Bayesian Geodesic Regression on Riemannian Manifolds (BGRM)

In this paper, we develop a generalized Bayesian geodesic regression on Riemannian manifolds, termed BGRM model. it can estimate the relationship between an independent scalar variable and a dependent manifold-valued random variable.

• By introducing a prior to the geodesic regression model, we can automatically select the number of relevant dimensions by driving unnecessary tangent vectors to zero.

• BGRM model is fully generative. The unnecessary dimensionality of the subspace will be automatically killed, and the principal models of variation can reconstruct shape deformation of individuals.

• We first validate our model using 2D and 3D synthetic data. We then use the human corpus callosum and mandible data to show the predicted shapes using our model. Our results indicate that the BGRM model shows reasonable shape variations with the increasing of age in a much lower-dimensional subspace.

Comparison between linear regression and geodesic regression

Results

(a): Geodesic regression on sphere using BGRM and PPGA model. The blue line is the ground truth geodesic. The red line is the estimated geodesic of BGRM model. The green line is the estimated geodesic of PPGA model. (b): The comparison of estimated geodesic of BGRM and PCA model. The red line is the estimated geodesic of BGRM in (a).

The shape variations of pentagon and reduced dimensionality using BGRM model. (a): The color from blue to red is the corresponding x of each shape. The shape shrinks with the increasing of x. (b): BGRM model automatically selects the first two dimensionalities out of 52 in total.

The regression results comparison of BGRM (a) and BLR (b) model using corpus callosum data. The estimated shapes are shown as the sequence from 1 (cyan) to 100 (pink). The color bar indicates the age in years. (c): The reduced dimensionality of BGRM model.

(a): The estimated human mandible shape using BGRM; (b): The comparison of original dimensionality and reduced dimensionality of BGRM model.

[Paper]

[Bibtex]

[Presentation]

Mixture Probabilistic Principal Geodesic Analysis

In this paper, we derive a mixture of PGA models as a natural extension of PPGA (MPPGA), where all model parameters including the low-dimensional factors for each data cluster is estimated through the maximization of a single likelihood function.

• Our model leads to a unified algorithm that well integrates soft data clustering and principal subspaces estimation on general Riemannian manifolds;

• In contrast to the two-stage approach mentioned above, our model explicitly considers the reconstruction error of principal modes as a criterion for clustering tasks;

• Our model provides a more powerful way to learn features from data in non-Euclidean spaces with multiple subpopulations.

Example MPPGA model with four clusters

Results

The comparison of our model MPPGA/MBPGA with K-means-PCA and MPPCA (after being projected from Eucliean space onto the sphere). We have three clusters marked in green, blue, and black. Yellow lines: ground truth geodesics; Red lines: estimated geodesics.

Corpus callosum shape variations (healthy k1 vs. control k2) along the first principal geodesic estimated by our model MPPGA and MBPGA.

Eigenvalues estimated by MPPGA/ MBPGA on corpus callosum data

2D examples of mandible shape data and shape variations (male vs. female) along the first principal geodesic estimated by MBPGA model.

[Paper]

[Bibtex]