Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Significance of the Gradient Vector

Significance of the Gradient Vector

A surface defined by a function of two variables can be understood by examining how it changes along specific directions. When one variable is held constant, the surface reduces to a curve that reflects variation in the other variable. For example, fixing one variable and moving parallel to a coordinate axis produces a cross-sectional curve. The slope of this curve at a given point represents how the function changes in that particular direction, providing a measure of local steepness.By...

Gradient Vectors and Their Applications

Gradient Vectors and Their Applications

Every point on a topographical map corresponds to a particular elevation, so the landscape can be modeled as a surface whose height depends on horizontal position. From any given location, a hiker may face infinitely many directions, but only one direction produces the fastest possible increase in elevation. This unique route is called the direction of steepest ascent, and in multivariable calculus, it is represented by the gradient vector of the elevation function.The gradient vector points...

Gradient and Del Operator

Gradient and Del Operator

In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The gradient is a mathematical operator that gives both the magnitude and direction of the maximum spatial rate of change. Consider a person standing on a mountain. The slope of the mountain at any given point is not defined unless it is quantified in a particular direction. For this reason, a "directional derivative" is defined, which is a vector...

Gradient Fields

Gradient Fields

A gradient field is a vector field derived from a scalar field. A scalar field assigns a single numerical value to every point in space, such as temperature, pressure, or electric potential. The gradient field describes how that value changes from point to point. It gives both the direction of the fastest increase and the rate of change in that direction.For a scalar field f(x, y), the gradient is written as\begin{equation*}\nabla f=\left\langle \jfrac{\partial f}{\partial x},\jfrac{\partial...

Maximizing the Directional Derivative

Maximizing the Directional Derivative

The directional derivative is a central concept in multivariable calculus that describes how a function changes at a given point when moving in a specified direction. This direction is represented by a unit vector, ensuring that only the orientation influences the rate of change. By varying the direction, different rates of change can be observed, demonstrating that the directional derivative depends strongly on the chosen direction.The directional derivative is computed using the gradient...

Directional Derivatives

Directional Derivatives

In multivariable calculus, partial derivatives describe how a function changes when movement is restricted to a single coordinate direction. For a surface represented by a function of two variables, one partial derivative measures the slope in the x-direction, while the other measures the slope in the y-direction. Although these quantities are useful for analyzing local behavior, most physical motion does not occur strictly parallel to the coordinate axes. Applications such as fluid flow, heat...

You might also read

Related Articles

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

Sort by

Same author

Cerebrovascular Reactivity Assessment with Breath-Hold Functional MRI in Patients with Moyamoya Angiopathy: Which Time Period to Analyze?

Diagnostics (Basel, Switzerland)·2026

Same author

Test-Retest Reliability and Inter-Scanner Reproducibility of Improved Spinal Diffusion Tensor Imaging.

Diagnostics (Basel, Switzerland)·2025

Same author

Assessing and Improving the Reproducibility of Cerebrovascular Reactivity Evaluations in Healthy Subjects Using Short-Breath-Hold fMRI.

Diagnostics (Basel, Switzerland)·2025

Same author

Comparison of ADMIRE, SAFIRE, and Filtered Back Projection in Standard and Low-Dose Non-Enhanced Head CT.

Diagnostics (Basel, Switzerland)·2025

Same author

Measuring cerebrovascular reactivity with breath-hold fMRI in patients with Moyamoya angiopathy: MR perfusion based delay correction significantly improves agreement to [<sup>15</sup>O]water PET.

Neuroradiology·2025

Same author

Fiber architecture in the human ventromedial striatum and its relation with the bed nucleus of the stria terminalis.

PloS one·2025

Search research articles

Related Experiment Video

Updated: Jun 26, 2026

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

Comparison of gradient encoding directions for higher order tensor diffusion data.

Sarah C Mang¹, Daniel Gembris, Wolfgang Grodd

¹Section Experimental MR of CNS, Diagnostic and Interventional Neuroradiology, University Hospital Tuebingen, Tuebingen, Germany. sarah.mang@med.uni-tuebingen.de

Magnetic Resonance in Medicine

|January 24, 2009

Summary

This summary is machine-generated.

Advanced diffusion models require optimized gradient encoding schemes. A force-minimizing scheme demonstrated superior performance for higher-order tensor estimations in diffusion MRI.

More Related Videos

Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging

Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging

Published on: November 8, 2012

Diffusion Imaging in the Rat Cervical Spinal Cord

Diffusion Imaging in the Rat Cervical Spinal Cord

Published on: April 7, 2015

Related Experiment Videos

Last Updated: Jun 26, 2026

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging

Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging

Published on: November 8, 2012

Diffusion Imaging in the Rat Cervical Spinal Cord

Diffusion Imaging in the Rat Cervical Spinal Cord

Published on: April 7, 2015

Area of Science:

Medical Imaging
Diffusion MRI
Computational Neuroscience

Background:

Higher-order tensors offer advanced representation of non-Gaussian diffusion.
Diffusion weighted image acquisition requires specific gradient encoding schemes for advanced models.

Purpose of the Study:

To evaluate the suitability of different gradient encoding schemes for higher-order tensor models.
To identify optimal gradient encoding for advanced diffusion MRI analysis.

Main Methods:

Investigated six different gradient encoding scheme types.
Utilized condition number of the estimation matrix and signal deviation on simulated data as quality measures.
Compared encoding schemes for higher-order tensor estimations.

Main Results:

A specific force-minimizing gradient encoding scheme yielded the best results.
Evaluated scheme performance using two distinct quality metrics.
Identified superior gradient encoding for advanced diffusion tensor imaging.

Conclusions:

Gradient encoding schemes significantly impact higher-order tensor estimation accuracy.
Force-minimizing schemes are well-suited for advanced diffusion MRI models.
The findings guide the selection of optimal gradient encoding for diffusion tensor imaging.