Digital reconstruction of calciﬁed early metazoans, terminal...

q 2001 The Paleontological Society. All rights reserved. 0094-8373/01/2701-0011/$1.00

Paleobiology, 27(1), 2001, pp. 159–171

Digital reconstruction of calcified early metazoans, terminalProterozoic Nama Group, Namibia

Wesley A. Watters and John P. Grotzinger

Abstract.—A method is presented for the digital reconstruction of weakly calcified fossils withinthe Nama Group, Namibia. These recently described fossils (Grotzinger et al. 2000) are preservedas calcitic void-fill in a calcite matrix, and individual specimens cannot be freed by conventionaltechniques. The technique presented here has several integrated steps: (1) the analysis of cross-sections of fossil specimens, (2) the construction of a three-dimensional ‘‘tomographic’’ model thatis assembled from the cross-sections, (3) the development of an idealized mathematical modelbased upon geometric parameters measured from the tomographic model, and (4) the visualizationof randomly oriented cross-sections through the mathematical model, which can be compared withfossil cross-sections in outcrop.

In this procedure, rocks containing the fossils are ground and digitally photographed at thick-ness intervals of 25 mm. A battery of image-processing techniques is used to obtain the contouroutlines of the fossils in serial cross-sections. A Delaunay triangulation method is then used toreconstruct the morphology from tetrahedrons which connect the contours in adjacent layers. Wefound that most of the fossils represent a single morphology with some well-defined charactersthat vary slightly among individual specimens. This fossil morphology was described by Grotzin-ger et al. (2000) as Namacalathus hermanastes. A mathematical description of the morphology is usedto obtain a database of randomly oriented synthetic cross-sections. This database reproduces thevast majority of cross-sections observed in outcrop.

Wesley A. Watters and John P. Grotzinger. Department of Earth, Atmospheric and Planetary Sciences, Mas-sachusetts Institute of Technology, Cambridge, Massachusetts 02139. E-mail: [email protected] [email protected]

Accepted: 24 July 2000

Introduction

The comparative analysis of shape and formis an essential tool for enabling synthesis inpaleontology. The study of fossil morpholo-gies provides an objective basis for taxonomicdescription and enables analysis of their func-tional behavior and phylogenetic affinities(Thompson 1942; Hofmann 1976; Benson et al.1982; Feldmann et al. 1989; Foote 1989). Theanalysis of biological form is the principal ap-proach used to determine the evolution of ho-mologous morphological characters throughtime.

Analysis of fossil shape can be improved bydevelopment of techniques that increase ac-curacy or speed, or that enable reconstructionof morphologies which might otherwise beimpossible to obtain (Feldman et al. 1989).Most morphological studies of fossils involveorganisms that produced well-developed skel-etons (e.g., Foote 1989), which can be liberatedfrom their matrix using conventional tech-niques (e.g., Grant 1989; Palmer 1989). In com-

parison, studies of organism impressions andother nonskeletal features are comparativelyrare (e.g., Seilacher 1989; Hofmann 1990) be-cause significant morphological information isnot preserved

In general, studies of morphology are basedupon an analogue rather than a digital orcomputerized approach. Exceptions includestudies by Chapman (1989) and Schmidtling(1995). Both of these studies make use of a se-rial-sectioning technique (Sandy 1989) that iscoupled with digital image analysis. Schmid-tling’s (1995) study was motivated by the needto quantify the distribution of hydrospiresmore accurately, for the purpose of calculatinghydrodynamic effects related to fluid circula-tion through spiracular openings.

The digital approach to fossil analysis isadopted here to quantify the form and taxo-nomic diversity of lightly calcified terminalProterozoic metazoan fossils. These fossilscannot be liberated from their rock matrix us-ing conventional techniques such as mechan-ical extraction (Palmer 1989) or acid dissolu-

160 WESLEY A. WATTERS AND JOHN P. GROTZINGER

tion (Grant 1989), and therefore serial grind-ing (Sandy 1989) is required. (Other nonin-vasive techniques such as nuclear magneticresonance tomography were attempted butproduced discouraging results.) Data are cap-tured digitally, which makes possible a rangeof image processing techniques (Fabbri 1984;Gonzalez 1992) to be applied in creating a ful-ly reconstructed virtual specimen. Renderedin this fashion, the virtual specimen can thenbe viewed in three dimensions using state-of-the-art computer graphics, and studied quan-titatively as a whole or in parts. Ultimately, themorphological parameters obtained from a setof reconstructed specimens can then be usedto develop a mathematical model of individ-ual fossil organisms. The model can then beanalyzed to more accurately assess the truetaxonomic diversity present within a popula-tion of two-dimensional cross-sections seen inoutcrop or hand samples.

Although we have applied this procedurespecifically to estimating taxonomic diversity,it also may be applied to problems in func-tional morphology and biomechanics, and tothe quantitative analysis of morphologicalvariability among related shapes, includinghomologous characters. A more detailed ex-planation of the procedure can be found inWatters 2000; the thesis and the software thatwas developed are available as freeware athttp://eaps.mit.edu/sedlab/projects/mitpcdpm.html.

Obtaining Serial Images

In overview, the process for obtaining serialimages is as follows. First, a small square slabof sufficient thickness containing suitable fos-sils is cut out from a hand sample. The surfaceof this sample is then photographed by a dig-ital camera to obtain an image. Then, a thinlayer measuring 25.4 mm in thickness is re-moved from the surface using a surface-grind-ing apparatus. The photographing and grind-ing are repeated in this way until the sampleis ground away, producing a set of severalhundred images obtained at 25.4-mm inter-vals.

The sample is installed using an epoxy ad-hesive within a square steel mount measuring10 cm on a side (Fig. 1A). The mount is made

of a durable metal so that the sample can betemporarily fastened to the magnetic platformof a surface-grinding apparatus (Fig. 1B), andso that its shape and the sample’s orientationwill not change with repeated grinding. Themagnetic platform slides back and forth in adirection perpendicular to the axle of thewheel and at a regular interval that can be ad-justed (Fig. 1B). As the platform slides backand forth, it also advances in a direction par-allel to the wheel’s axle at an adjustable rate.

After the surface of the sample is groundparallel to the base of the mount, six 1-mmholes are drilled through the whole sample,perpendicular to the surface and with regularspacing. These ‘‘registration holes’’ serve asmarks for correlating the positions of imagesat each layer of the sample. A Kodak DCS420digital camera with a Nikon N90 camera bodyand 55-mm Nikon macro lens was used fortaking images of the sample. The limiting col-or resolution is 24 bits per pixel, and the CCDarray has a total spatial resolution of 1524 31012 pixels. The camera is mounted on the car-riage of a copy stand whose height above thesample can be adjusted (Fig. 1C). The camera’sheight above the sample is adjusted so that thewhole surface can be captured in two images,and then the camera is rigidly fastened in thisposition.

The position of registration holes are chosenso that four holes appear in each image. Theregion of overlap between the two images iscomparable to the size of one fossil diameter,and it contains two registration holes (Fig.1A). Each pair of images for a given layer ofthe sample is referred to as a single ‘‘frame.’’In our case, just two images are used to scanthe entire surface, because the resolution ofeach scan is deemed adequate for resolvingthe smallest structures observed.

Summing the amount of time needed forcleaning the sample between grindings, fo-cusing the camera, and taking two images, itis possible to process six or seven frames perhour. Since the layers are being shaved at 25.4-mm (1/1000 inch) intervals, a sample that is2.54 cm in diameter will produce 1000 imagesin all and require 1000/6.5 ø 150 person-hours to process completely. From this onecan see clearly the main disadvantage of this

161DIGITAL RECONSTRUCTION OF EARLY METAZOANS

FIGURE 1. Apparatus for serial sectioning and image acquisition. A, Fossil and mount. B, Surface-grinding appa-ratus. C, Camera, sample, and copy stand.

approach. Many hours are required to processa single sample, and one sample may containonly several complete fossils that are well pre-served.

Obtaining Serial Contour Outlines

In overview, the frames are first examinedto select an interesting fossil for reconstruc-tion. The volume that contains the fossil isthen cut out from the ‘‘stack’’ of frames. This

cropped stack of frames is then studied to de-cide which structures belong to the fossil inorder to exclude those that do not. Then, im-age-processing techniques are used to isolaterecognizable shapes. In particular, each imagein the cropped stack is enhanced to exaggeratethe color contrast between fossil and matrixmaterial. Each frame is then converted to a bi-nary (black-and-white) image that shows thefossil in black and the matrix in white. The


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FIGURE 2. Image-processing procedures for serial sections. A, Intensity histogram for the gray-scale image. B, Ad-justed intensity histogram. C, Gray-scale image of the red color component, which is combined with the othercomponents to obtain a gray-scale image. D, High-contrast image, which results from adjusting the intensity his-togram. E, Thresholded image with threshold value T 5 0.25. F, Thresholded image with T 5 0.32. G, Thresholdedimage with T 5 0.55. H, Original binary image with selected ‘‘flood-filled’’ components. I, Binary image after ‘‘clos-ing’’ operation. J, Binary image with artifacts erased. K, Skeletons; left 5 original object, center 5 skeletons, right5 dilated and with spur pixels repeatedly removed until none remain. A different binary object from that in Figures2C–2H is shown here to supply a more dramatic demonstration of the effect, and because obtaining skeleton-refinedcontours is not recommended in all cases (see text for discussion).

pixels at the perimeter of the black regions arethen tabulated and stored as ‘‘contours.’’

It should be noted that all of the computeroperations described in this study are carriedout using software designed for a SiliconGraphics Indigo 2 computer and the IRIX 6.2operating system. The image processing iscarried out using scripts (Watters 2000) for usewith the Image Processing Toolbox withinMatlab Version 5.1 (MathWorks Co.). Thesescripts are available for public use as freewareand are contained in Appendix C of Watters2000.

The processing of image data occurs as fol-lows. First, a script is designed for the purposeof easily perusing all of the frames to identifyfossils for reconstruction. Another script isused to select a rectangular region of interest(ROI) in a frame near the center of the stack,and then to crop every image in the stack us-ing the same ROI to produce a cropped stackof frames, also called a ‘‘movie.’’ The movie isviewed to confirm that the fossil is entirelycontained within the volume. Also, these mov-ies are stored as a convenient and rough rep-resentation of the data, where time representsthe third spatial dimension. Before any fossilis processed for making a reconstruction, itsmovie is carefully studied to decide what fea-tures in the images belong to the fossil. For es-pecially complex shapes, notes are takenabout what features to exclude from the re-construction, and in which frames they reside.Such notes, as well as the spatial coordinatesof the ROIs for each fossil and the range offrame-numbers containing each fossil, are re-corded in an ASCII text file, along with a nu-meric label and short description. A system ofscripts, notes, descriptions, movies, and, later,fully reconstructed models, functions as a da-tabase for several kinds of data in different

stages of analysis (see Watters 2000 for furtherexplanation).

Many precautions are taken to ensure thatthe sample is always scanned in the same po-sition relative to the camera. Inevitably, slightdifferences will occur in the scale, rotation,and position of the sample within imagesfrom different frames. For this reason, once acollection of fossils has been selected for re-construction and before each one can be pro-cessed individually, the (x,y) positions of thefour registration holes in each image are tab-ulated. The proper corrections to scale, orien-tation, and position of each frame with respectto the first frame are calculated and are laterapplied to the contour outlines obtained fromeach frame.

Next, each 24-bit color image in a givenmovie is converted into a gray-scale image,where each pixel has an intensity value from0 (black) to 1 (white). To enhance the contrastbetween fossil and matrix material, a simpleintensity-histogram adjustment is performed,also known as ‘‘contrast stretching’’ (Gonza-lez 1992). A ‘‘binary’’ image, composed of pix-el values that are 1 (white) or 0 (black) is ob-tained from the intensity image using a sim-ple thresholding procedure. These steps areshown in Figure 2A–G. It is then possible toisolate the fossil by flood-filling its compo-nents with white pixels (1s) and then subtract-ing the original image. This process obtains animage with just the flood-filled region inwhite, against a black background (Fig. 2H).

There are a variety of so-called morpholog-ical operations that can be used to enhance abinary image. Many of these operations weretried, and just two were chosen that have theeffect of reducing the image ‘‘noise,’’ i.e., thepoints and gaps which contribute to theroughness of boundaries or which are proba-


FIGURE 3. Final contours of imaged fossil composed ofdiscrete points connected by straight lines. A, Completefossil cross-section. B, Enlargement of region in the boxof Figure 3A.

bly artifactual. The first operation is a ‘‘clos-ing,’’ (Fig. 2I,J), and this involves two simpleroperations, called ‘‘dilation’’ and ‘‘erosion’’(Fabbri 1984). The second morphological op-eration (Fig. 2K) that is sometimes used is thethinning of binary objects to obtain ‘‘skele-tons’’ and skeleton-refined objects (Fig. 2K,center). In effect, these are lines one pixel inthickness that trace the center-line of fossilwalls. A dilation is then used to thicken thewalls uniformly (Fig. 2K, right). Skeleton re-finement can be performed only for specimensthat have a constant, slender wall thickness inevery frame, and not for those, like that shownin Figure 2C, where the interior space of thestem is filled by calcite. An erasing tool wasused to eliminate unwanted features and ar-tifacts in order to produce a ‘‘cleaned’’ binaryimage (Fig. 2J). Such kinds of manual editingdevices are frequently a part of image pro-cessing projects, since minute and artificialfeatures inevitably arise (e.g., see Chapter 6 ofFabbri 1984).

The last step in the process is to extract con-tours from the enhanced binary image. In thiscase, the contours are obtained using a routinein Matlab that finds and tabulates points at theboundary between black and white regions ina binary image. Then, all that remains is to ap-ply the rotation, position, and scale correc-tions calculated earlier for the current frame.The contours that result from the whole pro-cess are displayed in Figure 3; note that thereare three contours that define this particular

fossil cross-section. The entire process is re-peated for every frame in the cropped stack,or for as many as the user decides are needed.The frames can be processed either in the or-der in which they appear in the stack or usingan order that obtains the entire reconstructionat each stage with increasing vertical resolu-tion.

Slight errors in the final position and ori-entation of contours result from errors in mea-suring the positions of registration holes for agiven frame. If the contour is badly mis-aligned, then the registration-hole positionsfor the frame can be checked and measuredagain. In the case of one sample, our registra-tion holes measured approximately 13 pixelsin diameter (total spatial resolution of 7.7 31025 m/pixel). Because the position of eachregistration hole is carefully measured usinga magnified image, an error of at most 3 pixelsin reckoning the center is expected. This in-troduces an error of about 0.2 mm in the ab-solute positions of each frame. (This amountsto about 3.3% of the average size of the Namaspecimens [0.6 cm].)

The Geiger-Delaunay ReconstructionMethod

All contour data are formatted for use withthe public domain software Nuages and arestored in a ‘‘contour file.’’ Nuages is used to ob-tain a three-dimensional reconstruction fromserial contour data using tetrahedrons and wasdesigned by Bernhard Geiger of INRIA inFrance (http://www-sop.inria.fr/prisme/logiciel/nuages.html.en). Geiger (1993) con-firmed the accuracy of his method by compar-ing the reconstructed model of a geometricalobject with its mathematical description. Inwhat follows, this is assumed to be adequatejustification for using Geiger’s method, andtherefore individual steps in the following ex-planation of his process are not motivated sep-arately. In his discussion, Geiger reduces theproblem of finding an overall reconstruction tothe problem of computing a solid slice betweenadjacent cross-sections P1 and P2 (i.e., framescontaining one or multiple contours).

The first step is to calculate the Voronoi di-agram and Delaunay triangulation for pointsin each contour of P1 and P2. Given a set S of


N points (S 5 {pi ∈ R2 z i 5 (1 . . . N)}) such thatno four points lie on a common circle, let V(i)be the set of points closer to the point pi thanto any other point in S. R2 is the set of coor-dinate pairs of all real numbers. V(i) is calledthe Voronoi cell associated with pi. The unionV(S) over all V(i) is called the Voronoi diagramof S. The Voronoi diagram is a subdivision ofthe plane into convex polygonal regions,which are the Voronoi cells. The boundaries ofthese regions are called Voronoi edges, andthe points at which (always three) Voronoiedges connect are called Voronoi vertices.Each Voronoi edge borders just two Voronoicells. The Delaunay triangulation of S is the setof points along straight lines that connect anytwo points in S whose Voronoi cells share anedge. These straight lines are called Delaunayedges.

The Voronoi diagram and Delaunay trian-gulation are calculated for the set of points inthe contours of P1 and P2. In the next step, theVoronoi diagrams and Delaunay triangula-tions are updated after the addition of pointsto contours and to the interiors of contours inboth cross-sections. Each of these additionsrepresents an important refinement to the fi-nal model (Geiger 1993). The final Delaunaytriangulation for two example contours, C1

(Fig. 4A) and C2 (Fig. 4C) in P1 and P2, re-spectively, are shown in Figure 4B and 4D,where these contours have been greatly sim-plified for the purposes of illustration (i.e.,these have many fewer points than a typicalcontour, such as that shown in Fig. 3).

A fossil is reconstructed by assembling a setof tetrahedral components in three dimen-sions. That is, tetrahedrons are used to con-nect points, triangles, and line segments in thefinal Delaunay triangulations of contours inthe adjacent cross-sectional planes P1 and P2.For each triangle t ∈ P1, the point p ∈ P2 isfound that is closest to the center of t. The tri-angle t is connected with p to form a tetrahe-dron of type t1 (Fig. 4E). The process is re-peated for triangles in P2 and points in P1,forming tetrahedrons of type t2 (Fig. 4F). Athird kind of tetrahedron (Fig. 4G), with justone edge in P1 and one edge in P2, is drawnwherever an intersection occurs in the unionof Voronoi diagrams for the two planes (i.e.,

the projection of the Voronoi diagram in P2

onto that for P1). That is, wherever two Voron-oi edges intersect, the corresponding Delau-nay edges are joined by a type t12 tetrahedron.Some of the ‘‘connecting lines’’—the sides ofthe tetrahedrons used to connect the contoursC1 and C2—are shown in Figure 4H and 4I.The fully rendered reconstruction withopaque triangular panels is shown in Figure4J.

A different vertical position is assigned toeach cross-section in the contour file before anaccurate reconstruction can be assembled. It ispossible to determine the scale of this posi-tion, since the vertical size of the fossil isknown (i.e., the number of frames that spanthe fossil, multiplied by 25.4 mm), and sincethe number of pixels spanning a physical unitof length can be measured from the horizontalsections.

A Mathematical Description ofNamacalathus

This section provides an overview of amathematical description for the morphologyof Namacalathus hermanastes. It summarizes auseful approach for constructing mathemati-cal models that have parameters which can be‘‘fitted’’ directly to tomographic models.

The main reason for constructing a mathe-matical model is to obtain an idealized formof the morphology, prior to diagenetic effectssuch as the thickening and bending of fossilwalls. Another purpose of such a model is toobtain a database of idealized cross-sectionsthrough random orientations of the morphol-ogy. This database can then be used to iden-tify specimens in the field, which in turnmakes possible the measurement of popula-tion statistics for fossils in a stack of serial im-ages (Grotzinger et al. 2000). A mathematicalmodel should reflect the features common toall of the tomographic reconstructions, withsome parameters assigned to characteristicsthat vary among individual specimens—suchas absolute dimension and proportion. In thestudy by Grotzinger et al. (2000), the completeor nearly complete reconstructed specimenshave a stem and an outward flaring cup (Fig.5). The cup has a broad circular opening at thetop with a lip that curves inward. The cup is


FIGURE 4. Explanation of the Geiger-Delaunay reconstruction technique. A, A contour C1 in the plane P1. B, FinalDelaunay triangulation of the original and added points in the contour C1. C, A contour C2 in the plane P2. D, FinalDelaunay triangulation of the original and added points in the contour C2. E, Tetrahedron of type t1. F, Tetrahedronof type t2. G, Tetrahedron of type t12. H, Reconstruction based upon contour C1 in plane P1 (lower) and contour C2

in plane P2 (top), where all unobstructed connecting lines (i.e., sides of tetrahedrons) on the exterior are shown. I,The same reconstruction except where all connecting lines on the exterior are shown. J, Fully rendered reconstruc-tion composed of opaque triangles, based upon contour C1 in plane P1 (lower) and contour C2 in plane P2 (top).


FIGURE 5. Tomographic reconstruction and mathematical model based upon the reconstruction. A, Several viewsof the tomographic model shown in Figure 9I of Grotzinger et al. 2000. B, Cross-sections and wire-frame diagramsof the tomographic model: first a cross-section showing regular holes sliced horizontally (relative to assumedgrowth position) through center of cup (top), vertically through assumed growth axis (upper-middle), and wire-frame views of circular opening at inferred top (lower-middle) and of the cup in profile (bottom). C, The mathe-matical model based upon the tomographic reconstruction.

perforated by six or seven holes (called ‘‘win-dows’’) of similar size and shape, which oftenhave slight or pronounced inward-curvinglips (called ‘‘window lips’’), and a matchingnumber of regular sides. Cups taper to a shal-low base from which a hollow cylindrical stemextends. Stems are commonly longer than the

maximum cup dimension and are open atboth ends.

For constructing the mathematical model itis assumed that these are universal propertiesof the morphology, while the characters thatare observed to vary among individual spec-imens are assigned parameters. The most im-


FIGURE 6. Horizontal cross-sections of cup, lip, andstem obtained from the mathematical description. Left:Spline-smoothed plots of cup cross-sections obtainedusing F1(c,z) for several sample values of the profile pa-rameters associated with the cup. Right: Plots of stemcross-sections obtained using F2(c,z) for sample valuesof profile parameters associated with the stem.

portant assumption is that the overall mor-phology in its original and undistorted formhas a regular symmetry according to the num-ber of windows in the cup. It is then possibleto construct a complete mathematical descrip-tion that specifies the shape of any horizontalcross-section (i.e., perpendicular to the cup’saxis of symmetry) at any vertical positionalong the model’s height. The technique de-scribed earlier can then be used to assemblethe reconstruction from tetrahedral compo-nents.

The Namacalathus model is composed ofthree geometrical objects: (1) a wall that formsthe cup, (2) a wall that forms the inward-curv-ing lip, and (3) a wall that forms the stem (Fig.6). Each wall is made up of an interior and ex-terior boundary, whose diameters differ bytwice the wall thickness T. The boundaries ofeach wall are described using just two kindsof two-dimensional vector-valued functions,called F1(c,z) (used to describe the cup walland the lip wall) and F2(c,z) (used to describethe stem wall). Each value of (c,z) yields aunique value of (x,y) in the case of each func-tion, where the angular parameter c variesover the entire domain [0,2p].

The functions F1(c,z) and F2(c,z) can beused to completely specify an individualmodel, according to a set of global parametersthat can be classified in two categories: (1) pa-rameters which specify the absolute dimen-sion of objects and features (called ‘‘global di-mension parameters’’) and (2) parameterswhich specify ‘‘relative’’ position quantities

and which are functions of vertical position(called ‘‘profiles’’). Each profile is expressed asa fraction of a global dimension parameter orsome quantity that has been derived from theglobal dimension parameters. The global di-mension parameters include the wall thick-ness, the maximum cup radius, the total cupheight, the total stem height, and the verticalspan of the inward-curving lip. The profilesinclude (as a function of vertical position) theradius of the cup wall, the window diameter,the size of window lips, the inclination of win-dow lips, the curvature of the cup wall (i.e.,flat or round), the radius of the cup lip, thestem radius, and the trajectory of the stem.When these parameters are defined, the vec-tor-valued functions F1(c,z) and F2(c,z) arecompletely determined and can be used todraw an entire model in three dimensions.Sample cross-sections generated using F1(c,z)and F2(c,z) are shown in Figure 6.

Database of Synthetic Cross-Sections

This section contains a database of syntheticcross-sections, obtained at random orienta-tions through the mathematical models shownin Figure 10B and 10D in Grotzinger et al.2000 and in Figure 5 of this paper. The data-base is nearly exhaustive insofar as it containsthe majority of cross-sections that are in someway ‘‘obviously distinctive.’’ That is, while aninfinite number of cross-sections can be gen-erated, only certain cross-sectional morphol-ogies are distinctive. The purpose of generat-ing such a database is so that assemblages ofNamacalathus can be easily identified in thefield, where only two-dimensional cross-sec-tions are visible. It should be emphasized thatthese synthetic cross-sections represent onlycomplete specimens, and that these are basedupon morphological parameters measuredfrom only two tomographic reconstructions.In outcrop, the majority of specimens are bro-ken or distorted in some way and exhibit awider range in the relative sizes and shapes ofsome features (actual specimens may havebroken stems, shorter stems, smaller cup lips,larger window lips, a different radial profile,etc.)

All cross-sections were generated using the‘‘SceneViewer’’ program for Silicon Graphics


FIGURE 7. Cross-sections through the mathematical model of an individual fossil at different orientations, showingisolated components that represent portions of a contiguous fossil wall. In this figure, several features of the Na-macalathus morphology are labeled.

FIGURE 8. Synthetic cross-sections, types 0–3. A, Type0 cross-sections of two mathematical models of Nama-calathus. Notice that the interior and exterior boundariesof the walls are clearly visible in several cases, such asin the lower left corner. The fossil wall appears to bethickest in oblique cross-sections that ‘‘graze’’ the sur-face, seen here in the tips of crescent-shaped cross-sec-tions in the lower left. B, Type 1 cross-sections. C, Type2 cross-sections. D, Type 3 cross-sections.

computers, normally used for viewing andmanipulating three-dimensional data setsstored in the ‘‘inventor’’ format. (The softwareprogram used to articulate the models in threedimensions [Nuages] is able to save the artic-ulated models in a variety of formats, includ-ing the inventor format.) The wire-frame‘‘draw-style’’ in SceneViewer is used to view aslice of the model between the near and far‘‘clipping planes.’’ These invisible parallelplanes conceal all portions of the model that

are nearer than the ‘‘near’’ plane and fartherthan the ‘‘far’’ plane. The model and both clip-ping planes can be translated and rotatedwithin a three-dimensional Cartesian coordi-nate space. When the clipping planes are set0.0001 distance units apart from each other, avery thin cross-section of the model is ob-tained. An independent screen-capture utilityis then used to store the cross-section in a jpegimage file. The model can be moved into ran-dom orientations to obtain the whole range ofpossible cross-sections.

When fully assembled, the model is com-posed of a single contiguous wall that connectsthe stem, cup, and inward-turning lip (i.e., theso-called cup lip). Cross-sections of this mor-phology are normally composed of one or sev-eral isolated components, which are in fact con-nected in the third spatial dimension. Severalexamples of these are shown in Figure 7, wheredifferent components of individual cross-sec-tions are labeled according to the structuresthey represent in the three-dimensional model.Cross-sections observed in outcrop are dividedinto nine types. Eight of these are numbered 0to 7, according to the number of components inthe cross-section (except in the case of Type 0),and the ninth type is called ‘‘Type F.’’ (The TypeF designation is explained later.) The eight num-bered types refer to cross-sections with thinwalls of approximately constant thickness. Syn-thetic versions of the eight numbered types thatwere generated from the mathematical modelsare shown in Figures 8 and 9. The database con-tains a nearly exhaustive set of these eight types,where each type is defined as follows. Type 0cross-sections consist of just one unbroken com-ponent that completely encloses a space (e.g.,ring-shaped cross-sections). Type 1 cross-sec-tions consist of just one component that does not


FIGURE 9. Synthetic cross-sections, types 4–7. A, Type4 cross-sections. B, Type 5 cross-sections. C, Type 6cross-sections. D, Type 7 cross-sections. Type 7 cross-sections are possible only in the case of specimens withseven ‘‘windows’’ (Grotzinger et al. 2000).

FIGURE 10. Type F cross-sections. A, Type F cross-sections in outcrop. B, Type F synthetic cross-sections.

completely enclose a space (e.g., goblets, cups,and broken rings). Type 2 through Type 7 referto cross-sections that have two through sevencomponents, respectively.

Type F (F 5 ‘‘filled’’) refers to cross-sectionsthrough parts of the interior of a Namacalathuscup that were not initially filled by sediment,

and which were eventually filled by void-fill-ing calcite crystals (i.e., skeletal grainstones).Figure 10A contains some examples of Type Fcross-sections observed in outcrop. Figure 10Bcontains some examples of synthetic Type Fcross-sections, obtained by projecting an outerwall of the mathematical model onto a cross-sectional plane. It is assumed that the spacebetween the walls and this plane was not filledby sediment because the walls prevented sed-iment from falling into the regions directly be-neath them. In fact, this method is only ap-proximate, because the sediment is not ex-pected to behave exactly in this manner. Moreaccurate cross-sections of Type F could begenerated by thinning or eroding the walls ofthe cross-sections shown in Figure 10B.

Type F cross-sections may have formed inways other than infilling of void space withcalcite spar. For example, these might be thecross-sections of fossil walls that have under-gone diagenetic ‘‘fattening’’ through neo-


morphic recrystalization (see Grotzinger et al.2000: Fig. 13). Digital images of serial cross-sections, however, show that Type F cross-sec-tions generally occur on just one side of an in-dividual specimen, and within a region that isoriented parallel to the bedding plane. Thisobservation favors the previous explanation,that these cross-sections represent interior re-gions not initially filled by sediment. Also,Type F cross-sections could be interpreted asexamples of a separate taxon if it were not forthree observations. First, if the interior struc-ture of a Type F cross-section is ‘‘erased,’’what remains can generally be matched witha cross-section in one of the eight numberedcategories. Second, individual specimens withType F cross-sections have been reconstruct-ed, and these are consistent with the mor-phology of a partially filled Namacalathus.Third, these occur in rocks containing abun-dant examples of the eight numbered types.Since the varieties of Type F cross-sections arepossibly very great in number, Figure 10Bprobably contains only a small subset.

Conclusions

The digital methods presented here areused to construct tomographic and mathe-matical models, which in turn are used to ob-tain a database of synthetic cross-sections thatcontains the majority of shapes observed inoutcrop. These digital methods offer severalimmediate and potential advantages.

1. The models can be used to find ‘‘signa-tures,’’ such as the database of synthetic cross-sections, which can be used to easily identifyan assemblage of fossils in outcrop. As seen inthis study, the database can be used to dem-onstrate that the vast majority of morpholog-ically diverse two-dimensional shapes ob-served in outcrop can be collectively ascribedto a single taxon (Grotzinger et al. 2000).

2. Diagenetic effects can be ‘‘simulated’’ tomake realistic representations of the fossil inalternative diagenetic contexts (e.g., Type Fcross-sections), or these effects can be sub-tracted to make idealized models of the orig-inal morphology (e.g., thin-walled mathemat-ical models with regular symmetry).

3. Digital models can be shared more easilythan actual specimens, so that morphologicaldata can be made available to a larger numberof scientists.

4. Standards for morphological measure-ment and description can be defined moreeasily in a digital medium, so that these canbe accomplished more or less automaticallyusing specialized software.

Acknowledgments

A.-C. Auclair is acknowledged for help inacquiring contour and image data. We aregrateful to C. Marshall for introducing us tothe unpublished thesis work of J. Schmidtling.R. Chapman and an anonymous reviewer pro-vided very helpful reviews of the manuscript.Support for this research was provided by Na-tional Science Foundation grant EAR-9628257.

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