Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Second Derivatives and the Shape of a Graph01:29

Second Derivatives and the Shape of a Graph

246
The second derivative of a function provides essential information about a graph's curvature and how it changes over an interval. It helps determine whether a function is concave upward or concave downward and identifies points where the curvature changes. These properties are fundamental in analyzing real-world scenarios, such as changes in road elevation, population growth, and economic trends.A function f(x) is considered concave upward on an interval if its graph lies above all its tangent...
246
Region of Convergence of Laplace Tarnsform01:20

Region of Convergence of Laplace Tarnsform

1.4K
The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
1.4K
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

2.8K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
2.8K
Properties of Laplace Transform-I01:15

Properties of Laplace Transform-I

1.3K
The Laplace transform is a powerful mathematical tool used to convert functions from the time domain into the frequency domain, greatly simplifying the analysis and solution of linear time-invariant systems. This transformation is facilitated by several universal properties: Linearity, Time-Scaling, Time-Shifting, and Frequency Shifting.
The Linearity property is foundational to the Laplace transform. It states that the transform of a linear combination of functions is equivalent to the same...
1.3K
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

4.6K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
4.6K
Properties of Laplace Transform-II01:16

Properties of Laplace Transform-II

658
Time differentiation, convolution, integration, and periodicity are fundamental concepts in analyzing functions and signals over time. Each concept provides a unique perspective on how functions evolve, interact, and repeat, offering essential tools for various scientific and engineering applications.
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...
658

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

CARL: A Framework for Equivariant Image Registration.

Proceedings. IEEE Computer Society Conference on Computer Vision and Pattern Recognition·2026
Same author

Inverse Consistency by Construction for Multistep Deep Registration.

Medical image computing and computer-assisted intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention·2026
Same author

LiftReg: Limited Angle 2D/3D Deformable Registration.

Medical image computing and computer-assisted intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention·2026
Same author

Erratum for: Prediction of Lobar Emphysema Progression with a CT-Based Foundational Model.

Radiology·2026
Same author

Longitudinal FreeSurfer with non-linear subject-specific template improves sensitivity to cortical thinning.

Proceedings of the International Society for Magnetic Resonance in Medicine ... Scientific Meeting and Exhibition. International Society for Magnetic Resonance in Medicine. Scientific Meeting and Exhibition·2026
Same author

Improved, rapid fetal-brain localization and orientation detection for auto-slice prescription.

Proceedings of the International Society for Magnetic Resonance in Medicine ... Scientific Meeting and Exhibition. International Society for Magnetic Resonance in Medicine. Scientific Meeting and Exhibition·2026

Related Experiment Video

Updated: Mar 27, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

687

Laplace-Beltrami Eigenvalues and Topological Features of Eigenfunctions for Statistical Shape Analysis.

Martin Reuter1, Franz-Erich Wolter2, Martha Shenton3

  • 1Dept. of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge; A.A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Boston.

Computer Aided Design
|February 18, 2010
PubMed
Summary

This study introduces spectral shape descriptors using Laplace-Beltrami and Laplace eigenvalues for brain structure analysis. These methods enable efficient, registration-free shape comparison, revealing differences in caudate nuclei between control and schizotypal subjects.

Keywords:
Brain structureCaudate NucleusEigenfunctionsEigenvaluesLaplace-Beltrami SpectraMorse-Smale complexNodal domainsReeb GraphSchizotypal personality disorder

More Related Videos

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

12.0K
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

43.8K

Related Experiment Videos

Last Updated: Mar 27, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

687
Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

12.0K
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

43.8K

Area of Science:

  • Computational anatomy
  • Medical image analysis
  • Differential geometry

Background:

  • Comparing complex shapes, such as brain structures, often requires computationally intensive pre-processing like registration.
  • Isometry-invariant shape descriptors can simplify shape analysis by reducing the need for alignment.
  • Laplace-Beltrami and Laplace eigenvalues offer a spectral approach to shape characterization.

Purpose of the Study:

  • To propose and evaluate surface-based Laplace-Beltrami and volumetric Laplace eigenvalues/eigenfunctions as robust shape descriptors.
  • To demonstrate the application of these spectral measures for comparing 2D surfaces and 3D solids.
  • To investigate their utility in statistical shape analysis of neuroanatomical structures.

Main Methods:

  • Utilized surface-based Laplace-Beltrami and volumetric Laplace eigenvalues and eigenfunctions as spectral shape descriptors.
  • Applied Dirichlet and Neumann boundary conditions, analyzing their properties, particularly in 3D.
  • Performed topological analyses using Morse-Smale complexes and Reeb graphs on eigenfunctions for feature localization.

Main Results:

  • Demonstrated the discriminatory power of 2D and 3D spectral methods on female caudate nuclei populations.
  • Showcased the advantages of Neumann boundary conditions over Dirichlet spectra in 3D shape analysis.
  • Identified novel shape descriptors from eigenfunctions capable of localizing geometric properties and detecting differences via topological features.

Conclusions:

  • Surface-based Laplace-Beltrami and volumetric Laplace eigenvalues provide isometry-invariant shape descriptors suitable for minimal pre-processing.
  • Neumann boundary conditions offer advantages for 3D spectral shape analysis.
  • The integration of topological features of eigenfunctions represents a novel approach for statistical shape analysis in 2D and 3D.