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Quantitative Shape Analysis with Weighted Covariance Estimates for Increased Statistical Efficiency.

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Background The introduction and statistical formalisation of landmark-based methods for analysing biological shape has made a major impact on comparative morphometric analyses. However, a satisfactory solution for including information from 2D/3D shapes represented by ‘semi-landmarks’ alongside well-defined landmarks into the analyses is still missing. Also, there has not been an integration of a statistical treatment of measurement error in the current approaches. Results We propose a procedure based upon the description of landmarks with measurement covariance, which extends statistical linear modelling processes to semi-landmarks for further analysis. Our formulation is based upon a self consistent approach to the construction of likelihood-based parameter estimation and includes corrections for parameter bias, induced by the degrees of freedom within the linear model. The method has been implemented and tested on measurements from 2D fly wing, 2D mouse mandible and 3D mouse skull data. We use these data to explore possible advantages and disadvantages over the use of standard Procrustes/PCA analysis via a combination of Monte-Carlo studies and quantitative statistical tests. In the process we show how appropriate weighting provides not only greater stability but also more efficient use of the available landmark data. The set of new landmarks generated in our procedure (‘ghost points’) can then be used in any further downstream statistical analysis. Conclusions Our approach provides a consistent way of including different forms of landmarks into an analysis and reduces instabilities due to poorly defined points. Our results suggest that the method has the potential to be utilised for the analysis of 2D/3D data, and in particular, for the inclusion of information from surfaces represented by multiple landmark points.
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Ragheb et al. Fron tiers in Zoology 2013, 10:16
http://www.frontiersinzoology.com/content/10/1/16
RESEARCH Open Access
Quantitative shape analysis with weighted
covariance estimates for increased statistical
efficiency
Hossein Ragheb1*,NeilAThacker
1,PaulABromiley
1, Diethard Tautz2and Anja C Schunke2
Abstract
Background: The introduction and statistical formalisation of landmark-based methods for analysing biological
shape has made a major impact on comparative morphometric analyses. However, a satisfactory solution for
including information from 2D/3D shapes represented by ‘semi-landmarks’ alongside well-defined landmarks into the
analyses is still missing. Also, there has not been an integration of a statistical treatment of measurement error in the
current approaches.
Results: We propose a procedure based upon the description of landmarks with measurement covariance, which
extends statistical linear modelling processes to semi-landmarks for further analysis. Our formulation is based upon a
self consistent approach to the construction of likelihood-based parameter estimation and includes corrections for
parameter bias, induced by the degrees of freedom within the linear model. The method has been implemented and
tested on measurements from 2D fly wing, 2D mouse mandible and 3D mouse skull data. We use these data to
explore possible advantages and disadvantages over the use of standard Procrustes/PCA analysis via a combination of
Monte-Carlo studies and quantitative statistical tests. In the process we show how appropriate weighting provides
not only greater stability but also more efficient use of the available landmark data. The set of new landmarks
generated in our procedure (‘ghost points’) can then be used in any further downstream statistical analysis.
Conclusions: Our approach provides a consistent way of including different forms of landmarks into an analysis and
reduces instabilities due to poorly defined points. Our results suggest that the method has the potential to be utilised
for the analysis of 2D/3D data, and in particular, for the inclusion of information from surfaces represented by multiple
landmark points.
Introduction
The introduction of geometric morphometrics has laid
the foundations for a quantitative description of shapes
and shape differences, thus revolutionising the century
old quest for comparing anatomical features of organisms
[1]. It is now also increasingly used to link quantita-
tive descriptions of shape with developmental processes
and associated genetic factors [2]. This process generally
involves the construction of a parametric model based
upon exemplar biological shape specimens, and the most
popular of these are linear models. These are used to
*Correspondence: hossein.ragheb@manchester.ac.uk
1Imaging Sciences, Faculty of Medical and Human Sciences, University of
Manchester, Manchester, UK
Full list of author information is available at the end of the article
quantify and predict the correlations in shape variation
between and within species. The objectives of this paper
are to improve the statistical efficiency of analysis tech-
niques used in the genetic interpretation of shape varia-
tion (morphometrics) and to broaden the scope of prob-
lems which can be tackled with shape analysis tools. In
particular we believe that much shape data is not suit-
able for use in current approaches, and ‘semi-landmarks’
(those poorly localised in one direction and the major-
ity of measurements for smooth 3D shape) cannot be
appropriately utilised [3,4].
Over a decade during the 70’s, bio-mathematical and
biometrical aspects of biological shape studies were
treated separately. This early work was later criticised
during the 80’s by Bookstein [5], Goodall [6] and
© 2013 Ragheb et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/2 .0), which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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Kendall [7]. Later, Bookstein [8] worked towards con-
verging notations from Goodall, Kendall and himself,
for the biometric analysis of landmark data in a bio-
mathematically interpretable framework of shape. As a
consequence of these efforts, the standard method for
analysis of variation in landmark position is generally
regarded as ‘Procrustes’. It comprises a least-squares align-
ment of a set of landmark features to a mean shape, and
this is often followed by eigenvector analysis of the lin-
ear correlations in variation around that mean. While
the technique is now very popular the approach has
several limitations with regard to the types of variation
with which it can deal. One of these limitations is due
to the assumption associated with taking least-squares
differences and eigenvector summaries of distributions.
Though many regard these as simply definitional, and in
particular associated with ‘shape’, any statistical interpre-
tation suggests that data are measures with homogeneous
noise. On the other hand, the Mantel test [9,10] has some-
times been used as an alternative to Procrustes distance
to compute correlation between distance matrices (usually
symmetric). Though many papers have been published in
this area, we are aware of no work in this, or any related,
area of point distribution modelling that has provided
a framework to allow data to be analysed according to
a measurement process.
Although landmarks are generally carefully chosen in
order to allow accurate measurements of position within
the image, problems will occur if ‘semi-landmarks’, mea-
sured from smooth curves or surfaces and only accu-
rately localised in one dimension, are input to the anal-
ysis. Landmarks with a high degree of variability can act
as outliers in the alignment stage, generating correlated
compensating shifts and rotations of the other points.
As PCA aims to describe the main sources of variation,
high levels of such correlated movement will then nec-
essarily contaminate the extraction of eigenvectors [11].
Figure 1 The geometry shows how to define a base-plane in 3D
(consistent with the base-line in 2D) using three landmark
points.
This contamination cannot be considered a generic vari-
ation, as it has occurred purely due to the uncertainty in
the measurement. This in turn follows from the subjec-
tive definition of the landmark leading to the view that
problems can be avoided via appropriate definition. The
mathematical concept of homology (and mapping) under-
lies many of the considerations behind much theoretical
work that is described with the mathematical formalisms
of isomorphism. Because of such restrictions on the def-
inition of landmarks, semi-landmarks were introduced
[12] in order to allow inclusion of other points which
are not homologous among the specimens. By this we
mean that a unique corresponding location can not be
Table 1 The algorithmic procedure for our new method
Step Process
1 Initialise each translation parameter tkusing the mean of
landmarks in each corresponding shape (k=1, 2, ..., K).
2 Initialise each rotation parameter Rkbased on the orientation
of each shape relative to the 2-point baseline in 2D or 3-point
reference plane in 3D (Figure 1).
3 Initialise scale parameters skas unity, i.e. original scales.
4 Initialise measurement covariance matrix as identity matrix.
5 Compute initial transformed shapes zk.
6 Compute the initial mean shape m(and adjust transformation
parameters so that the mean orientation is roughly aligned
with the reference baseline/plane).
7 Compute current transformed shapes zk.
8 Compute the current mean shape m(Eq. 1).
9 Compute the whitening matrix W.
10 Compute current ghost points gk.
11 Construct current models z
kbased on PCA and the number of
eigenvectors ejchosen J(Eq. 2).
12 Minimise the Mahalanobis distance corresponding to every
shape zk(Eq. 3) using simplex optimisation (where ejand W
are fixed while tk,Rkand sk,andso,zk,m,gkand z
kare varied).
13 Update current estimates of tk,Rkand skbased on the out-
come of the optimisation, and then update current estimates
of zk,m,gkand z
k.
14 Compute current estimate of the sample covariance matrix C
(Eq. 4).
15 Compute covariance correction term Cejdue to degrees of
freedom in the model (Eqs 5-6) for every eigenvector used ej
(J=1, 2, ..., J).
16 Skip this step for the first iteration (as it requires an esti-
mate of C); compute covariance correction term Cidue to
parameter orthogonalisation (Eqs. 7-8) for every direction vec-
tor icorrespondingto transformation parameters,i=1, 2, ..., I
(where I=4in2DandI=7in3D).
17 Compute current estimate of the measurement covariance
matrix C(Eq. 9).
18 Repeat steps 7 to 17 until convergence (typically 10 itera-
tions).
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defined. Measurement at these locations must be regu-
larised by a constraint, such as bending energy [12,13],
in order to recover the information missing due to the
nature of local structure.
From a statistical perspective a homology (in this con-
text) must be augmented by distributions indicative of
the extent to which a correspondence can be estab-
lished. The standard way to deal with inappropriate
weighting of data in a least-squares fit is to gener-
alise the least-squares cost to a Mahalanobis distance,
computed using measurement covariances. By avoid-
ing the requirement of specifying a unique homolo-
gous location, this has the advantage of accommodating
varying precision in measured data without having to
try to re-create missing data. There have been sev-
eral attempts in the literature to include measurement
errors for landmark points. For example, Fitzpatrick
et al. [14] worked on the relationship between locali-
sation error and registration error in rigid-body, point-
based registration. Chui and Rangarajan [15] proposed a
general framework for non-rigid point matching, where
outliers are effectively rejected. Rohlf and Slice [16],
and Walker [17] investigated how to estimate mea-
surement covariances in forms. However, Richtsmeier
et al. [18], Adams et al. [1] and Rohlf [19] all stated that
further research was needed in this area. Also, Walker [17]
and Lele [20] concluded that generalised Procrustes anal-
ysis (GPA) estimators of the variance-covariance matrix
are flawed. Despite the fact that some biologists have
noticed these problems, they seem to know of no avail-
able alternatives and continue to use GPA to estimate
covariances [21].
Text books [22] state that using weighted Procrustes
does not lead to a Kendall’s shape space. Claiming that
“statistical analysis cannot employ parametric models”,
they suggested that resampling-based methods must be
used instead. Another reason for rejecting the idea of
a weighted Procrustes was said to be a “lack of clear
criteria for determining appropriate weighting of semi-
landmarks”. These criticisms can only really be inter-
preted once a method for weighting is specified. Goodall
[23] suggested a method in which the same covariance
was used for all landmarks. By this we mean there was
no separate description of the perturbation of individ-
ual landmarks. It has been noted that such a matrix
is inestimable [24]. Goodall himself acknowledged that
“as a model of measurement error this is a drawback,
as the direction of greatest variation may vary consid-
erably between landmarks”. Despite this problem, later
work [25] generalised this idea to a Bayesian framework.
We believe that it makes sense instead to suggest an
approach which can support the process of landmark
location as measurement, with a covariance describing
the localisation of each landmark separately (see [26,27]
for example). Specifically, Rohr et al. [26] used covari-
ance matrices in a Mahalanobis distance form for non-
isotropic data, where covariances were estimated from
image data through landmark localisation, i.e. using grey-
value information from local pixels around each landmark
for matching an image area/volume structure to another
through optimisation of a cost function. The minimal
localisation uncertainty for each point were estimated
using the Cramer-Rao Bound (CRB). Also, smoothness
was included as the second term in their functional and
controlled using a regularisation parameter. To our knowl-
edge, they have been the first to provide a relatively
comprehensive approach for incorporating anisotropic
covariances into image registration using splines. How-
ever, here we only deal with pre-defined landmark data
and, unlike their method (and our recently published
method [28]), do not attempt to extract landmarks and
their corresponding covariances from image data. Specif-
ically, in [28], we have applied smoothing to local edge
data (where information is) prior to optimisation in order
to remove the effects of spatial noise and obtain mean-
ingful CRB estimates. However in our current study, the
only input data fed to our method are a number of shapes
represented by fixed landmark points. Hence, we do not
take into account any information about the local struc-
ture surrounding each landmark. This way, the task of
covariance estimation may be seen even more challeng-
ing. We are aware that in biological studies it is now
commonly accepted that for point-based shapes, extra
information about the local/global pixels in the image
plane/volume (for 2D/3D data) is usually available using
modern imaging equipment. However, here our observa-
tion is that geometric morphometrics should originally
be capable of dealing with the study of 2D/3D forms [18]
even for non-biological data or cases where information
about the local structure around each landmark is miss-
ing or difficult to access or process. It is worth mentioning
here that one reason why Procrustes still is popular is
that apart from the forms (shapes) represented by land-
mark points it does not require any further data such
as images from which the points have been originated.
Hence, even though the datasets we use in our experi-
ments are biological and one could also feed in the image
data, in this study we chose to start the process from
pre-defined landmarks only. Ideally, covariances extracted
from image data (using other methods such as ours [28])
could be fed to our current method and be used, for
instance, as initial estimates. This is however a subject for
future investigation.
There have been further publications on anisotropic
weighting, for instance in [29,30]. Mathematically, these
methods are all equivalent to our approach, in that they
use a Mahalanobis distance based upon anisotropic dis-
tributions of individual points. However, they do not have
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a well-defined mechanism for the estimation of these dis-
tributions. This is a key issue when applying these ideas
to shape samples. Our work provides such a mechanism
while incorporating corrections for estimation bias [7].
The basic concept can be implemented via a standard
technique used in pattern recognition, often referred to as
whitening [31]. For instance, in the context of shape anal-
ysis, the whitening transform and shape de-correlation
were used as a preprocessing step in PCA/ICA analy-
sis [32,33]. However, there is a difference between using
whitening methods to model the signal variation of data
(as used in these papers) and using the same technique
to better construct a likelihood function that accounts for
correlationinmeasureddata(aswedohere).Recently,
the technique has been applied to the within group bio-
logical covariances [34], but again not to the process of
noise on measurements. Here, we shall investigate pos-
sible generalisations of Procrustes along these lines, and
the different ways such a measurement covariance may
be estimated. As a key issue here is the computability
of these covariances, the stability of the resulting analy-
sis is an important question for investigation. The theory
presented here can thus be classified in the same cate-
gory as both Procrustes based shape analysis [35] and
active shape models [36]. The main difference, however,
being that our model is for a realisable system and self-
consistent estimation of the associated model parame-
ters.
There has been an ongoing discussion in the biology
literature regarding appropriate ways to deal with non-
homologous landmarks (points defined on smooth curves
and surfaces) during statistical analysis. For instance,
Klingenberg [37] has objected to Polly’s conclusions
[38] regarding the benefits of existing homology-free
approaches. He believes that these approaches all depend
critically on some sense of homology since they are not
really free of assumptions about the correspondence of
parts. Oxnard and O’Higgins [39] haverecommended that
it is biology that has to inform morphometrics in planning
the landmark configuration (mainly mathematical land-
marks, i.e. those computed using geometric constraints
based on the neighbouring true landmarks) in relation
to the hypothesis available. The approach to dealing with
semi-landmarks in the morphometric analysis of shape
currently seems to be divided between two alternatives,
both of which aim to adjust the position of these land-
marks by optimising a specific metric, before constructing
a linear model of variation about the mean. These metrics
are bending energy (BE) and Procrustes distance (PD) [3].
Arguments for and against these approaches are based
upon specific examples in biology. Although evidence
has been reported of utility [40], Slice [41] has stated that
the application of the BE approach to biomedical and
anthropological problems has been minimal. Vignon and
Pierre [4], and Prez et al. [42] have shown concern regard-
ing the observation that different methods for handling
semi-landmarks could result in different conclusions in a
discriminant analysis study. Gomez-Robles et al. [43] have
examined the advantages and disadvantages of differ-
ent novel methods in geometric morphometric analyses
including homology-free approaches, landmark-based
approaches, and combinations of both techniques.
Comparison between results from shape analysis and
genetics is an important research topic in evolutionary
Figure 2 Typical landmarks corresponding to sample images of fly wings (left) and mo use mandibles (right); for the fly wing d ata,
landmarks 1-15 correspond to the original data sets FL1, FL2, FR1 and FR2, while landmarks 16-19 were added later (to FL1) in order to
experiment with semi-landmarks (P-FL1).
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Figure 3 Typical landmarks for a sam ple volume (top-left) from the 3D mouse skull (MS) data when projected on the xy (top-right), zy
(bottom-left) and xz (bottom-right) planes (the 50 points are too close to display full numbering).
biology. For instance, Frederich et al. [44] have attempted
to estimate the statistical correlation between morpho-
logical, genetic and geographical distances. We offer an
alternative shape analysis method that tackles the existing
problem in the literature, so that well defined comparisons
become statistically valid and informative.
Methods
Suppose that there are Kshapes in our data-set and each
shape vector wkcontains Nlandmark points, i.e. wk=
[w1x,w1y,w2x,w2y, ..., wNx,wNy]kfor the case of 2D data.
We then apply a scale sk, a rotation Rkand a translation
tkto the original data to get an aligned version of the
Figure 4 Fly wing data (P-FL1): major eigenvalues of the error using our 1, 2 and 3 component models against those computed using a
repeatability test on four new semi-landmarks placed manually on a subset of data; the 2 component model gives closest agreement to
the expected localisation values; the two dashed lines show the ±2.8σrange.
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Figure 5 Mouse mandible data: major eigenvalues of the error estimated using our 5, 6 and 7 component models on MM1 data against
those computed using the corresponding repeatability test (MM1 and MM2); the two dashed lines show the ±2.8σrange.
data called zk,wherezk=[z1x,z1y,z2x,z2y, ..., zNx,zNy]k
and zk=skRk(wktk).
The mathematical description of the model so far can
accommodate any value of scale or orientation for the
definition of mean model. We therefore define the orien-
tation of mean shape so that the line between a specified
pair of points is horizontal. This also has the benefit
that initial estimates of alignment for sample kcan be
set according to the relative positions of these points.
We also use the average distance between these same
landmarks to rescale the mean shape at each iteration so
that scale remains numerically defined.
For 2D data, we assume a different but fixed 2×2covari-
ance matrix for each landmark derived from the measure-
ment process. These are composed into the matrix C.This
is a tri-diagonal matrix, the diagonal line of which con-
tains data for individual landmarks. Outside of the 2 ×2
covariances, the off diagonal elements of Care zero, i.e.
there are no correlations between landmarks. The use
of a fixed data covariance cancels out when taking the
weighted mean, to regenerate the conventional formula
for the mean;
m=1
K
K
k=1
zk(1)
where m=[m1x,m1y,m2x,m2y, ..., mNx,mNy]. This defini-
tion for mean shape has previously been shown to provide
unbiased estimates using Monte-Carlo re-sampling stud-
ies [19], which is to be expected for a valid likelihood
estimate of parameters.
The points zkdo not have uniform independent noise
distributions, which is one of the assumptions for the
application of PCA. However, this property can be
obtained via a whitening transformation. Although trans-
formation of data can be considered as a new space, it
can also be interpreted as an affine re-projection. The
points obtained by applying a whitening transforma-
tion are referred to here as ‘ghost points’. Ghost points
are accordingly defined in the original coordinate sys-
tem and, being scaled projections relative to the shape
centroid, are an alternative way to summarise the orig-
inal measurement relative to the observable structure.
This is an important philosophical issue for those who
believe that the original co-ordinate system is some-
how more meaningful as a description of biological vari-
ation than any linear re-projection (see Discussions).
The process amplifies the spatial variation in directions
which are well measured relative to those which are
not so that the resulting locations have isotropic errors
(as required). In turn, this allows accurately measured
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Figure 6 Mouse skull data: major eigenvalues of the error estimated using our 8, 10, 12 and 14 component models for the 3D MS data
against those computed using the corresponding repeatability test; the two dashed lines show the ±2.8σrange.
structuretobeencodedinthemostsignificanteigenvec-
tors (those with largest eigenvalues) of the linear model.
We transfo rm zkto ghost points gkusing the matrix Wso
that gT
k=W(zkm)T
.
By applying singular value decomposition to C1,i.e.
C1=UTVU, and making it equivalent to WTIW,we
find that the required whitening matrix is W=V1/2U.
Application of PCA to gkfollows for construction of the
shape covariance, giving the eigenvectors ejand eigenval-
ues μjfor the whitened space of ghost points as those
which minimise the unexplained variance for fixed J<N,
where Jis the number of eigenvectors used in the model.
Hence, for any specific shape example k, linear factors
λjk =ej.gkcan be computed to best approximate zkwith
the model z
k;
F=
K
k=1
gT
kgk
J
j=1
μjeT
jej,z
k=m+W1
J
j=1
λjkej(2)
A genuine likelihood should be based upon the variation
of the data around the assumed model. Failure to do this
results in residuals which cannot be meaningfully inter-
preted a. Using this argument, if we wish to align to the
mean shape we should use a covariance that is consistent
with the distribution around the model. In order to find
the best Rk,tk,skparameters for each k, we minimise a
Mahalanobis distance which is given by
log(Pkz)=(z
kzk)TC1(z
kzk)(3)
This is simply the likelihood estimate for the location of
the shape given the linear model and the assumed mea-
surement covariances and can be interpreted directly as
aχ2statistic. By replacing Cwith Iand z
kwith mthis
reduces to the least-squares function for standard Pro-
crustes. We can therefore interpret this as a generalisation
of the standard approach. However, we do not wish to
generalise further by using for example PPCA (probabilis-
tic principal component analysis) [45], as an additional
assumption of a Gaussian distribution over derived vari-
ables is generally invalidated in morphometric data sets.
Use of Eq. (3) requires an initial estimate of the model
and transformed data zk. By setting the initial estimates
of the measurement covariance Cto an identity matrix,
these parameters are given by the Procrustes result. We
can therefore use Procrustes to set up the initial trans-
formation estimates. To reach the best possible alignment
using our new method (anisotropic C), we iteratively re-
estimate Rk,tkand skusing the assumed ej,m,C1and
W1.Thisgivesusanewzk,andsoanewmand F
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Figure 7 Fly wing data (P-FL1): error eigenvalues estimated using the Monte-Carlo data (where mean shape, eigenvectors, and
measurement covariances are identical to the model which generated the simulated data) against the expected ones (Test A); using 2
model components; there is only marginal evidence of estimation bias before correction;the two dashed lines show the ±2.8σrange.
for construction of ej. For fixed covariances, convergence
can be monitored via construction of the total likelihood
log(P)=K
k=1log(Pk). One may use the final estimates
of zkand z
kto construct the sample covariance;
C=1
K
K
k=1
(z
kzk)(z
kzk)T(4)
For a well defined likelihood method this covariance
should be consistent with the assumed distribution C.
However, the use of free parameters during alignment
and model construction introduces biases that must be
addressed in an iterative analysis in order to avoid insta-
bilities, which will now be described.
Covariance correction
When attempting to estimate C, the use of free parame-
ters during model fitting reduces the sample covariance
obtained from residuals. This mechanism is precisely that
identified in [7], whereby the observable variation in any
single shape sample is reduced onto a manifold in 2N4
dimensions for a 2D shape defined for Npoints. This has
generally been considered as a bias in the overall data
distribution rather than being associated specifically with
a statistical estimation error (as here). A possible out-
come of this is the over weighting of landmarks leading
to a runaway convergence on one landmark, during itera-
tive estimation of C.However,thisbiaseffectisestimable
and therefore correctable, as will be illustrated by Monte-
Carlo simulation (below). For a single scale parameter
associated with an approximate linear vector fwe can
use error propagation to estimate the expected average
reduction in the covariance for each 2 ×2 landmark
component of the matrix arising from errors in parameter
estimates nCas
nC=fnTfn
fTC1f,C=
N
n=1
nC(5)
Note that the denominator is the change in χ2expected
due to a unit change in f,andfn=Dnfis the 2D com-
ponent of fcorresponding to landmark n(where Dnis
an operator which zeros all but those quantities associ-
ated with the nth landmark and is for outer product
between two vectors). For an eigenvector ejdefined in
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Figure 8 Mouse mandible data (MM1): error eigenvalues estimated using the Monte-Carlo data (where mean shape, eigenvectors, and
measurement covariances are identical to the 6-component model which generated the simulated data) against the expected ones (Test
A); for this number of parameters there is now evidence of a systematic underestimate of covariance (prior to correction); the two
dashed lines show the ±2.8σrange.
the whitened ghost space, this would suggest a total cor-
rection of
Cej=W1ejTW1ej
W1ejTC1W1ej=W1ejTW1ej
(6)
The known structure of the covariance can be enforced
by zeroing relevant off-diagonal terms. The parameters
of the linear model, including scale, rotation, transla-
tion and linear model weightings can also be treated in
this way. If irepresents one of the direction vectors
of these parameters (with 2Nelements for 2D data), it
follows that direction vectors corresponding to transla-
tion in x and y directions 1=[ 1, 0, 1, 0, ...] and 2=
[ 0, 1, 0, 1, ...] are orthogonal, i.e. 1.2=0. Similarly,
direction vectors 3=m=[m1x,m1y,m2x,m2y, ...] and
4=[m1y,m1x,m2y,m2x, ...] corresponding to scaling
and rotation are orthogonal, and so 3.4=0. Note that
mis identical to the mean vector defined in Eq. (1).
Strictly, Kendall’s definition of shape explicitly removes
aspects of object transformation before model construc-
tion. Joint estimation of shape and alignment parameters
is potentially unstable as estimated linear shape parame-
ters can correlate with transformation parameters. Here
we stabilise this process by removing first order correla-
tions from the data covariance Fprior to model construc-
tion.
Hence, to orthogonalise the model, we modify ghost
points as follows.
g
k=gk(gk.ˆ
YT
i)ˆ
YT
i,ˆ
YT
i=Wˆ
T
i(7)
where the unit vector ˆ
iis the normalised form of i.
The new gkis computed iteratively using each ˆ
iso
that any variation about the mean that could have been
described by an alignment parameter is removed from the
correlation matrix Fprior to model construction. The cor-
responding measurement covariance correction term is
hence given by
Ci=(ˆ
T
iC1ˆ
i)1(ˆ
T
i׈
i)(8)
Therefore, the measurement covariance is estimated
using
C=C+
J
j=1
Cej+
I=4
i=1
Ci(9)
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Figure 9 Mouse skull data (MS): error eigenvalues estimated using the Monte-Carlo data (where mean shape, eigenvectors, and
measurement covariances are identical to the 14-component model which generated the simulated data) against the expected ones
(Test A); the two dashed lines show the ±2.8σrange.
Using the above formula, the contribution to the χ2lost
by using a scaling parameter associated with each vector ej
and icontributes a value of unity to the χ2for each addi-
tional independent degree of freedom, totalling J+4. Our
method for covariance correction is therefore consistent
with a degree of freedom correction as described in con-
ventional analysis approaches [46]. As a consequence the
covariance estimation process can be considered equiv-
alent to the Expectation-Maximisation (EM) algorithm,
both in operation and parameter estimates, so that the
conventional proof of convergence is applicable [47].
Extension from 2D to 3D
Here we outline the mechanism we use to extend 2D
shape rotation analysis, and the extraction of corrected
anisotropic measurement covariances, to 3D. The meth-
ods are demonstrated in the analysis of 3D mouse skull
data, both as a test of the theory/software implementa-
tion and as an illustration of use for the identification
of outlier landmarks.
The extension to 3D data is mainly involved with the
mechanism of representing and estimating 3D shape rota-
tions. We define a fixed orientation co-ordinate system
from a set of 3D data-points based upon a selection
of three landmark points. We then represent a rotation
matrix in terms of three separate rotations about the co-
ordinate axes. Finally we compute the linear vectors which
approximate the first order shifts seen in the 3D points
due to these rotations. These are then used in the lin-
earised approximation for sample covariance correction,
as described earlier. These extensions are enough to sup-
port a quantitative analysis of 3D landmark data, for the
estimation of landmark accuracy and identification of out-
lier data. The mathematical model used is described in
detail here and in Appendix A. We provide quantitative
tests in the Results and discussion section which demon-
strate the numerical stability of the algorithms using
Monte-Carlo data.
Rotation matrix
Our first task is to define a co-ordinate system for a 3D
data-set, from which we can define certain basic prop-
erties of orientation for the mean shape, and so that
individual data samples can be approximately oriented
prior to optimisation during linear model construction. In
the2Dcasethisisdonebydefiningthelinebetweentwo
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Figure 10 Fly wing data (P-FL1): error eigenvalues estimated using the Monte-Carlo data against the expected ones (estimated using the
original data) which were used when generating the simulated data; independent models (Test B); using 2 model components; the two
dashed lines show the ±2.8σrange.
landmark points in the mean model as horizontal. In 3D,
in order to stay consistent with the 2D, we define 2 points
to establish a horizontal, and then a third to define the
vertical relative to the first two.
Given a 3D shape, we take three points P1,P2,P3,
with relatively large distance from each other (Figure 1)
to define the orientation plane for the shape. The rota-
tion matrix RTis hence found based on basic vector
calculations (see Appendix A).
Roll, pitch and yaw angles
Given the rotation matrix that brings a data set into align-
ment with the preferred co-ordinate system it is possible
to represent the rotation as a sequence of rotations about
three orthogonal axes. According to basic 3D rotation
formulas, and using α,β,andγas yaw, pitch, and roll
respectively, the 3D rotation matrix is defined as three
consecutive rotations around the z, y, and x coordinate
axes.
Rxyz =Rx(γ )Ry)Rz(α) (10)
By making the rotation matrix RTequivalent to Rxyz,
we find the yaw, pitch, and roll angles (see Appendix A).
Thus we can convert easily between the rotation matrix
and rotation parameters.
Orientation adjustments
We initialise the rotation angles, by computing the
RTmatrix for every original shape in the data set
based upon the three identified landmark points and
extracting the corresponding α,β,andγangles. These
are then further adjusted during iterative alignment
via optimisation of the anisotropic measurement-based
Mahalanobis distance. We perform orientation adjust-
ment on the mean shape following every iteration
over the set of shape samples. In this case the set of
yaw, pitch, and roll angles corresponding to the mean
shape are subtracted from the corresponding rotation
angles for each shape sample, so that the computed
mean shape complies with the three-point orientation
constraint.
Direction vectors
In order to correct the covariances due to alignment
parameters in 3D, we need the approximate linear direc-
tion vectors corresponding to translation, rotation and
scale. Computing these for translation and scale is
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Figure 11 Mouse mandible data (MM1): error eigenvalues estimated using the Monte-Carlo data against the expected ones (estimated
using the original data) which were used when generating the simulated data; independent 6-component models (Test B); for this
number of linear model components there is considerable er ror in the uncorrected estimates; the tw o dashed lines show the ±2.8σrange.
straightforward. If m=[m1x,m1y,m1z,m2x,m2y,m2z, ...]
is the vector corresponding to the 3D mean shape (with
3Nelements), then the direction vectors due to trans-
lation in x, y and z directions are simply given by
1=[1,0,0,1,0,0,...],2=[0,1,0,0,1,0,...] and 3=
[0,0,1,0,0,1,...].Also,thedirectionvectorduetoscaling
is 4=m.
For rotation, we compute the direction vector corre-
sponding to each individual rotations Rz,Ryand Rx.For
the mean shape mrotated by the yaw angle αaround the
zaxis,wehavem=Rz(α)m.Asαbecomes very small,
the tangential direction of movement in landmark point n
due to this rotation is
uα0=[mny,mnx, 0] (11)
By applying the same method, one can find the direc-
tion vectors due to rotation by the pitch angle βaround
the y axis and by the roll angle γaround the x axis (see
Appendix A). Hence we have
uβ0=[mnz,0,mnx] (12)
uγ0=[0,mnz,mny] (13)
It follows that 5=[m1y,m1x,0,m2y,m2x, 0, ...],
6=[m1z,0,m1x,m2z,0,m2x, ...] and 7=
[0,m1z,m1y,0,m2z,m2y, ...]. The set of vectors 1,2
and 3on one hand, and the set of vectors 5,6and 7
on the other hand are mutually orthogonal and orthog-
onal to the vector 4due to scaling. These direction
vectors now constitute the linearised parameterisations
needed for corrections to the sample covariance (where
I=7inEq.9).
Procedures
Here, in Table 1, we provide the step-by-step procedure
for our new shape analysis method that involves lin-
ear model construction, data alignment and anisotropic
covariance estimation and correction. Note that there is
an arbitrary order for the application of the transforma-
tion parameters which remains consistent throughout the
whole process. In fact, whatever this order, the net effect
of the covariance correction (Eqs. 7-9) is to subtract the
same total linear subspace.
Model selection
A method is needed to select appropriate linear model
order based upon the outputs from our analysis. If the
linear model is valid then estimated measurement covari-
ances will combine two processes of statistical fluctuation.
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Figure 12 Mouse skull data (MS): error eigenvalues estimated using the Monte-Carlo data against the expected ones (estimated using
the original data) which were used when generating the simulated data; independent 14-component models (Test B); the two dashed
lines show the ±2.8σrange.
The first of these will be measurement precision σr(our
ability to define homologous points reliably), and the
second will be due to random (unmodellable) biological
variation σb. So that the observed statistical variation seen
in a given direction vfor any landmark σvis
σ2
v=σ2
r+σ2
b(14)
Unfortunately we cannot know the expected value of
σbin advance. However, the first of these terms can
be estimated via reproducibility experiments and com-
pared to the measured directional covariances, using the
observation that σvσr. Thus if we observe individ-
ual estimates of measurement covariance which begin to
surpass the limiting accuracy known to be set by repro-
ducibility tests, then the model must be over-fitting the
data and therefore has too many parameters. We check
that for a given model order this inequality is satisfied
within statistical limits by considering the principle axes
of each landmark measurement distribution. We use a
1% confidence level to set the hypothesis test for over-
fitting. This test is expected to be most reliable for the
largest variances.
Monte-Carlo tests and outlier identification
As our method is based on likelihood, we require that the
assumed distribution matches the corrected covariance.
The standard way to validate this is through generating
Monte-Carlo (MC) data using the known distributions. In
what follows we experiment with MC data and display a
number of informative scatter plots for two forms of test;
Test A: When applying our method to the MC data, the
mean shape, eigenvectors and measurement covariances
used are identical to the ones used when generating the
simulated data.
Test B: All parameters are estimated using the MC data in
order to compare the measurement covariances estimated
using the simulated data with those expected, i.e. the ones
assumed when generating the MC data.
For Test A the covariances estimated using our method
are expected to be within statistical sampling limits of the
ones used when generating the MC data. Failure to do so
is taken as an indication of a problem with the data sample
(i.e. outliers). Outliers can be identified at early stages of
analysis as those points which have the largest normalised
residual errors.
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Figure 13 Fly wing data (P-FL1): error eigenvalues computed using the residuals after Procrustes alignment on the Monte-Carlo data,
against the expected ones which were used when generating the simulated data; for 2, 3 and 4 model components; the two dashed lines
show the ±2.8σrange.
We use 2.8 standard deviations of the error on the
sample variance (or being allowed to have 1% of data
falling outside the limits), where the error on the stan-
dard deviation σis σ/2(K1)with Kbeing the
number of samples [48]. Additional variance is expected
for Test B (beyond that seen in Test A), where the
linear model must also be estimated. Therefore, hav-
ing excluded the possibility of outliers using Test A,
we can interpret variations beyond the statistical lim-
its as due to instability in linear model construction
(specifically the mean and eigenvectors).
χ2test
A test is needed to confirm the equivalence of measure-
ment covariances computed during repeatability exper-
iments, in order to confirm that our methods generate
estimates which are consistent. This can also be done by
splitting the data into two separate groups if there are a
sufficient number of samples. We perform a modified χ2
test based upon the construction of corrected covariances
on one data set and then used for the calculation of χ2for
the second set. For large numbers of samples (K>30)
the resulting statistic when applied to each 2D landmark
is expected to be approximately Gaussian with mean 2K
and variance 4K. We set the statistical test for significant
difference on the basis of an allowable range of χ2/DoF
corresponding to ±2.8 S.D., i.e. [0.8, 1.2] for 200 samples.
The corresponding plot would confirm the stability of the
method if 99% of the χ2/DoF values fall inside the range
expected.
Fisher information
Fisher information (FI) is a concept for quantifying the
constraint on an estimated value associated with data.
It has the useful property that the amount of estimated
information is linear in the quantity of data. It is gen-
erally defined according to the second derivative of a
log-likelihood function, but from the association of this
function and the CRB we can also observe that, for good
model fits, it is proportional to the inverse variance. An
empirical estimate of the FI contained in data, and asso-
ciated with a particular model, can therefore be obtained
from the residual distributions following parameter esti-
mation.
We use this idea here to summarise the amount of infor-
mation that has been extracted from data for a specific
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Figure 14 Mouse mandible data (MM1): error eigenvalues computed using the residuals after Procrustes alignment on the Monte-Carlo
data, against the expected ones which were used when generating the simulated data; for 4, 6 and 8 model components; the two dashed
lines show the ±2.8σrange.
analysis. As this quantity scales linearly with the quan-
tity of data it allows us to make comparative statements
regarding the statistical efficiency associated with the esti-
mation process. For example, if the FI is seen to double
on the same dataset when applying an alternative analysis
then this is statistically equivalent to having four times as
much data to begin with. A poor analysis method might
need a lot more data to reach the same level of statistical
equivalence in a hypothesis test than a good method.
Ethics
The animal datasets used in this paper have been
approved according to German ethical standards. They
were registered under number V312-72241.123-34 (97-
8/07) and approved by the ethics commission of the
Ministerium für Landwirtschaft, Umwelt und ländliche
Räume on 27.12.2007.
Results and discussion
We have used example datasets to investigate the stabil-
ity of covariance weighted shape analysis and to com-
pare quantitative performance figures to the standard
approach using Procrustes. We have selected several
datasets in order to demonstrate behaviour with different
quantities of data, data dimensionality (i.e. 2D and 3D)
and model order.
As standard methods, even those including landmark
weighting, are not conventionally used in a way that
would support estimation of landmark variability we have
made some assumptions regarding what would be the
most straightforward approach. As mentioned earlier, in
this paper we are interested in analysing point-based
shape datasets without seeking to obtain extra knowl-
edge about local structures surrounding each landmark.
Hence, in conjunction with our method, we have not
used methods that estimate localisation errors from the
original image data such as those described in [26,28].
For Procrustes we use the residuals from the fitted models
to make an estimate of landmark measurement error
(although this is widely concluded in the literature not to
work [22]). For methods that would support anisotropic
weighting, we use a variation of our own method (incor-
porating iterative re-weighted alignment) to estimate the
resulting residuals during iterative analysis. The difference
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Figure 15 Mouse skull data (MS): error eigenvalues computed using the residuals after Procrustes alignment on the Monte-Carlo data,
against the expected ones which were used when generating the simulated data; for 12, 14 and 16 model components; the two dashed
lines show the ±2.8σrange.
between this and our preferred method is the lack of cor-
rection for degree of freedom biases, we therefore refer to
this as the “uncorrected” method.
Data
We experiment with two 2D data sets of manual mark-
ups(Figure2).Thefirstdataset,calledMM1,corre-
sponds to mouse mandible micro-CT images and con-
sists of 337 samples with 14 landmarks per sample.
Wealsohavearepeatdataset,calledMM2,forwhich
same mandible images have been used to mark-up the
points.
Next, we use some fly wing data in order to test the per-
formance of our method on semi-landmarks and also to
test the statistical stability of our method. There are four
original data sets available from left and right wings(L and
R) of 200 female flies, called FL1, FL2, FR1 and FR2 [49].
Two images of each wing were taken from slightly differ-
ent viewing positions (1 and 2), and used for marking-up
in order to perform reproducibility tests [49]. Each of
these four data sets has 200 samples with 15 landmarks
per sample. Further, as we had access to the fly wing
images,wehaveaddedfoursemi-landmarkstoeachsam-
ple of the original data set FL1. Once finished, we removed
Figure 16 Fly wing data (P-FL1): error bars (×20) estimated using our method (left), and computed from the residuals left using
Procrustes (right); 2-component models.
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Figure 17 Fly wing data (FL1): error bars (×20) estimated using our method (left), and computed from the residuals left using Procrustes
(right); 2-component models.
5% outliers and stored 189 samples with 19 landmarks
per sample. This resulting data set, which is called P-FL1,
plays an important role in our experiments with semi-
landmarks. In order to be able to test the repeatability
with these added semi-landmarks, we have repeated the
marking-up process only for the four new landmark points
and using a subset of the left fly wing images.
We also experiment with the mouse skull (MS) 3D data
of semi-automatic mark-ups (Figure 3) produced based
on training examples and the corresponding micro-CT
images. This makes a typical 3D data set of interest in evo-
lutionary biology research. We have used our automatic
tool to localise landmarks on these mouse skulls based on
few given manual mark-up examples [50,51]. This is based
on landmark localisation technique recently described in
[28] (where more details may be found). The automatic
tool also identifies outliers for manual correction and so
we do not expect any outlier in this data set. The mark-
up data set obtained this way (MS) consists of 42 samples
with 50 landmarks per sample. Further, there are two sub-
sets of repeat data based on manual mark-ups (on the
mouse skulls) each consisting of 12 samples to be used in
repeatability tests.
Model selection
In Figures 4, 5 and 6, we plot the eigenvalues corre-
sponding to the errors estimated against those computed
fromtherepeatdata.Thesearethemagnitudesofthe
errors in the direction of major eigenvectors. It can be
seen that while for the fly wing data the errors are com-
parable (Figure 4), for the mouse mandibles there are
several landmarks for which the error estimates are much
larger than expected (Figure 5). We cannot argue for an
increased model order as this then reduces other values to
well below the observed repeatability (over fitting). As the
additional variance seen is due to the inability of the model
to predict correlations in the data, our conclusion must
be that either this data is not well described by a linear
model, or the repeatability estimate systematically under-
estimates the true accuracy with which points can be
meaningfully located. This can happen if local image fea-
tures (which are themselves not well biologically related to
the main structures, such as the brightest pixel) are used to
identify locations. The plot for the more complex mouse
skull data (Figure 6) suggests that 14 components is about
the number needed by the linear model. Hence, when
experimenting with the 3D MS data we use 14 model com-
ponents, while using 6 components with the 2D MM data
and 2-3 components with the 2D FL and P-FL data. The
1% allowable range is set in accordance with 12 repeat data
samples.
Monte-Carlo tests
We show the Monte-Carlo plots for the Test A in
Figures 7, 8 and 9, while the Figures 10, 11 and 12 show
those for the Test B. The results for the Test A on 2D data
sets (Figures 7 and 8) indicate very little difference for the
low parameter fly wing data, and a more obvious system-
atic underestimate of covariance (as expected) for the 6
dimensional mouse mandible data (prior to correction).
Further, the results for the Test B (Figures 10 and
11) indicate that even for the mouse mandible data, the
values of covariance are significantly different, due to
the amplification of initial estimation bias during the
Figure 18 Mouse mandible data (MM1): error bars (×20) estimated using our method (left), and computed from the residuals left using
Procrustes (right); 6-component models.
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Figure 19 Fly wing data (P-FL1): error bars (×20) estimated using our method (left), and computed from the residuals left using
Procrustes (right), with the corresponding aligned data superimposed; 2-component models.
process of iterative linear model estimation. The cor-
rection process now removes these instabilities bring-
ing estimated covariances back close to the expected
sampling limits and symmetrically around the expected
correlation line.
Turning to the 3D MS data, for the Test A (Figure 9)
the eigenvalues fall inside the allowable range (dashed
lines). However for the Test B (Figure 12), the eigen-
values appear to fall under the lower bound. The
under-estimation seen is in accordance with a cor-
rection factor based upon the number of samples
and model complexity (KJ)/K.Unliketheear-
lier biases this under-estimation does not destabilise
the analysis, as a common multiplicative change on
all variance estimates leaves the estimated model
parameters unaffected.
Note that the equivalent residual distributions esti-
mated here from the conventional Procrustes analysis
have no associated correction process and (along with
uncorrected estimates from our own algorithm) are prob-
ably indicative of anything which could be attempted
based upon estimating sample covariances for existing
weighted methods.
Here, we compare the results obtained using Procrustes
(Figures 13, 14 and 15) to those shown earlier (Figures 10,
11 and 12) using our likelihood-based method. To pro-
duce such quantitative results, we apply Procrustes to
the same Monte-Carlo data which were generated based
on our corrected covariances. Following Procrustes align-
ment, eigenvalues are computed using the remaining error
residuals. These eigenvalues are then plotted against the
expected ones, where values on horizontal axis are iden-
tical to those used in Figures 10–12. Clearly, (and in
contrast to Figures 10–12) in all plots corresponding to
Procrustes the measured values are not within the pre-
dicted statistical limits (dashed lines). By inference, the
linear model vectors constructed using Procrustes are
contaminated by random errors associated with poorly
measured landmarks, as expected. When compared to the
plots from our weighted method with the correction pro-
cess switched off (e.g. Figure 13 compared to Figure 10),
eigenvalues extracted from Procrustes residuals are fur-
ther away from the expected values. As seen in the figures,
changing the number of degrees of freedom of the model
is also not sufficient to correct this issue. We can conclude
that Procrustes generates a linear model which is a less
efficient description of the true information contained in
the data.
Shape analysis
In Figures 16, 17 and 18, we show the anisotropic error
bars computed using the eigenvectors and eigenvalues of
the 2 ×2 covariance matrices for the 2D data sets. All
error bars are rescaled for visualisation purposes (see cap-
tions). Error bars for each landmark show the extent of
an elliptical (non-isotropic) distribution around the cor-
responding point in the mean shape. Such distributions
estimated using our method show exactly why we can-
not assume isotropic distributions for the data as assumed
in Procrustes. The P-FL1 data used in Figure 16 con-
sists of 19 landmarks (15 + 4) while the FL1 data used in
Figure 17 consists of 15 common landmarks only. Using
Figure 20 Mouse mandible data (MM1): error bars (×20) estimated using our method (left), and computed from the residuals left using
Procrustes (right), with the corresponding aligned data superimposed; 6-component models.
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Figure 21 Mouse skull data: error bars (×30) estimated using our covariance-based method (top); and those computed using Procrustes
residuals (bottom); 14-component models; projection planes: zy (left), xy (middle) and xz (right).
these plots, one can see that the semi-landmarks have
anisotropic covariances which match the expected local-
isation stability. Also we can see how after adding the
4 semi-landmarks the anisotropic errors estimated using
our method remain stable, while with Procrustes some
change both in orientation and in size, e.g. landmark
points 11 and 15 (see Figure 2 for landmark numbers).
These error bars are shown again in Figures 19 and
20 with the corresponding aligned data superimposed.
In these figures, the extent of error distributions illus-
trated by the error bars are not expected to match to
those illustrated by the alignment, as general biological
shape variation and measurement error are independent
processes. Although, localisation is determined by local
shape characteristics and measurement accuracy plays a
role in the overall distribution of landmarks around the
mean shape. As a consequence poorly measured land-
marks may have a variation about the mean that is
dominated by noise.
Turning to 3D data, in Figure 21, we have shown the
anisotropic error bars estimated using our method (sub-
figures on the top row) and those computed using Pro-
crustes residuals (sub-figures on the bottom row). In
Figure 22, we have shown the corresponding aligned data
using our method only, as these dense aligned data are
visually quite similar for the two methods. In order to dis-
play the 3D results we have used their projections on three
2D planes in the original coordinate system.
In both Figures 21 and 22, from the left to the right, we
show the projected results on zy, xy and xz planes respec-
tively. Using the mouse skull volume shown in Figure 3,
one can see how these viewing planes (zy, xy and xz)
correspond to the coronal, sagittal and transverse planes
respectively. In these 42 data sets, five had a marked asym-
metry of the nasal bones (affecting landmark 1), three
had a partially open frontal suture (affecting landmark
3), and one exhibited both of these effects. In Figure 21,
one can observe that the largest error bars estimated
using our method are for the landmarks 3 and 1. This
is consistent with the data clouds corresponding to these
landmarks in Figure 22 where in each case some points
standawayfromthemaincloudduetothedeforma-
tions mentioned above. This is not the case for Procrustes
where the error residuals left after alignment for land-
marks 3 and 1 show severe underestimation. This is due
to the fact that Procrustes translates strong shifts in one
Figure 22 Mouse skull data: aligned data obtained using a 14-component model based on our covariance-based method; projection
planes: zy (left), xy (middle) and xz (right).
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Figure 23 Fly wing data: error eigenvalues estimated using the likelihood method and those computed using the residuals after
Procrustes alignment, when each method is applied to P-FL1 (4 semi-landmarks (16-19) added to FL1) against those when applied to FL1;
the plot is for the 15 common landmarks (1-15); the Procrustes results have many points which are well beyond the expected limits,
suggesting that model parameters are not consistently determined upon inclusion of semi-landmarks; the two dashed lines show the
±2.8σrange.
landmark position into smaller shifts in all landmarks.
However, in this example the observed variation is largely
restricted to deformations of the nasal bones (landmark
1) and partially open frontal suture (landmark 3) without
displaying noticeable shape changes in other parts of the
skull. Hence the larger error bars of our method give in
this case a more accurate representation of the observed
biological variation. This is in agreement with the results
shown earlier in Figure 15 where for two landmarks Pro-
crustes residuals are much smaller than the expected
error values (standard deviations) with which the Monte-
Carlo data were generated. For our method, however,
estimated errors are all comparable to the expected ones
as shown earlier in Figure 12. In order to compare the
magnitude of errors estimated using our method to those
suggested by the repeat data, one should revisit Figure 6.
The figure again suggests comparable error estimations.
Finally, it is clear from the zy and xz projection planes
that expected symmetry is achieved to a large extent in
orientation and size for most corresponding error bars
(in either method).
Now we turn to further comparing our method to
Procrustes in a quantitative manner. The inconsistency
observed earlier in error bars corresponding to the resid-
uals left after applying Procrustes to the fly wing data
(Figures 16 and 17) is displayed more clearly using a
scatter plot in Figure 23. Here we plot the eigenvalues
corresponding to the 15 common landmarks after the
4 semi-landmarks are added against those without any
additional landmarks. We plot this for both the likeli-
hood and Procrustes methods. It is clear here that there
are departures from the permitted scatter region when
Procrustes is used. This indicates a significant change in
the unexplained variance following linear model construc-
tion, which itself implies differences in the linear model
itself, i.e. the Procrustes model is unstable following the
addition of poorly measured landmarks.
Further, we performed a χ2test based upon the
construction of corrected covariances on one data set
(FL1/FR1) and then used for the calculation of χ2for a
second data set (FL2/FR2). The corresponding plot for χ2
test in Figure 24 confir ms the stability of our method, as all
Ragheb et al. Fron tiers in Zoology 2013, 10:16 Page 21 of 24
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Figure 24 Fly wing data (15 original landmarks): the χ2/DoF ratios when our method is applied to two sets of repeat fly wing data, FL2
and FR2, using a 3-component model and fixed covariances (Figure 16: top-left) estimated earlier from the FL1 and FR1 data sets
respectively; the two dashed lines show the ±2.8σrange.
χ2/DoF values fall in the range expected. Further χ2tests
(not shown here) with different numbers of data samples
and combinations of data sets indicate the appropriate-
ness of the assumed linear model for the fixed number of
components.
In Table 2, we list the Fisher Information (FI) value for
the two methods and the three data sets studied. Again,
for Procrustes, variances used to compute the FI value are
obtained from the residuals left after alignment between
the data and the simulated linear model. Our method,
which is based on likelihood and measurement covari-
ance, gives FI values roughly between two and four times
those obtained using Procrustes. The largest difference
corresponds to the 3D MS data with 14 model compo-
nents. As FI is proportional to the quantity of data, this
demonstrates that the changes away from the isotropic
assumption inherent to Procrustes/PCA has a significant
effect on the efficacy of the model, equivalent to having
defined only a third as many landmarks from the out-
set. We can also see in this table the effect of adding 4
semi-landmarks to the 15 original landmarks. The num-
bers in the parentheses show the contributions from the
4 added landmarks to the total FI values. The reason for
the decrease in the FI values after adding 4 landmarks is
that we are using the same number of degrees of free-
dom (DoF) to describe correlations between more points.
Table 2 Fisher information (FI) values: listed for the Procrustes and our method when applied to the fly wing data (P-FL1),
mouse mandible data (MM1) and 3D mouse skull data (MS)
FI value Procrustes Likelihood
3D Mouse skull data (14-component model) 23.62 111.88
Mouse mandible data (6-component model) 6.60 19.55
Fly wing data (2-componentmodel) 15 points 17.36 29.68
Fly wing data (2-componentmodel) 15 (+4)points 13.81 (+0.88) 25.46 (+2.47)
Ragheb et al. Fron tiers in Zoology 2013, 10:16 Page 22 of 24
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Also these values are computed for uncorrected covari-
ance estimates, because correction is not available when
using Procrustes.
Finally, the PCA analysis shows that in fly wing data
3 components can account for about 65% of vari-
ance, while for mouse mandibles 6 components are
needed to achieve the same level. In both cases, the
model order preferred by our analysis is significantly
less than the heuristic limit of 90% used by some
researchers.
Conclusions
Our analysis approach has been driven by the require-
ments of statistical estimation, quantitation and self
consistency, i.e. distributions assumed during likelihood
construction match the data and estimated parameters
match those generating the data. From a more philo-
sophical standpoint we can consider what we are doing
when we identify landmark locations and attempt to com-
pare them between sets (shapes). We do not expect that
biology manipulates the locations of our chosen land-
marks directly, they simply appear to move around as
the net effect of distributed developmental and evolu-
tionary influences. Recent considerations of biology have
introduced the phrase “palimpsest” [52], as an analogy
with repeatedly erasing and rewriting text in an ancient
parchment, to describe the way that structures develop.
Notice that the initial choice of landmarks is subjective,
not only in terms of the features selected but also how
we chose to define their locations. A landmark is the
result of a localisation procedure (partly influenced by
multiple biological considerations) which has an associ-
ated positional uncertainty. In this work we have asso-
ciated the problems of working with semi-landmarks
in biological shape analysis as being a consequence of
the statistical assumptions implicit to analysis techniques
such as Procrustes/PCA. We have implemented a new
method which takes appropriate account of measure-
ment and landmark localisation stability in order to
obtain a new form of analysis which is consistent with a
likelihood-based definition of the alignment and model
building tasks. This method can be equivalently inter-
preted as a redefinition of the landmark location as ghost
points.
The conventional interpretation of Procrustes is that
the resulting linear model is a pure shape description
which can be directly associated with biological pro-
cesses. Some may argue that extending the approach
to weight data, even to accommodate semi-landmarks,
breaks with this tradition. However, it is our belief
that any distinction between the original landmark and
our definition of a ghost point, as locations which are
somehow true measurements of biology in one case
but not the other, is arbitrary. Re-weighting of data
using a covariance is statistically equivalent to modi-
fying the information available by changing the speci-
fied set of landmarks. Use of a least-squares measure
(which assumes isotropic errors) does not introduce
some absolute measurement of biology. Both approaches
need to be calibrated using known samples with iden-
tifiable biological cause in order to make any scientific
interpretation.
Now that we have a specific definition for how to weight
landmarkdata,wecanseethatusingghostpointsdoesnot
invalidate use of Kendall’s statistics as suggested in [22].
The use of these approaches follows due to scale normal-
isation of the shape data, it is not an intrinsic property of
the use of the original landmarks co-ordinates per-se. We
can also re-project scaled (whitened) shapes onto the tan-
gent space defined in the transformed ghost space if we
wish, in order to remove local curvature arising from scale
normalisation.
Far from there being no objective way to define these
covariances [22,24], there are at least three; a) one can
estimate them directly from repeatability of measure-
ments(e.g.see[48]);b)theycanbedirectlyestimated
via conventional statistical means when using likelihood-
based landmark location (CRB) (e.g. see [25,26,28]); c)
they can be estimated as the unexplained stochastic vari-
ation (residuals) in fitted data (as in this paper and e.g.
[27]). For the latter, when estimated using residuals of the
fittedshapemodel,wewillseecontributionsadditional
to the measurement process; this is the stochastic (there-
fore unmodellable) behaviour of the biology itself. Our
results indicate that measurement covariances can be reli-
ably estimated in our data for sample sizes at least as small
as 40.
Our result indicate that the new method summarises
the information content of the measured data better
(improved FI scores), and with more stability than Pro-
crustes/PCA (consistent models are generated following
the addition of new points). Although we have not pro-
vided empirical evidence in this paper, the expected the-
oretical advantages of this approach are several; a) as all
landmarks of fixed local structure have an associated mea-
surement covariance, the approach described provides
a consistent way of incorporating qualitatively different
forms of landmark (type I, type II, semi-landmarks, geo-
metric landmarks, etc.) into the analysis; b) provided that
landmark stability is well described by a Gaussian distri-
bution, our method removes the instabilities inherent in
the analysis due to poorly determined points; c) as the
parameters for the linear model are now self-consistently
estimated for an identifiable generative scheme (embod-
ied here via Monte-Carlo simulation) it affords the appli-
cation of an eigenvector analysis statistical rigour; d) it
offers the possibility of interpreting the linear modelling
process as a statistical approximation, with consequent
Ragheb et al. Fron tiers in Zoology 2013, 10:16 Page 23 of 24
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interpretations of the requirement for the number of lin-
ear model components; e) finally, generalisation of the
approach would seem to be possible which would support
the analysis of dense landmarks on surfaces and curves.
We have also demonstrated how linear model order
selection can be performed by comparing baseline repro-
ducibility errors with those estimated from the model.
Finally, we have shown how the use of repeated analysis
on matched samples can be used to confirm the stability
of the estimated anisotropic error. We believe that these
tools are sufficient to allow use of this technique in bio-
logical studies. More study is needed in order to develop
an understanding of the value of our new technique in a
greater range of biological analyses.
The methods described in this paper are freely available
from the TINA web site [51] via the Geometric Mor-
phometric toolkit, as a system for quality assessment and
validation of output data.
Endnote
aBookstein [53]: “Wherever there is partial registration
the true value of a (vector deformation) is inaccessible.
Appendix A
Rotation matrix
Based on the geometry shown in Figure 1, we first calcu-
late the vectors ˆ
va,ˆ
vband ˆ
vc.
ˆ
va=P2P1
P2P1,vb=(P3P1)[(P3P1).ˆ
va]ˆ
va,
ˆ
vc=ˆ
va×vb
vb
The rotation matrix RTis hence given by:
RT=
ˆ
vax ˆ
vay ˆ
vaz
ˆ
vbx ˆ
vby ˆ
vbz
ˆ
vcx ˆ
vcy ˆ
vcz
Roll, pitch and aw angles
The multiplication of the rotation matrices Rx(γ ),Ry(β)
and Rz(α) gives
Hence by enforcing RT=Rxyz , it is straightforward to
find the rotation angles α,βand γ.
Direction vectors
At each landmark point nwith mnx,mny and mnz as the
mean coordinates, the rotated vector by angle αaround
the z axis is
m=
(cos α)mnx (sin α)mny
(sin α)mnx +(cos α)mny
mnz
The first derivatives of this vector with respect to αgives
uα=
(sin α)mnx (cos α)mny
(cos α)mnx (sin α)mny
0
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HR undertook software and methods development, performed experiments,
and produced the final version of the manuscript and responses to reviews.
NAT conceived the new statistical methods for shape analysis and provided
technical project coordination. PAB developed the automatic landmarking
software and provided maintenance of software libraries, web pages and
infrastructure. DT provided overall scientific management and coordination of
the project. ACS participated in acquisition of datasets including
manual/automatic landmark identification. All authors read and approved the
final manuscript.
Acknowledgements
This work was funded by institutional resources of the Max-Planck Society. The
authors would like to thank Chris Klingenberg (at the University of Manchester)
and Louis Boell (at the Max-Planck Institute for Evolutionary Biology) for
providing the fly wing data and the mouse mandible data respectively.
Author details
1Imaging Sciences, Faculty of Medical and Human Sciences, University of
Manchester, Manchester, UK. 2Max-Planck Institute for Evolutionary Biology,
Plön, Germany.
Received: 20 December 2012 Accepted: 19 March 2013
Published: 2 April 2013
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doi:10.1186/1742-9994-10-16
Cite this articl e as: Raghebet al.:Qu antitative shape analysis with weighted
covariance estimates for increased statistical efficiency. Fronti ers in Zoology
2013 10:16.
... The semi-automatic landmark point localisation algorithm was implemented within the TINA Geometric Morphometrics toolkit, which also includes the TINA Manual Landmarking tool [8], and algorithms that perform quantitative shape analysis with weighted covariance estimates for increased statistical efficiency [54]. This package has been made available as free and open source software (FOSS) under the GNU General Public Licence (www.gnu.org), and can be obtained via the TINA web-site (www.tina-vision.net). ...
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