Estimating topology preserving and smooth displacement fields

Area of Science:

• Computational geometry
• Image analysis
• Scientific computing

Background:

• Displacement fields are crucial for quantifying shape transformations in various scientific domains.
• Existing methods often struggle with preserving topological consistency and smoothness, especially with noisy data.
• Robust morphometric analysis requires accurate and reliable displacement field estimation.

Purpose of the Study:

• To introduce a novel method for enforcing topology preservation and smoothness in displacement fields.
• To develop a framework for estimating the closest topology-preserving displacement field.
• To extend topology preservation to adaptive smoothing via Jacobian constraints.

Main Methods:

• Analysis of topology preservation conditions for 2D and 3D discrete displacement fields.
• Formulation of finding the closest topology-preserving field using its gradients.
• Solution via a cyclic projections framework.
• Adaptive smoothing formulated as an extension of topology preservation using Jacobian constraints.

Main Results:

• The proposed technique efficiently estimates topology-preserving displacement fields from noisy observations.
• The method demonstrates reliability in handling data that initially lacks topology preservation.
• Adaptive smoothing enhances robustness of morphometric analyses without losing critical morphological details.

Conclusions:

• The developed method provides a fast and reliable approach for topology-preserving displacement field estimation.
• The adaptive smoothing component significantly improves the robustness of noise-affected morphometric analyses.
• This work offers a valuable tool for accurate shape analysis in scientific research.