p. 336-355 In: Mondaini, R. (ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology. E-papers, Rio de Janeiro, 2004.

Geometric methods combining contour and landmark information in the statistical analysis of biological shape. Leandro R. Monteiro1 , Luis H. Guillermo2 , Luis A. Rivera2 , Ana P. M. Di Beneditto1 1

Laborat´ orio de Ciˆencias Ambientais, Centro de Biociˆencias e Biotecnologia, 2

Laborat´ orio de Ciˆencias Matem´ aticas, Centro de Ciˆencias e Tecnologia, Universidade Estadual do Norte Fluminense.

Av. Alberto Lamego 2000, Campos dos Goytacazes, RJ, cep 28013-600. E-mail: {lrmont, guillerm, rivera, anapaula}@uenf.br

Abstract

sagitta otolith shape variation in two related species of sciaenid fish (genus Stellifer) was related to fish size and species using shape information extracted by the spline relaxation algorithm. The application of such techniques for evolutionary, ecological and fisheries research is discussed.

The geometric methods for the statistical analysis of biological shape have traditionally been based on landmarks: points of reference in a biological structure that presented a degree of correspondence (or homology) within and across samples. After the morphometric synthesis and the geometric ”revolution” in morphometrics, the contours of biological structures were considered unsatisfactory for biological purposes, because the coordinates of points along the outline of an object lack the biological correspondence of landmarks. A recent development was the algorithm of spline relaxation or sliding semilandmarks (points of reference along a shape outline without known correspondence within and across samples) that allows one to incorporate outline information by defining directions (tangents to the outline) along which the contour points can be slid in order to reduce the bending energy criterion of the thin plate spline. After filtering out the tangential differences among individuals, the information regarding shape variation in semilandmarks is restricted to directions perpendicular to the object outline. This approach allows for the combination of landmarks (homologous reference points) with semilandmarks (non-homologous points) in the same statistical analysis and is a very powerful tool for the analysis of shape variation patterns in biological structures with few or no landmarks. As an example of the application of the technique, the

Keywords: Outline shape, Semilandmarks, Thin plate spline, Fish otoliths, Stellifer.

1

Introduction

The study of biological shape variation and its relation with causal factors have gone through a revolution during the last decade, after the so-called Morphometric Synthesis [2], which combined the Riemannian shape space of David Kendall [19], multivariate statistical analyses in a tangent space, and a set of visualization techniques for the results [3]. As a consequence, the focus of morphometric techniques shifted from outline data [29], where the individual points carrying shape information bear no biological correspondence among specimens, to landmark data, where the relevant information is contained in reference points that are biologically correspondent among specimens (operationally homologous). There is little debate around the fact that landmarks are a richer source of information for biological studies [2] than the contours or outlines of structures, however, it is also important to note that certain biological structures present few 1

or no landmarks, but have been successfully studied by outline methods, such as mollusc, crustacean and protozoan shells [12, 20], plant leaves, certain parts of the vertebrate skeleton [39], fish otoliths [16] and brain structures [4]. Furthermore, it has been shown empirically (by the application of image unwarping techniques) that, for most data sets, no landmark set is complete enough to depict all shape variation present in biological structures (see TPSsuper software [32]). The statistical analysis of shape using geometric methods is currently based on a powerful method, called Procrustes analysis, that performs a least squares superimposition of landmarks in corresponding positions. The superimposition generates a shape manifold (Kendall’s shape space) with a Riemannian metric called Procrustes distance [37, 11]. Compared to alternative methods, based on interlandmark distances and angles, the methods based on landmark coordinates are more powerful and allow for less biased estimates of average shapes [30, 33]. Part of the advantages of coordinate and superimposition based methods originate from the topology of Kendall’s shape space, which is a multidimensional sphere (manifold), whereas the techniques based on interlandmark distances generate spaces with odd topologies, with restrictions and boundaries that artificially constrain shape variation in certain directions [25, 31]. Given the advantages of the shape manifolds used for the study of landmark sets, it is desirable for outline methods to use them as well for the statistical study of contour shape. As a result, the focus of outline methods have shifted from the fitting of functions to object contours to approaches that used the Procrustes metric or at least incorporate contour information to existing landmark configurations. There were attempts to combine landmarks with outlines of biological structures by incorporating contour derivative information to landmark data [5], or by combining the Procrustes superimposition to eigenshape (using normal deviations in each point instead of tangent angle functions) analysis of outlines [35]. The algorithm of spline relaxation proposed by Bookstein [4] combines the powerful techniques based on the Procrustes superimposition with the localization of shape differences provided by the thin plate spline function to extract shape information from both landmarks and outlines in the

context of the same shape spaces currently used in the methods of the morphometric sinthesys. Our purpose in this study is to show the mathematical details of the spline relaxation algorithm and an application to the covariation of shapes and causal factors in a biologically relevant context.

2

Thin-plate splines semilandmarks

using

The general basic problem is: given k points P1 , ..., Pk ∈ IRn and h = (h1 , ..., hk ) ∈ IRk , we seek a function f , depending of the k + 1 + n coeficients (w, a0 , a) ∈ IRk × IR × IRn , of the form

n

IR 3 x → f (x) = a0 + a.x +

k X

wj U (x − Pj ),

j=1

where U : IRn → IR is defined by U (x) = ||x||2 ln ||x||, for all non zero vector x, such that 1) f (Pi ) = hi , 1 ≤ i ≤ k 2) lim||x||→∞ f (x) = constant R 3) IRn ||Hess(f )(x)||2 dx be minimum between such functions. Here 2

||Hess(f )(x)|| =

2 X  ∂2f (x) ∂xi ∂xj

1≤i,j≤n

is the trace of the square  of the Hessian matrix ∂2f of the function f at Hess(f )(x) = ∂xi ∂xj (x) the point x.

n×n

Such f is called thin-plate spline function. It is well known that the solution f for this problem is given by the solution of the following linear system: Pk j=1 wj U (Pi − Pj ) + a0 + a.Pi = hi , 1 ≤ i ≤ k Pk j=1 wj = 0 Pk j=1 wj Πj (Pi ) = 0, 1 ≤ i ≤ k (with Πj (x) = xj ∀x = (x1 , ..., xn ) ∈ IRn )

Or, in the matrix form, LW = H; where   . P .. Q   L =  ··· ···, . QT .. O 

 w  a0  , a

  h W = H  0  ∈ IRk+1+n ; with 0   .. P = (U (Pi − Pj ))k×k , Q = 11 . X k×(1+n) and the i-th line of the matrix X being the coordinates of the point Pi . In such way that the solution f is determined by W = L−1 H; which provide-us the minimum bending energy T T −1 cn W T H = (L−1 )T H = cn (L−1 k h) h = cn h Lk h;

where cn is a certain n-dimensional constant. Here L−1 k , the so called bending energy matrix, is the k×k upper left submatrix of the inverse of L. The interest in this work is for the case of the k planar points Pi = (xi , yi ) ∈ IR2 , 1 ≤ i ≤ k. In this case we have Z Z 1 T −1 2 2 2 h LK h = min [zxx +2zxy +zyy ]dxdy. 8π z=f (x,y) IR2 Here that minimum is taken on all functions z = f (x, y) =

k X

wj U (x−xj , y−yj )+a0 +a1 x+a2 y

j=1

whose coefficients LW = H above.

satisfy

the

linear

system

An interpolation in the Cartesian plane IR2 is couched as a pair (fx , fy ) of these functions based in L and for which fx uses a vector Hx of abscissas of a target form and fy uses a vector Hy of ordinates. The bending energy being minimized is now the quadratic form   Hx −1 T T ( H x Hy ) L k . Hy This is the way to extend the above method so that some of the target landmarks are freed to slide along lines. let be the vector Y 0 = (Yx0 , Yy0 ) ∈ IR2k ,

with Yx0 and Yy0 having the abscissas and ordinates, respectively, of the right handlandmarks Yj0 , 1 ≤ j ≤ k. In this context we seek the spline of one set of landmarks Xj , 1 ≤ j ≤ k onto another set of landmarks Yj , 1 ≤ j ≤ k of which the elements of a subset {Yji }i∈I are free to slide away from their nominal positions Yj0i along vectors ui ∈ IR2 . In order to minimize the bending energy   Yx −1 T T ( Y x Yy ) Lk Yy as that landmarks Yji range over the lines ti 7→ Yji + ti ui , we consider the parameter vector t = (t1 , ..., tm ) and the 2k × m matrix U obtained from the directions u1 , ..., um in the following form ( ujx if l = ij Ulj = ujy if l = k + ij 0 otherwise. Therefore the task here is minimize the quadratic form   Yx −1 T T Y T L−1 Y = ( Y Y ) L x y k k Yy over the hiperplane Y = Y 0 + U t. It is well known the following solution for this weighted least squares problem: −1 T −1 0 t = −(U T L−1 U Lk Y . k U)

The extraction of shape variables is performed by the decomposition of the bending energy matrix L−1 k using its eigenvectors and eigenvalues T L−1 k = EΛE ;

where the columns of E are the eigenvectors of L−1 k associated with the corresponding eigenvalues of the diagonal matrix Λ. The eigenvectors of the bending energy matrix are called principal warps and the eigenvalues are called bending energies. The magnitude of the bending energies can be used as an index of localization for the shape deformations depicted by each principal warp (smaller bending energies are associated with warps in larger scales). The projection of the target matrices (after sliding) on the space spanned by principal warps is accomplished by f = Y EΛ−α/2 . W

The resulting scores are called partial warps and can be used as variables in multivariate statistical analyses of shape variation.

3

Application to a biological problem: Ontogenetic and interspecific differences in otolith shape

The sagitta otoliths are calcium carbonate concretions in the inner ear of fish, that act as sound transducers and play an important role in fish hearing [13]. Because of its accretionary growth and species (sometimes population) specific shape, they can be used as tools in fish aging [14, 18] determination of stock relationships [9, 38], ecomorphological studies [41, 43], and identification of fish species in archaeological or fossil samples [6, 42] or in the stomach content samples of predators for dietary item identification [8, 10]. It is generally thought that the size of the otoliths can be influenced by environmental factors such as salinity, depth and temperature (by means of an indirect influence on fish growth), whereas otolith shape is species specific and show less variation among conspecific individuals (even in some ontogenetic series) [1]. The reason for such stability would be a biological constraint posed by its function as a sound transducer [13]. In this context, different shapes would result in different mechanical efficiencies for hearing at different frequencies [27]. Of particular importance, would be the relative sizes of otoliths and the underlying maculas (a pad of hairy cells that sense the otolith vibrations), which can be approximated by the relative size of the sulcus (Figure ??). Although the quantitative measure of shape variation of otoliths is considered an important index for species discrimination and for the testing of hypotheses related to function and the ecological significance of shape differences, most studies use low information measures of shape variation, such as ratios of areas or linear distances [13, 1, 43]. Only recently, more sophisticated attempts at shape quantification have been reported in the literature, mostly using Fourier decompositions of the otolith or sulcus outlines [41, 9, 16]. Given the ac-

Figure 1: Otolith in distal view showing the semilandmarks and landmarks digitized. Points 1 to 50 are semilandmarks equally spaced along the contour. Points 51 to 54 are landmarks located along the sulcus acusticus. cretionary nature of otolith growth by deposition of material from the endolymphatic fluid [26], and the close relationship of growth with check rings (localized ultrastructural discontinuities in otolith surface), a method that is sensitive to local changes normal (at right angle) to outline shape should be highly informative about the relationship of otolith shape with endogenous or exogenous causal factors. The algorithm extending the thin plate splines and warp analysis to sliding semilandmarks proposed by Bookstein [4] should fit this purpose and provide an importan means for extracting information from this rich biological structure. The fish species studied here (Stellifer brasiliensis and S. rastrifer ) are demersal (living close to the bottom of the sea) and are found in estuarine or shallow coastal areas, associated with muddy or sandy bottoms along the South American Atlantic Coast. Our sample (29 S. brasiliensis and 28 S. rastrifer ) was collected by bottom trawls in the North Shore of Rio de Janeiro State, Brazil, during the year of 1998. The otoliths were removed from the specimens and photographed (only the right otoliths in distal view) by a digital camera Pixera Professional connected to a Zeiss stereomicroscope. The images were thresholded and binarized for contour extraction by ImageJ software, developed by W. Rasband at the NIH (http://rsb.info.nih.gov/ij/). The outline of each specimen was stored as pixel coordi-

nates (reaching 1600 entries in the larger specimens) which were reduced to 50 points equally spaced (using contour length s as a parameter) along the contour, starting at the intersection of the major axis with the anterior margin of the otolith. In general, the program could compute k points uniformly sampled in segment of s/k. The axes are the eigenvectors of the covariance matrix of the initial k points sampled. Figure ?? shows the contour of the image with k = 50 points sampled and the main axes.

Figure 2: Points sampled on the contour and the main axes. After the automatic collection of contour data (semilandmarks), four landmarks, corresponding to reference points (biologically corresponding among specimens) along the sulcus and the collum were digitized and their Cartesian coordinates stored along with the semilandmarks for the thin plate spline analysis (Figure ??).The sliders for each semilandmark were defined (as explained above) as tangent vectors to the outline in the position of the point. This tangent was defined as the chord between the previous and next points to the semilandmark. The partial warps (the shape variables extracted by the thin plate spline) calculated after the sliding algorithm contained information on shape differences normal to the otolith contours. These shape variables were used in multivariate analyses of major axis of variation (relative warps) depicting between species differences and a within species analysis of size related shape variation (otolith allometry). The relative warps are principal components in the space of partial warps [2], a space tangent to Kendall’s shape space in the vicinity of the

mean shape configuration. The mean shapes calculated for each species is depicted in Figure ??. The allometric analysis was performed by partial least squares [34]. This technique solves the multivariate problem of finding linear combinations in a group of variables (the shape variables) that maximize covariation with a second group of variables (size, a single variable in this case). In matrix notation, this procedure can be written as S12 = F1 DFT2 , where the rows of S12 are the variables in set 1 and the columns are the variables in set 2. The F matrices are linear combinations of variable sets that provide the best least-squares approximation to the matrix of covariances S12 . D is a diagonal matrix of singular values that are proportional to the covariance between linear combinations in F1 and F2 . Statistical tests for significant associations between the two sets of variables were conducted using permutation tests, which were carried out repeating the analyses with 999 independent random permutations of specimen ordering in the two data sets. Shape changes associated with the linear combinations obtained by PLS (hereby referred as PLS shape vectors) were visualised as deformed grids.Detailed descriptions of the statistical foundations of the PLS procedure are given by Streissguth et al. [40] and Rohlf and Corti [34]. The amount of shape variation explained by each partial leas squares (PLS) vector was calculated by sums of Procrustes distances from observed to reconstructed specimens, relative to the sums of Procrustes distances from each specimen to the grand mean shape [24]. The size variable used was the standard length of the specimen in cm, measured from the tip of the snout to the base of the caudal fin.

4

Results

The analysis of otolith allometry showed significant results in both species. For Stellifer brasiliensis, the partial least squares vector (there is a single linear combination because there was a single variable in the second group - fish size) explained 9.28% of total otolith shape variation within species, and a statistically significant relationship as determined from

relative position of the sulcus and collum landmarks relative to the otolith outline. The first two relative warps explained 70.1% of total shape variation (Figure ??). The first axis depicts the shape differences between species, which is related to differences localized along the anterior and posterior margin (Figure ??). Stellifer brasiliensis presents a relatively smaller ostium (area between the collum and the anterior margin), and a relatively larger area posterior to the sulcus. Even considering the interindividual variation, the shape differences between species are so large that it is possible to assign all individuals to the correct species by shape alone. Because there is a mean size difference between species (S. rastrifer individuals are larger than S. brasiliensis), we examined the possibility that interspecific shape differences were associated with allometric patterns. Comparing the axis of interspecific shape variation (approximated by Relative Warp 1) with the standard length of the specimens (Figure ??), it is possible to see that there is no relationship between size and the shape changes depicted in Relative Warp 1.

5 Figure 3: Mean shapes of Stellifer brasiliensis and S. rastrifer calculated by Procrustes analysis and the sliding landmark relaxation.

the permutation tests (P = 0.01). This shape vector was highly correlated with the specimens’ standard length and depicted the pattern of ontogenetic variation (Figure ??). As the specimens increase in size, there is a relative growth of the anterodorsal margin, as well as a shift in the relative position and orientation of the sulcus and the collum. For S. rastrifer, the partial least squares vector explained 9.26% of total otolith shape variation within species and a statistically significant relationship as determined from the permutation tests (P = 0.02). The shape changes relative to fish size are depicted in Figure ?? and show a relative increase of the anteroventral margin, concomitant with a relative decrease (probably caused by negative allometric growth) of the posterodorsal margin, changing the

Discussion

Our results in the otolith analysis indicate that the extension of landmark geometric morphometric methods to outline data provided by warp analysis of semilandmarks have a great potential in the analysis of shape of structures that combine landmarks and outlines, and that mostly show localized shape changes. The form of accretionary growth in the otoliths [15, 26], and other biological structures with similar growth, such as mollusc shells [7] result in a pattern where the points in the surface of the structure do not show an explicit correspondence over an ontogenetic series, but where shape changes are accomplished by normal displacements on the surface. These normal displacements are exactly the only type of variation allowed for by the semilandmark extension of the thin plate spline [4]. The localization of biologically important shape differences caused by differential deposition of material (the check ring hypothesis of Gauldie [13]) also matches with the hierarchical nature of progressively localizable, decreasing scale of partial warps [2]. The eigenvalues of the bending energy

Figure 4: Partial least squares results for the correlation of shape variables and size (fish standard length) for S. brasiliensis. Upper grid shows predicted shape differences for small sized specimens (positive scores in PLS vector) relative to mean shape as grid deformations. Lower grid shows predicted shape differences for large sized specimens (negative scores in PLS vector). Scatterplot shows correlation between PLS vector scores for otolith shape and Fish size.

Figure 5: Partial least squares results for the correlation of shape variables and size (fish standard length) for S. rastrifer. Upper grid shows predicted shape differences for large sized specimens (positive scores in PLS vector) relative to mean shape as grid deformations. Lower grid shows predicted shape differences for small sized specimens (negative scores in PLS vector). Scatterplot shows correlation between PLS vector scores for otolith shape and Fish size.

Figure 6: Ordination of specimens in the space of the first two relative warps. matrix provide an index of scale for each partial warp, indicating the bending energy needed to accomplish the shape change depicted by a particular warp. By analogy with the metal plate modeled by the thin plate spline, the larger the scale, the less energy required for the shape change [3]. The size related shape variation observed within species was small, but significant. A number of papers have reported the absence of allometric shape changes in otoliths [13, 1]. However, this pattern is not found as a rule, for there might be allometric changes in the area ratios [21]. As an aside, the shape variables used by such papers are usually ratios of areas (i.e. macular area or sulcus area versus otolith area), which can be maintained constant over a range of shape changes. Studies using more sophisticated methods of statistical shape analysis show that there should be a detectable allometric effect in individual differences [36] as we found in the present study. The small allometric effect possibly arises from a biological constraint in otolith shape and function. Because of their function as sound transducers, otolith shape variation should influence the frequency sensitivities and directional hearing in fish. As a result, the within species shape variation should be conservative to maintain function [23]. The shape differences observed in relation to fish size indicate that for both species otolith growth is highly localized in the anterior or posterior margins, what could result in a pattern of negative allometric growth of otolith width, commonly

Figure 7: Shape changes depicted by the first relative warp. Upper grid shows changes related to positive scores (S. rastrifer ) relative to grand mean shape. Middle grid shows the grand mean shape (reference configuration used). Lower grid shows changes related to negative scores (S. brasiliensis) relative to grand mean shape.

observed in other species [22]. This pattern might arise from an anatomical constraint that prevents the deposition of material along the ventral margin [15]. The ontogenetic variation in the relative position of the sulcus observed in both species might generate functional differences in hearing for fishes of different sizes, because of the relative growth of anterior and posterior regions of the otoliths that act as levers in the mechanism of sound transduc-

otolith shape in the hearing function is still missing in the literature [27].

Figure 8: Scatterplot of relative warp scores versus fish standard length for the two species studied. tion [13]. Furthermore, the changes in relative position of the sulcus might indicate a change in orientation of the sensory epithelium, that could have an importance in localization of the sound source. Further research would be needed to clarify this issue, including bioacoustic analyses and a better description and quantitative analysis of diet and habitat use. The interspecific comparisons showed a large discontinuity between the two species in shape space, explaining around 60% of total variation, what confirms the tendency of within species otolith shape variation to be smaller than among species variation [16]. It is generally thought that otolith size is directly influenced by fish growth [26] although there can be a decoupling between somatic and otolith growth caused by environmental factors [17]. On the other hand, it is also thought that shape is mostly controlled by genetic factors [1], possibly arising from the functional constraints observed [13]. The shape differences observed between species are localized in the anterior and posterior margins and depict relative differences in size of the anterior portion (ostium) of the sulcus and the posterior region of the otolith. As a consequence there is a difference in the relative position of the collum, which is a fulcrum in the lever system of sound transduction [13]. It is hypothesized that such shape differences might have an effect in the frequency threshold heard by a fish species. However, a more complete understanding of the effect of

The ecology of Stellifer brasiliensis and S. rastrifer is poorly known. These species occur in soft (mud or sand) substrates along the Atlantic Coast of South America, where they feed on small bottom dwelling invertebrates, such as crustaceans [28]. Hearing is an important part of the sensory apparatus for these animals, since they probably use it to locate food items, conspecifics for mating and avoid predators by hearing the ultrasound produced by dolphins [27]. Whether the shape differences observed are indicative of functional differences in the hearing of sounds between species remains to be clarified by further research. Comparative ecological studies among these closely related species could be important in this issue, for the shape differences observed are large enough to provide evidence that such a pattern could be found in this case, as shown by comparisons of other closely related species [1].

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