The noncentral Bartlett decompositions and shape densities

In shape analysis, it is usually assumed that the matrix X:N-K of the co-ordinates of landmarks in K is isotropic Gaussian. Let Y:(N-1)-K be the centered matrix of landmarks from X so that Y ~ N([mu], [sigma]2I). Let Y=TT be the Bartlett decomposition of Y into lower triangular, T, and orthogonal, [Gamma], components. The matrix T denotes the size-and-shape of X. For N-1>=K (the usual case in multivariate analysis is N-1<K) and rank [mu]=0 or 1, we find the joint distribution of T by the method of random orthogonal transformations. We deduce the marginal distribution of the Procrustes distance between two shapes, and hence the joint shape density, which is the joint density of T/T. For rank [mu]>=2 the distribution of T is related to the noncentral Wishart distribution, an integral over the orthogonal group, [Gamma]=±1. To derive the distribution of T when [Gamma]=+1, so that [Gamma] is a rotation, we investigate extending the method of random orthogonal transformations, especially when rank [mu]=K>=2. The case K=2 is tractable, but the case K=3 is not. However, by a direct method we obtain the shape density when rank [mu]=K=3 and [Gamma]=1 as a computable double-series of trigonometric integrals. However, for K>3, the density is not tractable which is not surprising in view of the same problem for the standard non-central Wishart distribution.