Robust Diffeomorphic Mapping via Geodesically Controlled Active Shapes

Algorithm 2

Geodesically controlled diffeomorphic active shapes.

(A) Training.

 (1) Generate a template surface 𝑇 from 𝑁 surfaces 𝑆₁,…,𝑆_𝑁 using the method in [35].

 (2) Map 𝑇 to the 𝑁 surfaces and obtain the initial momenta 𝛼_𝑛,𝑛=1,…,𝑁. Compute the.

    mean momentum 𝛼 and perform PCA in the tangent space as described in Section 2.2.

 (3) Estimate the constant 𝜎², and other terms required for 𝐸₂ (e.g., estimate the density

    function 𝑝_int and 𝑝_ext for interior and exterior of ROI from training images as described in Section 3.4.1).

(B) Segmentation.

 (1) Initialize the template surface 𝑇, and register the target data to it (e.g., through a

    similitude matching, rigid motion × scale).

 (2) Run Algorithm 1 to optimize the initial momentum 𝛼.

 (3) The result of segmentation is the deformed surface 𝑆^𝛼. Apply the inverse of the

    transformation in step 1 if necessary.