IMG-02

MRI signal-to-noise

SNR ∝ Δx Δy Δz √(NEX Ny Ts). Voxel and averaging set the noise floor.

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Image qualityMRI SNR

Governing equation

\mathrm{SNR}\propto\Delta x\,\Delta y\,\Delta z\sqrt{N_{\mathrm{EX}}N_y T_s}

$\Delta z \in \left[0.5, 10\right]$

where

\Delta x: Pixel x (mm)
\Delta y: Pixel y (mm)
\Delta z: Slice (mm)
N_{EX}: Averages NEX (—)
N_y: Phase encodes (—)
T_s: Sampling time (ms)
SNR: Relative SNR (a.u.)

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

\mathrm{SNR}\propto\Delta x\,\Delta y\,\Delta z\sqrt{N_{\mathrm{EX}}N_y T_s}

$\Delta z \in \left[0.5, 10\right]$

$\Delta z = 0.5$ $SNR = 46.08$

Along the curve

②

Domain of the variable — blue

\Delta z_{\min} = 0.5,\quad \Delta z_{\max} = 10

$\Delta z \in \left[0.5, 10\right]$

$\Delta z = 0.5$ $\Delta z = 0.5$

Along the curve

③

Response curve — green

\mathrm{SNR}\propto\Delta x\,\Delta y\,\Delta z\sqrt{N_{\mathrm{EX}}N_y T_s}

$\Delta z \in \left[0.5, 10\right]$

$\Delta z = 0.5$ $SNR = 46.08$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}SNR}{\mathrm{d}\Delta z}

$\Delta z \in \left[0.5, 10\right]$

$\Delta z = 0.56$ $SNR^{\prime} = 92.16$

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-02 — MRI signal-to-noise) is the form associated with MRI SNR. Working symbols: $\Delta x$ , $\Delta y$ , $\Delta z$ , $N_{EX}$ , $N_y$ , $T_s$ $\rightarrow$ $SNR$ . Halving voxel volume halves SNR. Doubling NEX buys only √2. Bandwidth sits in Ts.

Purpose

Purpose: compute

SNR

from

\Delta x

\Delta y

\Delta z

N_{EX}

N_y

T_s

in Medical imaging via

\mathrm{SNR}\propto\Delta x\,\Delta y\,\Delta z\sqrt{N_{\mathrm{EX}}N_y T_s}

SNR ∝ Δx Δy Δz √(NEX Ny Ts). Voxel and averaging set the noise floor. 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

\Delta x = 1.200\,\mathrm{mm}

\Delta y = 1.200\,\mathrm{mm}

\Delta z = 4.000\,\mathrm{mm}

N_{EX} = 2.000\,\mathrm{—}

N_y = 256.000\,\mathrm{—}

T_s = 8.000\,\mathrm{ms}

, the governing relation

\mathrm{SNR}\propto\Delta x\,\Delta y\,\Delta z\sqrt{N_{\mathrm{EX}}N_y T_s}

yields

SNR = 368.64\,\mathrm{a.u.}

. A voxel, a noise cloud, an SNR bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Pixel x \Delta x1.2 mmPixel y \Delta y1.2 mmSlice \Delta z4 mmAverages NEX N_{EX}2 —Phase encodes N_y256 —Sampling time T_s8 ms

Outputs

Relative SNR SNR368.64 a.u.

Reading speed

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Narration of this film

A voxel, a noise cloud, an SNR bar.

Halving voxel volume halves SNR. Doubling NEX buys only √2. Bandwidth sits in Ts.

Reading speed

Watch on YouTube