Projects:GeodesicShapeRegression

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Ongoing Work (Updated 10/2014)

Geodesic Regression for Anatomical Shape Complexes

Shape regression is of crucial importance for statistical shape analysis. It is useful to find correlations between shape configuration and a continuous scalar parameter such as age, disease progression, drug delivery, or cognitive scores. When only few follow-up observations are available, regression is also a necessary tool to interpolate between data points and provide a scenario of continuous shape evolution over the parameter range. Longitudinal studies also require to compare such regressions across different subjects.

Multi-Object Shape Complexes

Medical imaging data is available in many formats. From volumetric imaging data, anatomical structures are extracted. The structures can be represented in several ways (in 2D or 3D): triangular meshes for surfaces, landmarks, point clouds, curves, etc. It is important to consider image data in anatomical context, which motivates regression on image and shape data in different combinations (a multi-object complex).

Geodesic Growth Model

The geodesic regression model leverages image and shape data together to estimate a single time-dependent deformation of the ambient space. By analogy with linear regression, we estimate a baseline shape configuration (intercept) and deformation parameters (slope) in order to minimize the distance between the model and the observed shapes.

The deformations in our model are diffeomorphisms (smooth, one-to-one, and invertible transformation of space) which act on shapes embedded in the ambient 2D or 3D space. The group of diffeomorphisms forms an infinite dimensional manifold. This is the space in which our model is geodesic, a minimal energy curve on the manifold where each point along the curve is a diffeomorphism.

We call this continuous curve of diffeomorphisms a geodesic flow of diffeomorphisms, which is conveniently parameterized through a tangent space representation. The geodesic is fully described by a vector field in Euclidean space (the tangent space at a point on the manifold), called the initial momenta. From initial momenta, we can travel along the geodesic and construct diffeomorphisms by solving a system of coupled ODEs. The flow of diffeomorphisms act continuously on the embedded shapes, starting with the baseline shape configuration, resulting in continuous shape trajectories.

Therefore, the model is fully parameterized by an initial baseline shape configuration (intercept) and initial momenta (slope). Model estimation consists of estimating a baseline shape configuration, initial momenta as well as their spatial locations, called control points. The control point formulation allows momenta to be located in areas which undergo the most dynamic changes, rather than being restricted by any particular shape representation, located at the voxels of the image for instance. A graphical overview of geodesic regression is shown in Fig. 1.

Fig. 1. Conceptual overview of geodesic regression on multi-object complexes containing both image and shape data. The framework estimates parameters at t = 0 which consist of the baseline image I0 and shape X0 along with the deformation model parameterized by control points c0 and initial momenta α0 such that overall distance between the deformed objects and the observations are minimal.

For details on the mathematical formulation and estimation of model parameters given observed data, see [1,2,3].

Literature

[1] Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic regression of image and shape data for improved modeling of 4D trajectories. IEEE International Symposium on Biomedical Imaging (ISBI '14)

[2] Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic image regression with a sparse parameterization of diffeomorphisms. Geometric Science of Information (GSI '13). LNCS vol 8085, pp. 95-102. (2013)

[3] Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic shape regression in the framework of currents. Proc. of Information Processing in Medical Imaging (IPMI '13). Vol 23, pp. 718-729. (2013)

Key Investigators

• Utah: James Fishbaugh, Marcel Prastawa, Guido Gerig
• INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman

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