical ordering of thewavelet coefficients, and network targeted wavelet splatting(WS), which uses a global ordering. We focus in particularon the low resolution images computed during user interac-tion. The final images, i.e. for which all wavelet coefficientsare used, are the same. We will use two example data sets,a Phantom and CT data set. The Phantom data set has 8 bitsper voxel, and was generated from a mathematical descrip-tion of ellipsoids. It is a 3-D variant of the Shepp and Loganhead phantom [3], which is used to assess the quality of to-mographic reconstruction algorithms. The CT head data has16 bits per voxel. Both data sets contain128 × 128 × 128voxels.For WS it is difficult to determine the minimum numberof coefficients necessary to generate a low resolution imagewhich is useful. The final sorting and merging removes thehierarchical ordering provided by the wavelet decomposi-tion. During the rendering phase, therefore, it is impossi-ble to speak about an approximation at a certain reconstruc-tion level. Since the low resolution image always involves amixture of approximation coefficientscMk and detail coeffi-cientsdj,τk , interpretation can become difficult. An exampleis shown in Fig.2. We used a two-level wavelet decompo-sition with the Haar wavelet. The number of approximationcoefficients for the Phantom data set is 11296 after removalof the coefficients that are zero. Figure.2(a) shows an imageobtained by TWS using the approximation coefficients only,and Fig.2(b) shows an image obtained by WS using the first11296 wavelet coefficients. The effect shown in Fig.2(b) isundesirable, since the object appears to be partially empty.The effect disappears by taking 1000 coefficients more.Figure3 shows a comparison between TWS and WS forlow resolution images computed for a fixed frame rate. We(a) TWS (b) WSFigure 2. Low resolution image renderings.used a two-level wavelet decomposition with the linear B-spline wavelet. Since TWS computes low resolution im-ages using all approximation coefficients, we determinedthe number of coefficients that can be visualized with WSat the same frame rate. Both data sets contain 32767 ap-proximation coefficients. The Phantom data set is renderedat 11 frames per second by TWS and the CT head data setat 8 frames per second on a Pentium II 400 MHz proces-sor. The same frame rates are reached by WS using the first7100 coefficients of the Phantom data set and the first 7200of the CT head data set. The Phantom data set indicates thatWS gives undesirable results, because only an outer shell ofthe data is shown (Fig.3(c)). In an interactive session, forinstance, the objects inside the ellipsoid appear suddenlywhen interaction is ceased. A similar effect can be seen inthe CT head data (Fig.3(d)), where large parts of the softtissue are missing in the low resolution image.(a) 32767 (b) 32767(c) 7100 (d) 7200Figure 3. Low resolution renderings by TWS(top row) and WS (bottom row). (a), (c): Phan-tom data; (b), (d): CT head data.3Although WS suffers from artefacts in approximationimages, it produces high-resolution images with less coef-ficients than TWS. This is an important advantage of theglobal ordering of the coefficients. An example is shownin Fig. 4(a) and Fig.4(b) for the Phantom data and the CThead data, respectively. When the number of coefficientsused is increased to 32767, the total number of approxima-tion coefficients used above by TWS, the resulting imagesshow more detail than those in Fig.3(a) and Fig.3(b).(a) 32767 (b) 32767Figure 4. Renderings by WS.3 Hierarchical Wavelet SplattingWe can now introduce a new variant calledhierarchicalwavelet splatting(HWS), which combines the strengths ofboth TWS and WS. Instead of sorting and merging approx-imation and detail coefficients like in WS, HWS generatestwo sequences of coefficients: a sequence containing onlyapproximation coefficients and another sequence containingall detail coefficients. Since the approximation coefficientscMk are all of the same decomposition level, we can use two-stage splatting to increase rendering speed. The sequencecontaining the detail coefficients is sorted in descending or-der, as in ordinary WS. Both sequences can be compressedby deltacoding, normalization, and quantization. While auser is interacting with the data, only the sequence contain-ing the approximation coefficients is used. When interac-tion ceases, the image is refined incrementally with the de-tail coefficients.4 Experimental resultsWe applied HWS and WS to the CT head data set toinvestigate their performance and visual quality. A two-level wavelet transform with the linear B-spline wavelet wasused. Table1 shows cumulative rendering times (in msec)of HWS and WS on a Pentium II 400 MHz processor. Thefirst row shows the rendering time of a low resolution im-age. The second row shows the time after which the relativeL2 norm (mean squared error) of the difference between thepartial reconstruction and the image at full resolution is lessthan0.0001. Both methods reach this point using 82767 co-efficients. Since HWS uses two-stage splatting when com-puting low resolution images, rendering time is significantlyshorter.Table 1. Cumulative rendering times (msec)of HWS and WS.HWS WSlow resolution (32767) 63 275L2 < 0.0001 (82767) 399 582Figure 5(a) shows an image rendered by HWS usingonly 4% of the wavelet coefficients, and Fig.5(b) showsthe full reconstruction. Differences between these imagesare hardly distinguishable, providing an extra motivation forthe use of wavelets.(a) 82767 (b) 2023421Figure 5. Comparison of a partial reconstruc-tion using only 4% of the coefficients (a) andthe full reconstruction (b).5 DiscussionWe formulated a new variant of wavelet splatting, calledhierarchical wavelet splatting. Image quality for low reso-lution images is higher in comparison to WS, since HWSuses all approximation coefficients to generate the image.In contrast, WS uses a mixture of approximation and detailcoefficients, which leads to images that are difficult to in-terpret when a small number of coefficients is used. SinceHWS uses only approximation coefficients to compute lowresolution images, it can make use of two-stage splatting toincrease rendering speed. Although HWS has less freedomto choose the number of coefficients used for rendering, itis faster than WS for similar image quality.4References[1] R. A. DeVore, B. Jawerth, and B. J. Lucier. Image compres-sion through wavelet transform coding.IEEE Transactionson Information Theory, 38(2):719–746, 1992.[2] R. A. Drebin, L. Carpenter, and P. Hanrahan. Volume ren-dering. Computer Graphics (SIGGRAPH ’88 proceedings),22(4):65–74, 1988.[3] A. C. Kak and M. Slaney.Principles of Computerized Tomo-graphic Imaging. IEEE Press, New York, 1988.[4] L. Lippert and M. H. Gross. Fast wavelet based volume ren-dering by accumulation of transparent texture maps.Com-puter Graphics Forum, 14(3):431–443, 1995.[5] L. Lippert, M. H. Gross, and C. Kurmann. Compression do-main volume rendering for distributed environments. Techni-cal Report 263, ETH-Z̈urich, 1997.[6] T. Malzbender. Fourier volume rendering.ACM Transactionson Graphics, 12(3):233–250, 1993.[7] J. Wang and H. K. Huang. Medical image compression byusing three-dimensional wavelet transformation.IEEE Trans-actions on Medical Imaging, 15(4):547–554, 1996.[8] M. A. Westenberg and J. B. T. M. Roerdink. X-ray render-ing through two-stage splatting. Technical Report 99–9–11,Institute for Mathematics and Computing Science, 1999.[9] L. A. Westover. Footprint evaluation for volume rendering.Computer Graphics, 24(4):367–376, 1990.5

X-ray volume rendering by hierarchical wavelet splatting

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X-ray volume rendering by hierarchical wavelet splatting

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