Technical SectionCovariance matrix of a shape population: A tale on spline setting
Graphical abstract
Introduction
The growing use of 3D shape acquisition tools and rapid advancement of shape modeling techniques have led to increasing interest in shape modeling of a population of objects. Computing the covariance matrix of a population of shapes is essential for identifying shape variation across the population and building statistical shape models (SSMs). This statistical shape model provides a compact characterization of the shape variability pattern in a set of shapes (training set). Computing the covariance matrix for modeling a population of objects has witnessed growing applications including image segmentation [1], [2], facial recognition [3], computer animation [4], medical diagnosis [5], [6], patient-specific modeling [7], [8], mass customization [9] and biological growth modeling, etc. Shape variation is usually extracted through a statistical technique, Principal Component Analysis (PCA). This process is performed by eigenvalue decomposition of the covariance matrix. The eigenvectors of the decomposition characterize the geometric variation pattern, and the corresponding eigenvalues represent the amount of such shape variation. The covariance matrix is usually computed from a discrete set of points (a.k.a. landmarks) sampled on each shape, leading to the classical Point Distribution Model (PDM) [10]. Fig. 1(a) displays discrete points sampled on a shape and such a point set from a collection of shapes displayed in Fig. 1(b) are then used to compute the covariance matrix.
In this paper, we propose the computing of the covariance matrix on a spline based continuous representation of shapes. More specifically, it involves the use of B-splines as a parametric shape representation (Fig. 1(c) and (d)). It also involves the use of B-splines to represent the reparameterization of parametric curves and surfaces. Note that B-spline based shape representation and B-spline based reparameterizations are independent and they can have different numbers of control points, knot vectors and degrees. The use of B-splines for shape representation and for reparameterization is motivated by the following considerations. B-splines provide a compact and flexible parametric shape representation. B-splines are also capable of representing various reparameterization functions due to their local modification property. More importantly, the dual use of B-splines makes it straightforward to compute directly from the formulations of the covariance matrix of continuous shapes. In these formulations of the covariance matrix of a shape population, the input for each matrix entry is two continuous shapes, rather than discrete points on the two shapes. We refer to these formulations as continuous formulations. Although the continuous formulations [11], [12], [13], [14] have been developed for some time by extending the discrete form from finite number of points to infinite number of points, the forms themselves are deemed hard to compute. In [15], the entries in the continuous form of the covariance matrix were approximated by finite points weighted by the area on the mean shape, and it was found that such an approximated continuous form is effective in ensuring faithful sampling of shapes and preventing the sampled point from moving away from “difficult” areas on the shapes.
In this paper, we apply our B-spline representations into two continuous formulations of the covariance matrix [13], [14]. We indicate that, with B-splines, these continuous forms can be either computed in an analytical form or approximated through numerical integration. Two common numerical integration scheme-based approaches, the mid-point and Gauss quadratures, are given for computing the covariance matrix. We also demonstrate that, with these two continuous formulations, one is amendable for analytical computing (without discretization), but is parameterization-dependent and the other is parameterization-independent and more stable in computing the shape correspondence for building SSM. We apply the B-spline based continuous formulations of the covariance matrix in the optimization of the correspondence across a shape population. This is achieved by reparameterizing each shape to minimize the description length (DL) of the shape population.
The contribution of this paper is that, with B-splines, these continuous forms of covariance matrix are readily computable. Both the closed-form and the quadrature based numerical procedure are efficient and accurate in the sense it would take a large number of sampled points with the usual discrete landmark points based formulation to converge to the same covariance matrix. When data points are parameterized with the chord length method in B-spline fitting, the resulting covariance matrix does not depend on the data sampling scheme.
The remainder of this paper is organized as follows. In Section 2, we review the usual discrete formulation for computing the covariance matrix and the continuous formulations from the literature. In Section 3, we apply the B-spline-based shape representation and reparameterization into two continuous formulations and derive closed-form and efficient quadrature methods for computing the covariance matrix. Section 4 presents closed-form and quadrature methods for computing the covariance matrix from reparameterized B-spline curves/surfaces. In Section 5, we briefly present the optimization formulation for minimizing the description length of a shape population with the computed covariance matrix. In Section 6, we present our numerical results, for which we compare the results from the two formulations and then apply the continuous formulation of the covariance matrix in optimizing shape correspondence for building SSMs. This paper is concluded in Section 7.
Section snippets
Review: discrete and continuous formulations of the covariance matrix
In this section, we briefly summarize the covariance matrix formulations that have been proposed in the literature. They can be categorized into discrete and continuous formulations.
Covariance matrix of spline curves and surfaces
For the above continuous formulations of the covariance matrix, (10), (12), we demonstrate that they can be computed efficiently and accurately with Bézier/B-spline based shape representation, either in closed-form or with quadrature methods.
Reparameterization via B-splines R(u)
In computing the continuous formulations of the covariance matrix of a shape population, either analytically or approximately, we have adopted the B-spline representation of the shapes. To study the influence of shape parameterization (i.e. how points are sampled or distributed) on the covariance matrix, we present a method below for computing the covariance matrix of shapes after B-spline reparameterization, i.e., where is the reparameterization function represented again in
Shape correspondence optimization via reparameterization
With the B-spline representation of reparameterization functions and the diffeomorphic conditions (32), (34) presented in 4.1.1 Reparameterization of curves, 4.1.2 Reparameterization of surfaces, we thus have the following optimization formulation for using B-spline based reparameterization for manipulating shape correspondence in B-spline curves/surfaces:In this formulation, b is the set of
Numerical examples
In this section, we compare the numerical results of computed covariance matrices from two continuous formulations under different discretization resolutions. To compare the analytical form of the covariance matrix (10) from continuous formulations and their approximations through mid-point or Gauss quadrature and their convergence, we compare the covariance matrix norm and its largest eigenvalue. The matrix norm used is the Frobenius norm of an matrix ()
Conclusion
In this paper, we have presented methods for accurately and efficiently computing continuous formulations of the covariance matrix in which B-splines are used both as a shape representation and as a form of reparameterization. We have indicated, with B-spline representation of the shapes, the formulation I is amenable to analytical computing without sampling or discretization. Numerical approaches based on mid-point and Gauss quadrature are developed for approximating both continuous
Acknowledgments
This work is supported in part by AFOSR Grant FA9550-12-1-0206 and NSF Grant 0900597.
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