IMG-03

Nyquist sampling

fs > 2 fmax. The sampling rate that avoids aliasing.

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SamplingNyquist

Governing equation

f_s>2f_{\max}

$f_{max} \in \left[1, 250\right]$

where

f_{max}: Max frequency (kHz)
f_s: Nyquist rate (kHz)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

f_s>2f_{\max}

$f_{max} \in \left[1, 250\right]$

$f_{max} = 1$ $f_s = 2$

Along the curve

②

Domain of the variable — blue

f_{max}_{\min} = 1,\quad f_{max}_{\max} = 250

$f_{max} \in \left[1, 250\right]$

$f_{max} = 1$ $f_{max} = 1$

Along the curve

③

Response curve — green

f_s>2f_{\max}

$f_{max} \in \left[1, 250\right]$

$f_{max} = 1$ $f_s = 2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f_s}{\mathrm{d}f_{max}}

$f_{max} \in \left[1, 250\right]$

$f_{max} = 2.566$ $f_s^{\prime} = 2$

Along the curve

Lecture brief

Historical brief

Larmor precession, CT Beer projections, SNR and Nyquist sampling are why MRI and CT images exist as numbers. The lab computes frequency, dose and resolution limits. This sheet (IMG-03 — Nyquist sampling) is the form associated with Nyquist. Working symbols: $f_{max}$ $\rightarrow$ $f_s$ . k-space step Δk = 1/FOV. Undersample k and wrap-around ghosts appear.

Purpose

Purpose: compute

f_s

from

f_{max}

in Medical imaging via

f_s>2f_{\max}

fs > 2 fmax. The sampling rate that avoids aliasing. Use it when a real medical imaging question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f_{max} = 32.000\,\mathrm{kHz}

, the governing relation

f_s>2f_{\max}

yields

f_s = 64.00\,\mathrm{kHz}

. A waveform, dots on it, a reconstructed sine. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Max frequency f_{max}32 kHz

Outputs

Nyquist rate f_s64.00 kHz

Reading speed

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Narration of this film

A waveform, dots on it, a reconstructed sine.

k-space step Δk = 1/FOV. Undersample k and wrap-around ghosts appear.

Reading speed

Watch on YouTube