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Biological shape analysis by digital curvature
Pattern Recognition 37 (2004) 515 – 524 www.elsevier.com/locate/patcog
Biological shape analysis by digital curvature Luciano da F. Costaa;∗ , S)ergio F. dos Reisb , Renata A.T. Arantesa , Ana C.R. Alvesc , GianCarlo Mutinaria a Cybernetic
Vision Research Group, Institute of Physics at S˜ao Carlos - IFSC - University of S˜ao Paulo, Caixa Postal 369, S˜ao Carlos, SP 13560-970, Brazil b Departamento de Parasitologia, Universidade Estadual de Campinas, 13083-970 Campinas, S˜ ao Paulo, Brazil c Programa de P. os-Graduac/a˜ o em Zoologia, Universidade Estadual Paulista, 13500-000 Rio Claro, S˜ao Paulo, Brazil Received 20 August 2002; accepted 30 July 2003
Abstract This paper reports the novel application of digital curvature as a feature for morphological characterization and classi2cation of landmark shapes. By inheriting several unique features of the continuous curvature, the digital curvature provides invariance to translations, rotations, local shape deformations, and is easily made tolerant to scaling. In addition, the bending energy, a global shape feature, can be directly estimated from the curvature values. The application of these features to analyse patterns of cranial morphological geographic di6erentiation in the rodent species Thrichomys apereoides has led to encouraging results, indicating a close correspondence between the geographical and morphological distributions. ? 2003 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. Keywords: Shape analysis; Digital curvature; Morphological and geographic distributions; Morphometric analysis; Morphological landmark
1. Introduction Biological shape is the result of endogenous and exogenous mechanisms that operate at di6erent scales of space and time and levels of organizational complexity. Endogenous mechanisms include molecular signalling between epithelial and mesenchymal cells and interactions between gene products that control gene expression and cell movement in the extracellular environment [1,2]. These molecular and cellular phenomena that underlie the dynamic processes of morphogenesis interface with exogenous ecological factors and deterministic and stochastic evolutionary forces to produce the observed complex morphologies in the 2nal shape of organisms. The description and measurement of biological shape is therefore of fundamental importance to assess variation in natural populations and also to infer its ecological and evolutionary causes. ∗ Corresponding author. Tel.: +55-16-273-9858; fax: +55-16-273-9879. E-mail address: [email protected] (L. da F. Costa).
Traditional approaches to the measurement of shape have relied on distances from which descriptors of shape are constructed as linear combinations of distances and coef2cients derived from principal components extracted from covariance matrices [3]. More recently, emphasis has been shifted to morphological landmarks as the primary source of data for the description of shape and variation in shape to address questions of ecological and evolutionary interest (e.g., [4–8]). Morphological landmarks are de2ned points in a biological structure, such as processes or sutures, which are assumed to be homologous across individuals and populations [9]. The position of morphological landmarks can be recorded as Cartesian rectangular coordinates thereby archiving the con2guration of the biological structures of interest [10]. Landmarks and their associated coordinates allow a more complete representation of the shape than afforded by distance measurements, and conceptually should provide a more natural approximation to the spatio-temporal process of morphogenesis which generates observed shapes. Landmarks are the data from which features must be extracted to provide variables for the measurement and
0031-3203/$30.00 ? 2003 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2003.07.010
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interpretation of shape and variations in shape. The Cartesian coordinates of each landmark are stored and used to extract characteristics such as the distance between each subsequent landmark in the shape. Ratios between successive distances along the shape landmarks are also considered as features. Here we introduce digital curvature as a morphometric procedure to generate features that can be used with multivariate statistical methods to analyse shape variation. Basically, the digital curvature can be understood as the di6erence between the angles of pairs of subsequent vectors de2ned by the landmark sequence (see in Section 2). As such, this measure associates an angle di6erence to every landmark point in such a way as to express the degree of sharpness of the respectively de2ned vertex. It is interesting to observe that such a methodology provides an alternative approach to traditional curvature analysis methods, where this important measure is numerically estimated along the object outline. However, instead of taking into account all curvature values, only the most prominent contour elements, corresponding to the highest curvature peaks, are considered for object representation, an approach that is closely related to previous concepts from visual psychophysics [11]. While Pavlidis [33] provides an interesting and comprehensive review of related shape analysis methods, the work by Mokhtarian et al. [34] addresses the important issue of multiscale curvature analysis for object characterization. Such an approach is particularly interesting because, in addition to being invariant to rigid body transformations (including translation, rotation and reGection), the curvature preserves almost all information about the shape under analysis. As a matter of fact, given the curvature at every point of the contour, it is possible to reconstruct the shape up to a rigid body transformation. In addition, the curvature magnitude provides a direct measurement of the saliency of the points along the contour. Thus, while straight portions of the object borders lead to near zero curvature values, abrupt variations of the contour orientation are clearly characterized by curvature peaks. Indeed, it is possible to use such approaches in order to obtain an automated identi2cation the landmark points de2ned by curvature peaks. The main motivation for adopting the digital curvature as a shape feature is that it inherits several of the unique features of the continuous curvature as a shape descriptor (e.g. [11,12]) while presenting some relevant advantages. First, it is invariant to translation and rotation, and can easily be modi2ed in order to become invariant to scale, e.g. by normalizing the shape by energy or diameter [13,14] or any other convenient normalization as recently pointed out by Llenonart [15]. Second, curvature is a local measure, in the sense that a small perturbation or shape variation at one point has limited spatial e6ect over the shape description. Moreover, and unlike its continuous counterpart, the digital curvature is also invariant to scaling. This measure has also been related to the salience of points along
the shape [11], and the sum of its squared values along the shape is known to be proportional to the bending energy stored into the shape, which can provide a useful global shape feature [16,17], also considered in this article. Finally, as shown in this paper, the digital curvature and the distance between landmarks are simple measures that can be easy and economically obtained by using elementary vector operations. Shape is a multidimensional component of variation in morphological form [18] which is expected to have a high information content regarding the evolutionary processes responsible for the observed diversity [19,20]. The description of patterns of variation in morphological shape within and among populations is therefore fundamental for de2ning the boundaries of independent evolutionary units in nature, and an important step in the recognition of such evolutionary units is the identi2cation of groups of populations that share morphological features of shape and geographic continuity over geographic space [21,22]. In this paper, we illustrate the use of digital curvature and bending energy with a description of patterns of geographic variation in cranial shape in Thrichomys apereoides (family Echimyidae), a rodent which ranges in distribution from northeastern, central, and southeastern Brazil into Paraguay and Bolivia. Throughout its distribution, T. apereoides inhabits xeric and rocky environments in caatinga and cerrado domains in Brazil and chaco in Paraguay [23–25]. Preliminary work on this species has indicated the existence of two geographic units, a northern and a southern group recognized for samples taken from a large area of its distribution, including northeastern, central and southeastern Brazil [26,27] . The question we ask is whether variation in cranial shape, measured by digital curvature, distance between landmarks and bending energy of cranial landmarks, is geographically structured and provides evidence for northern and southern groups of populations. This paper is organized as follows. The procedure of digital curvature is introduced and its mathematical foundation is presented and subsequently the formalism of digital curvature and bending energy is employed in combination with multivariate methods to analyse patterns of cranial morphological geographic di6erentiation in the rodent species T. apereoides. The results obtained are discussed in terms of the adequacy of digital curvature as a descriptor of shape and the evolutionary and systematic biology of T. apereoides. 2. Digital curvature-based shape measures Let the ith of the N landmark points in a shape be represented by the vector ˜vi , de2ned with respect to an arbitrarily positioned orthogonal coordinate system (see Fig. 1). The order of such landmarks is established by convention and marked either by the operator or in automated fashion by following the object contour starting from some standard reference point and moving in a speci2c sense. As
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Fig. 1. A generic shape represented in terms of N landmark points and the vectors ˜ai and ˜bi and the angle i de2ned at each landmark i.
and ˜aN =
˜vN − ˜vN −1 ; ˜vN − ˜vN −1
˜bN = ˜v1 − ˜vN : ˜v1 − ˜vN
Fig. 2. Morphological landmarks de2ned for the lateral view of the skull of Thrichomys apereoides.
successive landmarks are found along the shape outline, successive integer numbers are assigned to them (see Fig. 2). In the present method the landmarks were marked by an operator, starting always from the same reference point and moving in clockwise fashion. It should be observed that the order and starting point reference do not a6ect the further analysis considered in this work. This is so because not only the digital curvature is invariant to such choices, but also because the order of the features, namely the angle di6erence and length ratio between segments de2ned by pairs of successive landmark points, has no e6ect over the statistical methods used in discriminant analysis. At each landmark i = 2; 3; : : : ; N − 1, it is possible to de2ne two vectors ˜ai and ˜bi as indicated in Eqs. (1) and (2), respectively ˜vi − ˜vi−1 ; ˜vi − ˜vi−1
(1)
˜bi = ˜vi+1 − ˜vi : ˜vi+1 − ˜vi
(2)
˜ai =
In particular, ˜a1 =
˜v1 − ˜vN ; ˜v1 − ˜vN
˜b1 = ˜v2 − ˜v1 ˜v2 − ˜v1
(3) (4)
(5) (6)
The smallest angle i between the vectors ˜ai and ˜bi is therefore given in terms of their scalar product, i.e. ˜ai · ˜bi −1 : (7) i = cos ˜ai ˜bi This angle expresses the sharpness of the shape at each landmark point, in the sense that the larger the value of i , the sharper the vertex. As this measurement, which is always non-negative, corresponds to the smallest angle between the two vectors, the vector product between ˜ai and ˜bi has to be incorporated in order to orient the angle, i.e. ki = sgn(˜ai × ˜bi )i ;
(8)
where sgn is the signum function, returning 1 for positive arguments and −1 for negative arguments. Now the value ki de2ned at each landmark point i can be understood as the digital curvature of the shape at that point [28]. In case the landmarks are marked in clockwise fashion along the shape, convex and concave vertices will imply negative and positive values of ki , respectively (the opposite is veri2ed for counterclockwise sense). While the integral of the squared curvature values along a continuous contour is known to be proportional to the bending energy stored into the shape, a directly similar measure, hence the digital bending energy , can be obtained for a landmark shape, being de2ned by the sum of the squared digital curvature values, i.e. =
N
ki2 :
(9)
i=1
It is important to observe that the digital curvature is invariant to translation, rotation and even scaling. Furthermore, it is substantially more robust to noise and distortions
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than the traditional curvature. Such a property can be readily appreciated by considering that small perturbations (e.g. a glitch) added along the object contour can generate pronounced peaks of curvature (which expresses the variation of local orientation of the curve). By operating at larger scales (i.e. assuming that the landmark points are reasonably well-separates along the object contour), and given its intrinsic characteristics, the digital curvature is much less sensitive to such perturbations. More speci2cally, the variation of values obtained by Eq. (8) as consequence of small perturbations on the position of the respective landmark can be veri2ed to be much less intense than it would be otherwise obtained by using traditional curvature-based methods. This can be immediately appreciated by recalling that the digital curvature reGects the angle between the segments connected to the landmark, which changes little as consequence of small repositioning of the landmark. This tendency is experimentally con2rmed in the results section of this work. This property consists an additional advantage of the proposed approach. 3. Distance measures Let us de2ne the following two distances with respect to each landmark (xi ; yi ). a = (xi+1 − xi )2 + (yi+1 − yi )2 ; (10) b=
(xi−1 − xi )2 + (yi+1 − yi )2 :
(11)
Now, we de2ne the following ratio between distances for (xi ; yi ): a C= : (12) b Observe that a coarse representation of the original shape, or an exact representation of the polygon connecting landmark points (assuming no noise), can be recovered from the angles and ratios. 4. Materials and methods 4.1. Biological specimens The 428 samples of T. apereoides specimens examined in this study are housed in the mammal collections of the Museu Nacional, Rio de Janeiro, Brazil. The samples used in this study represent collecting sites in northeastern, central and western Brazil (see Fig. 3), indexed by geographical coordinates given below, along with sample size (n): state of Cear)a—Itapag)e (sample a; 3◦ 41 S, 39◦ 34 W; n = 24), Campos Sales (b; 7◦ 04 S, 40◦ 23 W; n = 36), and Crato (c; 7◦ 14 S, 39◦ 24 W; n = 25); state of Para)Lba—Princesa Isabel (d; 7◦ 44 S, 38◦ 00 W; n = 13); state of Pernambuco— Bodoc)o (e; 7◦ 47 S, 39◦ 55 W; n = 32), Triunfo (f; 7◦ 50 S, 38◦ 07 W; n = 25), Caruaru (g; 8◦ 17 S, 35◦ 38 W; n = 27),
Pesqueira (h; 8◦ 22 S, 36◦ 41 W; n = 37), Alagoinha (i; 8◦ 27 S, 36◦ 46 W; n=9), and Floresta (j; 8◦ 36 S; 38◦ 34 W; n = 11); state of Alagoas—Santana 22 S, 36◦ 14 W; n = 16), and Palmeira dos )Indios (m; 9◦ 25 S, 36◦ 37 W; n = 14); state of Bahia—Barreiras (n; 12◦ 09 S, 45◦ 59 W; n = 10), Feira de Santana (o; 12◦ 15 S, 38◦ 57 W; n = 38), Palmeiras (p; 12◦ 31 S, 41◦ 34 W; n = 28), and Bom Jesus da Lapa (q; 13◦ 15 S, 43◦ 25 W; n=5); state of Goi)as—Serra da Mesa (r; 14◦ 01 S, 48◦ 18 W; n = 6); state of Minas Gerais—Ja)Lba (s; 15◦ 51 S, 43◦ 03 W; n = 12); Salinas (t; 16◦ 10 S, 42◦ 17 W; n = 25). 4.2. Morphological landmarks Twelve morphological landmarks, assumed to be homologous, were de2ned for the lateral view of the skull and are shown in Fig. 2. For the collection of landmarks, each skull was placed parallel to the focal plane under a Pixera (Pixera Corporation, Los Gatos, California) digital camera system and the x and y coordinates of each landmark for the lateral view of the skull were obtained landmark de2nition see [26,27]. 4.3. Multivariate data analysis Two multivariate data analysis approaches have been considered in the present work: hierarchical clustering and canonical variate analysis (CVA), which are reviewed in the following. Hierarchical approaches to cluster analysis [29,30] are based on the progressive agglomeration of individuals and sub-clusters while taking into account some distance or dispersion measures. Here we concentrate attention on the hierarchical approach known as Ward’s method, which performs agglomeration in such a way as to progressively minimize intra-cluster dispersion. The result is a tree structure, know as dendrogram, expressing the relationship between the several sub-clusters and observations. The classes de2ned for all possible number of classes can be conveniently obtained by cutting the dendrogram at speci2c height points. In the present case, two classes are obtained from the dendrograms, corresponding to northern and southern populations. As the labeling of these classes is arbitrary, two comparisons between the original and obtained clusters are tried, and the one leading to the larger number of correct classi2cations is taken as result. The achieved performance can be suitably characterized in terms of classi2cation matrices. CVA is a multivariate method that incorporates external information about groups de2ned a priori, where group structure arises naturally from the data as a result of sampling di6erent populations [3]. Essentially, given p variables and g groups the ith group (i = 1; : : : ; g) is assumed to be a sample of size ni , ( gi=1 ni = n), from a multivariate normal distribution Np (; ), it is possible to calculate the group mean xi (known as the [ith] sample centroid), the overall sample mean (x) O and the within- and among-groups sample
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Fig. 3. Sample localites for populations of T. apereoides from northeastern, central, and southeastern Brazil: 1, Itapag)e; 2, Campos Sales; 3, Crato; 4, Princesa Isabel; 5, Bodoc)o; 6, Triunfo; 7, Caruaru; 8, Pesqueira; 9,Alagoinha; 10, Floresta; 11, ViPcosa; 12, Santana do Ipanema; 13, Palmeira dos )Indios; 14, Barreiras; 15, Feira de Santana; 16, Palmeiras; 17, Bom Jesus da Lapa; 18, Serra da Mesa; 19, Ja)Lba; 20, Salinas.
covariance matrices W and B, respectively [31]. Then the s = min(g − 1; p) sample canonical axes ak for k = 1; : : : ; s are chosen successively to maximize aTk Bak =aTk Wak . The position of the [ith] sample centroid xi projected onto the kth sample canonical axis ak is given by zOik = aTl (xOi − x) O and this information is used to infer the nature of the di6erences among groups. Con2dence regions around population centroids for canonical axes were constructed using parametric bootstrap theory [31]. The parametric bootstrap procedure ˆ distribution where xi is the vector is based on the N (xOi ; ) of means for the ith population sample and ˆ is the unbiased estimate of the pooled within-group covariance matrix. The Cholesky factorization was to applied to ˆ in the simulation of 1000 replicate data matrices [32] and CVA was performed to yield bootstrap eigenvalues and eigenvectors. The 2 distribution was used to construct 95% con2dence regions by applying formulae provided by Krzanowski [3]. Digital curvatures and one global measure of shape, the digital bending energy, were used with the clustering procedures and CVA to assess the structure of geographic variation among samples from di6erent localities of the rodent species T. apereoides. The analyses were implemented in MATLAB. 5. Results The 428 individuals of T. apereoides were processed in order to extract the digital curvatures (DC) and bending
energies (E), and the following feature vectors were considered: (i) all digital curvatures and bending energies; (ii) all digital curvatures; and (iii) bending energies only, and (iiii) all digital curvatures and ratios. Fig. 4 shows the dendrogram for 30 clusters and respective regions obtained by using the above characterized methodology. CVA of 12 digital curvatures and one bending energy shows that canonical axes 1 and 2 explained 63% of the variation, with the 2rst axis accounting for a major portion of the variation (46%). This axis consistently discriminates two groups of populations: one northern group composed of populations a − m from the other southern including populations n − t. The only exception is population h which occupies an intermediate position between these two major clusters of populations (Fig. 5). An identical ordination was obtained in a analysis excluding the measure of bending energy (results not shown). A better separation was obtained by considering the ratio and digital curvature. (Fig. 6). The proposed methodology has also been experimentally evaluated with respect to the addition of increasing degrees of perturbation (noise) to the landmark points. More speci2cally, the landmarks for each object were displaced from their original position by a perturbation vector p ˜ = ( x; y), where x and y are random numbers in the interval −0:5 6 x; y ¡ 0:5 , ¿ 0, so that the choice of determines the intensity of the perturbation. Fig. 7 shows the original shape of 2ve samples (a) – (c), choosen at random out of the 428 cases, with respective distorted versions
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Fig. 4. Dendrogram of 30 clusters obtained by using the Ward method and Euclidean distances.
Fig. 5. Bivariate plot of centroids (denoted as crosses) and 95% con2dence regions for the 2rst 2 axes derived from a CVA digital curvatures for the lateral view of the skull of 20 populations of Thrichomys apereoides from northeastern, central, and southeastern Brazil (letters near centroids refer to the localities listed in Section 4.1).
Fig. 6. CVA considering ratio and digital curvatures for two groups of populations: one northern group and other southern.
considering = 0:1 (d) – (f), = 0:2 (g) – (i), = 0:3 (j) – (l) and = 0:4 (m) – (o). The respective canonical analysis results are shown in Fig. 8. It is clear from such set of plots that the proposed methodology, at least for the speci2c data set considered in this investigation, is robust for perturbations up to 0.2, with substantial overlap between the two classes being observed for larger perturbation values.
This work has reported the use of the digital curvature as a geometrical feature for the analysis of biological shapes. More speci2cally, this simple feature has been used for the morphological characterization of T. apereoides specimens collected from several localities in northeastern, central and southeastern Brazil. Geographic patterns of morphological variation were assessed using Ward’s clustering method and canonical variate analysis. The obtained results indicate a
6. Concluding remarks
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Fig. 7. The original shapes of 2ve samples (a) – (c), chosen at random out of the 428 cases, with respective distorted versions considering = 0:1 (d) – (f), = 0:2 (g) – (i), = 0:3 (j) – (l) and = 0:4 (m) – (o).
well- de2ned correspondence between the geographical and morphological distributions, substantiating the digital curvature as a suitable shape feature. The two groups uncovered can be diagnosed on the basis of shape features of the lateral view of skull and geographic continuity. The descrip-
tion of commonality of shape parameters revealed by the digital curvature and the spatial distribution of populations is a 2rst step in the de2nition of independent evolutionary units in nature at the species level, which is the fundamental unit of biodiversity. The simplicity of shape features derived
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Fig. 8. Results of CVA with respect to the distorted versions showed in Fig. 7.
from the formalism of digital curvature associated with linear multivariate statistical methods provide a promising approach to the study of morphological variation and its systematic and evolutionary implications.
7. Summary The current article describes a new feature for biological shape characterization and comparison, namely the digital curvature, and illustrates its application with respect to rodent cranial characterization in terms of the respective geographical origin. Homology points in biological shape are often identi2ed as curvature peaks along the respective outlines. The approach presented here assign to each homology point the vector connecting this point to the next, in such a way that the digital curvature at each point is obtained as the di6erence between the angles of two successive vectors. Therefore, such as the continuous curvature, the therefore de2ned digital counterpart also expresses the variation of the angle along the tangent to the contour. In
order to complement this information, the ratio between the magnitude of the two successive vectors is also estimated and used as a feature for shape classi2cation. In addition to its inherent simplicity and conceptual interpretation, the adopted morphological characterization also allows local deformations to the original shape to have limited e6ects, acting only over a few homology points. The potential of the proposed features for morphological representation is fully illustrated and corroborated with respect to rodent cranial outlines. More speci2cally, it is veri2ed that the canonical analysis distributions obtained by using the proposed features present a well-de2ned relationship with the geographical regions where the individuals were captured.
Acknowledgements Luciano da F. Costa is grateful to CNPq (301422/92-3), FINEP-RECOPE (77.97.0575.00) and FAPESP (96/05497-3 and 99/12765-2) for 2nancial support. Work by S.F. dos Reis is supported by FAPESP (99/06845-3).
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About the Author—LUCIANO DA FONTOURA COSTA (b. in Brazil in 1962) holds B.Sc.’s in Eletronic Engineering and Computer Science, and M.Sc. in Applied Physics, and a Ph.D. in Eletronic engineering (King’s College University of London). He founded the Cybernetic Vision Research Group at the University of S˜ao Paulo, which has become one of the most active and highly regarded Brazilian groups
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in the areas of natural and arti2cial vision and computational neuroscience. He has acted as member of editorial board of several international journals and is Associate Editor of Integrative Neuroscience (World Scienti2c). His scienti2c production includes over 150 reviewed articles. His main interest include natural and arti2cial vision, shape analysis, pattern recognition, computational neuroscience, computational biology, and bioinformatics. About the Author—RENATA ANTONIA TADEU ARANTES holds a B.Sc. degree from Universidade do Sagrado CoraPca˜ o, in 1994, in System Analysis, and M.Sc. in Electrical Engineering, and is currently a Ph.D. student in computational physics at Cybernetic Vision Research Group at the University of S˜ao Paulo, Brazil. Her research interest involves 2D shape analysis, digital curvature, pattern recognition, mathematical morphology, morphological and geographic distributions, morphometric analysis, morphological landmark. ) About the Author—SERGIO FURTADO DOS REIS holds a Ph.D. degree in Zoology from Michigan State University. He has done postdoctoral studies in molecular systematics at the University of California, Berkeley. His research involves the statistical analysis of shape and mathematical models of population dynamics. About the Author—ANA CLAUDIA REIS ALVES holds a B.Sc. degree in Biological Sciences and a M.Sc. degree in Zoology from Universidade Federal do Rio de Janeiro and is currently working towards her Ph.D. degree in Biological Systematics at the Universidade Estadual Paulista. Her main interests lie in molecular systematics, morphological evolution and methods for the description and interpretation of biological shape. About the Author—GIANCARLO MUTINARI is currently a B.Sc. degree student from University of S˜ao Paulo, in Informatics, and scienti2c initiation student at Cybernetic Vision Research Group at the University of S˜ao Paulo, Brazil. His research interest involves arti2cial vision, continuous and digital curvature, pattern recognition, morphological and geographic distributions, morphometric analysis.
Biological shape analysis by digital curvature Luciano da F. Costaa;∗ , S)ergio F. dos Reisb , Renata A.T. Arantesa , Ana C.R. Alvesc , GianCarlo Mutinaria a Cybernetic
Vision Research Group, Institute of Physics at S˜ao Carlos - IFSC - University of S˜ao Paulo, Caixa Postal 369, S˜ao Carlos, SP 13560-970, Brazil b Departamento de Parasitologia, Universidade Estadual de Campinas, 13083-970 Campinas, S˜ ao Paulo, Brazil c Programa de P. os-Graduac/a˜ o em Zoologia, Universidade Estadual Paulista, 13500-000 Rio Claro, S˜ao Paulo, Brazil Received 20 August 2002; accepted 30 July 2003
Abstract This paper reports the novel application of digital curvature as a feature for morphological characterization and classi2cation of landmark shapes. By inheriting several unique features of the continuous curvature, the digital curvature provides invariance to translations, rotations, local shape deformations, and is easily made tolerant to scaling. In addition, the bending energy, a global shape feature, can be directly estimated from the curvature values. The application of these features to analyse patterns of cranial morphological geographic di6erentiation in the rodent species Thrichomys apereoides has led to encouraging results, indicating a close correspondence between the geographical and morphological distributions. ? 2003 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. Keywords: Shape analysis; Digital curvature; Morphological and geographic distributions; Morphometric analysis; Morphological landmark
1. Introduction Biological shape is the result of endogenous and exogenous mechanisms that operate at di6erent scales of space and time and levels of organizational complexity. Endogenous mechanisms include molecular signalling between epithelial and mesenchymal cells and interactions between gene products that control gene expression and cell movement in the extracellular environment [1,2]. These molecular and cellular phenomena that underlie the dynamic processes of morphogenesis interface with exogenous ecological factors and deterministic and stochastic evolutionary forces to produce the observed complex morphologies in the 2nal shape of organisms. The description and measurement of biological shape is therefore of fundamental importance to assess variation in natural populations and also to infer its ecological and evolutionary causes. ∗ Corresponding author. Tel.: +55-16-273-9858; fax: +55-16-273-9879. E-mail address: [email protected] (L. da F. Costa).
Traditional approaches to the measurement of shape have relied on distances from which descriptors of shape are constructed as linear combinations of distances and coef2cients derived from principal components extracted from covariance matrices [3]. More recently, emphasis has been shifted to morphological landmarks as the primary source of data for the description of shape and variation in shape to address questions of ecological and evolutionary interest (e.g., [4–8]). Morphological landmarks are de2ned points in a biological structure, such as processes or sutures, which are assumed to be homologous across individuals and populations [9]. The position of morphological landmarks can be recorded as Cartesian rectangular coordinates thereby archiving the con2guration of the biological structures of interest [10]. Landmarks and their associated coordinates allow a more complete representation of the shape than afforded by distance measurements, and conceptually should provide a more natural approximation to the spatio-temporal process of morphogenesis which generates observed shapes. Landmarks are the data from which features must be extracted to provide variables for the measurement and
0031-3203/$30.00 ? 2003 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2003.07.010
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interpretation of shape and variations in shape. The Cartesian coordinates of each landmark are stored and used to extract characteristics such as the distance between each subsequent landmark in the shape. Ratios between successive distances along the shape landmarks are also considered as features. Here we introduce digital curvature as a morphometric procedure to generate features that can be used with multivariate statistical methods to analyse shape variation. Basically, the digital curvature can be understood as the di6erence between the angles of pairs of subsequent vectors de2ned by the landmark sequence (see in Section 2). As such, this measure associates an angle di6erence to every landmark point in such a way as to express the degree of sharpness of the respectively de2ned vertex. It is interesting to observe that such a methodology provides an alternative approach to traditional curvature analysis methods, where this important measure is numerically estimated along the object outline. However, instead of taking into account all curvature values, only the most prominent contour elements, corresponding to the highest curvature peaks, are considered for object representation, an approach that is closely related to previous concepts from visual psychophysics [11]. While Pavlidis [33] provides an interesting and comprehensive review of related shape analysis methods, the work by Mokhtarian et al. [34] addresses the important issue of multiscale curvature analysis for object characterization. Such an approach is particularly interesting because, in addition to being invariant to rigid body transformations (including translation, rotation and reGection), the curvature preserves almost all information about the shape under analysis. As a matter of fact, given the curvature at every point of the contour, it is possible to reconstruct the shape up to a rigid body transformation. In addition, the curvature magnitude provides a direct measurement of the saliency of the points along the contour. Thus, while straight portions of the object borders lead to near zero curvature values, abrupt variations of the contour orientation are clearly characterized by curvature peaks. Indeed, it is possible to use such approaches in order to obtain an automated identi2cation the landmark points de2ned by curvature peaks. The main motivation for adopting the digital curvature as a shape feature is that it inherits several of the unique features of the continuous curvature as a shape descriptor (e.g. [11,12]) while presenting some relevant advantages. First, it is invariant to translation and rotation, and can easily be modi2ed in order to become invariant to scale, e.g. by normalizing the shape by energy or diameter [13,14] or any other convenient normalization as recently pointed out by Llenonart [15]. Second, curvature is a local measure, in the sense that a small perturbation or shape variation at one point has limited spatial e6ect over the shape description. Moreover, and unlike its continuous counterpart, the digital curvature is also invariant to scaling. This measure has also been related to the salience of points along
the shape [11], and the sum of its squared values along the shape is known to be proportional to the bending energy stored into the shape, which can provide a useful global shape feature [16,17], also considered in this article. Finally, as shown in this paper, the digital curvature and the distance between landmarks are simple measures that can be easy and economically obtained by using elementary vector operations. Shape is a multidimensional component of variation in morphological form [18] which is expected to have a high information content regarding the evolutionary processes responsible for the observed diversity [19,20]. The description of patterns of variation in morphological shape within and among populations is therefore fundamental for de2ning the boundaries of independent evolutionary units in nature, and an important step in the recognition of such evolutionary units is the identi2cation of groups of populations that share morphological features of shape and geographic continuity over geographic space [21,22]. In this paper, we illustrate the use of digital curvature and bending energy with a description of patterns of geographic variation in cranial shape in Thrichomys apereoides (family Echimyidae), a rodent which ranges in distribution from northeastern, central, and southeastern Brazil into Paraguay and Bolivia. Throughout its distribution, T. apereoides inhabits xeric and rocky environments in caatinga and cerrado domains in Brazil and chaco in Paraguay [23–25]. Preliminary work on this species has indicated the existence of two geographic units, a northern and a southern group recognized for samples taken from a large area of its distribution, including northeastern, central and southeastern Brazil [26,27] . The question we ask is whether variation in cranial shape, measured by digital curvature, distance between landmarks and bending energy of cranial landmarks, is geographically structured and provides evidence for northern and southern groups of populations. This paper is organized as follows. The procedure of digital curvature is introduced and its mathematical foundation is presented and subsequently the formalism of digital curvature and bending energy is employed in combination with multivariate methods to analyse patterns of cranial morphological geographic di6erentiation in the rodent species T. apereoides. The results obtained are discussed in terms of the adequacy of digital curvature as a descriptor of shape and the evolutionary and systematic biology of T. apereoides. 2. Digital curvature-based shape measures Let the ith of the N landmark points in a shape be represented by the vector ˜vi , de2ned with respect to an arbitrarily positioned orthogonal coordinate system (see Fig. 1). The order of such landmarks is established by convention and marked either by the operator or in automated fashion by following the object contour starting from some standard reference point and moving in a speci2c sense. As
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Fig. 1. A generic shape represented in terms of N landmark points and the vectors ˜ai and ˜bi and the angle i de2ned at each landmark i.
and ˜aN =
˜vN − ˜vN −1 ; ˜vN − ˜vN −1
˜bN = ˜v1 − ˜vN : ˜v1 − ˜vN
Fig. 2. Morphological landmarks de2ned for the lateral view of the skull of Thrichomys apereoides.
successive landmarks are found along the shape outline, successive integer numbers are assigned to them (see Fig. 2). In the present method the landmarks were marked by an operator, starting always from the same reference point and moving in clockwise fashion. It should be observed that the order and starting point reference do not a6ect the further analysis considered in this work. This is so because not only the digital curvature is invariant to such choices, but also because the order of the features, namely the angle di6erence and length ratio between segments de2ned by pairs of successive landmark points, has no e6ect over the statistical methods used in discriminant analysis. At each landmark i = 2; 3; : : : ; N − 1, it is possible to de2ne two vectors ˜ai and ˜bi as indicated in Eqs. (1) and (2), respectively ˜vi − ˜vi−1 ; ˜vi − ˜vi−1
(1)
˜bi = ˜vi+1 − ˜vi : ˜vi+1 − ˜vi
(2)
˜ai =
In particular, ˜a1 =
˜v1 − ˜vN ; ˜v1 − ˜vN
˜b1 = ˜v2 − ˜v1 ˜v2 − ˜v1
(3) (4)
(5) (6)
The smallest angle i between the vectors ˜ai and ˜bi is therefore given in terms of their scalar product, i.e. ˜ai · ˜bi −1 : (7) i = cos ˜ai ˜bi This angle expresses the sharpness of the shape at each landmark point, in the sense that the larger the value of i , the sharper the vertex. As this measurement, which is always non-negative, corresponds to the smallest angle between the two vectors, the vector product between ˜ai and ˜bi has to be incorporated in order to orient the angle, i.e. ki = sgn(˜ai × ˜bi )i ;
(8)
where sgn is the signum function, returning 1 for positive arguments and −1 for negative arguments. Now the value ki de2ned at each landmark point i can be understood as the digital curvature of the shape at that point [28]. In case the landmarks are marked in clockwise fashion along the shape, convex and concave vertices will imply negative and positive values of ki , respectively (the opposite is veri2ed for counterclockwise sense). While the integral of the squared curvature values along a continuous contour is known to be proportional to the bending energy stored into the shape, a directly similar measure, hence the digital bending energy , can be obtained for a landmark shape, being de2ned by the sum of the squared digital curvature values, i.e. =
N
ki2 :
(9)
i=1
It is important to observe that the digital curvature is invariant to translation, rotation and even scaling. Furthermore, it is substantially more robust to noise and distortions
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than the traditional curvature. Such a property can be readily appreciated by considering that small perturbations (e.g. a glitch) added along the object contour can generate pronounced peaks of curvature (which expresses the variation of local orientation of the curve). By operating at larger scales (i.e. assuming that the landmark points are reasonably well-separates along the object contour), and given its intrinsic characteristics, the digital curvature is much less sensitive to such perturbations. More speci2cally, the variation of values obtained by Eq. (8) as consequence of small perturbations on the position of the respective landmark can be veri2ed to be much less intense than it would be otherwise obtained by using traditional curvature-based methods. This can be immediately appreciated by recalling that the digital curvature reGects the angle between the segments connected to the landmark, which changes little as consequence of small repositioning of the landmark. This tendency is experimentally con2rmed in the results section of this work. This property consists an additional advantage of the proposed approach. 3. Distance measures Let us de2ne the following two distances with respect to each landmark (xi ; yi ). a = (xi+1 − xi )2 + (yi+1 − yi )2 ; (10) b=
(xi−1 − xi )2 + (yi+1 − yi )2 :
(11)
Now, we de2ne the following ratio between distances for (xi ; yi ): a C= : (12) b Observe that a coarse representation of the original shape, or an exact representation of the polygon connecting landmark points (assuming no noise), can be recovered from the angles and ratios. 4. Materials and methods 4.1. Biological specimens The 428 samples of T. apereoides specimens examined in this study are housed in the mammal collections of the Museu Nacional, Rio de Janeiro, Brazil. The samples used in this study represent collecting sites in northeastern, central and western Brazil (see Fig. 3), indexed by geographical coordinates given below, along with sample size (n): state of Cear)a—Itapag)e (sample a; 3◦ 41 S, 39◦ 34 W; n = 24), Campos Sales (b; 7◦ 04 S, 40◦ 23 W; n = 36), and Crato (c; 7◦ 14 S, 39◦ 24 W; n = 25); state of Para)Lba—Princesa Isabel (d; 7◦ 44 S, 38◦ 00 W; n = 13); state of Pernambuco— Bodoc)o (e; 7◦ 47 S, 39◦ 55 W; n = 32), Triunfo (f; 7◦ 50 S, 38◦ 07 W; n = 25), Caruaru (g; 8◦ 17 S, 35◦ 38 W; n = 27),
Pesqueira (h; 8◦ 22 S, 36◦ 41 W; n = 37), Alagoinha (i; 8◦ 27 S, 36◦ 46 W; n=9), and Floresta (j; 8◦ 36 S; 38◦ 34 W; n = 11); state of Alagoas—Santana 22 S, 36◦ 14 W; n = 16), and Palmeira dos )Indios (m; 9◦ 25 S, 36◦ 37 W; n = 14); state of Bahia—Barreiras (n; 12◦ 09 S, 45◦ 59 W; n = 10), Feira de Santana (o; 12◦ 15 S, 38◦ 57 W; n = 38), Palmeiras (p; 12◦ 31 S, 41◦ 34 W; n = 28), and Bom Jesus da Lapa (q; 13◦ 15 S, 43◦ 25 W; n=5); state of Goi)as—Serra da Mesa (r; 14◦ 01 S, 48◦ 18 W; n = 6); state of Minas Gerais—Ja)Lba (s; 15◦ 51 S, 43◦ 03 W; n = 12); Salinas (t; 16◦ 10 S, 42◦ 17 W; n = 25). 4.2. Morphological landmarks Twelve morphological landmarks, assumed to be homologous, were de2ned for the lateral view of the skull and are shown in Fig. 2. For the collection of landmarks, each skull was placed parallel to the focal plane under a Pixera (Pixera Corporation, Los Gatos, California) digital camera system and the x and y coordinates of each landmark for the lateral view of the skull were obtained landmark de2nition see [26,27]. 4.3. Multivariate data analysis Two multivariate data analysis approaches have been considered in the present work: hierarchical clustering and canonical variate analysis (CVA), which are reviewed in the following. Hierarchical approaches to cluster analysis [29,30] are based on the progressive agglomeration of individuals and sub-clusters while taking into account some distance or dispersion measures. Here we concentrate attention on the hierarchical approach known as Ward’s method, which performs agglomeration in such a way as to progressively minimize intra-cluster dispersion. The result is a tree structure, know as dendrogram, expressing the relationship between the several sub-clusters and observations. The classes de2ned for all possible number of classes can be conveniently obtained by cutting the dendrogram at speci2c height points. In the present case, two classes are obtained from the dendrograms, corresponding to northern and southern populations. As the labeling of these classes is arbitrary, two comparisons between the original and obtained clusters are tried, and the one leading to the larger number of correct classi2cations is taken as result. The achieved performance can be suitably characterized in terms of classi2cation matrices. CVA is a multivariate method that incorporates external information about groups de2ned a priori, where group structure arises naturally from the data as a result of sampling di6erent populations [3]. Essentially, given p variables and g groups the ith group (i = 1; : : : ; g) is assumed to be a sample of size ni , ( gi=1 ni = n), from a multivariate normal distribution Np (; ), it is possible to calculate the group mean xi (known as the [ith] sample centroid), the overall sample mean (x) O and the within- and among-groups sample
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Fig. 3. Sample localites for populations of T. apereoides from northeastern, central, and southeastern Brazil: 1, Itapag)e; 2, Campos Sales; 3, Crato; 4, Princesa Isabel; 5, Bodoc)o; 6, Triunfo; 7, Caruaru; 8, Pesqueira; 9,Alagoinha; 10, Floresta; 11, ViPcosa; 12, Santana do Ipanema; 13, Palmeira dos )Indios; 14, Barreiras; 15, Feira de Santana; 16, Palmeiras; 17, Bom Jesus da Lapa; 18, Serra da Mesa; 19, Ja)Lba; 20, Salinas.
covariance matrices W and B, respectively [31]. Then the s = min(g − 1; p) sample canonical axes ak for k = 1; : : : ; s are chosen successively to maximize aTk Bak =aTk Wak . The position of the [ith] sample centroid xi projected onto the kth sample canonical axis ak is given by zOik = aTl (xOi − x) O and this information is used to infer the nature of the di6erences among groups. Con2dence regions around population centroids for canonical axes were constructed using parametric bootstrap theory [31]. The parametric bootstrap procedure ˆ distribution where xi is the vector is based on the N (xOi ; ) of means for the ith population sample and ˆ is the unbiased estimate of the pooled within-group covariance matrix. The Cholesky factorization was to applied to ˆ in the simulation of 1000 replicate data matrices [32] and CVA was performed to yield bootstrap eigenvalues and eigenvectors. The 2 distribution was used to construct 95% con2dence regions by applying formulae provided by Krzanowski [3]. Digital curvatures and one global measure of shape, the digital bending energy, were used with the clustering procedures and CVA to assess the structure of geographic variation among samples from di6erent localities of the rodent species T. apereoides. The analyses were implemented in MATLAB. 5. Results The 428 individuals of T. apereoides were processed in order to extract the digital curvatures (DC) and bending
energies (E), and the following feature vectors were considered: (i) all digital curvatures and bending energies; (ii) all digital curvatures; and (iii) bending energies only, and (iiii) all digital curvatures and ratios. Fig. 4 shows the dendrogram for 30 clusters and respective regions obtained by using the above characterized methodology. CVA of 12 digital curvatures and one bending energy shows that canonical axes 1 and 2 explained 63% of the variation, with the 2rst axis accounting for a major portion of the variation (46%). This axis consistently discriminates two groups of populations: one northern group composed of populations a − m from the other southern including populations n − t. The only exception is population h which occupies an intermediate position between these two major clusters of populations (Fig. 5). An identical ordination was obtained in a analysis excluding the measure of bending energy (results not shown). A better separation was obtained by considering the ratio and digital curvature. (Fig. 6). The proposed methodology has also been experimentally evaluated with respect to the addition of increasing degrees of perturbation (noise) to the landmark points. More speci2cally, the landmarks for each object were displaced from their original position by a perturbation vector p ˜ = ( x; y), where x and y are random numbers in the interval −0:5 6 x; y ¡ 0:5 , ¿ 0, so that the choice of determines the intensity of the perturbation. Fig. 7 shows the original shape of 2ve samples (a) – (c), choosen at random out of the 428 cases, with respective distorted versions
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Fig. 4. Dendrogram of 30 clusters obtained by using the Ward method and Euclidean distances.
Fig. 5. Bivariate plot of centroids (denoted as crosses) and 95% con2dence regions for the 2rst 2 axes derived from a CVA digital curvatures for the lateral view of the skull of 20 populations of Thrichomys apereoides from northeastern, central, and southeastern Brazil (letters near centroids refer to the localities listed in Section 4.1).
Fig. 6. CVA considering ratio and digital curvatures for two groups of populations: one northern group and other southern.
considering = 0:1 (d) – (f), = 0:2 (g) – (i), = 0:3 (j) – (l) and = 0:4 (m) – (o). The respective canonical analysis results are shown in Fig. 8. It is clear from such set of plots that the proposed methodology, at least for the speci2c data set considered in this investigation, is robust for perturbations up to 0.2, with substantial overlap between the two classes being observed for larger perturbation values.
This work has reported the use of the digital curvature as a geometrical feature for the analysis of biological shapes. More speci2cally, this simple feature has been used for the morphological characterization of T. apereoides specimens collected from several localities in northeastern, central and southeastern Brazil. Geographic patterns of morphological variation were assessed using Ward’s clustering method and canonical variate analysis. The obtained results indicate a
6. Concluding remarks
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Fig. 7. The original shapes of 2ve samples (a) – (c), chosen at random out of the 428 cases, with respective distorted versions considering = 0:1 (d) – (f), = 0:2 (g) – (i), = 0:3 (j) – (l) and = 0:4 (m) – (o).
well- de2ned correspondence between the geographical and morphological distributions, substantiating the digital curvature as a suitable shape feature. The two groups uncovered can be diagnosed on the basis of shape features of the lateral view of skull and geographic continuity. The descrip-
tion of commonality of shape parameters revealed by the digital curvature and the spatial distribution of populations is a 2rst step in the de2nition of independent evolutionary units in nature at the species level, which is the fundamental unit of biodiversity. The simplicity of shape features derived
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Fig. 8. Results of CVA with respect to the distorted versions showed in Fig. 7.
from the formalism of digital curvature associated with linear multivariate statistical methods provide a promising approach to the study of morphological variation and its systematic and evolutionary implications.
7. Summary The current article describes a new feature for biological shape characterization and comparison, namely the digital curvature, and illustrates its application with respect to rodent cranial characterization in terms of the respective geographical origin. Homology points in biological shape are often identi2ed as curvature peaks along the respective outlines. The approach presented here assign to each homology point the vector connecting this point to the next, in such a way that the digital curvature at each point is obtained as the di6erence between the angles of two successive vectors. Therefore, such as the continuous curvature, the therefore de2ned digital counterpart also expresses the variation of the angle along the tangent to the contour. In
order to complement this information, the ratio between the magnitude of the two successive vectors is also estimated and used as a feature for shape classi2cation. In addition to its inherent simplicity and conceptual interpretation, the adopted morphological characterization also allows local deformations to the original shape to have limited e6ects, acting only over a few homology points. The potential of the proposed features for morphological representation is fully illustrated and corroborated with respect to rodent cranial outlines. More speci2cally, it is veri2ed that the canonical analysis distributions obtained by using the proposed features present a well-de2ned relationship with the geographical regions where the individuals were captured.
Acknowledgements Luciano da F. Costa is grateful to CNPq (301422/92-3), FINEP-RECOPE (77.97.0575.00) and FAPESP (96/05497-3 and 99/12765-2) for 2nancial support. Work by S.F. dos Reis is supported by FAPESP (99/06845-3).
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About the Author—LUCIANO DA FONTOURA COSTA (b. in Brazil in 1962) holds B.Sc.’s in Eletronic Engineering and Computer Science, and M.Sc. in Applied Physics, and a Ph.D. in Eletronic engineering (King’s College University of London). He founded the Cybernetic Vision Research Group at the University of S˜ao Paulo, which has become one of the most active and highly regarded Brazilian groups
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in the areas of natural and arti2cial vision and computational neuroscience. He has acted as member of editorial board of several international journals and is Associate Editor of Integrative Neuroscience (World Scienti2c). His scienti2c production includes over 150 reviewed articles. His main interest include natural and arti2cial vision, shape analysis, pattern recognition, computational neuroscience, computational biology, and bioinformatics. About the Author—RENATA ANTONIA TADEU ARANTES holds a B.Sc. degree from Universidade do Sagrado CoraPca˜ o, in 1994, in System Analysis, and M.Sc. in Electrical Engineering, and is currently a Ph.D. student in computational physics at Cybernetic Vision Research Group at the University of S˜ao Paulo, Brazil. Her research interest involves 2D shape analysis, digital curvature, pattern recognition, mathematical morphology, morphological and geographic distributions, morphometric analysis, morphological landmark. ) About the Author—SERGIO FURTADO DOS REIS holds a Ph.D. degree in Zoology from Michigan State University. He has done postdoctoral studies in molecular systematics at the University of California, Berkeley. His research involves the statistical analysis of shape and mathematical models of population dynamics. About the Author—ANA CLAUDIA REIS ALVES holds a B.Sc. degree in Biological Sciences and a M.Sc. degree in Zoology from Universidade Federal do Rio de Janeiro and is currently working towards her Ph.D. degree in Biological Systematics at the Universidade Estadual Paulista. Her main interests lie in molecular systematics, morphological evolution and methods for the description and interpretation of biological shape. About the Author—GIANCARLO MUTINARI is currently a B.Sc. degree student from University of S˜ao Paulo, in Informatics, and scienti2c initiation student at Cybernetic Vision Research Group at the University of S˜ao Paulo, Brazil. His research interest involves arti2cial vision, continuous and digital curvature, pattern recognition, morphological and geographic distributions, morphometric analysis.