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

1Laboratorio de Ciencias Ambientais, Centro de Biociencias e Biotecnologia,2Laboratorio de Ciencias Matematicas, Centro de Ciencias 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

The geometric methods for the statistical analysisof biological shape have traditionally been based onlandmarks: points of reference in a biological struc-ture that presented a degree of correspondence (orhomology) within and across samples. After themorphometric synthesis and the geometric ”revolu-tion” in morphometrics, the contours of biologicalstructures were considered unsatisfactory for bio-logical purposes, because the coordinates of pointsalong the outline of an object lack the biological cor-respondence of landmarks. A recent developmentwas the algorithm of spline relaxation or slidingsemilandmarks (points of reference along a shapeoutline without known correspondence within andacross samples) that allows one to incorporate out-line information by defining directions (tangents tothe outline) along which the contour points can beslid in order to reduce the bending energy crite-rion of the thin plate spline. After filtering out thetangential differences among individuals, the infor-mation regarding shape variation in semilandmarksis restricted to directions perpendicular to the ob-ject outline. This approach allows for the combi-nation of landmarks (homologous reference points)with semilandmarks (non-homologous points) inthe same statistical analysis and is a very powerfultool for the analysis of shape variation patterns inbiological structures with few or no landmarks. Asan example of the application of the technique, the

sagitta otolith shape variation in two related speciesof sciaenid fish (genus Stellifer) was related to fishsize and species using shape information extractedby the spline relaxation algorithm. The applicationof such techniques for evolutionary, ecological andfisheries research is discussed.

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

1 Introduction

The study of biological shape variation and its rela-tion with causal factors have gone through a revolu-tion during the last decade, after the so-called Mor-phometric Synthesis [2], which combined the Rie-mannian shape space of David Kendall [19], multi-variate statistical analyses in a tangent space, anda set of visualization techniques for the results [3].As a consequence, the focus of morphometric tech-niques shifted from outline data [29], where theindividual points carrying shape information bearno biological correspondence among specimens, tolandmark data, where the relevant information iscontained in reference points that are biologicallycorrespondent among specimens (operationally ho-mologous). There is little debate around the factthat landmarks are a richer source of informationfor biological studies [2] than the contours or out-lines of structures, however, it is also important tonote that certain biological structures present few

1

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

or no landmarks, but have been successfully stud-ied by outline methods, such as mollusc, crustaceanand protozoan shells [12, 20], plant leaves, certainparts of the vertebrate skeleton [39], fish otoliths[16] and brain structures [4]. Furthermore, it hasbeen shown empirically (by the application of im-age unwarping techniques) that, for most data sets,no landmark set is complete enough to depict allshape variation present in biological structures (seeTPSsuper software [32]).

The statistical analysis of shape using geometricmethods is currently based on a powerful method,called Procrustes analysis, that performs a leastsquares superimposition of landmarks in corre-sponding positions. The superimposition generatesa shape manifold (Kendall’s shape space) with aRiemannian metric called Procrustes distance [37,11]. Compared to alternative methods, based oninterlandmark distances and angles, the methodsbased on landmark coordinates are more powerfuland allow for less biased estimates of average shapes[30, 33]. Part of the advantages of coordinate andsuperimposition based methods originate from thetopology of Kendall’s shape space, which is a mul-tidimensional sphere (manifold), whereas the tech-niques based on interlandmark distances generatespaces with odd topologies, with restrictions andboundaries that artificially constrain shape varia-tion in certain directions [25, 31]. Given the ad-vantages of the shape manifolds used for the studyof landmark sets, it is desirable for outline meth-ods to use them as well for the statistical study ofcontour shape. As a result, the focus of outlinemethods have shifted from the fitting of functionsto object contours to approaches that used the Pro-crustes metric or at least incorporate contour infor-mation to existing landmark configurations.

There were attempts to combine landmarks withoutlines of biological structures by incorporatingcontour derivative information to landmark data[5], or by combining the Procrustes superimposi-tion to eigenshape (using normal deviations in eachpoint instead of tangent angle functions) analysisof outlines [35]. The algorithm of spline relaxationproposed by Bookstein [4] combines the powerfultechniques based on the Procrustes superimpositionwith the localization of shape differences providedby the thin plate spline function to extract shape in-formation from both landmarks and outlines in the

context of the same shape spaces currently used inthe methods of the morphometric sinthesys. Ourpurpose in this study is to show the mathemati-cal details of the spline relaxation algorithm and anapplication to the covariation of shapes and causalfactors in a biologically relevant context.

2 Thin-plate splines using

semilandmarks

The general basic problem is: given k pointsP1, ..., Pk ∈ IRn and h = (h1, ..., hk) ∈ IRk, we seeka function f , depending of the k + 1 + n coeficients(w, a0, a) ∈ IRk × IR × IRn, of the form

IRn ∋ x → f(x) = a0 + a.x +

k∑

j=1

wjU(x − Pj),

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

3)∫

IRn ||Hess(f)(x)||2dx be minimum betweensuch functions.

Here

||Hess(f)(x)||2 =∑

1≤i,j≤n

[∂2f

∂xi∂xj(x)

]2

is the trace of the square of the Hessian matrix

Hess(f)(x) =(

∂2f∂xi∂xj

(x))

n×nof the function f at

the point x.

Such f is called thin-plate spline function. It iswell known that the solution f for this problem isgiven by the solution of the following linear system:

∑kj=1 wjU(Pi − Pj) + a0 + a.Pi = hi, 1 ≤ i ≤ k

∑kj=1 wj = 0

∑kj=1 wjΠj(Pi) = 0, 1 ≤ i ≤ k (with Πj(x) =

xj ∀x = (x1, ..., xn) ∈ IRn)

Or, in the matrix form, LW = H; where

L =

P... Q

· · · · · ·

QT... O

,

W =

wa0

a

, H

h00

∈ IRk+1+n; with

P = (U(Pi − Pj))k×k, Q =(

11... X

)k×(1+n)

and the i-th line of the matrix X being the coordi-nates of the point Pi.

In such way that the solution f is determined byW = L−1H; which provide-us the minimum bend-

ing energy

cnWT H = (L−1)T H = cn(L−1k h)T h = cnhT L−1

k h;

where cn is a certain n-dimensional constant. HereL−1

k , the so called bending energy matrix, is the k×kupper left submatrix of the inverse of L.

The interest in this work is for the case of the kplanar points Pi = (xi, yi) ∈ IR2, 1 ≤ i ≤ k. In thiscase we have

1

8πhT L−1

K h = minz=f(x,y)

∫ ∫

IR2

[z2xx+2z2

xy+z2yy]dxdy.

Here that minimum is taken on all functions

z = f(x, y) =

k∑

j=1

wjU(x−xj , y−yj)+a0+a1x+a2y

whose coefficients satisfy the linear systemLW = H above.

An interpolation in the Cartesian plane IR2 iscouched as a pair (fx, fy) of these functions basedin L and for which fx uses a vector Hx of abscissasof a target form and fy uses a vector Hy of ordi-nates. The bending energy being minimized is nowthe quadratic form

(HTx HT

y )L−1k

(Hx

Hy

).

This is the way to extend the above method sothat some of the target landmarks are freed to slidealong lines. let be the vector Y 0 = (Y 0

x , Y 0y ) ∈ IR2k,

with Y 0x and Y 0

y having the abscissas and ordinates,respectively, of the right handlandmarks Y 0

j , 1 ≤j ≤ k. In this context we seek the spline of one setof landmarks Xj , 1 ≤ j ≤ k onto another set oflandmarks Yj , 1 ≤ j ≤ k of which the elements ofa subset {Yji

}i∈I are free to slide away from theirnominal positions Y 0

jialong vectors ui ∈ IR2.

In order to minimize the bending energy

(Y Tx Y T

y )L−1k

(Yx

Yy

)

as that landmarks Yjirange over the lines ti 7→

Yji+ tiui, we consider the parameter vector t =

(t1, ..., tm) and the 2k ×m matrix U obtained fromthe directions u1, ..., um in the following form

Ulj =

{ujx

if l = ijujy

if l = k + ij0 otherwise.

Therefore the task here is minimize the quadraticform

Y T L−1k Y = (Y T

x Y Ty )L−1

k

(Yx

Yy

)

over the hiperplane Y = Y 0 + Ut. It is well knownthe following solution for this weighted least squares

problem:

t = −(UT L−1k U)−1UT L−1

k Y 0.

The extraction of shape variables is performedby the decomposition of the bending energy matrixL−1

k using its eigenvectors and eigenvalues

L−1k = EΛET ;

where the columns of E are the eigenvectors of L−1k

associated with the corresponding eigenvalues of thediagonal matrix Λ. The eigenvectors of the bend-ing energy matrix are called principal warps andthe eigenvalues are called bending energies. Themagnitude of the bending energies can be used asan index of localization for the shape deformationsdepicted by each principal warp (smaller bendingenergies are associated with warps in larger scales).The projection of the target matrices (after sliding)on the space spanned by principal warps is accom-plished by

W = Y EΛ−α/2.

The resulting scores are called partial warps andcan be used as variables in multivariate statisticalanalyses of shape variation.

3 Application to a biological

problem: Ontogenetic and

interspecific differences in

otolith shape

The sagitta otoliths are calcium carbonate concre-tions in the inner ear of fish, that act as sound trans-ducers and play an important role in fish hearing[13]. Because of its accretionary growth and species(sometimes population) specific shape, they can beused as tools in fish aging [14, 18] determination ofstock relationships [9, 38], ecomorphological studies[41, 43], and identification of fish species in archae-ological or fossil samples [6, 42] or in the stomachcontent samples of predators for dietary item identi-fication [8, 10]. It is generally thought that the sizeof the otoliths can be influenced by environmen-tal factors such as salinity, depth and temperature(by means of an indirect influence on fish growth),whereas otolith shape is species specific and showless variation among conspecific individuals (evenin some ontogenetic series) [1]. The reason for suchstability would be a biological constraint posed byits function as a sound transducer [13]. In this con-text, different shapes would result in different me-chanical efficiencies for hearing at different frequen-cies [27]. Of particular importance, would be therelative sizes of otoliths and the underlying macu-las (a pad of hairy cells that sense the otolith vibra-tions), which can be approximated by the relativesize of the sulcus (Figure ??).

Although the quantitative measure of shape vari-ation of otoliths is considered an important in-dex for species discrimination and for the testingof hypotheses related to function and the ecolog-ical significance of shape differences, most studiesuse low information measures of shape variation,such as ratios of areas or linear distances [13, 1,43]. Only recently, more sophisticated attempts atshape quantification have been reported in the lit-erature, mostly using Fourier decompositions of theotolith or sulcus outlines [41, 9, 16]. Given the ac-

Figure 1: Otolith in distal view showing the semi-landmarks and landmarks digitized. Points 1 to 50are semilandmarks equally spaced along the con-tour. Points 51 to 54 are landmarks located alongthe sulcus acusticus.

cretionary nature of otolith growth by depositionof material from the endolymphatic fluid [26], andthe close relationship of growth with check rings(localized ultrastructural discontinuities in otolithsurface), a method that is sensitive to local changesnormal (at right angle) to outline shape should behighly informative about the relationship of otolithshape with endogenous or exogenous causal factors.The algorithm extending the thin plate splines andwarp analysis to sliding semilandmarks proposed byBookstein [4] should fit this purpose and providean importan means for extracting information fromthis rich biological structure.

The fish species studied here (Stellifer brasilien-

sis and S. rastrifer) are demersal (living closeto the bottom of the sea) and are found in es-tuarine or shallow coastal areas, associated withmuddy or sandy bottoms along the South Ameri-can Atlantic Coast. Our sample (29 S. brasilien-

sis and 28 S. rastrifer) was collected by bottomtrawls in the North Shore of Rio de Janeiro State,Brazil, during the year of 1998. The otoliths wereremoved from the specimens and photographed(only the right otoliths in distal view) by a dig-ital camera Pixera Professional connected to aZeiss stereomicroscope. The images were thresh-olded and binarized for contour extraction by Im-ageJ software, developed by W. Rasband at theNIH (http://rsb.info.nih.gov/ij/). The out-line of each specimen was stored as pixel coordi-

nates (reaching 1600 entries in the larger specimens)which were reduced to 50 points equally spaced (us-ing contour length s as a parameter) along the con-tour, starting at the intersection of the major axiswith the anterior margin of the otolith. In general,the program could compute k points uniformly sam-pled in segment of s/k. The axes are the eigenvec-tors of the covariance matrix of the initial k pointssampled. Figure ?? shows the contour of the imagewith k = 50 points sampled and the main axes.

Figure 2: Points sampled on the contour and themain axes.

After the automatic collection of contour data(semilandmarks), four landmarks, corresponding toreference points (biologically corresponding amongspecimens) along the sulcus and the collum weredigitized and their Cartesian coordinates storedalong with the semilandmarks for the thin platespline analysis (Figure ??).The sliders for eachsemilandmark were defined (as explained above) astangent vectors to the outline in the position of thepoint. This tangent was defined as the chord be-tween the previous and next points to the semiland-mark. The partial warps (the shape variables ex-tracted by the thin plate spline) calculated after thesliding algorithm contained information on shapedifferences normal to the otolith contours. Theseshape variables were used in multivariate analysesof major axis of variation (relative warps) depict-ing between species differences and a within speciesanalysis of size related shape variation (otolith al-lometry). The relative warps are principal compo-nents in the space of partial warps [2], a space tan-gent to Kendall’s shape space in the vicinity of the

mean shape configuration. The mean shapes calcu-lated for each species is depicted in Figure ??. Theallometric analysis was performed by partial leastsquares [34]. This technique solves the multivariateproblem of finding linear combinations in a groupof variables (the shape variables) that maximize co-variation with a second group of variables (size, asingle variable in this case). In matrix notation,this procedure can be written as

S12 = F1DFT2 ,

where the rows of S12 are the variables in set 1and the columns are the variables in set 2. TheF matrices are linear combinations of variable setsthat provide the best least-squares approximationto the matrix of covariances S12. D is a diago-nal matrix of singular values that are proportionalto the covariance between linear combinations inF1 and F2. Statistical tests for significant associ-ations between the two sets of variables were con-ducted using permutation tests, which were carriedout repeating the analyses with 999 independentrandom permutations of specimen ordering in thetwo data sets. Shape changes associated with thelinear combinations obtained by PLS (hereby re-ferred as PLS shape vectors) were visualised as de-formed grids.Detailed descriptions of the statisti-cal foundations of the PLS procedure are given byStreissguth et al. [40] and Rohlf and Corti [34].The amount of shape variation explained by eachpartial leas squares (PLS) vector was calculated bysums of Procrustes distances from observed to re-constructed specimens, relative to the sums of Pro-crustes distances from each specimen to the grandmean shape [24]. The size variable used was thestandard length of the specimen in cm, measuredfrom the tip of the snout to the base of the caudalfin.

4 Results

The analysis of otolith allometry showed significantresults in both species. For Stellifer brasiliensis, thepartial least squares vector (there is a single linearcombination because there was a single variable inthe second group - fish size) explained 9.28% of totalotolith shape variation within species, and a statis-tically significant relationship as determined from

Figure 3: Mean shapes of Stellifer brasiliensis andS. rastrifer calculated by Procrustes analysis andthe sliding landmark relaxation.

the permutation tests (P = 0.01). This shape vec-tor was highly correlated with the specimens’ stan-dard length and depicted the pattern of ontogeneticvariation (Figure ??). As the specimens increase insize, there is a relative growth of the anterodor-sal margin, as well as a shift in the relative posi-tion and orientation of the sulcus and the collum.For S. rastrifer, the partial least squares vector ex-plained 9.26% of total otolith shape variation withinspecies and a statistically significant relationship asdetermined from the permutation tests (P = 0.02).The shape changes relative to fish size are depictedin Figure ?? and show a relative increase of theanteroventral margin, concomitant with a relativedecrease (probably caused by negative allometricgrowth) of the posterodorsal margin, changing the

relative position of the sulcus and collum landmarksrelative to the otolith outline.

The first two relative warps explained 70.1%of total shape variation (Figure ??). The firstaxis depicts the shape differences between species,which is related to differences localized along theanterior and posterior margin (Figure ??). Stel-

lifer 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, theshape differences between species are so large thatit is possible to assign all individuals to the correctspecies by shape alone. Because there is a mean sizedifference between species (S. rastrifer individualsare larger than S. brasiliensis), we examined thepossibility that interspecific shape differences wereassociated with allometric patterns. Comparing theaxis of interspecific shape variation (approximatedby Relative Warp 1) with the standard length ofthe specimens (Figure ??), it is possible to see thatthere is no relationship between size and the shapechanges depicted in Relative Warp 1.

5 Discussion

Our results in the otolith analysis indicate thatthe extension of landmark geometric morphometricmethods to outline data provided by warp analy-sis of semilandmarks have a great potential in theanalysis of shape of structures that combine land-marks and outlines, and that mostly show localizedshape changes. The form of accretionary growthin the otoliths [15, 26], and other biological struc-tures with similar growth, such as mollusc shells[7] result in a pattern where the points in the sur-face of the structure do not show an explicit cor-respondence over an ontogenetic series, but whereshape changes are accomplished by normal displace-ments on the surface. These normal displacementsare exactly the only type of variation allowed forby the semilandmark extension of the thin platespline [4]. The localization of biologically importantshape differences caused by differential depositionof material (the check ring hypothesis of Gauldie[13]) also matches with the hierarchical nature ofprogressively localizable, decreasing scale of partialwarps [2]. The eigenvalues of the bending energy

Figure 4: Partial least squares results for the cor-relation of shape variables and size (fish standardlength) for S. brasiliensis. Upper grid shows pre-dicted shape differences for small sized specimens(positive scores in PLS vector) relative to meanshape as grid deformations. Lower grid shows pre-dicted shape differences for large sized specimens(negative scores in PLS vector). Scatterplot showscorrelation between PLS vector scores for otolithshape and Fish size.

Figure 5: Partial least squares results for the cor-relation of shape variables and size (fish standardlength) for S. rastrifer. Upper grid shows predictedshape differences for large sized specimens (posi-tive scores in PLS vector) relative to mean shapeas grid deformations. Lower grid shows predictedshape differences for small sized specimens (nega-tive scores in PLS vector). Scatterplot shows corre-lation between PLS vector scores for otolith shapeand Fish size.

Figure 6: Ordination of specimens in the space ofthe first two relative warps.

matrix provide an index of scale for each partialwarp, indicating the bending energy needed to ac-complish the shape change depicted by a particularwarp. By analogy with the metal plate modeled bythe thin plate spline, the larger the scale, the lessenergy required for the shape change [3].

The size related shape variation observed withinspecies was small, but significant. A number of pa-pers have reported the absence of allometric shapechanges in otoliths [13, 1]. However, this patternis not found as a rule, for there might be allomet-ric changes in the area ratios [21]. As an aside,the shape variables used by such papers are usuallyratios of areas (i.e. macular area or sulcus area ver-sus otolith area), which can be maintained constantover a range of shape changes. Studies using moresophisticated methods of statistical shape analysisshow that there should be a detectable allometriceffect in individual differences [36] as we found inthe present study. The small allometric effect pos-sibly arises from a biological constraint in otolithshape and function. Because of their function assound transducers, otolith shape variation shouldinfluence the frequency sensitivities and directionalhearing in fish. As a result, the within species shapevariation should be conservative to maintain func-tion [23]. The shape differences observed in relationto fish size indicate that for both species otolithgrowth is highly localized in the anterior or poste-rior margins, what could result in a pattern of neg-ative allometric growth of otolith width, commonly

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

observed in other species [22]. This pattern mightarise from an anatomical constraint that preventsthe deposition of material along the ventral margin[15]. The ontogenetic variation in the relative po-sition of the sulcus observed in both species mightgenerate functional differences in hearing for fishesof different sizes, because of the relative growth ofanterior and posterior regions of the otoliths thatact as levers in the mechanism of sound transduc-

Figure 8: Scatterplot of relative warp scores versusfish standard length for the two species studied.

tion [13]. Furthermore, the changes in relative posi-tion of the sulcus might indicate a change in orien-tation of the sensory epithelium, that could have animportance in localization of the sound source. Fur-ther research would be needed to clarify this issue,including bioacoustic analyses and a better descrip-tion and quantitative analysis of diet and habitatuse.

The interspecific comparisons showed a large dis-continuity between the two species in shape space,explaining around 60% of total variation, what con-firms the tendency of within species otolith shapevariation to be smaller than among species vari-ation [16]. It is generally thought that otolithsize is directly influenced by fish growth [26] al-though there can be a decoupling between somaticand otolith growth caused by environmental fac-tors [17]. On the other hand, it is also thoughtthat shape is mostly controlled by genetic factors[1], possibly arising from the functional constraintsobserved [13]. The shape differences observed be-tween species are localized in the anterior and pos-terior margins and depict relative differences in sizeof the anterior portion (ostium) of the sulcus andthe posterior region of the otolith. As a consequencethere is a difference in the relative position of thecollum, which is a fulcrum in the lever system ofsound transduction [13]. It is hypothesized thatsuch shape differences might have an effect in thefrequency threshold heard by a fish species. How-ever, a more complete understanding of the effect of

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

The ecology of Stellifer brasiliensis and S. ras-

trifer is poorly known. These species occur in soft(mud or sand) substrates along the Atlantic Coastof South America, where they feed on small bot-tom dwelling invertebrates, such as crustaceans [28].Hearing is an important part of the sensory appara-tus for these animals, since they probably use it tolocate food items, conspecifics for mating and avoidpredators by hearing the ultrasound produced bydolphins [27]. Whether the shape differences ob-served are indicative of functional differences in thehearing of sounds between species remains to beclarified by further research. Comparative ecologi-cal studies among these closely related species couldbe important in this issue, for the shape differencesobserved are large enough to provide evidence thatsuch a pattern could be found in this case, as shownby comparisons of other closely related species [1].

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