Visualizing Shape Deformations with Variation of Geometric Spectrum.

Area of Science:

• Computational geometry
• Differential geometry
• Medical imaging analysis

Background:

• Quantifying non-isometric deformations in 3D surfaces is crucial for various scientific fields.
• Existing methods often struggle with multi-scale and non-isometric shape variations.
• Spectral geometry offers a promising avenue for robust shape analysis.

Purpose of the Study:

• To present a novel spectral geometry approach for quantifying and visualizing non-isometric deformations of 3D surfaces.
• To determine multi-scale, non-isometric deformations by analyzing variations in the Laplace-Beltrami spectrum.
• To validate the method's accuracy and effectiveness using synthetic and real-world data.

Main Methods:

• Mapping two manifolds using spectral geometry.
• Calculating multi-scale, non-isometric deformations via Laplace-Beltrami spectrum variation.
• Employing a scale function defined on each vertex, derived through quadratic programming and spectrum variation theorems.
• Solving for the scale function by integrating derivations from iterative steps.

Main Results:

• The method successfully quantifies non-isometric deformations by computing a scale function.
• Experiments on synthetic data demonstrate the method's accuracy.
• Applications to epilepsy and Alzheimer's patient data show effective quantification of hippocampal shape variations.
• Comparison with non-rigid Iterative Closest Point (ICP) and voxel-based methods highlights the proposed approach's advantages.

Conclusions:

• The proposed spectral geometry method provides a robust and accurate way to quantify non-isometric 3D surface deformations.
• The technique is effective for analyzing shape variations in medical imaging, including patient-specific brain data.
• This approach offers significant advantages over traditional spatial registration methods for deformation analysis.