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Parametric Surfaces

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...

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Updated: Jul 7, 2026

Three-Dimensional Shape Modeling and Analysis of Brain Structures

Three-Dimensional Shape Modeling and Analysis of Brain Structures

Published on: November 14, 2019

Topology preserving deformable image matching using constrained hierarchical parametric models.

O Musse¹, F Heitz, J P Armspach

¹Laboratoire des Sciences de l'Image de l'Informatique et de la télédetection, Strasbourg 67085, France. olivier.musse@ensps.u-strasbg.fr

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society

|February 6, 2008

Summary

This study introduces a new method for deformable image matching that guarantees topology preservation. The approach ensures accurate mapping, preventing distortions in medical image analysis.

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Area of Science:

Medical image analysis
Computer vision
Computational anatomy

Background:

Deformable image matching is crucial for comparing medical images.
Maintaining topology (e.g., preventing holes) during image deformation is a significant challenge.
Existing methods may fail to preserve topology under large deformations.

Purpose of the Study:

To present a novel approach for topology-preserving deformable image matching.
To ensure the mapping is globally one-to-one, preserving image structures.
To develop a fast and robust algorithm for tracking large nonlinear deformations.

Main Methods:

A constrained hierarchical parametric approach is utilized.
Deformation is parameterized across multiple scales using multiresolution subspaces.
The Jacobian of the transformation is controlled to ensure global one-to-one mapping.
A fast nonlinear constrained optimization algorithm is developed.

Main Results:

The proposed method successfully preserves topology in deformed images.
The algorithm effectively tracks large nonlinear deformations.
Experimental results demonstrate efficacy on both simulated and real medical image data.

Conclusions:

The novel constrained hierarchical parametric approach ensures topology preservation in deformable image matching.
This method offers a robust solution for analyzing complex anatomical changes in medical imaging.
The developed algorithm is efficient and applicable to real-world medical image analysis tasks.