IMG-06

Beer projection

p = −ln(I/I0) = μ L. The line integral a CT inverts.

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CTBeer–Lambert

Governing equation

p=-\ln(I/I_0)=\mu L

$L \in \left[1, 50\right]$

where

I_0: Entrance intensity (—)
\mu: Linear attenuation (1/cm)
L: Path length (cm)
I: Transmitted (—)
p: Projection (—)

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

p=-\ln(I/I_0)=\mu L

$L \in \left[1, 50\right]$

$L = 1$ $I = 0.2$

Along the curve

②

Domain of the variable — blue

L_{\min} = 1,\quad L_{\max} = 50

$L \in \left[1, 50\right]$

$L = 1$ $L = 1$

Along the curve

③

Response curve — green

p=-\ln(I/I_0)=\mu L

$L \in \left[1, 50\right]$

$L = 1$ $I = 0.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}L}

$L \in \left[1, 50\right]$

$L = 1.308$ $I^{\prime} = 0.2$

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-06 — Beer projection) is the form associated with Beer–Lambert. Working symbols: $I_0$ , $\mu$ , $L$ $\rightarrow$ $I$ , $p$ . Polychromatic beams beam-harden: the effective μ drops along the path. Water correction is the first fix.

Purpose

Purpose: compute

I

p

from

I_0

\mu

L

in Medical imaging via

p=-\ln(I/I_0)=\mu L

p = −ln(I/I0) = μ L. The line integral a CT inverts. 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

I_0 = 1.000\,\mathrm{—}

\mu = 0.200\,\mathrm{1/cm}

L = 20.000\,\mathrm{cm}

, the governing relation

p=-\ln(I/I_0)=\mu L

yields

I = 0.0183\,\mathrm{—}

p = 4.000\,\mathrm{—}

. A slab, a ray, a log-attenuated projection. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Entrance intensity I_01 —Linear attenuation \mu0.2 1/cmPath length L20 cm

Outputs

Transmitted I0.0183 —
Projection p4.000 —

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Narration of this film

A slab, a ray, a log-attenuated projection.

Polychromatic beams beam-harden: the effective μ drops along the path. Water correction is the first fix.

Reading speed

Watch on YouTube