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Evolutionary Ecology of the coral genus Madracis - an illustration of the nature of species inscleractinian corals

Vermeij, M.J.A.

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Citation for published version (APA):Vermeij, M. J. A. (2002). Evolutionary Ecology of the coral genus Madracis - an illustration of the nature ofspecies in scleractinian corals. Ponsen en Looijen BV.

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Download date: 17 Dec 2020

Chapter 7

Three-dimensional morphological analysis of growth forms of Madracis mirabilis (preliminary

results)

Vermeij M.J.A., Kaandorp J.A., Bak R.P.M, and L.E.H. Lampmann

Chapter 7 Three dimensional analysis of growth forms 109

In the previous section (pp. 77-86 in Kaandorp and Kuebler, 2001) the morphological analysis of two-dimensional images of growth forms of the three species, collected along environmental gradients, was discussed in detail. The morphological analysis only partly works using two-dimensional images; for complex-shaped growth forms with many overlapping branches, as shown in Figure 1, a two-dimensional analysis fails and a three-dimensional analysis is required. In this section we will present some preliminary results on how similar methods, as discussed in the previous section (pp. 77-86 in Kaandorp and Kuebler, 2001), can be extended towards a three-dimensional analysis. Since a complete set of measurements is not yet available (will be described in Vermeij et al. in prep.), we will focus on the three-dimensional data acquisition and give a short description of the thinning procedure, the construction of morphological skeletons, and the measurements based on these skeletons, for the three-dimensional case.

Figure 1. Original images of CT scans of Madracis mirabilis (a and c) with corresponding images generated with a surface rendering technique (b and d). (a and b) represents a shallow water (10m) growth form of Madracis mirabilis and (c and d) a deep water morph (30m).

Chapter 7 Three dimensional analysis of growth forms 110

Figure 2. Visualization of a branching object, tessellated with a triangulated mesh. (b). The object in (a) is mapped onto a discrete solid representation in a S144A3$ lattice, (c). The morphological skeleton is constructed using the lattice representation in (b). (d). The junctions, displayed in green, are determined in the skeleton and can be used to calculate branching angles, (e). At four of the branching points of the morphological skeleton maximum spheres are constructed; the radius of each sphere gives an estimation of the thickness of the branching object.

Chapter 7 Three dimensional analysis of growth forms

For the morphological analysis we have looked at growth forms of the stony coral Madracis mirabilis (Figure la,b,c,d). For the three-dimensional data acquisition we have used X-ray CT (Computed Tomography) scanning techniques. The CT scan data was stored in DICOM format (a general data format used for medical images). The CT scan data consists o f 5 1 2 x 5 1 2 x Z (we used z= 50) three-dimensional pixels, the so-called "voxels". The slice thickness of the CT scan data is 2.5 mm in the XY-direction. Each voxel represents a density value between 0 and 2A12, where 0 is the lowest density (the air around the coral skeleton), while high values indicate the calcium carbonate of the coral skeleton. Within the coral skeleton these density values vary (see Figure la,c); in a number of cases, growth layers within the skeleton can be distinguished using these density variations. In Figure lc and d two images are shown of CT scans of the Madracis mirabilis growth forms shown in Figure la and b. In this figure the data sets are visualized using a volume rendering (see Upson, 1991). In this figure all density values, including those for some of the surface structures (for example corallites), are visualized. The same technique can be used to visualize density variations in the skeleton, showing growth layers in sections through the data set. The surface rendering in Figure lb and d was constructed with the marching cube technique discussed in Lorensen and Cline (1987). The surface is constructed, approximately, at the boundary between air and the calcium carbonate skeleton of the coral. With this technique, using the original data set of 512 x 512 x z voxels, an image is reconstructed with an equal resolution in X, Y, and Z directions (see for details Schroeder et al. 1997). In these images only the surface of the coral is visualized, without any surface structures such as corallites. On the voxels representing the surface of the corals a triangulated mesh was constructed using this surface rendering technique. This triangulated surface representation can be used to map the form onto a lattice of 512A3 voxels, with an equal resolution in the X, Y, and Z directions. An example is shown of a simple branching object, which is visualized using a surface rendering technique where the surface is tessellated with a triangular mesh. Figure 2b is obtained by mapping the object shown in Figure 2a onto a 144A3 lattice. The original surface is now converted into a solid and discrete lattice representation. This three-dimensional lattice representation of the branching object corresponds to the two-dimensional discrete images used in the previous section (pp. 77-86 in Kaandorp and Kuebler, 2001),where a voxel in the state T represents the object, while a voxel in state '0' represents the environment. This lattice representation can be used as the input data set for a three-dimensional version of the thinning algorithm applied in the previous section (pp. 77-86 in Kaandorp and Kuebler, 2001). An overview of three-dimensional

Chapter 7 Three dimensional analysis of growth forms 112

thinning techniques can be found in Jonker and Vossepoel (1995). In theory these techniques construct the medial axis or medial plane in a three-dimensional object. Figure 2c shows a slice made through a lattice obtained by mapping the surface shown in Figure 2a onto a 512A2 lattice. For every point in the coral, the shortest distance to the environment, is measured by constructing a sphere in every voxel in the coral, which is extended until the surface of the sphere reaches the exterior of the coral. The radius of this sphere is used to estimate the shortest distance from eveiy coral voxel to the environment. The medial axis can now, as in the two-dimensional case, be defined as a curve connecting the local maxima in the three-dimensional "distance map". In three dimensions the ridge of local maxima might be arranged along an axis (for example when thinning a cylindrical object) or consist of a plane of maxima (for example when thinning a flattened object). Slices of the discrete representation, obtained by mapping the surface shown in Figure, lb onto a 512A3 lattice at respectively the level j = 151$ (a) and j = 251$ (b), approximately through the middle of the colony are shown in Figure 3. The color gradient indicates, for every point in the coral, the shortest distance to the environment. Blue indicates the points which are relatively close to the environment, while the white points are more remote from the environment. In Figure. 2c the morphological skeleton of the solid object shown in Figure 2a. is constructed by applying the thinning algorithm described by Tsao and Fu (1981). As in the measurements done in section (pp. 77-86 in Kaandorp and Kuebler, 2001), this skeleton can used to measure several morphological properties, such as the thickness of branches and the determination of branching points, branching angles, branch spacing etc. In the next stage, shown in Figure 2d, the branching points (junctions) are determined in the skeleton. In Figure 2e the result of Figure 2d, is used to construct maximum spheres, where the centers of the spheres are located at the branching points in the morphological skeleton. The radius of these spheres gives an estimation of the maximal thickness of the branches and corresponds to the da measure discussed in section (pp. 77-86 in Kaandorp and Kuebler, 2001). The thinning algorithm described by Tsao and Fu (1981)gives reasonable results for relatively simple branching objects as shown in Figure 2a. One of the major pitfalls in many of the thinning techniques, for example in the algorithm applied above, is the generation of new branches due to small perturbations at the surface of the object. This phenomenon is demonstrated in Figure 4. A morphological skeleton consists of one medial axis, and in Figure 4 the addition of a small perturbation results in the formation of a new side branch. Especially for large three-

Chapter 7 Three dimensional analysis of growth forms 113

Figure 3. Distance of the various positions in the colony of the Madracis mirabilis colony from (Fig lb) to the environment are visualized by a color gradient. The slices at j= 151$ (a) and j= 215$ (b) are shown.

dimensional objects, such as the three-dimensional lattice representation of the object shown in Figure lb, the result is a highly complex ("bushy") skeleton. For practical measurements this complicated structure is virtually useless. In a forthcoming paper (Vermeij et al. in prep.), we are planning to analyze three-dimensional data sets obtained from CT scans of various Madracis species by using an improved version of the thinning technique, producing less "bushy" skeletons which are more suitable for carrying out measurements such as those shown in Figure 4.

Figure 4. Morphological skeleton resulting from applying the thinning procedure to the three-dimensional lattice representation of the coral shown in Figure lb.

Acknowledgements We thank Sean Spicer (Stanford University) for the program that allowed us to make Figure 3. Robert Belleman and Rafael Garcia Leiva (Section Computational Science, Universiteit van Amsterdam) helped with the production of the other figures.

Chapter 7 Three dimensional analysis of growth forms 114

CITED LITERATURE Jonker PP and Vossepoel AM (1995) Mathematical morphology in 3D images:

comparing 2D and 3D skeletonization algorithms. In: Wojciechowski K ed. BENEFIT Summerschool School on Morphological Image and Signal Processing, Zakopane, pp. 83-108, Silesian Technical University, ACECS, Gliwice, Poland, 1995.

Kaandorp JA and Kuebler J (2001) The Algorithmic Beauty of Seaweeds, Sponges and Corals. Springer Verlag, Heidelberg, New York. pp. 193.

Lorensen WE and Cline HE (1987) Marching cubes: a high resolution 3D surface construction algorithm. ACM Computer Graphics 21(4): 163-169.

Schroeder WK, Martin K and Lorensen B (1997) The Visualization Toolkit: An Object-Oriented Approach to 3D Graphics. 2"d Edition, Prentice Hall, New Jersey.

Tsao YF and Fu KS (1981) A parallel thinning algorithm for 3-D pictures. Computer Graphics and Image Processing 17: 315-331.

Upson C (1991) Volumetric visualization techniques. In: Rogers DF and Earnshaw RA cds. State of the art in Computer Graphics: 313-350. Springer-Verlag, Berlin.

Vermeij MJA, Kaandorp JA, Bak RPM, Lampmann LEH and Sloot PMA (In preparation) Three dimensional analysis of growth forms of Madracis species.

Chapter 7 Three dimensional analysis of growth forms 115

Chapter 7 Three dimensional analysis of growth forms 116