J. Embryol. exp. Morph. 90, 363-377 (1985)
363
Printed in Great Britain © The Company of Biologists Limited 1985
Measurement of biological shape: a general method
applied to mouse vertebrae
D. R. JOHNSON, P. O'HIGGINS, T. J. McANDREW,
L. M. ADAMS
Department of Anatomy, Medical School, University of Leeds, Leeds LS2 9JT,
U.K.
AND R. M. FLINN
Centre for Computer Studies, University of Birmingham, U.K.
SUMMARY
A method is described for recording and analysing the projected shape of mouse vertebrae.
The image of the shape is captured by a television camera, cleaned, digitized and subjected to
mathematical analysis. A visual representation is obtained by reconstructing the shape in polar
coordinates about its centre of area. Further statistical analysis of the whole shape is performed
after a Fourier transform. This allows the shape to be represented by and reconstructed from 15
numbers. The method does not rely on homologous points or expert opinion and allows mean
shapes to be constructed. It successfully distinguished between 92 % of the test data, Tl and T2
vertebrae from two strains of mice.
INTRODUCTION
A common problem facing the morphologist is the comparison of the shapes of
complex biological structures in a manner that allows full account to be taken of
natural variation. The traditional answer to this problem has been to derive simple
quantitative data which are suitable for univariate or multivariate statistical analysis. Traditionally these data have been in the form of linear and angular measurements taken between defined homologous points, or ratios of such measurements.
In addition to the problems inherent in defining homologous points, this approach
suffers from the additional disadvantages that It ignores the frequently large
intervening regions and that it produces measurements which may be disconnected
from each other. In consequence so much information is lost that the original
shape, or even an approximation to it, cannot be reconstructed from the data.
This paper describes a system which retains as much of the information present
in a shape as possible for mathematical analysis and does not depend on
homologous points. The description falls into two parts, first the process of image
capture and storage and secondly a discussion of the possible methods of analysis
of the data so produced.
Key words: shape, mouse, vertebrae, computer, Fourier transform.
364
D. R. JOHNSON AND OTHERS
y
\
Fig. 1. The digitizing apparatus. A mouse second thoracic vertebra (T2) is placed on
the stage of the dissecting microscope (centre) fitted with a black and white television
camera. The image of the bone appears on the monitor (left). A digital image of the
shape of the bone has been generated by the microcomputer and interface and is
displayed on the computer screen (right).
MATERIALS AND METHODS
1. Image capture, processing and storage
Our first attempts to capture outlines of bones relied on a digitizing pad. The outline of a
shape produced via a camera lucida or from a photographic print was digitized by tracing around
it with a cursor. The accurate tracing of an outline proved to be excessively slow and laborious
while the transfer of an outline to a tracing and thence to a computer multiplied errors.
We found that the process of image capture could be speeded appreciably by the use of a
simple video camera interface (VCI, Educational Electronics, 30 Lake St, Leighton Buzzard,
Beds LU7 8RX, England). This low-cost unit (less than £200) allows a standard video signal to
be digitized by a microcomputer.
As examples of biological shapes we have chosen the anteroposterior projections of the first
and second thoracic vertebrae (Tl and T2 respectively) of two strains of mice: (1) the multiple
recessive strain (REC) is homozygous for the genes short ear (se), vestigial tail (vt), non-agouti
(a), brown (b), dilute (d), pink eye (p), chinchilla (cc/l) and waved-2 (wa-2); (2) the F] (DOM) of
two inbred strains C57BL and C3H which carries dominant alleles at all these loci. The papaindigested skeletons used are part of the material of Griineberg & McLaren (1972) and were
loaned by the British Museum (Natural History).
The mouse vertebra is placed on the illuminated base of a Wild M5a dissecting microscope
(Fig. 1) and back lit. The microscope carries a standard C mount and is equipped with a black
and white video camera. The image of the bone, suitably magnified to fill the screen and
reversed black/white (for the convenience of the operator) appears on a monitor screen and is
also fed to the VCI, which is in turn connected to a BBC (Acorn) microcomputer.
Once the synchronization and gain controls of the interface have been set to match the camera
an image can be captured and displayed on the computer screen (Fig. 2). This is achieved by
Measurement of shape
365
repeated sampling of the video signal and takes about four seconds. The BBC microcomputer
works in various display modes. The software supplied with the VCI allows an image consisting
of 160x256 pixels (in 4 shades) to be digitized.
Once the image has been captured the standard software allows it to be dumped to printer or
to disc. We have added further programs which allow additional manipulations:
(i) Clean up. This program converts the four shades of the original image to two (i.e. produces
a black and white image) and sharpens the edge by an averaging method similar to that used in
many computer-enhancement programs. The screen memory is searched for a shade change and
when one is detected the new shade is compared with that of its immediate neighbours and reset
to conform to the majority. As well as cleaning the outline this program (which takes 45 sees)
removes the image of dust from the background. We found accidentally that the image of a
human hair laid across the vertebra was also removed.
(ii) Digitizing. This program finds the edge of a cleaned up image by sampling the screen
diagonally. A vertebra is made up of two outlines, an outer representing the edge of the bone
and an inner representing the border of the neural canal. The program allows the two outlines to
be digitized automatically from one object by default. A 'wandering probe' starts at the bottom
left corner of the screen and runs diagonally sampling locations 1,1; 2,2; 3,3 etc. until it locates a
shade change at the edge of the shape. It then follows the edge of the outline until it has returned
to the start point. A second iteration of this program starts at the centre of the screen by default
(but can be preset anywhere). If the preset is within a foramen the outline of the latter is found
and digitized as before. The outer outline is digitized clockwise and the inner anticlockwise to
aid later identification (Fig. 3). More foramina can be identified by further defaults, or the
'wandering probe' which is visible on screen can be set by means of a joystick (50 sees for two
outlines).
The data are stored on a disc as pairs of Cartesian coordinates then transferred to the
mainframe computer for further analysis.
Fig. 2. The reconstructed image of the vertebra captured by the apparatus as it
appears on the computer screen.
366
D. R. JOHNSON AND OTHERS
IISST2 IIS iMir 123S M t r Hilts.
NESS 'W FM KM N 'f FN m i
• •• •• v
Fig. 3. The inner and outer outlines of the vertebra after digitization. The location of
each spot on the screen is stored as a pair of Cartesian coordinates in afixedorder on a
computer disc.
2. Mainframe manipulation of the data
Superimposition of outlines and fitting
In order to achieve a mean outline from a series of individual outlines the latter must be
superimposed. We do this in three stages. The outlines are first scaled to a standard area. The
centre of area (centroid) of each is then found by integration and the shape re-expressed as 128
polar coordinates centred on this point (128 is a perfect square and a perfect square is needed for
the fast Fourier transform, see below). This process allows superimposition of outlines upon
their centroids.
Because the bones have been placed in no special orientation upon the microscope stage it is
also necessary to rotate them relative to each other. In practice this is done by making a least
squares fit comparison for each outline against a standard. The outline is rotated by one polar
coordinate at a time and the fit, relative to the standard, repeated. After complete rotation one
orientation will be found to give a minimum value for the sum of the differences of the squares
on each polar radius; this is designated best fit. The method used is a modification of the least
squares fit described by Sneath (1967). Sneath took as his measurement of fit the size of the sum
of the squared distances between a series of 'homologous landmarks' expressed in Cartesian
coordinates. We have used as a measure of fit the residual area when shapes are superimposed
upon their centroids. This is a function of the sum of the differences on each individual radius
squared:
r-iN
Residual area oc >
V(r n 2 -R n 2 ) 2
where rn, Rn are corresponding polar coordinates on two shapes, N = number of polar
coordinates used and n is large.
Measurement of shape
367
Standard shapes were chosen by trial and error according to symmetry. A circle is obviously
inappropriate because the residual area is equal in all orientations, but a semicircle is a
possibility. In practice we used a simple polygon derived from a vertebral outline. Fig. 4
illustrates the process of fitting. The reference axis was taken as the midline of the standard
shape and the start point for Fourier analysis (see below) as the point at which the ventral side of
the outer outline of the vertebra crosses this axis.
Errors
Errors in the procedure described above may arise from two sources, videodigitization and
manipulation of data. To ascertain the size of these errors a test of technique was performed.
This consisted of sequentially capturing 20 images of the same bone which was removed from the
microscope stage and replaced in a different position and orientation after each capture. The 20
sets of Cartesian coordinates so generated were passed to the Amdal and fitted to a standard
shape as described above.
Fig. 4. Deriving the best fit. (Top) The image of the vertebra, reconstituted from the
coordinate stream and re-expressed in polar coordinates about its centre of area lies in
no particular orientation. Upon it is superimposed a standard shape of equal area
orientated about X and Y axes. (Bottom) After a least squares fit has been performed
the image of the vertebra is rotated to the position of best fit.
368
D. R. JOHNSON AND OTHERS
Further manipulation
Once all outlines in a group have been orientated with respect to a standard in this way the
mean value of each polar coordinate is calculated and a mean outline generated (Fig. 5).
Comparison of group means
Group mean outlines can now be plotted on top of each other. The significance of the
difference at any polar radius can be estimated by a t test, or an estimate of total shape similarity
can be made.
Fitting an unknown outline to group means
A single individual of unknown provenance can be assessed for fit to any number of group
means. A bone having a good fit (low total sum of differences of squares) with one group and a
high total sum of differences of squares with another is likely to be a member of the first group.
Fourier transforms
The mathematical techniques applied to the data so far have been extremely simple. The
opened out graph of polar coordinates can, however, be regarded as a waveform and sophisticated techniques already developed for waveform analysis in physical sciences and telecommunications applied. Jean Baptiste Fourier (1768-1830) described a method which splits a
complex waveform into a series of sine and cosine components of varying amplitude. Lestrel
(1974) applied the series to biological shapes. The general Fourier series can be represented as:
F(0) = a0 + ax cos 6 + bi sin 6 + SL2 COS20 + b2 sin20... an cosnd + bn sin nd
where ao is a constant, ai-a n are known as cosine components, bi-b n are known as sine
components, and F(0) is the magnitude of a polar radius r.
Because the sine and cosine components are 90° out of phase the Fourier series can describe
highly irregular waveforms by means of a series of numbers (Lestrel, 1982). An alternative
notation uses amplitude and phase lag instead of sine and cosine components.
[
R n c o s ( n 0 + <j>n)
n=l
where Ro-Rn are known as amplitude components and 0i-0 n are known as phase lag
components. The amplitude/phase lag notation has the advantage over sine/cosine notation
that the amplitude coefficients are independent of the start point of the waveform. It does not,
however, allow shape reconstruction.
A simple shape is adequately described by the early harmonics of the Fourier series: a more
complex one will require more components to describe it accurately. A corollary of this is that
early components of the series describe gross features of the shape and later ones fine detail. In
practice the 'fine detail' may represent noise in the measurement process and may often be
discarded without detriment to shape analysis. Since we are dealing with several pairs of
components, a multivariate statistical approach, which considers all variates simultaneously, is
appropriate. We used the DISCRIM procedure within SAS (Statistical Analysis System; SAS
users guide 1982) which calculates the generalized squared distances between each test individual and the calibration groups, and classifies them on the basis of 'nearest group'.
Thefirstpair of Fourier coefficients (a0, b ^ are excluded because thefirstcosine component is
constant (since areas are equalized, Lestrel, 1974) and the first sine component is always zero.
The first fifteen pairs of coefficients referred to hereafter are therefore coefficient numbers 2-31
inclusive.
RESULTS
Errors
Capture
The unit of resolution of the television screen is the illuminated dot, the pixel. If
the image of a bone occupies any part of a pixel the latter will be illuminated. This
Measurement of shape
369
is obviously the source of a small error. With the magnification used (x50
objective plus xO-3 correcting lens on the microscope, 16" colour monitor) the
image of a test T2 vertebra had a maximum height of 152 mm and width of
172mm 1 pixel measured 0-5 mm highxl-Omm wide. The maximum error due to
this source was thus less than 1 %.
Computer rounding error
The test of technique on a single vertebra allows us to measure the sum of
capture and computing errors. The 128 mean values produced by the test of
technique had a mean variance ratio
/standard deviation
\
mean
Vertebral comparisons
Fig. 5 shows the computed mean outer outlines for 22 Tls and 14 T2s from the
dominant strain. The computer-generated plot gives a polar reconstruction of
vertebral shape (left) and an opened out linear plot (right). The trace represents
the mean outline ± 10 standard errors of the mean. The more usual representation
of mean ± 2 standard errors plots as a single line.
Fig. 6 shows comparison plots of Tls (upper) and T2s (lower) from dominant
and recessive strains. Significant differences (P< 0-05) are present in many areas.
Table 1 shows the results of fitting individual Tls to group mean shapes. 44 out
of 53 bones (83 %) fitted best to their 'correct1 group. A similar comparison
amongst the T2s gave 32 out of 34 (94 %). Overall 87 % were correctly classified.
The Fourier series will adequately describe a shape in less than 128 variables
(the number of polar coordinates chosen) and so potentially simplifies the data.
Fig. 7 shows reconstructions of a T2 vertebra based on 5-60 sine/cosine coefficient
pairs. It can be seen that 15 pairs subjectively appear to describe the shape
adequately and further coefficients add little to definition.
Univariate analysis of the first 15 coefficient pairs was undertaken. Examples of
bar charts showing the upper and lower 95 % confidence limits and mean ± 2 S.E.M.
for each population (group) are reproduced in Fig. 8. All cosine components
showed significant differences between group means at the level of P< 0-0001
(Table 2), some discriminating between Tl and T2 and some between DOM and
REC. Only 4 of 15 sine components were significant at this level. No single
coefficient split all 4 groups unequivocally.
An objective test of the decision to analyse only 15 pairs of coefficients is to
perform a discriminant function analysis which compares all variates simultaneously. For this a random sample of 10 % of all vertebrae was removed from
the data set and an attempt made to classify them with respect to the remainder.
This was repeated 10 times using 5,10,13,15,18 and 20 coefficient pairs. The best
classification of this data set (92 % correct) was obtained using 15 sine/cosine pairs
(Fig. 9). If fewer or more pairs are used definition suffers; below 15 pairs the
370
D. R. JOHNSON AND OTHERS
50
100
150
200
Angle (degrees)
250
300
350
Fig. 5. Mean vertebral shapes for 21 first (Tl) and 14 second (T2) thoracic vertebrae
from the DOM strain of mice. Thick line, mean: thin lines, mean ± IOXS.E.M.
100
150
200 250
Angle (degrees)
300
350
Fig. 6. Comparison plots of means of 22 DOM Tls and 14 T2s (thick lines)
superimposed upon 31 REC Tls and 20 T2s (thin lines). The horizontal bars show the
areas where the shapes differ significantly at the level P< 0-05.
shapes are poorly defined, above 15 pairs the effects of sample size and noise
intrude. 79 out of 87 (91 %) of shapes were correctly classified using 15 cosine
components only.
Measurement of shape
371
Table 1. Fits of individual Tl vertebrae against group means
Bone
number
DM95 Tl
DM97 Tl
DM98 Tl
DM99 Tl
DM100 Tl
DM101 Tl
DM102 Tl
DM103 Tl
DM104 Tl
DM106 Tl
DM107 Tl
DM108 Tl
DM109 Tl
DM110 Tl
DM111 Tl
DM112 Tl
DM113 Tl
DM114 Tl
DM115 Tl
DM116 Tl
DM117 Tl
DM119 Tl
REC20T1
REC23T1
REC25T1
REC34T1
REC35T1
REC36T1
REC37T1
REC38T1
REC39T1
REC40T1
REC41 Tl
REC42T1
REC43T1
REC44T1
REC45T1
REC46T1
REC47T1
REC49T1
REC50T1
REC51T1
REC52T1
REC53T1
REC54T1
REC55T1
REC57T1
REC58T1
REC59T1
REC60T1
REC62T1
REC63T1
REC65T1
Fit to recessive mean
(total sum of squares)
1009
2640
1959
2307
1931
667
2609
1344
1232
1514
908
1926
2251
773
2187
1975
1687
1508
2611
4632
2224
1251
702
377
738
643
1338
8653
529
600
1317
778
Fit to dominant mean
(total sum of squares)
911
855
848
826
1520
923
1089
611
519
706
522
1000
813
1389
572
962
953
698
1199
2493
855
1129
1950
996
1203
1154
1536
7056
475
2252
3477
819
348
1439
2479
5340
5340
2543
1878
1248
1835
3644
1522
1566
1174
1706
5988
1123
1231
1188
2300
3703
* Denotes misclassification.
876
2480
2480
924
1021
877
568
1593
1721
846
1239
689
2744
1534
1602
1304
880
1478
664
854
724
1683
Decision
D
D
D
D
D
R*
D
D
D
D
D
D
D
R*
D
D
D
D
D
D
D
D
R
R
R
R
R
D*
D*
R
R
R
R
R
R
R
R
R
R
R
R
D*
R
D*
R
R
D*
D*
D*
R
R
R
R
372
D. R. JOHNSON AND OTHERS
DISCUSSION
The final shape attained by a bone must be dependent upon a host of factors,
both genetical and environmental. The almost universal occurrence of pleiotropy
(multiple effects of genes on characters) has led to the hypothesis that total
phenotype is acted upon by selection and that it is this which evolves rather than
individual characters or genes (Wright, 1968; Cheverud, 1982). Integrated systems
are now emphasized in morphogenesis (Waddington, 1957; Leamy, 1977; Riedl,
1978; Lande, 1979; Atchley, Rutledge & Cowley, 1981; Cheverud, 1982; Bonner,
1982). If we view a bone as an integrated system then we must ask how best to
measure its total shape.
In the conventional methods of comparing bone shapes homologous points are
defined in such a manner as to permit measurements which reflect individual
features thought to be of biological significance and which can be taken quickly
and consistently. In practice we suspect that the latter consideration often outweighs the former. Thus Festing (1972) chose 13 measurements of the mouse
mandible which could be read off 'as quickly as they could be recorded by an
assistant', Atchley (1983) used eight traits 'chosen because they are easily
measured and the measurements are highly repeatable' (rat mandible) and Leamy
& Atchley (1983) used 19 scapular measurements 'taken from well defined
landmarks to optimise repeatability'. Multivariate analysis will remove correlations between such measurements so that they are mathematically respectable,
but their biological significance must remain in doubt.
The technique described here does not rely upon homologous points. No start
point for the coordinate stream is specified: the only defined point is the centre of
area, which is a relatively neutral property of the shape. The number of polar
coordinates generated depends upon the video system used. More than 128 points
20
Fig. 7. A mouse T2 vertebra (REC60 T2) reconstructed from 5-60 pairs of sine/cosine
coefficients.
Measurement of shape
0-68
2-21
-5-03
Cos 2
- DOMT1
—
- RECT1
DOMT2 RECT2
017
-r-
2-72
-20-6
19-7
-6-96
14-63
Cos 3
-108-8
9-77
Cos 4
Fig. 8. Bar charts showing mean ± 2 S.E.M. (this line) and 95 % confidence limits (thin
line) of the first three pairs of cosine and sine coefficients of the Fourier series. Note
that the cosine components are better discriminators than the sines.
could, of course, be generated from a video system giving higher resolution. These
are easily obtainable, but expensive.
Because the system generates a reconstructed shape rather than a series of
numbers we need to think about analysis of results in a different way, deriving
functions which relate to the shape as a whole rather than arbitrary measurements
within it. Our simple polar plot superimposition allows variation to be taken into
account, provides acceptable discrimination betv/een shapes and tells us whether
the difference is significant at a particular point or in a particular area. Using polar
coordinates homologous points (the tips of the transverse processes, for example)
will not necessarily map on the same radius in two compared shapes. The nth polar
374
D. R. JOHNSON AND OTHERS
Table 2. Variance ratio (F) values for the first fifteen pairs of Fourier coefficients and
associated probabilities (T)
cosine
sine
Pair
F
P
F
P
2
3
4
5
6
7
8
9
10
291-97
1224-70
378-92
59-88
273-11
365-94
135-97
69-02
269-93
52-77
66-41
101-97
62-85
8-08
72-01
< 0-0001
< 0-0001
< 0-0001
< 0-0001
< 00001
< 0-0001
< 0-0001
< 0-0001
< 0-0001
< 00001
< 0-0001
< 00001
< 00001
< 0-0001
< 0-0001
6-03
1-28
0-85
2-33
8-40
7-16
0-98
5-85
11-81
5-34
2-85
1419
13-77
2-61
2-85
< 0-0010
< 0-2780
< 0-4750
< 00794
< 0-0001
< 0-0003
< 0-4066
< 0-0012
< 0-0001
< 00022
< 0-0414
< 0-0001
< 0-0001
< 0-0588
< 0-0415
11
12
13
14
15
16
radius, encompassing the tip of the transverse process in shape A, should not,
therefore, be compared blindly with the nth polar radius in shape B which misses
it. If the nth radii between two shapes differ, then the shapes differ.
It can be seen from the data of Table 1 that a good fit to the dominant strain does
not necessarily indicate a poor fit to the recessive shape and vice versa. This is
because the shapes are highly irregular and a bone fitting the dominant shape well
in some areas may fit a recessive shape well in others. Because of this further
statistical analysis (such as probability of group membership) using this routine
was not attempted. We also suspect that the populations of shapes may overlap in
some cases, so that some individuals could be a member of either population.
Moore & Mintz (1972) however were able to identify coded bones from C3H and
C57BL mice with 85-100% accuracy. The variation in inbred strains would, of
course, be lower than in our material.
Each Fourier component, unlike each polar radius, represents a property of the
whole outline. The problem of homologies is thus minimized in that each Fourier
component is dependent only on the centroid (and in the case of sine/cosine
components on the start point which is derived from our fitting routine, not
arbitrarily specified by eye). Fifteen Fourier coefficient pairs can classify the shape
as well as 128 polar coordinates and fifteen cosine components as well as fifteen
sine/cosine pairs. The number 15 is probably a function of the particular shapes
used and the variance within the data set: other sets of data describing bones of
different shapes might well need more or fewer components to best describe them.
Since the Fourier procedure is a simple transformation of the original data its
discrimination cannot exceed that of the former. Less than 15 pairs of components
will simplify the shape (Fig. 7) and thus inhibit discrimination. The use of more
than 15 component pairs cannot increase the discrimination further, but should
375
Measurement of shape
not diminish it. In fact more pairs reduce discrimination a little: we suggest that
this is due to noise, i.e. cumulative errors in the system.
It is a property of the Fourier series that sine components describe axial
asymmetry (Zahn & Roskies, 1972; Lestrel, 1974): since the vertebral shapes are
essentially symmetrical about their midline they can be reconstructed minus any
asymmetry from the first 15 cosine components alone (Fig. 10, cf. Fig. 7).
Fourier components as used in this context are based upon a centroid and thus
reflect the disposition of the shape about this point relative to the start point. Use
of amplitude coefficients only would remove the start point dependency, but not
that on the centroid. Because we are dealing with similar shapes the effects of
centroid dependency are minimized and differences in Fourier components reflect
differences in shape. The Fourier analysis of an edge-based decomposition of a
shape (e.g. the tangent/angle function, Bookstein, 1977; Zahn & Roskies, 1972)
would remove centroid dependency also.
We suggest that an ideal system for shape measurement should conform to the
following criteria:
1. It should be practical and practicable.
2. It should accurately measure the form or any part of it.
3. It should allow reconstruction of the original shape (i.e. the derived
measures should be related to the shape by a determinable function).
4. The data should be suitable for statistical analysis, so that biological
variation can be accommodated.
5. Measurements of size should be independent of shape and vice versa.
6. Measurements of shape should, if required, be independent of any necessity
to define 'homologous points'.
10-
—i
15
10
1
1
r
20
No. of variate pain;
Fig. 9. Regression curve of percentage of bones misclassified against number of
coefficient pairs used.
376
D . R. J O H N S O N AND OTHERS
20
Fig. 10. A mouse T2 vertebra (REC60T2) reconstructed from 5-20 Fourier cosine
coefficients.
The Fourier method described here conforms to thefirstfiveof these desiderata: Fourier analysis of a curvature function would conform to all six.
Any new method of measuring shape must offer significant advantages over
existing methods. The system described here is of the same order of accuracy as
existing methods and, we think, offers considerable advantages. For further
comparison of traditional and more modern methods of analysis the reader is
referred to Ashton, Flinn, Moore & O'Higgins (in preparation).
The ultimate test of shape measurement must be its ability to 'recognize'
unknown shapes. The Fourier method described here classified 92 % of the sample
of outlines correctly. 15 coefficients describe total shape with a high measure of
accuracy, with no reliance upon expert opinion or 'homologous points'. We
suggest that it should now be possible, using suitable material, to partition size and
shape allowing the complex interrelations of these properties to be studied in a
biological context.
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{Accepted 29 July 1985)