Zelditch M.L. (и др.) Geometric Morphometrics for Biologists: a primer

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398 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Figure 15.7 Points digitized for an analysis of squirrel scapula shape. Landmarks are indicated

by black circles and semilandmarks by white circles. Semilandmarks are digitized in equal angular

increments, as shown in Figure 15.6C.

(A) (B)

Figure 15.8 Analysis of scapula shape variation with semilandmarks treated as equivalent to land-

marks: (A) GLS superimposition of all specimens; (B) shape change associated with positive scores

on PC1.

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BEYOND TWO-DIMENSIONAL CONFIGURATIONS OF LANDMARKS 399

equivalent, the semilandmarks have more influence on the result than is justified by the

number of degrees of freedom they represent. In our example, each semilandmark on the

anterodorsal corner plays as large a role in determining the optimal alignment as the land-

mark on the posterodorsal corner, which seems inappropriate because the semilandmarks

represent less information than the landmarks. This raises the question: would the changes

in curvature of the anterior edge be regarded as the dominant feature of variation if the

semilandmarks had less influence on the result?

Differential weighting of landmarks and semilandmarks

One approach to reducing the influence of semilandmarks on the superimposition is to

downweight them – i.e. construct a weighted Procrustes distance and use it as the criterion

for optimal superimposition. This superimposition would be computed using essentially

the same mathematics as the conventional superimposition, but applying scalar multiples

to reduce the influence of semilandmarks on the computations of the superimposition and

the shape differences.

This approach has the advantage of recognizing that there is a difference between

landmarks and semilandmarks, but it also has the disadvantage that the steps of the super-

imposition do not lead to configurations in Kendall’s shape space. At first this might not

seem like such a bad thing. We do not want to ignore the semilandmarks, and we do not

want to ignore the difference between landmarks and semilandmarks. This would sug-

gest that we do not want the Kendall’s shape space for just the landmarks, and we do

not want the Kendall’s shape space for configurations of landmarks plus semilandmarks.

The disadvantage lies in the consequences of not having configurations in Kendall’s shape

space – namely a lack of information about the shape of the space that the configurations

occupy or the properties of the distance metric for that space. An additional implication

of these uncertainties is that statistical analysis cannot employ conventional parametric

models; resampling-based methods must be used.

Another disadvantage of this approach is the lack of clear criteria for determining the

appropriate weighting of semilandmarks. We can be fairly confident that semilandmarks

represent less information than landmarks, especially if we know the rule used to select

the semilandmarks. Rules like those described earlier imply that each semilandmark has

only one degree of freedom. This may serve as a reasonable estimate of the information

represented by each semilandmark, but it should be regarded as an estimate with a high

degree of uncertainty. If curvature is simple (few reversals, as in the squirrel scapula exam-

ple), a small subset of semilandmarks may be sufficient to characterize the curve, which

implies that most of the semilandmarks do not contribute additional information. On the

other hand, if the curvature has a high but consistent complexity (like an oak leaf with a

fixed number of lobes), the information provided by landmarks and some semilandmarks

might be nearly equivalent. Should you decide to take this approach, we recommend you

try several different weighting schemes to insure that your conclusions are robust (i.e. not

dependent on a particular scheme).

Sliding semilandmarks to minimize bending energy

In this approach, developed by Green (1996) and Bookstein (1997a), the first step is

a conventional Procrustes superimposition (treating landmarks and semilandmarks as

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400 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(A) (B)

Figure 15.9 Tangents to the anterior edge at the locations of semilandmarks: (A) idealized esti-

mates based on the curvature of the edge at the semilandmark; (B) enlarged view of part of the edge,

showing estimation of the tangents at the semilandmarks from segments connecting adjacent semi-

landmarks. The line through the semilandmark is parallel to the line connecting the adjacent

semilandmarks.

equivalent) to compute a mean configuration and align the targets to it. This is followed by

moving the semilandmarks of each target to minimize the bending energy of the thin-plate

spline describing the deformation of the reference to that target. The semilandmarks are

not free to move in any direction; each is confined to “slide” along the line tangent to the

curve at that semilandmark (Figure 15.9A). The shape of the curve is not actually known,

so the tangent is estimated as the line parallel to the segment connecting adjacent landmarks

or semilandmarks (Figure 15.9B). After sliding, the superimposition is recomputed; if the

new mean configuration differs from the previous mean, the sliding and superimposition

are reiterated until they converge on a solution. The justification for this sliding tech-

nique is that differences in relative positions of semilandmarks along the curve cannot be

informative because this spacing was defined arbitrarily (that is, extrinsically). Thus, sliding

to minimize the bending energy of the deformation adjusts the spacing of the semiland-

marks to minimize the implication that there are shape changes due to differences in that

spacing.

Figure 15.10A shows the same data set used earlier, superimposed after sliding to

minimize bending energy. Several semilandmarks have ellipses of variation that imply

displacements along the anterior edge, particularly the ones in the ventral half. PC1 indi-

cates that these semilandmarks undergo correlated displacements toward the dorsal end

as the anterodorsal corner is squared out (Figure 15.10B). Thus there is little change in

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BEYOND TWO-DIMENSIONAL CONFIGURATIONS OF LANDMARKS 401

(A) (B)

Figure 15.10 Analysis of scapula shape variation after sliding semilandmarks to minimize bending

energy: (A) superimposition of all specimens; (B) shape change associated with positive scores on PC1.

the positions of these semilandmarks relative to each other, and therefore little localized

change along the anterior edge. Most of the localized change in the anterior edge occurs at

the corner. As before, the change in shape of the anterior edge is inferred to be the domi-

nant component of shape change. Displacements of the landmarks are generally slight; the

exceptions are the ventral displacements of the most dorsal landmarks, which are involved

in the general flattening of the dorsal edge and squaring of the anterodorsal corner.

Compared to both options discussed above, sliding semilandmarks to minimize bend-

ing energy has the advantage that it does not ignore the difference between landmarks and

semilandmarks. Compared to weighting, this sliding technique has the further advantage

of having a clear criterion for the optimal superimposition. The principal disadvantage of

this approach is that the semilandmarks are in new positions relative to the landmarks and

the other semilandmarks. However, this may not be the devastating flaw that it seems to

be. The underlying premise of sliding is that semilandmarks are not equal to landmarks.

As pointed out above, semilandmarks do not represent the same amount of independent

information as landmarks because semilandmarks are constrained to lie along the curve at

arbitrary intervals. Put another way, if moving semilandmarks does not alter the informa-

tion about the shape of the curve, then the configuration of landmarks and semilandmarks

after sliding might be considered to have the same shape as the configuration before sliding.

Sliding semilandmarks to perpendicular alignment on the reference

This approach, suggested by Sampson et al. (1996), differs from the previous one only

in the criterion used to determine how far the semilandmarks slide. The normal to the

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402 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Figure 15.11 Perpendicular alignment sliding. This enlarged view of part of the anterior edge of

the scapula shows the semilandmarks for a reference (white circles) and a target (gray circles) before

sliding. The semilandmarks of the target are slid toward the lines that are perpendicular to the edge

at the corresponding semilandmark of the reference.

curve (perpendicular to the tangent) is estimated for each semilandmark of the reference

configuration (Figure 15.11). Each semilandmark of the target slides along its tangent to

align with the perpendicular of the corresponding semilandmark of the reference. As in the

other method, sliding is justified by the argument that positions of semilandmarks along

the curve are uninformative because they are arbitrary, but here the conclusion from that

argument is that semilandmarks can be informative only about the bowing of the curve.

Sliding to perpendicular alignment extracts the information about bowing (displacement

perpendicular to the tangent) and minimizes the inference that the semilandmarks moved

along the curve (along the tangent).

Compared to the first two options discussed above (no adjustment and weighting),

sliding to perpendicular alignment has the same advantages and disadvantages as sliding

to minimize bending energy. In both cases, the final configurations do not have the same

shapes as the original configurations, but they are in Kendall’s shape space. Both sliding

methods recognize the difference between landmarks and semilandmarks, and both imply

that some displacements of semilandmarks do not constitute changes in shape. Because

the two sliding methods have not been subject to extensive evaluation, it is not clear if one

is better than the other.

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BEYOND TWO-DIMENSIONAL CONFIGURATIONS OF LANDMARKS 403

Ordinations and statistical analyses

Regardless of the methods of selecting and superimposing semilandmarks, a curve will be

represented by a large number of semilandmarks better than it will be by a smaller number

(at least until the density of semilandmarks exceeds the resolution permitted by digitizing

error). If either sliding method is used, it will also be improved by increasing the number

of landmarks because that will tend to produce more accurate estimates of the tangents at

the semilandmarks. However, increasing the number of semilandmarks on a curve means

it will play a larger role in superimposition and comparison. In fact, it is easy to imagine

cases in which the curve will exert a stronger influence on the results than all the features

represented by landmarks. In addition, increasing the number of semilandmarks increases

the number of coordinates in the data set, the number of spline coefficients computed from

those coordinates, and the discrepancy between these numbers and the number of degrees

of freedom.

One possible way to address the unbalanced representation of the curve and the other

features is to reduce the number of points representing the curve. This will also reduce

discrepancy in the number of degrees of freedom. Unfortunately, it is not clear how far

to reduce the number of semilandmarks or what criteria might be used to select which

semilandmarks to eliminate. In any event, the attendant reduction of the accuracy of the

reconstruction makes the idea of cutting semilandmarks unpalatable. Making the density of

semilandmarks inversely proportional to the general complexity of the curve can moderate

the loss of accuracy. Tangents estimated from chords connecting adjacent points will

deviate most from the true tangent in regions where the curvature changes most rapidly;

closer spacing of semilandmarks will more accurately estimate the spatial extent of a sharp

bend (Figure 15.12). Changes in the shape of the curve may include changes in the locations

of sharp bends (as in the squirrel scapulae), so variability of the curvature should also be

a consideration in determining the number of semilandmarks to digitize.

Another possible solution to unbalanced representation of curves and other features

is downweighting the semilandmarks, as discussed above. This approach would reduce

the influence of semilandmarks on computation of distances between shapes, so they

would play a smaller role in testing for patterns of shape variation within groups, or

differences in mean shape between groups. There would still be more coordinates and

shape variables than degrees of freedom (for K landmarks and H semilandmarks there

would still be 2(K +H) coordinates and 2(K +H) −4 shape variables, but only about

2K −4 +H degrees of freedom – depending on how degrees of freedom are attributed to

semilandmarks).

A compromise between removing some semilandmarks and downweighting them is

to use a large number of semilandmarks in the superimposition and sliding, then cull

semilandmarks before performing ordinations or statistical tests. Having a large number

of semilandmarks at the beginning maintains the accuracy of tangents and sliding; culling

several of them after the final superimposition reduces discrepancies in weighting and

degrees of freedom. Of course, the problem of choosing which points to cull is the same

as the problem of selecting which points to exclude from the beginning.

Regardless of which approach you select, it is always possible to perform a sensitivity

analysis to determine the effects of these choices. By comparing results obtained with

different combinations of semilandmarks, you can determine whether the choices influence

the conclusion. At the very least, this would provide reassuring information about the

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404 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(A)

(B)

Figure 15.12 Effect of semilandmark density on estimation of tangents: (A) wide spacing; (B) closer

spacing.

robustness of the conclusion. In addition, it may also reveal which parts of the curve are

most relevant to the conclusion (for example, which semilandmarks covary with each other

but are independent of the major pattern of landmark covariation).

Whatever method you use to evaluate or moderate the influence of semilandmarks on

the results of an analysis, there remains the problem of determining the correct number of

statistical degrees of freedom for conventional statistical tests and making the appropriate

adjustments in sample size and criterion for rejecting the null hypothesis (α-level). If you

digitize more points, you will have more coordinates and will need more specimens to

maintain the same α-level. If you need to digitize a lot of semilandmarks to characterize a

curve, you may need an unobtainable number of specimens just to have more individuals

than coordinates (which is required for a conventional statistical analysis). Even if you

can correct the p-value to reflect the correct number of degrees of freedom for a given

combination of landmarks and semilandmarks, it may still be difficult to acquire enough

specimens to permit as many semilandmarks as you might like. Furthermore, determining

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BEYOND TWO-DIMENSIONAL CONFIGURATIONS OF LANDMARKS 405

the correct number of degrees of freedom may not be as simple as adding the number of

semilandmarks to twice the number of landmarks and subtracting the four degrees of free-

dom lost in the Procrustes superimposition; sliding may further reduce the independence of

semilandmarks. Fortunately, all of these problems can be avoided by using the resampling

methods discussed in Chapter 8. More specimens would still be better (as always), but

the criteria for “enough” would no longer depend on the number of coordinates; more

importantly, the risk of an erroneous conclusion due to a mistaken estimate of the number

of degrees of freedom would be eliminated.

Summary

Analysis of three-dimensional configurations of landmarks can be performed within the

same theoretical framework as analysis of two-dimensional configurations. The mathe-

matics has been worked out and published. There is only a limited amount of readily

available software, mostly due to lack of demand. The main obstacle to analyzing three-

dimensional shapes is the difficulty of displaying the results on static two-dimensional

media like journal and book pages.

Some techniques for analyzing objects without landmarks lie completely outside the

theoretical framework used to analyze configurations of landmarks. These techniques do

not employ the definitions of size and shape used in landmark-based analyses, so the

configurations described by these methods do not lie in the shape space described by that

theory. In addition, these descriptions of curves are incommensurate with shape variables

computed from Procrustes superimposed landmarks. Therefore, PLS can be used to test

for correlations, but the two kinds of description cannot be combined into a single shape

analysis.

Techniques for analyzing features without landmarks in the same theoretical framework

as configurations of landmarks are not fully developed. Methods have been developed to

use extrinsic criteria to locate comparable points along a curve (semilandmarks) and then

analyze the semilandmarks simultaneously with landmarks in the theoretical framework

for landmarks. Criteria for choosing among these methods are not well established, so we

recommend that prospective users try multiple methods and several options within those

methods (including different combinations of points on the curve). The number of degrees

of freedom associated with the semilandmarks is also not well established, although it is

clear that each semilandmark can contribute no more than one additional degree of free-

dom (in contrast to two additional degrees of freedom for a landmark). Consequently,

the number of degrees of freedom contributed by semilandmarks cannot be more than the

number of semilandmarks. Because the number of degrees of freedom is unclear, but defi-

nitely much less than the number of coordinates, we recommend that resampling methods

be used in statistical tests. This will alleviate demands for excessively large samples and

avoid errors due to mistaken estimates of the number of degrees of freedom.

Software

As mentioned above, software tools for performing the analyses discussed in this chapter

are not as well developed as the other software tools we have presented. Accordingly, we

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406 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

briefly review some of the tools that are available for three-dimensional configurations of

landmarks and for semilandmarks, but do not discuss their operational details in depth.

Three-dimensional configurations of landmarks

The IMP package currently offers three programs for analysis of three-dimensional con-

figurations of landmarks. The program simple3D takes raw data from a TPS-like format

(one landmark per row), performs Procrustes superimposition, computes centroid size and

produces output files in a more conventional datafile format (one specimen per row –

, Y

, Z

...CS); it can also perform Goodall’s F-test and a bootstrapped F-test for sig-

nificant differences between two groups. The other two programs perform regression

(ThreeDRegress6) and principal components analysis (ThreeDPCA6) on files that have

been processed by simple3D; both are modeled on the programs used for analyses of two-

dimensional configuration. All three programs require a textfile describing a wireframe

that will be used in illustrations (data analyses can proceed without the wireframe, but the

only graphics that will function are the displays of the sample mean and the superimposed

specimens). The wireframe file is simply a list of landmarks to be connected by each seg-

ment (e.g. 1 2 3, indicating that the first segment connects landmarks 2 and 3 with one

row for each segment).

Semilandmarks

In the IMP package, two programs, MakeFan and SemiLand, have been designed for semi-

landmark processing. MakeFan provides several options for drawing fans (rays at equal

angular intervals) or combs (perpendiculars from a reference line) that can be used as guides

for digitizing semilandmarks. Pictures with fans can be digitized in MakeFan, or saved and

opened for digitizing in TPSDig. MakeFan saves coordinates in the TPS format, so they

must be converted in CoordGen before further processing. SemiLand takes the converted

files, slides the semilandmarks to perpendicular alignment on the reference configuration,

and then deletes the semilandmarks selected for culling (called helpers). Output from this

program (superimposed configurations of landmarks and retained semilandmarks) can be

input into any of the analytic programs in the IMP package.

In the TPS package, tpsrelw has been modified to permit analyses of semilandmarks

and landmarks. This requires a “sliders” file in NTS format that identifies which points

in the data file are semilandmarks. After loading both the data file and the sliders file, the

option Relax semilandmarks will be available in the Options pull-down menu. Selecting

Relax semilandmarks will invoke sliding to minimize bending energy.

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