Local and global energies for shape analysis in medical imaging

Area of Science:

• Computational geometry
• Solid mechanics
• Differential geometry

Background:

• Previous work introduced the TPS (Thin Plate Spline) space for shape statistics, equipped with a Riemannian metric and a flat, torsionful connection.
• This connection enables parallel transport of deformations, preserving Γ-energy (related to elastic strain energy) and decomposing deformations into spherical, deviatoric, and non-affine components.

Purpose of the Study:

• To introduce a new approach for calculating body bending energy within physical boundaries.
• To propose a novel metric for preservation during TPS direct transport, enhancing the mechanical coherence of shape change transport.
• To further analyze the geometry of TPS space for better representation of energy relationships.

Main Methods:

• Development of a new procedure to calculate body bending energy.
• Proposal and application of a new metric to be preserved during TPS direct transport.
• Geometric analysis of TPS space to relate Γ-energy, strain energy, and bending energy densities.

Main Results:

• A method to calculate body bending energy restricted to physical object boundaries is presented.
• The proposed new metric allows for shape change transport that is more consistent with the mechanical meaning of deformation.
• The study discusses the TPS space geometry to clarify the relationships between different energy densities.

Conclusions:

• The new approach provides a more mechanically grounded method for analyzing bending energy in shape deformations.
• Preserving a specific metric during TPS transport improves the physical interpretation of shape changes.
• Further geometric insights into TPS space aid in understanding complex energy distributions in deformed bodies.