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What is Variation?01:14

What is Variation?

Apart from the measures of central tendency, distribution, outliers, and the changing characteristics of data with time, an important characteristic of any data set is its variation or spread. In some data sets, the data values are concentrated closely near the mean; in others, the data values are more widely spread out from the mean.
The range, standard deviation, standard error, and variance are the different measures of variation.
Range: The range is the difference between its maximum and...
Variation01:19

Variation

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation, which is the square root of variance.
When independent and dependent variables are plotted on a scatter plot, the slope of a line is a value that describes the rate of change between the two...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
Coefficient of Variation01:10

Coefficient of Variation

The coefficient of variation measures the dispersion of the data points or distribution around the mean. Using the coefficient of variation, we can compare two data series with drastically different means or different units of measurement. The coefficient of variation for a sample and a population is expressed as a percentage of the ratio of standard deviation to the mean.
The coefficient of variation is a practical statistical tool in finance. It allows investors to assess the volatility or...
Variation: Normal Distribution, Range, and Standard Deviation02:32

Variation: Normal Distribution, Range, and Standard Deviation

In the field of psychology, there are several ways to organize measurements of a trait, feature, or characteristic (i.e., variables). Qualitative data, such as ethnicity, can be tabulated into a frequency count to provide information about the proportion, as well as the variety of groups in a sample or population. On the other hand, researchers can perform a wider set of calculations on quantitative data. The mean, mode, and median, for instance, are central tendency measures to identify a...
Change of Variables in Multiple Integrals01:30

Change of Variables in Multiple Integrals

Multiple integrals are often used to evaluate areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, however, the original region may have complicated curved boundaries when expressed in Cartesian coordinates. These complex boundaries can make the limits of integration difficult to describe and the overall calculation cumbersome. To simplify the evaluation process, a change of variables is introduced that transforms the...

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Related Experiment Video

Updated: Jul 7, 2026

Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities
07:13

Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities

Published on: October 27, 2023

Total variation regularization of matrix-valued images.

Oddvar Christiansen1, Tin-Man Lee, Johan Lie

  • 1Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Bergen, Bergen 5008, Norway.

International Journal of Biomedical Imaging
|February 8, 2008
PubMed
Summary
This summary is machine-generated.

This study extends the total variation restoration model to diffusion tensor images (DTIs). The new method ensures tensor positive definiteness, crucial for accurate DTI regularization and analysis.

Related Experiment Videos

Last Updated: Jul 7, 2026

Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities
07:13

Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities

Published on: October 27, 2023

Area of Science:

  • Medical Imaging
  • Image Processing
  • Computational Neuroscience

Background:

  • The Rudin, Osher, and Fatemi (1992) total variation model is a cornerstone for image restoration.
  • Diffusion Tensor Imaging (DTI) requires specialized regularization techniques due to its matrix-valued nature.
  • Existing color total variation models (Blomgren and Chan, 1998) provide a foundation for extending to higher-order tensor data.

Purpose of the Study:

  • To generalize the total variation restoration model for matrix-valued data, specifically Diffusion Tensor Images (DTIs).
  • To ensure positive definiteness of diffusion tensors during the regularization process, a critical requirement for DTI analysis.
  • To evaluate the performance of the proposed model on synthetic and real-world 3D human brain DTI data.

Main Methods:

  • Generalization of the total variation restoration model to handle matrix-valued data.
  • Implicit representation of the diffusion tensor D as D = LL(T), where L is the variable.
  • Regularization flow applied to the elements of L to maintain tensor positive definiteness.
  • Numerical experiments on synthetic and 3D human brain DTI datasets.

Main Results:

  • Successful generalization of the total variation model to DTI data.
  • Demonstrated ability to maintain positive definiteness of diffusion tensors during regularization.
  • Quantitative evaluation of the model's performance on both synthetic and real DTI datasets.

Conclusions:

  • The proposed generalized total variation model is effective for regularizing Diffusion Tensor Images.
  • The method ensures essential tensor positive definiteness, improving DTI data integrity.
  • The approach shows promise for enhancing the analysis of neuroimaging data like DTIs.