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Sampling Theorem

Sampling Theorem

In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.

Downsampling

Downsampling

When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...

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Related Experiment Video

Updated: Mar 27, 2026

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

Q-space truncation and sampling in diffusion spectrum imaging.

Qiyuan Tian^1,2, Ariel Rokem³, Rebecca D Folkerth⁴

¹Department of Electrical Engineering, Stanford University, Stanford, California, USA.

Magnetic Resonance in Medicine

|January 15, 2016

Summary

This summary is machine-generated.

Diffusion Spectrum Imaging (DSI) q-space sampling affects spin-displacement probability density functions (PDFs) and orientation distribution functions (ODFs). Optimal DSI requires specific sampling densities to minimize artifacts like Gibbs ringing for accurate brain imaging.

Keywords:

Gibbs ringing artifacts aliasing artifacts diffusion spectrum imaging ex vivo imaging high b-value q-space truncation and sampling

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Area of Science:

Neuroimaging
Diffusion Spectrum Imaging (DSI)
Quantitative MRI

Background:

Diffusion Spectrum Imaging (DSI) enables the characterization of water diffusion in biological tissues.
The accuracy of DSI-derived metrics, such as spin-displacement probability density functions (PDFs) and orientation distribution functions (ODFs), is influenced by q-space sampling and truncation.
Understanding these influences is crucial for reliable neuroimaging analysis.

Purpose of the Study:

To investigate the impact of q-space truncation and sampling on the spin-displacement probability density function (PDF) in Diffusion Spectrum Imaging (DSI).
To establish optimal parameters for DSI data acquisition and reconstruction to minimize artifacts.

Main Methods:

Acquired DSI data in ex vivo and in vivo human brains with varying bmax values and grid resolutions.
Reconstructed PDFs and ODFs using different q-space filtering and PDF integration lengths.
Analyzed the effects of data down-sampling on PDF and ODF reconstruction.

Main Results:

Gibbs ringing was observed in PDFs for both ex vivo and in vivo data, significantly impacting ODF reconstruction.
Down-sampled data led to interference between PDF replicas and ringing, obscuring orientations in ODFs.
Identified the 11x11x11 grid as suitable for both ex vivo and in vivo DSI.

Conclusions:

Minimum q-space sampling density should correspond to a field-of-view approximately twice the mean displacement distance (MDD).
To mitigate Gibbs ringing, ODFs should be reconstructed from unfiltered q-space data.
Constraining PDF integration length to around the MDD is recommended for accurate ODF reconstruction.