FR33 · Ultrasound

Ultrasound Pulse-Echo Imaging: Building a Picture from Echo Times

How does an ultrasound scanner turn echo arrival times into a picture, and what limits its sharpness? An ultrasound scanner sends a short pulse and records when echoes return to each element of its array. For every pixel it adds the samples that a reflector there would produce, so true echoes add up and others cancel. A shorter pulse sharpens depth; a wider array or a shorter wavelength sharpens the sideways detail.

A 5 MHz plane pulse hits three thin wires and a plastic plate. The echo times on 64 elements become an image. Drag a wire and see its blur.

Depth blur of a wire (FWHM) $0.224\,\mathrm{mm}$

Sideways blur of a wire (FWHM) $0.498\,\mathrm{mm}$

Brightest sidelobe $-18.5\,\mathrm{dB}$

Pressure reflection at the plate $0.246$

Choose the pulse and the array

Centre frequency

5

f_0\;(\mathrm{MHz})

Fractional bandwidth at −6 dB

0.6

B

Receive elements

64

N

Element pitch

Receive weighting

Plate material

Drag a white wire on the stage; the image is acquired again as it moves.

Receiver noise seed

Seed of the receiver noise

3311

Model, units and sources

$g(t)=e^{-t^2/2\sigma^2}\cos(2\pi f_0t),\quad \sigma=\frac{2\sqrt{2\ln2}}{2\pi Bf_0}$

Gaussian pulse whose spectrum falls to half amplitude at the band edges. Channel data are sampled at 100 MHz; the image is the envelope of the delay-and-sum output on a grid of 160 lateral and 360 depth points. Wires and plate faces return independent echoes; the plate is a fluid layer and the elements are points.

Jensen, Imaging of Medical Ultrasound (DTU notes, 2024), Eqs. 3.9–3.17 · Barnkob et al. (2012), Table II · Settnes & Bruus (2012), Table I

How echo times become an image

Model assumptions

Two-dimensional water tank imaged by a linear array of point elements. One plane pulse at normal incidence, 12.8 mm wide, is sent; the active elements receive. Wires are ideal point scatterers with cylindrical spreading, and the plate is treated as a fluid layer, so shear waves are omitted. Echoes add linearly: no multiple scattering between targets, no attenuation, no element directivity and no transmit diffraction. The image is a delay-and-sum reconstruction of simulated channel data with seeded receiver noise, not measured data. Water and polystyrene follow Barnkob et al. (2012); pyrex follows Settnes and Bruus (2012).

What the elements recorded

The image made from those echoes

Sideways cut through the selected wire

More numbers and how they are checked

Wavelength in water: $0$
Pulse envelope duration at half height: $0$
Depth blur expected from the pulse: $0$
Receive aperture width: $0$
Depth over aperture at the selected wire: $0$
First sideways minimum, simulated pulse: $0$
First sideways zero, single frequency: $0$
First sideways zero, paraxial formula: $0$
Grating-lobe angle for this pitch: $0$
Pressure transmission into the plate: $0$
Reflected plus transmitted energy flux: $0$
Back-face echo amplitude: $0$
Back face in the image: $0$
Back face, true depth: $0$

The yellow profile comes from the simulated broadband channel data. The dashed green curve sums the elements at one frequency without any time signals, and the white markers are the paraxial first zero. A short pulse averages over many wavelengths, which fills the zeros and lowers the sidelobes. The plate coefficients are checked separately against a one-dimensional wave simulation that never uses the formulas.

Start with the essentials

Focus question: How does an ultrasound scanner turn echo arrival times into a picture, and what limits its sharpness?
One-sentence intuition: For every pixel, the scanner works out how long sound would take to go there and come back to each element, picks those samples and adds them. Echoes from that pixel add in step, everything else partly cancels. Depth sharpness is set by the pulse length; sideways sharpness by the wavelength, the depth and the width of the receiving aperture.

Core mathematical model

Delay for one pixel and one element

\tau_j(x,z)=\frac{z+\sqrt{(x-x_j)^2+z^2}}{c}

The plane pulse reaches depth $z$ after $z/c$ ; the echo then travels back to element $j$ at $x_j$ . The image value is the envelope of the weighted sum of all channels at these delays.

Reflection and transmission at a flat face

R=\frac{Z_2-Z_1}{Z_2+Z_1},\qquad T=\frac{2Z_2}{Z_2+Z_1},\qquad R^2+\frac{Z_1}{Z_2}T^2=1

Pressure and normal velocity are continuous at the face, and $Z=\rho c$ is the characteristic impedance. The last relation says reflected plus transmitted energy flux equals the incident flux.

Depth resolution

\Delta z\approx\frac{c\,\tau_{\mathrm{FWHM}}}{2}

The echo envelope lasts $\tau_{\mathrm{FWHM}}$ at half height; the factor one half comes from the round trip. A wider bandwidth means a shorter pulse.

Sideways resolution

x_{\mathrm{null}}\approx\frac{\lambda z}{L}

The first zero beside a point, for a uniformly weighted aperture of width $L$ focused at depth $z$ (Jensen, Eq. 3.10). Its first sidelobe is near $2/(3\pi)$ , about −13 dB, for a single frequency.

Common difficulties

Depth comes from time, not from sight

Typical misconceptionThe scanner measures where things are, so the plate back face appears at its true depth.

Better mental modelThe image assumes one sound speed. Sound crosses the plastic faster than water, so the back-face echo returns early and the 2 mm plate looks about 1.3 mm thick.

Run the experiment

01
Follow one pulse
Start with Three wires and a plate and watch the stage.
What to observe: The flat pulse passes each wire, which sends back a circular echo. The array elements light up as the echoes arrive, and the channel data fill in curved traces.
02
Shorten and lengthen the pulse
Choose A long pulse smears depth, then slide the bandwidth back up.
What to observe: Each wire stretches along the depth axis when the bandwidth is small; the sideways width hardly changes.
03
Narrow the array
Choose A narrow array blurs sideways, then drag the middle wire deeper.
What to observe: The wire spreads sideways, and more so the deeper it sits, as the ratio of depth to aperture predicts.

Ultrasound Pulse-Echo Imaging: Building a Picture from Echo Times

An image made of echo times

Core mathematical model

Delay for one pixel and one element

Reflection and transmission at a flat face

Depth resolution

Sideways resolution

Common difficulties

Depth comes from time, not from sight

Run the experiment

Follow one pulse

Shorten and lengthen the pulse

Narrow the array

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