Genus-one surface registration via teichmüller extremal mapping.

Area of Science:

• Computational geometry
• Medical imaging
• Differential geometry

Background:

• Shape registration is crucial in medical imaging for analyzing anatomical structures.
• Genus-1 surface matching with landmarks is challenging due to uniqueness and bijectivity requirements.
• Conformal transformations naturally model anatomical deformation and growth.

Purpose of the Study:

• To develop a novel algorithm for landmark-based genus-1 surface registration.
• To achieve unique and bijective surface matching minimizing conformality distortion.
• To leverage Teichmüller maps for accurate anatomical structure registration.

Main Methods:

• Proposed an iterative algorithm: Quasi-conformal (QC) iteration.
• Utilized Teichmüller maps for genus-1 surface registration.
• Employed Beltrami coefficients (BCs) to represent diffeomorphisms on covering spaces.

Main Results:

• The QC iteration algorithm efficiently finds Teichmüller extremal mappings.
• The method ensures stable registration of genus-1 surfaces with numerous landmarks.
• Successful application in registering vertebral bones with feature curves.

Conclusions:

• The proposed QC iteration algorithm offers an efficient and stable solution for landmark-based genus-1 surface registration.
• This method advances the field of medical image analysis by enabling precise anatomical shape matching.
• The algorithm's effectiveness is demonstrated through numerical experiments and practical application on vertebral bones.