C. G. Small, The Statistical Theory of Shapes, Springer-Verlag, New York, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....at di#erent elevations, is needed. 1.2 Past Work in Shape Analysis Historically, there have been many exemplary e#orts in characterization and quantification of object shapes. Thompson [29] was among the first to quantify shape di#erences. More recently, the works of Kendall and his colleagues [12, 26], Bookstein [2] Mardia [5] and Kent [14] have resulted in an elegant statistical theory of shapes. A common theme here is that objects are represented using a finite number of salient points or landmarks (points in a Euclidean space, and one establishes equivalences with respect to the ....

.... n landmarks in for planar shapes) is a finite dimensional Riemannian manifold, often called a shape manifold; di#erent shapes correspond to elements of this space and quantification of shapes di#erences is achieved via a Riemannian metric on this space (for example, the Procrustean metric [26]) An important aspect of this work is its maturity to statistical frameworks. Researchers have defined probability distributions on these shape manifolds and have sought statistical approaches for shape estimation. In Grenander s formulation [7] shapes are considered as points on some ....

Christopher G. Small. The Statistical Theory of Shape. Springer, 1996.

....to deform one biological structure to another closely related structure is D arcy Thompson in his classical book On Growth and Form [109] where he deformed the skulls of human and primates, and other biological structures using deformable grids. Unlike classical morphometry in shape analysis [12, 13, 40, 62, 101] , the deformation based mor phometry tries to avoid anatomical landmarks in characterizing morphological changes. An anatomical landmark is a point assigned by an expert that corresponds between organism in some biologically meaningful way [40] However, it is very hard to identify such ....

C.G. Small. The Statistical Theory of Shape. Springer, New York, 1996.

....in characterization and quantification of object shapes. D Arcy Thompson [16] studied shape variations in, among other things, florets, fishes, mountains, and heights of schoolboys. In recent years, the credit for initiating a mathematical theory of shapes goes to David Kendall and his colleagues [9, 5, 15, 11]. In addition, a rich analysis of shapes exists due to the independent works of Bookstein [2] Mardia, Kent and colleagues [4] and many others. They study shapes using finite landmarks (points in Euclidean space) and establish equivalences with respect to shape preserving transformations, i.e. ....

Christopher G. Small. The Statistical Theory of Shape. Springer, 1996.

....K(#P i #)#u du i #P i K(#P i #) #u i j ) which makes points traveling in the same direction attract each other and points going in opposite directions repel each other. This space leads to a non linear version of the theory of landmark points and shape statistics of Kendall [Sm] and has been developed by Younes [Yo] A similar treatment can be made for the space of shapes n,1 , where n,1 is the stabilizer of the unit sphere. Geodesics on come from solutions of the TME for which t is supported on the boundary of the shape and perpendicular to it. Even though ....

C Small, The Statistical Theory of Shape, Springer, 1996.

....locally one to one. Let # and # be the singular values of A. The transformation h takes a unit circle to an ellipse with major and minor axes of length # and #. The value log(# #) is called the log anisotropy of h and is commonly used as a measure of how far h is from a similarity transform (see [15]) We use the log anisotropy measure to assign a deformation cost for each a#ne map (and let the cost be infinity if the a#ne map is not orientation preserving) The deformation costs are combined with a data cost that attracts the shape boundary to locations in the image that have high gradient ....

C. Small. The Statistical Theory of Shapes. Springer-Verlag, 1996.

....and quantification of object shapes. Darcy Thompson [20] studied the shape variations in, among other things, florets, fishes, mountains, spirals, and heights of schoolboys. In recent years, the credit for initiating a mathematical theory of shapes goes to David Kendall and his colleagues [10, 18]. In addition, a rich statistical analysis of shapes exists due to the independent works of Bookstein [1] Mardia [4] Kent [12] and others. A common characteristic of this work is that objects are represented using # Department of Mathematics, Florida State University, Tallahassee, FL 32306 ....

.... in IR for planar shapes) is a finite dimensional Riemannian manifold, often called a shape space or shape manifold; di#erent shapes correspond to elements of this space and a quantification of shapes di#erences is accomplished using a metric on this space (for example, the Procrustean metric [18]) Another related but independent idea comes from Grenander s formulation of a theory of patterns [6, 5] In this approach, the shapes (and more generally patterns) are considered as points on some infinite dimensional di#erentiable manifold, and the variation between the shapes are modeled by ....

Christopher G. Small. The Statistical Theory of Shape. Springer, 1996.

....set. The circle shows the equal error rate condition. of the standard deviation of the points of the training set that correspond to a point of the prototype. The third distance is the correlation between the prototype and the test signature after having performed a Procrustes transformation [70, 21] on them. This Procrustes transformation provides the optimal, in the least squares sense, translation, rotation and scaling between the prototype and the test signature, given the correspondence between their samples. The harmonic mean of these di erent distances is used as the classi cation ....

C.G. Small. The statistical theory of shape. Springer-Verlag, Inc., 1996.

....respect to a 32 CHAPTER 2. A THEORY OF SIMILARITY MEASURES A B g(A) g(B) Figure 2.1: Invariance given transformation group, an orbit can be interpreted as the shape of a pattern, and the orbit set can be seen as a class of shapes. This is a generalisation of the notion of shape used by Small [114], who defines shape as the orbit set of a pattern under the group of similarity transformations. Example 2.2.3. Under the group of a#ne transformations, the set of all triangles, and the set of all ellipses are shapes. Example 2.2.4. For projective transformations, the set of all quadrangles is ....

C. G. Small. The Statistical Theory of Shapes. Springer Series in Statistics. Springer-Verlag, New York, N. Y., 1996.

....close as possible to the bed. After one of the point sets has been transformed to match the other, the sum of squared di erences of the coordinates between them is called the Procrustes distance, while the shape instance de ned by the average of their coordinates is called Procrustes average shape [1 3, 14]. 2 An approach that uni es the correspondence estimation with the alignment problem for sets of points with unknown correspondences (called the Softassign Procrustes Matching Algorithm) was presented in [16] 5 Figure 2: Expert de ned pseudo landmarks (yellow squares) on the three shapes in ....

....than 10 of the object scale. The scale of a shape instance A = f(x i ; y i )g i=1: n is de ned as Scale(A) q (max i=1: n (x i ) min i=1: n (x i ) 1) max i=1: n (y i ) min i=1: n (y i ) 1) 4 4 This de nition of scale is di erent from that employed by most previous studies [1 3, 14] where Scale(A) 26 We enforce this upper bound when we set a point correspondence in Step 2.3 of Algorithm 2. In practice, if there exists a proper matching between two sets, most of the links are actually shorter than half of this upper bound. We also found it to be convenient to work in scale ....

C. G. Small, The Statistical Theory of Shape. Berlin: Springer-Verlag, 1996.

....(1993) give a general overview of the field and Bookstein (1991) supplies a more technical account. Marcus et al. 1996) include both introductory material and many examples of applications to biology and medicine. Dryden and Mardia (1998) give a comprehensive coverage of shape statistics andSmall (1996) covers some of the important properties of shape spaces. Rohlf (1999b) gives an overview of some of the relationships between shape statistics and the shape spaces on which they are based. The fundamental advances of geometric morphometrics over traditional approaches include the way one ....

....or great circle distance, r. These Procrustes distances are related as r = F H I K 2 2 1 sin d P , with 02rp radians. Kendall (1984) worked out some of the geometric properties of the space implied by the use of this distance as a metric (the space is now called Kendall shape space, Small 1996). While the GLS procedure and the methods discussed below can easily be carried out for three Geometric morphometrics in systematics 04 18 00 4 dimensional coordinates, their geometry is more complicated and will not be discussed here. Dryden and Mardia (1993) and Small (1996) address some of the ....

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Small, C. G. 1996. The statistical theory of shape. Springer, New York.

....aims of shape analysis are to measure and describe differences between shapes, to estimate a mean population shape from a random sample of objects and to describe the structure of shape variability in a dataset. Some reviews of the field have been given by Bookstein (1991, 1997) Kendall (1989) Small (1996) and Dryden and Mardia (1997) In this paper we consider the resistant shape analysis of point set configurations. For example, the points might be biological landmarks on a skull and we would like our analysis to be resistant to some outlier landmarks. 1.1 General shape Let M k be the space ....

Small, C. G. (1996). The statistical theory of shape. New York: Springer.

.... (left) and hieroglyphs retrieved from database, from [VV99] 2 Approaches Matching has been approached in a number of ways, including tree pruning [Ume93] the generalized Hough transform or pose clustering [Bal81] Sto87] geometric hashing [WR97] the alignment method [HU87] statistics [Sma96] deformable templates [SP95] relaxation labeling [RR80] Fourier descriptors [Lon98] wavelet transform [JFS95] curvature scale space [MAK96] and neural networks [Gol95] Without being complete, in the following subsections we will describe and group a number of these methods together. 2.1 ....

Christopher G. Small. The Statistical Theory of Shapes. Springer Series in Statistics. Springer, 1996.

....all along) and, furthermore, how it is the only such metric that satisfies certain reasonable symmetries arising from the symmetries of Euclidean distance in the original Cartesian image space. For more on the resulting smooth Riemannian manifold, which is quite a remarkable geometric object see [17]. In this Kendall formulation, which is now standard in all other applications of Procrustes methods, Procrustes distance is not determined as the sum of squares of the Euclidean distances between the two centered and scaled point sets. While it happens to agree (approximately) with the minimum ....

C. Small. The Statistical theory of shape. Springer-Verlag, 1996.

....(1977, 1984, 1989) and F.L. Bookstein (1978, 1986, 1989, 1991) Some references and reviews include Goodall (1991) Le and Kendall (1993) Kent (1994, 1995) Dryden and Mardia (1993) Small (1988) Stoyan et al. 1995) Stoyan and Stoyan (1994) and Mardia (1995) Two recent books on the topic are Small (1996) and Dryden and Mardia (1997) In Section 2 we describe the matching of two configurations using regression, making connections with general shape spaces and shape distances. In particular, we consider shape matching and affine shape matching. A particular application of matching electrophoresis ....

Small, C. G. (1996). The statistical theory of shape. Springer, New York.

....to look. Obviously we could compare the volume of a large number of standard brain regions, but this might not detect small local changes in the shapes of the regions. Some sort of overall shape analysis is clearly more desirable. Traditional shape analysis methods in the statistics literature (Small, 1996) rely on landmarks, but in the human brain such landmarks are di cult to identify reliably. Instead, researchers are now assessing brain shape through the continuous 3D deformations required to map the brain to an atlas standard. In e ect, this provides us with continuous landmark data, which in ....

Small, C.G. (1996) The Statistical Theory of Shapes. Springer, New York.

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C. G. Small, The Statistical Theory of Shapes, Springer-Verlag, New York, 1996.

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C.G. Small. The statistical theory of shape. Springer-Verlag, Inc., 1996.

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S. Small, editor. The statistical theory of shape. Springer, 1996.

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C. Small, The Statistical Theory of Shape. Springer, New York, 1996.

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C.G. Small. The Statistical Theory of Shape. Springer-Verlag, 1996.

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Small, C. (1996). The Statistical Theory of Shapes. Springer series in statistics. Springer.

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C.G. Small, The Statistical Theory of Shape, Springer, New York, 1996.

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C. G. Small. The Statistical Theory of Shape. Springer, 1996.

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C.G. Small. The Statistical Theory of Shapes. Springer, 1996.

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C.G. Small. The Statistical Theory of Shape. Springer-Verlag, 1996.

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Small, C. (1996). The Statistical Theory of Shapes. Springer series in statistics. Springer.

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