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Related Concept Videos

Dimensional Analysis01:23

Dimensional Analysis

Dimensional analysis is a powerful tool that is used in physics and engineering to understand and predict the behavior of physical systems. The basic idea behind dimensional analysis is to express physical quantities in terms of fundamental dimensions such as the mass, length, and time. Derived dimensions like the velocity, acceleration, and force are derived from the combinations of these fundamental dimensions.
Dimensional analysis allows us to analyze and compare physical quantities on a...
Dimensional Analysis02:19

Dimensional Analysis

The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that mathematical equations must meet the following two rules. The first rule is that, in an equation, the expressions on each side of the equal sign must have the same dimensions. This is fairly intuitive since we can only add or subtract quantities of the same type (dimension). The second rule states that, in an equation, the arguments of any of the...
Dimensional Analysis03:40

Dimensional Analysis

Dimensional analysis, also known as the factor label method, is a versatile approach for mathematical operations. The main principle behind this approach is: the units of quantities must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex and multi-step calculations involving several different quantities and their units.
Conversion Factors and Dimensional Analysis
The unit...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Acceleration Vectors01:30

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, Δv, therefore the greater the acceleration, the greater the change in velocity over a given time. Since velocity is a vector, it can change in magnitude, direction, or both. Thus acceleration is a change in speed or direction, or both. For example, if a runner traveling at 10 km/h due east slows to a stop, reverses direction, and continues their run at 10 km/h due...
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.

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Related Experiment Video

Updated: May 16, 2026

Assessing Cerebral Autoregulation via Oscillatory Lower Body Negative Pressure and Projection Pursuit Regression
11:26

Assessing Cerebral Autoregulation via Oscillatory Lower Body Negative Pressure and Projection Pursuit Regression

Published on: December 10, 2014

Accelerating MR parameter mapping using sparsity-promoting regularization in parametric dimension.

Julia V Velikina1, Andrew L Alexander, Alexey Samsonov

  • 1Department of Medical Physics, University of Wisconsin-Madison, Madison, Wisconsin, USA.

Magnetic Resonance in Medicine
|December 6, 2012
PubMed
Summary
This summary is machine-generated.

This study introduces parametric Compressed Sensing (p-CS) to accelerate Magnetic Resonance (MR) parameter mapping. The novel method accurately estimates quantitative MR maps from undersampled data, overcoming SNR limitations in parallel MRI.

Keywords:
T1/T2 relaxometrycompressed sensingmcDESPOTmodel-based reconstructionmyelin water fractionparallel MRI

Related Experiment Videos

Last Updated: May 16, 2026

Assessing Cerebral Autoregulation via Oscillatory Lower Body Negative Pressure and Projection Pursuit Regression
11:26

Assessing Cerebral Autoregulation via Oscillatory Lower Body Negative Pressure and Projection Pursuit Regression

Published on: December 10, 2014

Area of Science:

  • Magnetic Resonance Imaging (MRI)
  • Medical Physics
  • Image Reconstruction

Background:

  • MR parameter mapping is crucial for quantitative imaging but suffers from long scan times.
  • Parallel MRI accelerates data acquisition but introduces signal-to-noise ratio (SNR) penalties.
  • Regularization is necessary to mitigate SNR loss and ensure accurate parameter estimation.

Purpose of the Study:

  • To develop a novel regularization strategy for accelerated MR parameter mapping.
  • To improve the accuracy and precision of quantitative MR maps from undersampled data.
  • To overcome the limitations of existing reconstruction methods in quantitative MRI.

Main Methods:

  • Proposed a parametric Compressed Sensing (p-CS) framework incorporating signal smoothness in the parametric dimension.
  • Applied p-CS to undersampled data for MR parameter mapping.
  • Compared p-CS performance against image space total variation and model-based reconstruction.

Main Results:

  • p-CS effectively suppressed noise amplification inherent in parallel MRI.
  • Accurate and precise parametric maps were achieved from undersampled data.
  • Demonstrated favorable performance in variable flip angle T1 mapping compared to existing methods.

Conclusions:

  • The proposed p-CS regularization enables efficient acceleration of quantitative MRI techniques.
  • This method preserves parameter mapping accuracy without relying on explicit analytical signal models.
  • Facilitates quantitative MRI for complex signal models or deviations from analytical predictions.