US Speckle Models — Part 1: Rayleigh Distribution

Ultrasound speckle isn't random noise. It's physics.

The bright-and-dark grainy texture you see in every B-mode image is not a sensor artifact and not electronic noise. It's the deterministic interference pattern of many sub-resolution acoustic scatterers, imaged through a finite-aperture point spread function. If you scanned the exact same tissue twice with the exact same probe position and settings, you'd get the exact same speckle. It's a signal, not a nuisance.

The reason speckle looks random is that the underlying scatterer configuration is unknown to us — we can't resolve individual scatterers, and small changes in probe position redistribute the interference pattern in ways we can't track. So we treat it statistically, and the first statistical model anyone reaches for is Rayleigh.

This is Part 1 of a series on statistical models for ultrasound speckle. Here I'll cover where the Rayleigh model comes from, why it's a natural starting point, and the three specific places it stops holding up in real tissue.

Where Rayleigh comes from

Consider a single resolution cell containing many independent scatterers. Each scatterer contributes a complex-valued backscatter with some amplitude and phase. The total received signal is the coherent sum of all these contributions.

If we split that sum into in-phase (I) and quadrature (Q) components:

$$S = I + jQ = \sum_{k=1}^{N} a_k e^{j\phi_k}$$

where the scatterers have random amplitudes and random phases uniformly distributed on

$$[0, 2\pi]$$

, and N is large.

By the Central Limit Theorem, both I and Q converge to zero-mean Gaussian distributions with equal variance

$$\sigma^2$$

. The envelope amplitude — what the B-mode reconstruction actually visualizes — is the magnitude:

$$A = \sqrt{I^2 + Q^2}$$

The magnitude of two independent zero-mean Gaussians with equal variance follows a Rayleigh distribution:

$$p(A;\, \sigma) \;=\; \frac{A}{\sigma^2}\, \exp!\left(-\frac{A^2}{2\sigma^2}\right), \qquad A \geq 0$$

One parameter,

$$\sigma$$

, related to the underlying scatterer population. Mathematically elegant, physically motivated, and analytically tractable. This is why the model has stayed in textbooks for decades.

Where it breaks

Real tissue satisfies the CLT assumption in some places, and violates it in others. Three specific ways this shows up in practice:

1. Scatterer count isn't always "many"

The CLT argument assumes N is large enough for the Gaussian limit to hold — in practice, on the order of 10 or more scatterers per resolution cell. In homogeneous parenchyma at sufficient depth, this is a reasonable assumption. At lesion boundaries, cyst walls, tissue interfaces, and structured collagen, it isn't.

When N is small, the sum in the equation above doesn't converge to a Gaussian, and the envelope doesn't converge to Rayleigh. What you get instead is a sub-Rayleigh distribution — broader, with heavier tails and more variance in the low-amplitude region. The Rician distribution captures the case of few strong scatterers plus diffuse background; the Nakagami distribution generalizes further and covers both sub- and super-Rayleigh cases with a single shape parameter.

This isn't a small correction. Regions where Rayleigh breaks are exactly the regions clinicians care about — boundaries, lesions, structural changes — and forcing a Rayleigh fit there systematically misrepresents the tissue statistics.

2. Envelope detection has a built-in offset

The mean of a Rayleigh distribution is:

$$\mathbb{E}[A] \;=\; \sigma\sqrt{\frac{\pi}{2}} \;\approx\; 1.253\,\sigma$$

This creates a subtle but important issue for quantitative work. If you want to recover the underlying scatterer strength parameter

$$\sigma$$

from an image, you can't just take the local pixel mean and use it directly — the envelope detection operation inflates the mean by a factor of

$$\sqrt{\pi/2}$$

relative to the parameter that actually characterizes the tissue.

For qualitative B-mode viewing this doesn't matter — the image still looks like tissue. For quantitative analysis, it matters a lot. Any metric that treats pixel intensity as a direct proxy for scatterer density carries this systematic offset. It's not measurement error, it's not noise — it's a consequence of how the envelope operator maps the underlying complex signal to a positive real amplitude.

You can compensate for it if you know Rayleigh holds locally. You can't compensate for it cleanly when Rayleigh doesn't hold — because the relationship between pixel mean and scatterer strength changes with the underlying distribution.

3. Pixels aren't independent samples

Every scatterer's contribution is spread across a small spatial neighborhood by the point spread function. Neighboring pixels aren't independent samples of the same distribution — they're weighted combinations of overlapping scatterer sets. Even if the underlying distribution were perfectly Rayleigh at every location, the observed pixels would be correlated.

Standard estimators for

$$\sigma$$

— or for any distribution parameter, Rayleigh or otherwise — assume i.i.d. samples. When pixels are correlated, those estimators become biased, and their variance is systematically underestimated. Confidence intervals shrink, false-discovery rates rise, and the resulting parameter maps carry the PSF footprint mixed in with the tissue signal.

Most published QUS work on Rayleigh-based analysis quietly assumes independence and hopes the correlation is small enough not to matter. Sometimes it is. Often it isn't. This is a topic in its own right, and I'll come back to it in a later post in the series.

Where the series goes from here

Rayleigh isn't wrong. It's the right model for a specific regime — many independent scatterers, homogeneous tissue, no interest in absolute quantitative recovery of scatterer strength. That regime exists, and Rayleigh describes it accurately. What it isn't is a general model for ultrasound speckle.

Later parts of the series work through what to reach for when the Rayleigh assumptions break:

• Part 2 — Nakagami as a generalization, its shape parameter, and what each term in the PDF actually does
• Part 3 — Local m estimation and the shift from noise modeling to tissue characterization
• Part 4 — Correlated speckle, the PSF signature in the autocorrelation function, and what has to go into the observation model to handle it

Below are two visuals from the original discussion — Rayleigh compared to Nakagami PDFs at a few shape parameters, and a stylized picture of the "fully developed speckle" regime the CLT argument assumes.

---

References

• My paper on Bayesian restoration for B-mode ultrasound, including the correlated speckle noise model previewed in point (3) above: Ultrasonic Imaging, 2019
• ORCID (all publications): 0000-0003-2633-9371

---

I take on selective remote consulting on ultrasound image quality, quantitative ultrasound, and restoration problems — particularly on data from commercial hospital scanners rather than research-grade RF. Reach out via LinkedIn if that's the kind of problem you're working on.