Canonical morphometry versus statistical treatment of outlines through Fourier shape analysis: an empirical comparison

2007·4.7 MB·English

by  Francesco Brusoni

Boll. Malacol., 43 (9-12): 131-138(2007)
Canonical morphometry versus statistical treatment of outlines
through Fourier shape analysis: an empirical comparison
Francesco Brusoni* & Daniela Basso# (ED)
* ViaS. MariadelSole 10, Abstract
26900 Lodi, Italy, Theanalysisofoutlines isconsidered a powerfultool for morphometricstudies, sincetheabsenceofassump-
[email protected] tionson characters to be selected and measured reduces subjectivity. Fouriershape analysis converts each di-
gitalized outline Into a suite of numerical descriptors(the Fouriercoefficients)thatdescribecurves. Thecoeffi-
# Dip.todi Sc. cients are the input to multivariate statistical analyses, that elaborates a numerical comparison of shapes. A
Geologiche
eGeotecnologie, friendly, easily accessible and practical method for a statistical approach to shape analysis and the sensitivity
Universitàdegli Studi of both Elliptic FourierAnalysis and Fast FourierTransform have been tested here on a setof Goodallia trian-
di Mllano-Bicocca, gularis shells. Shape analysis results to be faster and more reliable than canonical morphometry. In the com-
Piazza della Scienza4, parison ofthe Principal ComponentAnalyses based on thetwo different Fouriertransformations, Fast Fourier
20126 Milano, Italy, Transform producesthesharpestseparation ofthe shell outlines into clusterswhile Elliptic FourierAnalysis re-
[email protected], sultsarecomparably lessclear. Both techniqueseasilyseparatethetwoformsof Goodallia triangularis.
(s)Corresponding
Author
Riassunto
La descrizione quantitativa delle forme biologiche viene tradizionalmente affidata alla morfometria. Nella
routine, la tecnica morfometrica viene tuttavia applicata a un limitato numero di caratteri misurabili, scelti
in modo più o meno arbitrario, riconducendo quindi la variabilità registrata a pochi parametri. Molte carat-
teristiche morfologiche vengono tradizionalmente descritte in modo qualitativo (ad esempio la convessità
della spira nei gasteropodi), senza quindi assegnare alcun parametro matematico di misura che possa suc-
cessivamente consentire un confronto su base statistica-quantitativa.
Per evitare che la scelta arbitraria di alcuni caratteri rispetto ad altri influenzi il risultato dello studio morfo-
logico-comparativo sarebbe necessario considerare la totalità dei caratteri misurabili. Il tentativo di avvici-
narsi a tale situazione ideale, evitando una eccessiva semplificazione, richiede tuttavia un enorme dispen-
dio di tempo. A questo si aggiunga la percentuale di errore dovuta alla misurazione manuale e l'aumento
delle difficoltà quando le dimensioni degli oggetti di studio sono molto ridotte.
La trasformazione del profilo di una struttura biologica in una serie di parametri matematici (shape analysis)
in luogo delle misure morfometrichetradizionali, consente di ridurre notevolmente la componentesoggettiva
edi sostituire o affiancare, in molti casi, il metodotradizionale di misurazione. Nellashapeanalysis, laforma è
ricondotta ad un unica computazione, che, a suavolta, riassume in sétutte le misurequali altezza, larghezza,
curvature ed angoli. Si ottiene quindi quella che si potrebbe definire la "formula geometrica" che descrive
l'esemplare nella sua globalità, che puòessereconvenientementetrattata in analisi statistichesuccessive.
Il metodo è applicabile a strutture di qualunque genere, tenendo sempre presente la necessità di operare
confronti tra punti tra loro omologhi. Allo stato attuale della tecnologia comunemente disponibile, le con-
chiglie dei molluschi bivalvi sembrano prestarsi meglio a questo tipo di analisi, in quanto più facilmente ri-
conducibili ad un profilo bl-dimenslonale di punti tra loro omologhi.
In questo lavoro si confrontano i risultati della morfometria tradizionale (peso, lunghezza, altezza e spesso-
re di ciascuna valva) e di alcuni metodi facilmente accessibili di shapeanalysis, basati sull'analisi di Fourier,
applicati ad uno stesso gruppo di 39 valve destre di Goodallia triangularis.
Sono stati utilizzati softwares differenti per trasformare le immagini digitali degli oggetti in parametri ma-
tematici. Una volta acquisita l'esperienza sufficiente, il tempo necessario per passare dall'immagine digita-
lizzata alla serie di Fourier pronta per l'elaborazione statistica è di pochi minuti per ciascuna valva. Per pas-
sare dall'acquisizione delle immagini digitali al risultato finale dell'analisi statistica multivariata sono state
impiegate complessivamente poche ore.
Il dendrogramma e la PCA eseguite sui profili delle 39 valve di G. triangularis mostrano una separazione in
due gruppi, corrispondenti alle due forme di Goodallia triangularis: una con profilo tendenzialmente trian-
golare e una di forma più allungata, in accordo con precedenti conclusioni di altri autori.
Nel caso considerato, la shape analysis è risultata notevolmente più rapida e ha prodotto risultati piu at-
tendibili e di più semplice interpretazione rispetto ai metodi morfometrici tradizionali.
Key words
Fouriershape analysis, bivalve, outline, methods.
Introduction in many diverse disciplines such as palaeontology, bio-
geography, ecology and palaeoecology and their appli-
Identification of shelled species based on external mor- cations. Biological structures and shapes are often very
phology is still the basis of all subsequent elaborations complex and their morphometric description triggers at 131
least four major problems: 1) the inadequacy of the use Material e methods
of a limited number of linear variables to describe a
three-dimensional object; 2) the time-consuming job re- Shell size is here intended as a proxy for age, even
quired when trying to use as many linear variables as though the direct correlation between size and age is
possible; 3) the arbitrary choice ofcharacters to be mea- notequivalentforall specimens (Jones, 1988).
sured, that commonly leads to difficulties in comparing Digital images of 39 right valves of the small bivalve
results from different researchers; 4) in small-sized, the Goodallia triangularis (Montagu, 1803) were obtained
difficulty to achieve a quantitative description of their from a microscope Olympus BX 50 equipped with a di-
morphology and the incidence of operator precision in gital camera Camedia C 2000z and software Camedia
the accuracy ofmeasurements. DP-soft. The digital pictures have been transformed in-
Recent advances in information technology made avai- to simple shell outlines (profiles), and then into a series
lable some user-friendly softwares for image manipula- of mathematical parameters to be analysed by multiva-
tion and multivariate statistical analysis. In some in- riate statistics (Fig. 1).
Francesco stances these softwares are freely accessible from the Presently, the entire procedure requires four different
web and possessawidepool ofpotential users. soft-wares. A common graphic software to extract the
Brusoni The opportunity to use low-cost and friendly technolo- profile; a software for image manipulation and photo
& gies with the aim to reduce subjectivity (in morphologi- touch that can produce the co-ordinates of each pixel of
cal data acquisition) and loss of time (for sample and the profile; a common work-sheet to store the co-ordi-
Daniela data manipulation), has been approached since the late nates and their Fourier transform; "PAST", a free soft-
eighties. It has been shown that Fourier shape analysis ware for statistical analyses in biology and palaeonto-
Basso (FSA) is particularly useful for numerical description of logy (Hammer et al, 2005) allowing to standardize data
diverse organisms or biological structures such as coc- and to perform multivariate statistical analysis of profi-
coliths (Garratt & Swan, 1997), leaves (White et al., les (similarity matrix, dendrogram and Principal Com-
1988), foraminifers (Healy-Williams, 1984; Belyea & ponent Analysis). FFT coefficients have been obtained
Thunell, 1984), bivalves (Ferson et al., 1985; Crampton, with the programs Hmatch and Hangle (Crampton &
1996; Crampton & Maxwell, 2000), trilobites (Foote, Haines, 1996) and subsequently elaborated with PAST
1989; Cronier et al., 1998), ostracodes (Burke et al., 1987; in order to perform the analysis of similarity and the
Foster & Kaesler, 1988), blastoid echinoderms (Waters, PCAordination.
1977),bryozoans (Anstey & Delmet, 1973) and fishotho- The valves have been randomly extracted from the 250
liths (Gauldie & Crampton, 2002; Lombarteetal.,2006). right valves on which a morphometric analysis based
FSA takes an outline contour, described as a polygon of on weight (as a proxy of shell thickness), length, height
digitized XY-coordinates, and decomposes it into a and width (as synthetic descriptors of shape) of each
spectrum of harmonically related trigonometric curves. valve was already performed (Basso, 1992). This ap-
Principles of FSA have been developed into different proach allows the comparison of the results obtained
transfomations. In particular, the Elliptic FourierAnaly- with the traditional method and with the digital-profi-
sis (EFA; Kuhl and Giardina, 1982) and the Fast Fourier les method. Procedural details are listed inappendix.
Transform (FFT) have been used within DOS-based
softwares devoted to shape analysis of fossils (Cramp- Results
ton, 1995).
The multivariate statistical analysis of outlines of 39 G.
The analysis of outlines is considered a powerful tool
for morphometric studies, especiallywhen homologous triangularis right valves produced graphic results (den-
drogram and PCAscatter plots, Figs. 2-5) in which each
structures are lacking and consequently, landmark-ba-
point represents one right valve profile. The statistical
sed measurements do not apply (Harper, 1999). Fourier
analysis produced different results for EFA- and FFT-ba-
shape analysis converts each digitalized outline into a
sed shape analysis. Some representative outlines are re-
suite of numerical descriptors (the Fourier coefficients)
that describe curves (Crampton & Maxwell, 2000). The portedover thecorrespondingnumberin thePCAplot.
coefficients are the input to multivariate statistical ana-
Elliptic Fourier Shape Analysis
lyses, that elaborates a numerical comparison ofshapes.
Haines & Crampton (2000) observed that EFA yelds a The dendrogram (Euclidean distance, paired groups.
relatively large number of Fourier coefficients that are Fig. 2) and thePCA(Fig. 3)based onthe39profiles ofG.
not computationally independent ofeach other. This re- triangularis right valves show three clusters (A to C).
dundancy would hamper and compromise the statisti- ClusterA is made of ventrally rounded valves (G. trian-
cal multivariate analysis of coefficients. To overcome gularis forma subtrigona), cluster B contains valves with
this problem, it has been proposed another method of more triangular shape (G. triangularis forma typica) (Fig.
FSAcalled FastFourierTransform (FFT;Crampton, 1995). 2). Cluster C contains valves with intermediate shapes
The aim of this paper is to suggest a friendly, easily ac- and the shell outlines at its border could be included in
cessible and practical method for a statistical approach the firstand second clusters. In the third cluster, outlines
to shape analysis and to explore the sensitivity of both no. 37 and 5 correspond to intermediate shapes. Results
EFA and FFT by comparison with results of traditional of PCA (Fig. 3) show that the first two components re-
132 morphometries. presentsabout91%ofthe totalvariability.
Canonical
morphometry
Fig. 1. Schematic procedure to perform
multivariatestatistical analysisofshell out- versus
lines. Details intheAppendix.
Fig. 1. Schema della procedura peresegui-
statistical
re un'analisi statistica multivariata su profili
diconchiglie. Dettagli InAppendice.
treatment
Fast Fourier Transform ratio are concentrated between values of 0.02 and 0.01 of
of
the firstcomponentandbetween0 and 0.04 ofthe second
The dendrogram based on coefficients produced by FFT component. The intermediate shell outlines that were outlines
(Euclidean similarity, paired groups, Fig. 4) shows only 2 plotted in cluster C (EFA plot) are here splitted into clu-
clusters(Aand B),representedalsointhePCAscatterplot sters A and B. In particular, outlines no. 5, 12 and 37 are through
(Fig. 5). Cluster A includes G. triangularis form subtrigona assigned to cluster B, close to the boundary with cluster
and cluster B encompasses specimens of G. triangularis A.Asa result,FFTproducesasharpseparationoftheout-
form typica. The shells with the maximum lenght/height linesintoclusters. Fourier
shape
lOcCMsiCvO-Tc-oCcOoTa-r1r-^CoMrio-oCoMo<oNLCnM^CiM-CrM^CcMoTCM-cCMOCOvfCO^C¡O-h-CNCO05oCO^t--oCiOt-IDiCO^t--otc-mCiOorc-oCcMot^-cCdOoCoO analysis:
an
empirical
comparison
Fig. 2. Dendrogram produced by hierar-
chical clustering of trigonometric coeffi-
cients (corresponding to the outlines) ob-
tainedwith EFA. ClustersAtoC are obtai-
ned at the - 0,015 level of Euclidean dis-
tance. Clustering calculated on unweigh-
ted pair-groupaverage(UPGMA).
Fig. 2. Dendrogramma prodotto dal clu-
stering gerarchico agglomerativo dei coef-
ficienti trigonometrici (corrispondenti ai
profili)ottenuti con l'Analisi Ellittica di Fou-
rier(EFA). I gruppi da Aa C si separanoal-
la distanza euclidea = -0,015. Il clustering
ècalcolatosul legamemedio(UPGMA). 133
Francesco
Brusoni
&
Daniela
Basso Fig. 3. PCAscatterplotbasedonthesame
coefficients (EFA) and Euclidean distance of
Fig. 2.TheAtoCclustersareindicated.
Fig. 3. Grafico PCA basato sugli stessi
coefficienti (EFA)edistanzeeuclideedi Fig.
2. Sonoindicati igruppi daAaC.
Discussion typica, prosocline in the subtrigona). Unlike "traditional"
morphometries, shape analysis considers only profiles
In order to study the morphometric variability of Goo- and disregards dimensions. Therefore, unlike Basso
dallia triangularis occurring in a Mediterranean shell as- (1992) our results show juvenile shells within all three
semblage (= thanatocoenosis). Basso (1992) faced the dif- clusters of EFA-based dendrogram, to indicate a possi-
ficulty to measure and deal with valves that rarely ex- bleearly divergencein shell morphology.
ceeded 3.3 mm in length. It was required a time-consu- When a convenientnumber of specimens is available, it
ming job and the use of high-precision calibres and ba- is possible to study the reciprocal position of the
lance. Subsequently, bivariate and multivariate statisti- "cloud" of points in the PCA plot, along possible gra-
cal analyses of the measured variables allowed the reco- dients ofshapevariability.
gnitiion of two ecomorphotypes of G. triangularis, one Itis ofa paramount importance toverify that the samples
more triangular, representing the Atlantic, cold-water are photographed with exactly the same position. Al-
form and now occurring in the relict sediment of the though PAST contains a standardization procedure allo-
Mediterranean shelf-break; the other more elongate, wing to deal with some variability in sample orientation,
which is the Mediterranean living form (form subtri- itisbettertohomogenize theorientation inorderto avoid
gona) (Fig. 6). These conclusions were laterconfirmedby confusion or subjective bias in the subsequent phases.
Giribet & Peñas (1999) in the framework of the revision Moreover, it must be considered that possible displace-
of the genus Goodallia. The occurrence of intermediate ments ofabivalve shell are in oneplane only and thatis
,
specimens, as observed by Basso (1992), prevents the se- quite easily managed with PAST standardization. Major
paration of these eco-morphotypes into different spe- problemsareexpectedtoarisefromgastropodshellorien-
cies. In the samepaper (Basso, 1992), mostofthese inter- tation, since rotation along the long axis ofthe shell or bi-
mediate specimens, interpreted asjuveniles on the basis tingwillresult ina differentprofileontheplane, and thus
of their very small size, were positioned at the left end the same shell variably oriented will be recognized as a
of the PCA scatter plot (Basso, 1992, Fig. 6). The largest differentobject. Thiskindofbiased imageacquisitioncan-
specimens were spread along the right end of the PCA not be solved by PAST standardization (Hammer et al,
plot, with the longest and relatively light valves at the 2005).
bottom right (form subtrigona), and the highest, relati- Most difficulties arise in establishing an homology bet-
vely heavy valves at the top right of the plot (form ween points ofprofiles; points mustbe philogenetically
typica;Basso, 1992, Fig. 6). homologous in order to perform a meaningful statisti-
The complete outline of the shells of the two subspecies cal analysis. (Booksteinetal, 1982; Read & Lestrel, 1986;
of Goodallia triangularis summarizes the general shape, Temple, 1992). In many cases it is possible to identify
the slope of the margins and the roundness, and the homologous points in a biological specimen. These
more or less equilaterality of the bivalve on thebasis of points are taken as landmarks in traditional morpho-
134 the position of the umbo (orthocline in the subspecies metry. Thousands of literature examples cover very di-
ccnojclootoc\jCtN—JCDh-a>^co- cnj^coooo-v—ccoococcooocoaC>Ovr--^v-CNJt—c<o—ccnojccoo^C\lcto—cCnNjJlC\oliTo-at—>oCN>Jc0o4cCNoI^co-CoNJcto—
Canonical
Fig. 4. Dendrogram produced by hierar-
chical clustering of trigonometric coeffi-
cients (corresponding to the outlines) ob-
tained with FFT. Clusters A and B are ob-
tained at the - 0,017 level of Euclidean morphometry
distance. Clustering calculated on un-
weighted pair-groupaverage(UPGMA).
Fig. 4. Dendrogramma prodotto dal clu- versus
steringgerarchicoagglomerativodeicoeffi-
cienti trigonometrici (corrispondenti ai pro-
fili) ottenuti con Trasformata Veloce di statistical
Fourier(FFT). I gruppiAe Bsiseparanoalla
distanza euclidea di-0,017. Il clusteringè
calcolatosul legamemedio(UPGMA). treatment
verse subjects such as trilobites, molluscan shells or the ther than a precise point (Crampton, 1995; Crampton & of
biometry of human skull (for example: Hughes & Haines, 1996). This problem represent a possible source
Chapman, 1995; Lacroce & Repetto, 2004). In other in- of uncertainty in manual measurements, and is negligi- outlines
stances these points that are easily identified from a ble only for large-sized specimens. Similarly, it is diffi-
biological point of view, are difficult to locate with pre- cult to locate homologous points on the spire of gastro- through
cision in thepractice. This canbe the case of the bivalve pods, with the exception of the apex (protoconch). The
umbo that corresponds to a small area on the shell ra- use of a complete outline, as in Fourier shape analysis, Fourier
shape
analysis:
an
empirical
comparison
Fig. 5. PCAscatterplotbasedonthesame
coefficients (FFT) and Euclidean distance of
Fig. 4. The A and B clusters are indicated
(from Basso, 1992).
Fig. 5. Grafico PCA basato sugli stessi -0 02-0,01 0 0 010 020 030 040 050 060 070 080,09
, , , , , , , , ,
coefficienti (FFT) e distanze euclidee della
Component
Fig. 4. Sono indicati gruppi Ae B(da Bas- 1
i
so, 1992). 135
4
A
1.0
1
.8
Wt
.6
W
.4
.2
-.2
H
-.4
-.6
L
Francesco -.8
i.0
Brusoni
& CM
c
0c)
Daniela o B
Q.
E
Basso oo 2.
1 .8
Fig. 6. A. Principal component analysis
for L= length; H = height; W=width; Wt
= weight. B. component scores plot (from
Basso, 1992).
Fig. 6. A Analisi dei componenti princi-
pali (PCA) per L = lunghezza; H = altezza;
W = spessore; Wt = peso. B. Grafico dei
componentscores{da Basso, 1992).
corresponds to the use of all homologous points at the required few hours (Fig. 1). With some practice, only
same time, therefore the identification of selected land- few minutes are necessary to produce the Fourier series
marks is no more necessary (Foote, 1989). However, all from the digitalpicture.
specimen outlines must be obtained with the same spe-
cimen orientation, in order to allow comparison among Conclusions
series ofhomologouspoints.
Haines & Crampton (2000) observed that EFA yields a This paperisaimed to compare theperformance oftradi-
relatively large number of Fourier coefficients that are tional morphometries and of FSA in the description of
notcomputationally independent ofeach other. This re- small bivalves. In agreement with Davoli (1982), the best
dundancy would hamper and compromise the statisti- practice of a morphologic-comparative study should be
cal multivariate analysis of coefficients. Actually, FFT of theuseofobservationsand measurements ofall morpho-
our Goodallia outlines produces the sharpest separation logical variables, avoiding the use of few, arbitrarily cho-
ofthe shells into clusters, which is an obvious advantage sencharacters ofclassification, hr thecaseofGoodallia, the
for interpretation. EFA results are comparably less clear. very small size and the fragility of shells prevent the col-
Interestingly, the same outlines (15 and 36) are at the op- lection offurtherlinearmeasurementsinaddition to shell
positeends along component 1 in both PCA, while outli- length, height and width considered by Basso (1992).
nes23and3, thatareoppositealongcomponent2inEFA Moreover, some morphologic features are traditionally
(Fig. 3),arealsosplitbycomponent 1 inFFT(Fig. 5). defined ina qualitativeway (such as thesymmetry ofthe
The procedure used here is much faster and simple in bivalve shell and its roundness), preventing their quanti-
comparison with traditional morphometries and allows fication,objectivedefinitionandstatisticaltreatment.
an easier graphic interpretation of results. Despite the The first advantage of FSA is the absence of assump-
use of several different softwares, the entire procedure tions on characters to be selected and measured. The
136 from digital photographs to dendrogram and PCAplots transformation of the outline of a biological structure
3)
into a suite of mathematical parameters reduces subjec- Transforming the outline graphic file into a suite
tivity since outlines are the synthesis of many measure- of co-ordinates (X, Y)
ments (height, length, width, slopes, convexity, etc.). It Most common softwares will "read" the outline the occi-
can be argued that some features cannotbe detected by dental way: from left to right, from top to bottom, as a
FSA of outer side of bivalve shells, such as internal written text. On the contrary, the outline transformation
structures, shell thickness or ornamentations. However, intoco-ordinatesrequirestoread itcounterclockwise,each
most internal features of bivalves correspond to major pixel only once. Therefore it is necessary to divide each
supraspecific diagnostic characters, such as the type of valveprofileintohalf(rightand left) thewillbedealtwith
palliai line (integer or with a sinus) and the hinge struc- separately. Successively the two halves will be converted
ture, that are efficiently and quite rapidly described in inASCII(.dat)filecontainingthepointsco-ordinates.
the traditional qualitative approach. Moreover, thick or- TheASCII filecontains the descriptionoftheprofilegiven
namentation such coarse ridges and costae, spines etc. by the co-ordinates of all the composing points. Opening
areexpected tobedetectablealong the outline. the electronic sheet of each half profile we will find three
Canonical
Multivariatestatistical elaborationhad analogousbearing columns: the first for abscissas, the second for ordinates,
in respect to the traditional morphometry, leading to an the third for the RGB colour of each point. Using filters it
easyseparation of thetwoformsofGoodalliawitha much is possible to separate the profile points from the back-
fasterand lesssubjectiveprocedure. The sameresults,ob- groundsimplyselectingthecolour (blackpoints=RGB0). morphometry
tained with the canonical morphometry based on linear After this stage a new file should be created containing
measurements (Basso, 1992), required electronic instru- only the twocolumns ofco-ordinates.
ments for precision mechanics and a rime-consuming The sameprocedure is repeated for the otherhalfprofile versus
specimen manipulation. The second advantage of FSA is that must be put upside down, in order to allowjoining
theobjectivenessandtherapidityofdataacquisition. the first half reading the whole profile counter clockwi- statistical
In the comparison of the PCA results based on the two se. This way the table will show the co-ordinates of
different Fourier transformations, FFT produces the points of the profile starting from the uppermost point
sharpest separation of the outlines into clusters while 4o)f the left half-profile, down to the lowermost point of treatment
EFAresultsarecomparably less clear (Figs 2-3). the same left half-profile, up in a counter clockwise sen- of
At the present state of the art, FSA is not expected to se from the lowermost point of the right half-profile to
substitute all canonical morphometric methods. A criti- theuppermostpointofthe same right half-profile. outlines
cal pointremains the choice oftopics thatcanbe succes- All couples of co-ordinates (the first for abscissas, the
sfully afforded through the comparison ofbi-dimensio- second for ordinates) pertaining to all outlines should through
nal profiles ofhomologous points, as mostbivalves are. be saved in the samefile.
Gastropods could be eligible for FSA, provided that a
Fourier
constant orientation of the specimens is ensured. Example:
Among bivalves, the analysis of similarity among diffe- XlrtYlrt XIbY1b whereeach lettera, b, ... shape
rent species of the same genus or among infraspecific X2aY2n X2bY2b identifies an outline
taxa appears particularlypromising. X3aY3a X3bY3b
analysis:
The method described in this paper involves the use of
different softwares. Work is in progress to merge and in- an
tegratethem intoonesinglesoftwarethatwouldbea step Transformation of co-ordinates into coefficients
forwardsthedivulgationofsuchapowerfultechnique. Two different pathways are followed depending on the empirical
selected type of Fourier transform: Elliptic Fourier
Appendix Analysis orFastFourierTransform.
comparison
Procedure:
Elliptic Fourier Analysis (EFA)
PASTprograms
1) Digital photographs
Specimens must be complete and broken margins must Theco-ordinates file is copied into PAST (Hammeretai.,
be avoided, since they will be interpreted by FSA as a 2005). It is now necessary to transpose the matrix of co-
dramatic change of outline. The color of the specimen ordinatesona line, alternatingabscissas and ordinates:
background shouldbe selected tocontrast the shell outli-
Example:
neandenhanceseparationwhileworkingatpixellevel.
XlflYlflX2a Y2a X3flY3fl
2) Obtaining outlines XlbY1bX2bY2bX3bX3b
All digital pictures to be dealt with must have standard
size (pixels) in order to obtain black outlines, composed This can be obtained with the PAST function "grouped
of the same number of points, on a white background columns TO MULTIVAR", edit menu) indicating two di-
(
(colours can be inverted). This condition must be sati- mensions. Then the transpose function can apply.
sfied when entering the PAST procedures for similarity Inorder tostandardizeprofiles insizeandrotation onthe
analysis. Common graphic softwares canbe used to ob- plane,itisnecessaryto usethePASTfunctionPROCRUSTES.
tain the simple shell outlines and "clean" them from Tliis function can be lengthybut is necessary to give pro-
possible "noise" pixel generatedby dust, shadowsetc. files the same orientation. Now profiles are converted 137
,
with the function Elliptic fourier shape analysis in a se- Davoli E, 1982. Cancellariidae (Gastropoda). In: E. Monta-
quenceoftrigonometriccoefficients (Fersonetal., 1985). naro Gallitelli (ed.) Studi monografici sulla malacologia
miocenica modenese. Parte I - I molluschi tortoniani di
Fast Fourier Transform (FFT) Montegibbio. PaleontographiaItalica,72: 5-73.
HANGLE, HMATCH programs Ferson S., Rohlf F.J. & Koehn R.K., 1985. Measuring shape
variation of two-dimensional outlines. Systematic zoology,
The co-ordinates file is the input for the program Han-
34: 59-68.
gle (Crampton & Haines, 1996) that output the Fourier FooteM., 1989. Perimeter-basedFourieranalysis: anewmor-
coefficients. Successively, the Hmatch program normali- phometric method applied to the trilobite cranidium. Jour-
zes the Fourier coefficients of each outline for orienta- nalofpaleontology 63: 880-885.
tionand startingpositionofthe trace. FosterD.W. &Kaesler R.L., 1988. Shapeanalysis. Ideasfrom
the Ostracoda. 53-69. In: McKinney M.L. (ed.) Heterochrony
P5)ASMTulctainvaarlisaotbeestuastiesdtifcoarl calnaaslsyisfiiscation and ordination i7.nPevloelnutuimonP:raesmsu,lNtiediwscYioprlkin,a3ry48appppr.oach. Topics in Geobiology,
Francesco of coefficients obtained with EFA or FFT (cluster analy- Garratt J.A. & Swan A.R.H., 1997. Fourier transforms of
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