Image analysis – Image transformation or preprocessing – Changing the image coordinates
Reexamination Certificate
2000-02-17
2004-11-09
Do, Anh Hong (Department: 2624)
Image analysis
Image transformation or preprocessing
Changing the image coordinates
C382S291000, C382S293000, C382S295000
Reexamination Certificate
active
06816632
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The field of art to which this invention relates is motion analysis, and more particularly to geometric methods for analyzing shape changes resulting from repetitive motions using Elliptic Fourier Analysis (EFA).
2. Description of the Related Art
Geometric morphometric methods characterize the shape of a configuration of landmarks in such a way as to retain all of the relative spatial information encoded in the original data throughout an analysis (See, Slice, D. E., F. L. Bookstein, L. F. Marcus, and F. J. Rohlf. 1996. A glossary for geometric morphometrics. In L. F. Marcus, M. Corti, A. Loy, G. J. P. Naylor and D. E. Slice (eds.),
Advances in Morphometrics
, pp. 531-551. Plenum, N.Y.). Often this information is represented as a matrix of the coordinates of points, called landmarks, that are presumed to capture the shape of the structure and to be homologous across specimens within a study. Such raw data cannot be analyzed directly since the values of the coordinates are a function of the location and orientation of the specimen, with respect to the digitizing device, at the time the data were collected. Geometric morphometric analyses register all specimens to a common coordinate system to remove such effects. This variation is usually further decomposed into components of size and shape differences.
Specifically, let X
i
be a p×k matrix of the k coordinates of the p landmarks describing the shape of the ith specimen. The prior art geometric morphometric analysis of such data usually begins by fitting the model:
X′
i
=&agr;
i
(
X
C
+D
i
)
H
i
+1&tgr;
i
where &agr;
i
is a scalar scale factor, H
i
is an orthonormal k×k rotation matrix, 1 is a p×1 vector of 1s, &tgr;
i
is a 1×k vector of translation terms, and D
i
is a p×k matrix of deviations from a p×k consensus configuration X
C
. The parameters of the model are estimated for each specimen so as to minimize the trace of (X′
i
−X
C
)
t
(X′
i
−X
C
) subject to the constraints that X′
i
t
1=0, where 0 is a k×1 vector of 0s, and tr(X′
i
t
X′
i
)=1. The constraints simply mean that each X′
i
is centered at the origin and that the sum of squared distances of the landmarks from the origin equals unity. The criterion being minimized is the sum of squared Euclidean distances from each landmark on X′
i
to the corresponding landmark on X
C
. Since X
C
is unknown, it and the parameters of the model are estimated by an iterative process using one of the X
i
as an initial estimate of X
C
(See, Gower, J. C. 1975. Generalized Procrustes analysis.
Psychometrika
, 40:33-51; and Rohlf, F. J., and D. E. Slice. 1990. Extensions of the Procrustes method for the optimal superimposition of landmarks.
Systematic Zoology
, 39:40-59).
The original data required pk parameters to represent variation in each of the k coordinates at each of the p points in a configuration. The fitting and associated constraints, however, impose a certain structure on the data that cannot be ignored during subsequent analyses. One degree of freedom in the sample variation is lost due to the estimation of the scale parameter, k degrees of freedom are lost due to translation, and k(k−1)/2 due to rotation. Though the superimposed data are still represented by pk values, their variation has, at most, pk-k-k(k−1)/2−1 degrees of freedom. Furthermore, this reduced shape space is non-Euclidean (See, Kendall, D. G. 1984. Shape manifolds, Procrustean metrics, and complex projective spaces.
Bull. Lond. Math. Soc
., 16:81-121; and Kendall, D. G. 1985. Exact distributions for shapes of random triangles in convex sets.
Adv. Appl. Probab
., 17:308-329), i.e. it is curved, thus precluding the direct application of standard linear statistical analyses. To address the latter problem, data are usually projected into a linear space tangent to shape space at the point of X
C
. Such an operation provides the best linear approximation to the curved shape space (See, Rohlf, F. J. 1996. Morphometric spaces, shape components, and the effects of linear transformations. In L. F. Marcus, M. Corti, A. Loy, G. J. P. Naylor and D. E. Slice (eds.),
Advances in Morphometrics
, pp. 117-129. Plenum, N.Y.).
A collection of triangles in two dimensions provides a simple, low-dimensional example. Each triangle can be completely described by the x,y coordinates of its three vertices and represented as a point in a p×k=2×3=6 dimensional space of all triangles. Superimposing the sample on their mean configuration using the above procedure results in a loss of k-k(k−1)/2−1=4 degrees of freedom, leaving two degrees of freedom for shape variation. For triangles, this curving 2D space embedded within the original six dimensional space can be visualized as the surface of a hemisphere centered at the origin. For small amounts of variation, the projection of the scatter onto a tangent plane touching the surface of this hemisphere at the point representing X
C
can be used as a linear approximation of the variation in shape space. Such an approximation of a curving surface by a planar one is analogous to using points on a flat map to represent positions on the curving surface of the earth. For configurations of more 2D points, shape space is a complex (p−2)-torus. The situation for configurations of higher dimension is comparable, but the structure of the shape space and the associated mathematics are much more complicated (See, Goodall, C. R. 1992. Shape and image analysis for industry and medicine. Short course. University of Leeds, Leeds, UK).
Geometric morphometrics provides a sophisticated suite of methods for the processing and analysis of shape data (see, Bookstein, F. L. 1991
. Morphometric Tools for Landmark Data. Geometry and Biology
. Cambridge University Press, New York; Rohlf, F. J., and L. F. Marcus. 1993. A revolution in morphometrics.
Trends in Ecology and Evolution
, 8:129-132; and Marcus, L. F., and M. Corti. 1996. Overview of the new, or geometric morphometrics. In L. F. Marcus, M. Corti, A. Loy, G. J. P. Naylor and D. E. Slice (eds.),
Advances in Morphometrics
, pp. 1-13. Plenum, N. Y.). Most geometric morphometric analyses to date have been oriented toward assessing group differences and the covariation of shape with extrinsic variables (see Marcus, L. F., M. Corti, A. Loy, G. J. P. Naylor, and D. E. Slice. 1996. Advances in Morphometrics,
NATO ASI Series A: Life Sciences
, pp. xiv+587. Plenum Press, N.Y. for numerous examples). For many questions, though, such “static” analyses are inadequate. The study of feeding, flying, walking, swinging, or swimming, for instance, require methods capable of characterizing dynamic, repetitive changes in the shape of a single set of structures within each specimen.
For practical applications, complex motions involving many landmarks can be analyzed. However, a simple data set can be used to illustrate the inadequacy of standard methods and the efficacy of the new procedure for the analysis of motion-related shape change.
FIGS. 1A-1C
show triangular configurations associated with three individuals of three hypothetical “species” used to model the changes in the relative locations of points associated with a particular motion. It is hypothesized that during the course of the motion, point C moves with respect to points A and B. Which points move is irrelevant in shape analysis, and, in fact, cannot be determined from the shape differences alone.
In the species
brevistrokus
, shown in
FIG. 1A
, the “idealized” motion has point C moving a short distance back and forth at a right angle to the line segment connecting points A and B. In the second species,
longistrokus
, shown in
FIG. 1B
, point C moves in approximately the same direction, but the range of motion is twice that of
brevistrokus
. In the last species,
elliptistrokus
, shown in
FIG. 1C
, the motion of point C is in the same ge
Do Anh Hong
Wake Forest University Health Sciences
LandOfFree
Geometric motion analysis does not yet have a rating. At this time, there are no reviews or comments for this patent.
If you have personal experience with Geometric motion analysis, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Geometric motion analysis will most certainly appreciate the feedback.
Profile ID: LFUS-PAI-O-3330828