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Related Concept Videos

Linear time-invariant Systems01:23

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Gradient Fields01:27

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...
Significance of the Gradient Vector01:27

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 Applications01:19

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...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

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Related Experiment Video

Updated: Jun 19, 2026

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

Characterizing and correcting gradient errors in non-cartesian imaging: Are gradient errors linear time-invariant

Ethan K Brodsky1, Alexey A Samsonov, Walter F Block

  • 1Department of Radiology, University of Wisconsin-Madison, Madison, Wisconsin 53705, USA. brodskye@cae.wisc.edu

Magnetic Resonance in Medicine
|October 31, 2009
PubMed
Summary
This summary is machine-generated.

Rapid calibration methods for non-Cartesian imaging are sensitive to scanner imperfections. This study shows gradient errors are linear but can change quickly, requiring calibration data matched to scan data for accurate multiecho 3D projection reconstruction (3DPR).

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Sample Drift Correction Following 4D Confocal Time-lapse Imaging
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Sample Drift Correction Following 4D Confocal Time-lapse Imaging

Published on: April 12, 2014

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Last Updated: Jun 19, 2026

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

Sample Drift Correction Following 4D Confocal Time-lapse Imaging
10:04

Sample Drift Correction Following 4D Confocal Time-lapse Imaging

Published on: April 12, 2014

Area of Science:

  • Magnetic Resonance Imaging (MRI)
  • Medical Imaging Physics
  • Biomedical Engineering

Background:

  • Non-Cartesian and rapid MRI sequences are susceptible to scanner imperfections like gradient delays and eddy currents.
  • These imperfections hinder the adoption of advanced imaging techniques and reduce image quality due to k-space trajectory deviations.
  • Existing calibration methods can be time-consuming, impacting clinical workflow.

Purpose of the Study:

  • To examine a rapid gradient calibration procedure for multiecho 3D projection reconstruction (3DPR) acquisitions.
  • To assess the validity of the linear and time-invariant (LTI) assumptions for gradient errors in rapid calibration.
  • To emphasize the importance of using calibration data synchronized with scan data.

Main Methods:

  • A rapid gradient calibration procedure was integrated into every multiecho 3DPR scan.
  • Trajectories for orthogonal gradient axes were measured, and oblique projection trajectories were synthesized as linear combinations.
  • The linearity and time-invariance of gradient errors were investigated.

Main Results:

  • The assumption of linearity for gradient errors was found to be reasonable.
  • Gradient errors were observed to vary over short time scales due to factors like gradient coil temperature.
  • The study confirmed that rapid calibration is feasible but requires careful consideration of error dynamics.

Conclusions:

  • Rapid gradient calibration is effective for 3DPR, but the assumption of time-invariance is not always valid.
  • Gradient errors can fluctuate, necessitating the use of calibration data acquired concurrently with the scan data.
  • Accurate calibration is crucial for maintaining image quality in non-Cartesian MRI techniques.