Robust Diffeomorphic Mapping via Geodesically Controlled Active Shapes

Algorithm 1

Geodesically controlled diffeomorphic segmentation algorithm.

In order to find an initial momentum 𝛼 minimizing (6)

 (1) Initialize with 𝑥(0) being vertices of template 𝑇 and 𝛼=𝛼.

 (2) For the current system, update matrices 𝐽(𝑡) and 𝐽(𝑡)^∗ according to (B.1) and (19).

 (3) Compute the gradient 𝜕𝐸₂/𝜕𝑥^𝛼, for example using (14), (16), or (21).

 (4) Solve system (B.3) backward in time with initial condition 𝜂_𝑥(1)=𝜕𝐸₂/𝜕𝑥^𝛼 and 𝜂_𝛼(1)=0.

 (5) Replace 𝑘_𝑛 by 𝑘_𝑛−𝜖(2𝑘_𝑛/𝜆_𝑛+1/𝜎²𝑢_𝑛⋅𝜂_𝛼(0)), 𝜖 being optimized with a line search.

 (6) Update the initial momentum 𝛼=𝛼+∑^𝐷_𝑛=1𝑘_𝑛𝑢_𝑛, and solve (2) initialized at

    𝛼, to obtain the updated surface 𝑆^𝛼.

 Iterate steps 2 to 6 until numerical convergence or reaching the maximum iteration limit.