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Related Concept Videos

Properties of DTFT II01:24

Properties of DTFT II

In the study of discrete-time signal processing, understanding the properties of the Discrete-Time Fourier Transform (DTFT) is crucial for analyzing and manipulating signals in the frequency domain. Several properties, including frequency differentiation, convolution, accumulation, and Parseval's relation, offer powerful tools for signal analysis.
The frequency differentiation property is illustrated by considering a DTFT pair and differentiating both sides with respect to ω. Multiplying by j...
Properties of DTFT I01:24

Properties of DTFT I

In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...
Discrete-time Fourier transform01:26

Discrete-time Fourier transform

The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.
One of the notable...
Relation of DFT to z-Transform01:20

Relation of DFT to z-Transform

The Discrete Fourier Transform (DFT) is a crucial tool for analyzing the frequency content of discrete-time signals. It converts a sequence of N samples from the time domain into its corresponding sequence in the frequency domain, where each sample represents a specific frequency component.
To understand how the DFT works, it's helpful to consider the z-transform, which is a method for representing discrete sequences in the complex frequency domain. The z-transform involves summing the terms of...
Discrete-Time Fourier Series01:20

Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...
Discrete Fourier Transform01:15

Discrete Fourier Transform

The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...

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Related Experiment Video

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Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging
17:06

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

Published on: November 8, 2012

DWI filtering using joint information for DTI and HARDI.

Antonio Tristán-Vega1, Santiago Aja-Fernández

  • 1Laboratory of Image Processing, University of Valladolid, E.T.S. Telecomunicaciones, 47011 Valladolid, Spain. atriveg@lpi.tel.uva.es

Medical Image Analysis
|December 17, 2009
PubMed
Summary

Filtering Diffusion Weighted Images (DWI) together improves diffusion tensor estimation. Joint filtering methods, like joint LMMSE, offer accuracy comparable to UNLM with reduced computational cost, especially for HARDI data.

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Related Experiment Videos

Last Updated: Jun 17, 2026

Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging
17:06

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Published on: November 8, 2012

DTI of the Visual Pathway - White Matter Tracts and Cerebral Lesions
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Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases
09:33

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

Area of Science:

  • Medical Imaging
  • Neuroimaging
  • Diffusion MRI

Background:

  • Diffusion Weighted Images (DWI) filtering is crucial before diffusion tensor or Orientation Distribution Function (ODF) estimation.
  • Ignoring filtering or filtering gradients separately introduces unrecoverable errors in tensor field estimation.

Purpose of the Study:

  • To develop a novel methodology for filtering DWI data by leveraging joint information across all gradient images.
  • To adapt and evaluate this joint filtering approach using Linear Minimum Mean Squared Error (LMMSE) and Unbiased Non-Local Means (UNLM) filters.

Main Methods:

  • A new methodology that filters all DWI gradient images simultaneously, exploiting shared first and second-order information.
  • Adaptation of LMMSE and UNLM filters to process DWI data jointly.
  • Testing the proposed filters on both synthetic and real diffusion MRI datasets.

Main Results:

  • The joint filtering approach significantly improves the accuracy of diffusion tensor and ODF estimation.
  • Joint LMMSE demonstrates accuracy comparable or superior to UNLM, particularly for High Angular Resolution Diffusion Imaging (HARDI).
  • The joint LMMSE method offers a substantial reduction in computational load compared to UNLM.

Conclusions:

  • Jointly filtering DWI volumes is superior to individual gradient filtering for accurate diffusion modeling.
  • The proposed joint LMMSE filter is an efficient and accurate method for DWI data processing, especially beneficial for HARDI.
  • This approach enhances the reliability and reduces the computational burden of diffusion MRI analysis.