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Size and shape analysis techniques for design
Size and shape analysis techniques for design M V Ratnaparkhi, M M Ratnaparkhi and K M Robinette Wright State University, Dayton and Armstrong Laboratory, WPAFB, Ohio, USA A major problem in designing highly specialized e q u i p m e n t such as oxygen masks, respirators, etc, is that the effectiveness of the equipment depends on its appropriateness for the size and shape of the body part for which it is designed. However, in general, among the individuals who are likely to be using this e q u i p m e n t , there is considerable heterogeneity in size and shape of the body parts. One solution is to use the available data to form h o m o g e n e o u s clusters of the population and then make separate designs for each cluster, commonly referred to as sizing. Current sizing practices are hindered by a p r o b l e m t e r m e d 'observer-inherence'; in other words, the positioning and orientation of the reference axis system can affect the size and shape .groupings more than size and shape themselves do. The impact of observer-inherence ~s felt most on systems that require the most stringent fit, such as helmets with optical systems. For such systems, traditional measures and analysis practices can be almost useless. In this paper, an analysis technique is introduced which should be observerinvariant in the three-dimensional case. The m e t h o d is illustrated using points selected along a horizontal cross-section. The points are first transformed into values called curvatures which are subsequently transformed into a series of Fourier coefficients. These are then used for arriving at shape clusters or groupings. The shape differences (and similarities) within and between clusters are examined graphically and discussed. The technique developed here can be extended to form clusters using the curvatures of a surface instead of that of a cross-section (ie, can be extended to the three-dimensional case) and methods for doing so are discussed.
Keywords: Anthropometry, contours, curvatures, Fourier transforms, clustering Introduction In the past, the primary measures of the human body used by anthropologists have been measures between so called 'homologous' points such as a specific ear point or nose point, which can be located 'consistently' from one individual to the next based upon biological features. Shape in these studies has meant the location at, and relative distances between, these points. This type of information has been used in the design of equipment. However, for certain complex designs, this information, while useful, is insufficient. For example, for the design of an oxygen mask, the information regarding the contour of the face is needed. Further, for increasing the efficiency of such masks, it becomes imperative that variability in such contours for different individuals be incorporated in the design. Until recently, advanced computer based techniques for collecting and analyzing the data for such purposes Vol 23 No 3 June 1992
have not been available. During the last few years the measuring technology has advanced to the state that it is now possible to digitize the surface of biological forms faster and in much finer detail than was previously possible. For example, data are now being collected on the head and face with 1 mm and greater resolution. In one current effort, 130 000 points are being collected on the head and face and put into digital form in less than 15 s. However, statistical summary and clustering methods for such data have not yet been developed. Many statistical methods are in use for classifying size and shape in biological systems. Some of these methods are: trend surface analysis 1, Fourier analysis2, finite element analysis3 and resistant fitting techniques4. Analyses using traditional data share a problem which has to do with the definition of the reference axis system and its effect on differences observed between subjects. This has been termed 'observer-inherence.' Cheverud et aP describe this as a ' . . . different means
0003-6870/92/03 0181-05 $03.00 (~) 1992 Butterworth-Heinemann Ltd
181
Size and shape analysis techniques for design of registration, which consists of standardizing the coordinate system of the objects to be compared, thus eliminating whole-body linear displacements relative to an arbitrary origin, and therefore all of the measurements are observer-inherent and the results dependent on the form of registration rather than the biological forms analysed.' This means that regardless of the axis system used, or which landmarks are used to define the axis system, the shapes will not be aligned. For example, if the traditional anatomical axis system is used, the points which define the X-Y plane are left and right Tragions (points on the ear) and right infraorbitale (a point below the eye.) Since the location of this plane is dependent upon the location of the ears, a variation of 25 mm in the ears vertically will rotate the plane, and thus the orientation of the head, by as much as 10-15 degrees. As a result, the eye of subject A may be mistakenly aligned with the brow ridge of subject B. This paper describes a method for summarizing and clustering forms which should resolve this 'observerinherence' difficulty when applied to three-dimensional surfaces. In other words, it should not be affected by the reference axis system or the registration, yet will group according to surface contour data. It is easier to develop, to describe and to illustrate a methodology for two-dimensional cases, such as the size and shape of contours or cross-sections in a plane. For this reason, in this paper only the data for an arbitrarily chosen single horizontal cross section of the head at the level of a homologous point called pronasale (the most anterior point of the nose) are used. Future research efforts will extend the method to three dimensions. It should be noted that the resolution of the observerinherence problem will not be accomplished until it is extended to the three-dimensional case. The objective of this paper is to introduce the approach without getting into detailed mathematical or statistical theory. The approach consists of three steps: (1) defining the curvatures as descriptive statistics for measuring shape of a cross-section, calculating these measures using the available data, (2) transforming the curvatures into corresponding Fourier descriptors (FDs), and (3) using these FDs for forming the clusters of the observed pronasale cross-sections mentioned above. These three steps are described and discussed below with an emphasis on the usefulness of curvatures and FDs in such studies. The actual results obtained by applying this methodology to the available data are presented afterwards, followed by a brief discussion of these results.
Methodology Description of sample data Stereophotometric data were collected on a sample of 20 US Army male personnel in the 1970s (unpublished). These 20 subjects were the only ones out of a set of over 100 whose heads were aligned so that the tragion (an ear point) and infraorbitale (an eye point) were roughly horizontal. This is a standard reference plane referred to as the Frankfurt plane. The section at pronasale was selected as the point at which to extract a cross-section because it has some extreme 182
shape changes which would suit our purpose.
illustrative
A plot of cross-section at pronasale is shown in Figure 1. The original observations were connected using a cubic spline curve fit. There were approximately 60 points on each cross-section. The origin of the crosssection (denoted by O) was set at the mid-point of the tragions. The straight line joining the origin to pronasale (denoted by A) is the polar axis. The observations on point P(0, r) are: 0, the angle AOP, and r, the radial distance from O to point P. To simplify the equations, evenly spaced observations were required and to accomplish this the values of 0 were set at 5° intervals and the r at the intersection with the curve was determined. This resulted in 72 P(0, r) 'observations.' This is a construct, again for the purpose of demonstration, and it should be noted that this is oversampling the original data set and should not be done in practice. New measurement technology is now providing better (ie, denser) data sets for use with this methodology in the future.
Curvature Intuitively, the term curvature refers to departure from straightness. Obviously, the curvature of a straight line is always zero and non-zero curvature indicates departure from straightness. In formal mathematical terminology, the curvature at a point on a curve is measured by the rate of change in the direction of the tangent line at that point (see Figure 1). In this study the curvatures at different points of the cross-section were used as descriptive statistics for the shape of that cross-section. Note that the characteristics of a shape of a curve, such as sharp changes in the shape, are always associated with sharp changes in the corresponding magnitudes (values) of curvature. For example, in Figure 1 there are sharp changes in the region of the ear. Such changes are described quanti-
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Figure 1 Polar plot of radii for pronasale cross-section Applied Ergonomics
M V RATNAPARKHI~ M M R A T N A P A R K H I A N D
tatively by the curvatures at corresponding points. The curvature C(t) at the t'th point on a curve is given by: C(t) = ( r 2 - rr" - 2 r'Z)/ ( r z + r'2) 3/2
where r(t) denotes the radius at the t'th point, r'(t) = [ r ( t + l ) - r ( t ) ] / d and r"(t) = [r'(t+l) - r'(t)]/d. The above formula is a discrete version of the formula for C(t) given in text books. In our study, t = 1, 2 . . . . 72, and d = 5° (representing the interval at which the radii were calculated). The important issue is that while the radii magnitudes are relative to the origin of the cross-section, the magnitudes of the curvatures do not depend directly on the choice of origin. This is the first step towards removing the dependency on the choice of axis system - in other words, towards removing the observerinherence. However, the 72 curvatures (C(t)) at 72 points are calculated using the radii at neighbouring points and hence are correlated. Therefore, without an additional transformation of the curvatures, the statistical analyses of C(t) will have to be considered as a 72dimensional multivariate analysis problem with unknown correlation structure among values of C(t). Any such analysis will be extremely difficult even though an appropriate methodology can be found. To avoid this the curvatures for each cross-section were transformed into a series of numbers which are commonly known as the discrete Fourier descriptors (FD) or Fourier coefficients. These FDs are obtained by using the discrete Fourier transform of sampled data points. This transformation is described briefly below. Discrete Fourier descriptors The Fourier transform5 is a standard mathematical technique that is used in a wide variety of applications. This transform has been used previously with human body measures and a brief discussion of such transforms related to human body measurements can be found in Park and Lee 6. However, this is the first known effort to use the transforms on curvature rather than distance or radii type body variables.
In this study the series of 72 curvatures for each cross-section were transformed into corresponding Fourier coefficients, commonly known as Fourier descriptors (FD). The Fourier transform (or the discrete Fourier transform) is used in the study of periodic functions or series of sampled data which are of periodic nature. In particular, the Fourier transform of a series of observed sample data is a functional representation of the data in terms of sine and cosine functions. Two basic components of the Fourier transform are the Fourier coefficients (pairs of sine and cosine coefficients) and the frequencies associated with them. The Fourier coefficients are also known as the Fourier Descriptors in many applications. For the data in this study, since there were 72 points in the input sample there are 37 pairs of these functions - ie, 37 frequencies and corresponding pairs of FDs denoted by (A k, Bk), k=0, 1, 2 . . . . 36. These FDs collectively represent the shapes of the cross-sections. Each coefficient pair indicates the relative contribution of the corresponding sine and cosine functions at Vol 23 No 3 June 1992
K M ROBINETTE
the associated frequency in describing the shape of an individual cross-section. In other words, they indicate the contribution of the particular curve (in this case, a cross-section) to a particular frequency. The order of the frequency increases as k increases, so the lower order frequencies are reflected in the first FD pairs, the higher order frequencies in the last. For example, the first FD pair suggests a certain circular shape as an approximation to the shape of the cross-section, the last FD pair suggests the occurrence of very sharp peaks. One important aspect of the Fourier transform for shape summarization is the orthogonality property of the Fourier coefficients. Transforming the related curvatures using the Fourier transform results in variables (the FDs) which for statistical analysis purposes are treated as uncorrelated. Further, the FDs under appropriate reparameterization of closed curve can be shown to be not dependent on the initial point on the curve that is used for such parameterization7. This is the second step toward removing dependency on the axis system. It is hypothesized at this point that the data will be observer-invariant in the case of the threedimensional surface. This would also be true in the 2-D cross-section case if the shape of the slice was not dependent upon the location of the slice - for example, if the slices were taken from a group of cylinders that remain the same vertically but which have different shapes horizontally. For demonstration purposes, we are treating the slices as if they are entities taken from such cylinders. Another desirable result of the orthogonality property of the FDs is that the statistical analysis is comparatively straightforward. As noted by Zahn and Roskois 7, the Euclidean distance between the FDs corresponding to the cross-sections for which the shapes look alike is smaller than the distance between the FDs corresponding to the cross-sections which do not look alike. This observation justifies the use of FDs for discriminating between individuals.
Clustering technique A clustering technique was applied to the data to demonstrate the utility of the FDs for distinguishing shapes. If the FDs are useful in a practical sense, then the cross-sections which cluster together should look alike and those which do not cluster (or group) should look different. For forming such clusters the Fourier descriptors of a cross-section are used as the characteristic measurements of shape. As mentioned above, the FDs, because of their orthogonality property, are suitable for use in cluster analysis. Also as mentioned before, in the space of FDs the Euclidian distance between any two sets of FDs provides a measure of dissimilarity between corresponding shapes. Further, large distances are associated with a higher degree of dissimilarity between the shapes. This observation is used for obtaining the clusters of cross-section using a statistical methodology known as cluster analysis. A number of methods are available for forming the clusters of individuals in a given sample and a discussion of such methods may be found in Johnson and
183
Size and shape analysis techniques for design Wichern 8. A brief description of the cluster analysis procedure used here is given below• Suppose that a number of items are to be grouped according to their characteristics and that these characteristics are the variables that are observed for each of the items. For example, in this study the cross-sections are the items and the FDs are the characteristics of the cross-sections• Using these characteristics, a measure of distance between items is obtained• Here, the Euclidean distance defined by the square root of the sum of squares of the differences between the FDs for any two cross-sections provides such a measure. More formally, let dij denote the distance between the (i, j)th pair of cross-sections• Then dii = [Z (Aki -- Akj) 2 q- Y. (Uki -- Bkj) 2 ],/2,
where the summation is over k. The 20 × 20 matrix D ={dii} corresponding to the 20 cross-sections used in this study is the dissimilarity matrix for the shapes of 20 cross-sections. For convenience these 20 cross-sections are denoted by Sl, S2, • . • S20• The distances calculated by using the above formula were clustered using the average linkage method. With this method each cross-section is treated as a cluster to start with. Then the search on matrix D is carried to find most similar (nearest) cross-sections (shapes). These two cross-sections are merged to form a cluster, CL~ say. Then the distance matrix D is updated by deleting the row and column corresponding to CL~ and then by adding a row and column corresponding to the distances between cluster CL~ and the remaining clusters. The above steps are repeated 19 times to obtain the clusters in the final form. This was accomplished with the Statistical Analysis System (SAS) software package 9.
Results The resulting data sets of radii, curvatures and Fourier coefficients are large and therefore only a small portion of these are presented in Tables 1 and 2 to illustrate the appearance of the data for different crosssections. Table 1 lists some of the radii and curvatures for one subject. The shape of the cross-section changes dramatically in the ear and nose regions. Therefore, the curvatures calculated for points in these regions show sharp changes. Table 2 lists the first three FDs for one subject as well as the frequencies associated with them. The clustering procedure produced three clusters. These clusters are shown in Table 3, where the numbers in the table refer to the respective code numbers that were assigned to the subjects. The number of subjects in the three clusters were found to be 12, 7 and 1. To assess the efficiency of the procedure used for clustering, pairwise comparisons of the cross-sections were carried out. The polar coordinate plots of the radii for two subjects from the same cluster are shown in Figures 2(a) and 2(b). Similarly, Figure 3 provides a comparison of polar coordinate plots of subjects from cluster 3 and cluster 1. As these examples illustrate, it appears that plot shapes within cluster 1 184
Table 1 Sample data: Radii (in inches (mm)) and curvatures for a typical cross-section Angle (degrees)
Radius (in)
(mm)
Curvature
0 5
4.72 4.44
120 113
1.46 -0.17
90 95 100
2.57 2.59 3.65
65 66 93
-19.69 0.44 -0.13
Table 2 Sample data: Fourier Descriptors (Ak, Bk) for a cross-section at k-th frequency Frequency
Ak
Bk
0.0000 0•0139 0•0278
0•0162 -0•0253 0.2543
0.0000 -0.3147 0•0393
Table 3 Clusters of 20 pronasale cross-sections Cluster number
Subject number
1 2 3
1, 2, 5, 8, 10, 11, 14, 16, 17, 18, 19, 20 3,4,6,7,9,13,15 12
are very similar but that plot shapes from clusters 3 and 1 are very different, particularly in the nose region•
Discussion One difficulty with this method is in selection of the FDs which are most useful in discriminating shapes• In many applications, a satisfactory representation can be obtained in terms of the first few coefficients and corresponding frequencies• This may not be true for human shapes, however• The shape of particular features, such as the nose, would contribute to some higher-order pair. There are many statistical techniques available which would allow one to determine which Fourier coefficients are those that discriminate between shapes. These will probably have to be employed in order for this method to be useful. Another difficulty is the fact that very different entities, such as the ear and the nose, may contribute to the same FD on one person and not on another• For this reason, the use of the method will probably have to be regional, for a portion of the surface such as the face, or the cranium. Further, to decide if it is necessary to use the regionwise analysis based on FDs, a close examination of the FDs and corresponding frequencies and their interpretations in terms of shape is advisable. Applied Ergonomics
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If these limitations in the method are accounted for, and if it is extended to the three-dimensional surface case, then it should be useful for discriminating between people who will achieve a fit in a helmet or other piece of equipment and those who will not. It should also be useful for such clustering requirements as separating male bones from female in forensic anthropology, or in separating different species. Future research planned includes the extension of the technique to three-dimensions with a better data set now available for evaluations of the sensitivity and
Vol 23 No 3 June 1992
practical utility of the method, and the development of methods for interpretation of any shape differences that can be identified.
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The work of Makarand V Ratnaparkhi and Milind M Ratnaparkhi was supported by Anthropology Research Project Inc, Yellow Springs, Ohio 45387, when they were working as consultants for Air Force Contract F33615-85-C-0531 (Task 4.1).
References 1 Sneath, P H 'Trend surface analysis of transformation grids' J Zool, Proc Zoological Soc of London Vol 151 (1967) pp 65-122 2 Lestrel, P 'Some problems in the assessment of morphological size and shape differences' Physical Anthropol Vol 18 (1974) pp 140-162 3 Cheverud, J, Lewis, J L, Banchrach, W and Lew W D 'The measurement of form and variation in form: an application of three-dimensional morphology by finite-element methods' Amer J Physical Anthropol Vol 62 (1983) pp 152-165 4 Siegel, A F and Benson, R H 'A robust comparison of biological shapes' Biometrics Vol 38 (1982) pp 341-350 5 Weaver, J H Applications of discrete and continuous Fourier analysis John Wiley & Sons (1983) 6 Park, K S and Lee, S N 'A three-dimensional Fourier descriptor for human body representation/reconstruction from serial cross-sections' Computers and Biomed Res Vol 2 (1987) pp 125-140 7 Zahn, C T and Roskois, R Z 'Fourier descriptors for plane closed curves' IEEE Trans on Computers Vol C-21 No 3 (1972) pp 269-281 8 Johnson R A and Wichern, D W Applied multivariate statistical analysis Prentice Hall (Chap 12) (1988) 9 Anon SAS User's Guide SAS Institute Inc (1982)
185
Keywords: Anthropometry, contours, curvatures, Fourier transforms, clustering Introduction In the past, the primary measures of the human body used by anthropologists have been measures between so called 'homologous' points such as a specific ear point or nose point, which can be located 'consistently' from one individual to the next based upon biological features. Shape in these studies has meant the location at, and relative distances between, these points. This type of information has been used in the design of equipment. However, for certain complex designs, this information, while useful, is insufficient. For example, for the design of an oxygen mask, the information regarding the contour of the face is needed. Further, for increasing the efficiency of such masks, it becomes imperative that variability in such contours for different individuals be incorporated in the design. Until recently, advanced computer based techniques for collecting and analyzing the data for such purposes Vol 23 No 3 June 1992
have not been available. During the last few years the measuring technology has advanced to the state that it is now possible to digitize the surface of biological forms faster and in much finer detail than was previously possible. For example, data are now being collected on the head and face with 1 mm and greater resolution. In one current effort, 130 000 points are being collected on the head and face and put into digital form in less than 15 s. However, statistical summary and clustering methods for such data have not yet been developed. Many statistical methods are in use for classifying size and shape in biological systems. Some of these methods are: trend surface analysis 1, Fourier analysis2, finite element analysis3 and resistant fitting techniques4. Analyses using traditional data share a problem which has to do with the definition of the reference axis system and its effect on differences observed between subjects. This has been termed 'observer-inherence.' Cheverud et aP describe this as a ' . . . different means
0003-6870/92/03 0181-05 $03.00 (~) 1992 Butterworth-Heinemann Ltd
181
Size and shape analysis techniques for design of registration, which consists of standardizing the coordinate system of the objects to be compared, thus eliminating whole-body linear displacements relative to an arbitrary origin, and therefore all of the measurements are observer-inherent and the results dependent on the form of registration rather than the biological forms analysed.' This means that regardless of the axis system used, or which landmarks are used to define the axis system, the shapes will not be aligned. For example, if the traditional anatomical axis system is used, the points which define the X-Y plane are left and right Tragions (points on the ear) and right infraorbitale (a point below the eye.) Since the location of this plane is dependent upon the location of the ears, a variation of 25 mm in the ears vertically will rotate the plane, and thus the orientation of the head, by as much as 10-15 degrees. As a result, the eye of subject A may be mistakenly aligned with the brow ridge of subject B. This paper describes a method for summarizing and clustering forms which should resolve this 'observerinherence' difficulty when applied to three-dimensional surfaces. In other words, it should not be affected by the reference axis system or the registration, yet will group according to surface contour data. It is easier to develop, to describe and to illustrate a methodology for two-dimensional cases, such as the size and shape of contours or cross-sections in a plane. For this reason, in this paper only the data for an arbitrarily chosen single horizontal cross section of the head at the level of a homologous point called pronasale (the most anterior point of the nose) are used. Future research efforts will extend the method to three dimensions. It should be noted that the resolution of the observerinherence problem will not be accomplished until it is extended to the three-dimensional case. The objective of this paper is to introduce the approach without getting into detailed mathematical or statistical theory. The approach consists of three steps: (1) defining the curvatures as descriptive statistics for measuring shape of a cross-section, calculating these measures using the available data, (2) transforming the curvatures into corresponding Fourier descriptors (FDs), and (3) using these FDs for forming the clusters of the observed pronasale cross-sections mentioned above. These three steps are described and discussed below with an emphasis on the usefulness of curvatures and FDs in such studies. The actual results obtained by applying this methodology to the available data are presented afterwards, followed by a brief discussion of these results.
Methodology Description of sample data Stereophotometric data were collected on a sample of 20 US Army male personnel in the 1970s (unpublished). These 20 subjects were the only ones out of a set of over 100 whose heads were aligned so that the tragion (an ear point) and infraorbitale (an eye point) were roughly horizontal. This is a standard reference plane referred to as the Frankfurt plane. The section at pronasale was selected as the point at which to extract a cross-section because it has some extreme 182
shape changes which would suit our purpose.
illustrative
A plot of cross-section at pronasale is shown in Figure 1. The original observations were connected using a cubic spline curve fit. There were approximately 60 points on each cross-section. The origin of the crosssection (denoted by O) was set at the mid-point of the tragions. The straight line joining the origin to pronasale (denoted by A) is the polar axis. The observations on point P(0, r) are: 0, the angle AOP, and r, the radial distance from O to point P. To simplify the equations, evenly spaced observations were required and to accomplish this the values of 0 were set at 5° intervals and the r at the intersection with the curve was determined. This resulted in 72 P(0, r) 'observations.' This is a construct, again for the purpose of demonstration, and it should be noted that this is oversampling the original data set and should not be done in practice. New measurement technology is now providing better (ie, denser) data sets for use with this methodology in the future.
Curvature Intuitively, the term curvature refers to departure from straightness. Obviously, the curvature of a straight line is always zero and non-zero curvature indicates departure from straightness. In formal mathematical terminology, the curvature at a point on a curve is measured by the rate of change in the direction of the tangent line at that point (see Figure 1). In this study the curvatures at different points of the cross-section were used as descriptive statistics for the shape of that cross-section. Note that the characteristics of a shape of a curve, such as sharp changes in the shape, are always associated with sharp changes in the corresponding magnitudes (values) of curvature. For example, in Figure 1 there are sharp changes in the region of the ear. Such changes are described quanti-
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Figure 1 Polar plot of radii for pronasale cross-section Applied Ergonomics
M V RATNAPARKHI~ M M R A T N A P A R K H I A N D
tatively by the curvatures at corresponding points. The curvature C(t) at the t'th point on a curve is given by: C(t) = ( r 2 - rr" - 2 r'Z)/ ( r z + r'2) 3/2
where r(t) denotes the radius at the t'th point, r'(t) = [ r ( t + l ) - r ( t ) ] / d and r"(t) = [r'(t+l) - r'(t)]/d. The above formula is a discrete version of the formula for C(t) given in text books. In our study, t = 1, 2 . . . . 72, and d = 5° (representing the interval at which the radii were calculated). The important issue is that while the radii magnitudes are relative to the origin of the cross-section, the magnitudes of the curvatures do not depend directly on the choice of origin. This is the first step towards removing the dependency on the choice of axis system - in other words, towards removing the observerinherence. However, the 72 curvatures (C(t)) at 72 points are calculated using the radii at neighbouring points and hence are correlated. Therefore, without an additional transformation of the curvatures, the statistical analyses of C(t) will have to be considered as a 72dimensional multivariate analysis problem with unknown correlation structure among values of C(t). Any such analysis will be extremely difficult even though an appropriate methodology can be found. To avoid this the curvatures for each cross-section were transformed into a series of numbers which are commonly known as the discrete Fourier descriptors (FD) or Fourier coefficients. These FDs are obtained by using the discrete Fourier transform of sampled data points. This transformation is described briefly below. Discrete Fourier descriptors The Fourier transform5 is a standard mathematical technique that is used in a wide variety of applications. This transform has been used previously with human body measures and a brief discussion of such transforms related to human body measurements can be found in Park and Lee 6. However, this is the first known effort to use the transforms on curvature rather than distance or radii type body variables.
In this study the series of 72 curvatures for each cross-section were transformed into corresponding Fourier coefficients, commonly known as Fourier descriptors (FD). The Fourier transform (or the discrete Fourier transform) is used in the study of periodic functions or series of sampled data which are of periodic nature. In particular, the Fourier transform of a series of observed sample data is a functional representation of the data in terms of sine and cosine functions. Two basic components of the Fourier transform are the Fourier coefficients (pairs of sine and cosine coefficients) and the frequencies associated with them. The Fourier coefficients are also known as the Fourier Descriptors in many applications. For the data in this study, since there were 72 points in the input sample there are 37 pairs of these functions - ie, 37 frequencies and corresponding pairs of FDs denoted by (A k, Bk), k=0, 1, 2 . . . . 36. These FDs collectively represent the shapes of the cross-sections. Each coefficient pair indicates the relative contribution of the corresponding sine and cosine functions at Vol 23 No 3 June 1992
K M ROBINETTE
the associated frequency in describing the shape of an individual cross-section. In other words, they indicate the contribution of the particular curve (in this case, a cross-section) to a particular frequency. The order of the frequency increases as k increases, so the lower order frequencies are reflected in the first FD pairs, the higher order frequencies in the last. For example, the first FD pair suggests a certain circular shape as an approximation to the shape of the cross-section, the last FD pair suggests the occurrence of very sharp peaks. One important aspect of the Fourier transform for shape summarization is the orthogonality property of the Fourier coefficients. Transforming the related curvatures using the Fourier transform results in variables (the FDs) which for statistical analysis purposes are treated as uncorrelated. Further, the FDs under appropriate reparameterization of closed curve can be shown to be not dependent on the initial point on the curve that is used for such parameterization7. This is the second step toward removing dependency on the axis system. It is hypothesized at this point that the data will be observer-invariant in the case of the threedimensional surface. This would also be true in the 2-D cross-section case if the shape of the slice was not dependent upon the location of the slice - for example, if the slices were taken from a group of cylinders that remain the same vertically but which have different shapes horizontally. For demonstration purposes, we are treating the slices as if they are entities taken from such cylinders. Another desirable result of the orthogonality property of the FDs is that the statistical analysis is comparatively straightforward. As noted by Zahn and Roskois 7, the Euclidean distance between the FDs corresponding to the cross-sections for which the shapes look alike is smaller than the distance between the FDs corresponding to the cross-sections which do not look alike. This observation justifies the use of FDs for discriminating between individuals.
Clustering technique A clustering technique was applied to the data to demonstrate the utility of the FDs for distinguishing shapes. If the FDs are useful in a practical sense, then the cross-sections which cluster together should look alike and those which do not cluster (or group) should look different. For forming such clusters the Fourier descriptors of a cross-section are used as the characteristic measurements of shape. As mentioned above, the FDs, because of their orthogonality property, are suitable for use in cluster analysis. Also as mentioned before, in the space of FDs the Euclidian distance between any two sets of FDs provides a measure of dissimilarity between corresponding shapes. Further, large distances are associated with a higher degree of dissimilarity between the shapes. This observation is used for obtaining the clusters of cross-section using a statistical methodology known as cluster analysis. A number of methods are available for forming the clusters of individuals in a given sample and a discussion of such methods may be found in Johnson and
183
Size and shape analysis techniques for design Wichern 8. A brief description of the cluster analysis procedure used here is given below• Suppose that a number of items are to be grouped according to their characteristics and that these characteristics are the variables that are observed for each of the items. For example, in this study the cross-sections are the items and the FDs are the characteristics of the cross-sections• Using these characteristics, a measure of distance between items is obtained• Here, the Euclidean distance defined by the square root of the sum of squares of the differences between the FDs for any two cross-sections provides such a measure. More formally, let dij denote the distance between the (i, j)th pair of cross-sections• Then dii = [Z (Aki -- Akj) 2 q- Y. (Uki -- Bkj) 2 ],/2,
where the summation is over k. The 20 × 20 matrix D ={dii} corresponding to the 20 cross-sections used in this study is the dissimilarity matrix for the shapes of 20 cross-sections. For convenience these 20 cross-sections are denoted by Sl, S2, • . • S20• The distances calculated by using the above formula were clustered using the average linkage method. With this method each cross-section is treated as a cluster to start with. Then the search on matrix D is carried to find most similar (nearest) cross-sections (shapes). These two cross-sections are merged to form a cluster, CL~ say. Then the distance matrix D is updated by deleting the row and column corresponding to CL~ and then by adding a row and column corresponding to the distances between cluster CL~ and the remaining clusters. The above steps are repeated 19 times to obtain the clusters in the final form. This was accomplished with the Statistical Analysis System (SAS) software package 9.
Results The resulting data sets of radii, curvatures and Fourier coefficients are large and therefore only a small portion of these are presented in Tables 1 and 2 to illustrate the appearance of the data for different crosssections. Table 1 lists some of the radii and curvatures for one subject. The shape of the cross-section changes dramatically in the ear and nose regions. Therefore, the curvatures calculated for points in these regions show sharp changes. Table 2 lists the first three FDs for one subject as well as the frequencies associated with them. The clustering procedure produced three clusters. These clusters are shown in Table 3, where the numbers in the table refer to the respective code numbers that were assigned to the subjects. The number of subjects in the three clusters were found to be 12, 7 and 1. To assess the efficiency of the procedure used for clustering, pairwise comparisons of the cross-sections were carried out. The polar coordinate plots of the radii for two subjects from the same cluster are shown in Figures 2(a) and 2(b). Similarly, Figure 3 provides a comparison of polar coordinate plots of subjects from cluster 3 and cluster 1. As these examples illustrate, it appears that plot shapes within cluster 1 184
Table 1 Sample data: Radii (in inches (mm)) and curvatures for a typical cross-section Angle (degrees)
Radius (in)
(mm)
Curvature
0 5
4.72 4.44
120 113
1.46 -0.17
90 95 100
2.57 2.59 3.65
65 66 93
-19.69 0.44 -0.13
Table 2 Sample data: Fourier Descriptors (Ak, Bk) for a cross-section at k-th frequency Frequency
Ak
Bk
0.0000 0•0139 0•0278
0•0162 -0•0253 0.2543
0.0000 -0.3147 0•0393
Table 3 Clusters of 20 pronasale cross-sections Cluster number
Subject number
1 2 3
1, 2, 5, 8, 10, 11, 14, 16, 17, 18, 19, 20 3,4,6,7,9,13,15 12
are very similar but that plot shapes from clusters 3 and 1 are very different, particularly in the nose region•
Discussion One difficulty with this method is in selection of the FDs which are most useful in discriminating shapes• In many applications, a satisfactory representation can be obtained in terms of the first few coefficients and corresponding frequencies• This may not be true for human shapes, however• The shape of particular features, such as the nose, would contribute to some higher-order pair. There are many statistical techniques available which would allow one to determine which Fourier coefficients are those that discriminate between shapes. These will probably have to be employed in order for this method to be useful. Another difficulty is the fact that very different entities, such as the ear and the nose, may contribute to the same FD on one person and not on another• For this reason, the use of the method will probably have to be regional, for a portion of the surface such as the face, or the cranium. Further, to decide if it is necessary to use the regionwise analysis based on FDs, a close examination of the FDs and corresponding frequencies and their interpretations in terms of shape is advisable. Applied Ergonomics
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If these limitations in the method are accounted for, and if it is extended to the three-dimensional surface case, then it should be useful for discriminating between people who will achieve a fit in a helmet or other piece of equipment and those who will not. It should also be useful for such clustering requirements as separating male bones from female in forensic anthropology, or in separating different species. Future research planned includes the extension of the technique to three-dimensions with a better data set now available for evaluations of the sensitivity and
Vol 23 No 3 June 1992
practical utility of the method, and the development of methods for interpretation of any shape differences that can be identified.
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Figure 3 Pronasale cross-sections for subjects 1 and 12 (subjects from different clusters)
The work of Makarand V Ratnaparkhi and Milind M Ratnaparkhi was supported by Anthropology Research Project Inc, Yellow Springs, Ohio 45387, when they were working as consultants for Air Force Contract F33615-85-C-0531 (Task 4.1).
References 1 Sneath, P H 'Trend surface analysis of transformation grids' J Zool, Proc Zoological Soc of London Vol 151 (1967) pp 65-122 2 Lestrel, P 'Some problems in the assessment of morphological size and shape differences' Physical Anthropol Vol 18 (1974) pp 140-162 3 Cheverud, J, Lewis, J L, Banchrach, W and Lew W D 'The measurement of form and variation in form: an application of three-dimensional morphology by finite-element methods' Amer J Physical Anthropol Vol 62 (1983) pp 152-165 4 Siegel, A F and Benson, R H 'A robust comparison of biological shapes' Biometrics Vol 38 (1982) pp 341-350 5 Weaver, J H Applications of discrete and continuous Fourier analysis John Wiley & Sons (1983) 6 Park, K S and Lee, S N 'A three-dimensional Fourier descriptor for human body representation/reconstruction from serial cross-sections' Computers and Biomed Res Vol 2 (1987) pp 125-140 7 Zahn, C T and Roskois, R Z 'Fourier descriptors for plane closed curves' IEEE Trans on Computers Vol C-21 No 3 (1972) pp 269-281 8 Johnson R A and Wichern, D W Applied multivariate statistical analysis Prentice Hall (Chap 12) (1988) 9 Anon SAS User's Guide SAS Institute Inc (1982)
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