US Speckle Models — Part 2: The Nakagami Distribution

Part 1 argued that Rayleigh is the right model for one specific regime — many independent scatterers, homogeneous tissue, no interest in absolute quantitative recovery — and breaks in three important ways when tissue doesn't satisfy those conditions. This part is about the generalization that fixes the first two of those problems (scatterer count assumptions and the fixed-shape constraint) while explicitly not fixing the third (pixel correlation). That last one waits until Part 4.

The generalization is the Nakagami distribution.

Where it comes from

The Nakagami distribution wasn't invented for ultrasound. Minoru Nakagami developed it in the 1940s to model the amplitude of received radio signals in HF propagation, where the received envelope showed variability the Rayleigh model couldn't capture — sometimes more, sometimes less. He wanted one flexible family that could sit anywhere along the spread-vs-concentration axis, with a single shape parameter to move between regimes.

Medical ultrasound picked it up several decades later for the same reason: the envelope of the backscattered signal doesn't always look Rayleigh, and a single flexible family beats switching between separate models for boundaries, parenchyma, and fibrous tissue.

The PDF:

$$p(x;\, m,\, \omega) \;=\; \frac{2 m^m}{\Gamma(m)\, \omega^m}\; x^{2m-1}\, e^{-m x^2 / \omega}, \qquad x \geq 0$$

Two parameters, one normalizing constant, one exponential kernel. Each piece is doing specific work.

Reading the terms

x — envelope amplitude. The variable being modeled. This is what the scanner reconstructs from the beamformed complex signal and hands off as a pixel intensity, up to whatever post-processing sits between envelope detection and display. In the raw envelope domain, x is real and non-negative. In the display domain, x is the pixel value after log compression, gain, and gamma — related to the true envelope amplitude but not identical to it. Part 3 gets into what that difference costs.

m — shape parameter. This is the term that made Nakagami interesting for tissue characterization. It's dimensionless, non-negative, and controls how concentrated the distribution is around its mode:

• Low m (m < 1) → distribution spreads out, heavy tail, mode close to zero. Physically: few scatterers per resolution cell, high uncertainty in the envelope amplitude. Boundaries, interfaces, sparsely-scattering regions.
• m = 1 → Rayleigh case (details below). Fully developed speckle regime.
• High m (m > 1) → sharp peak, tight distribution, mode moves away from zero. Physically: many scatterers, more organized structure, predictable statistics.

m is not fixed by physics — it's estimated from the data. Give it a homogeneous parenchyma window and it comes back near 1. Give it a boundary or cyst wall and it comes back below 1. Give it dense fibrous tissue and it comes back above 1. The distribution adapts to what's actually there.

ω — scale parameter. Equal to

$$\mathbb{E}[x^2]$$

, the mean squared envelope amplitude. It sets the intensity level of the distribution — think of it as absorbing depth-dependent attenuation, gain settings, and any global scaling of the image. When you compare m between regions, ω is what you normalize by to strip out those global effects.

Γ(m) — the gamma function. The normalizing constant. It ensures the PDF integrates to 1 across all valid x. For integer m,

$$\Gamma(m) = (m-1)!$$

; for non-integer m it interpolates smoothly. Its role here is structural — nothing physical, just what makes the math a proper probability distribution.

exp(−mx²/ω) — the exponential kernel. This is where the shape work happens. As m increases, this term penalizes large deviations from the mean more aggressively, which is what makes high-m distributions look tightly concentrated. As m decreases, the penalty softens, and the distribution spreads. All the intuition about "few scatterers → high uncertainty → wide distribution" and "many scatterers → predictable → tight distribution" is carried by this single term.

The prefactor

$$x^{2m-1}$$

shapes the low-x behavior — pulls the distribution away from zero as m grows, allows density at zero when m is small.

Rayleigh as a special case

The claim that Nakagami reduces to Rayleigh at m = 1 is often stated but rarely shown. It's worth doing explicitly, because the substitution reveals how

$$\omega$$

and Rayleigh's

$$\sigma$$

relate.

Set m = 1 in the Nakagami PDF. Since

$$\Gamma(1) = 1$$

:

$$p(x;\, 1,\, \omega) \;=\; \frac{2}{\omega}\, x\, \exp!\left(-\frac{x^2}{\omega}\right)$$

Compare to the Rayleigh PDF from Part 1:

$$p(x;\, \sigma) \;=\; \frac{x}{\sigma^2}\, \exp!\left(-\frac{x^2}{2\sigma^2}\right)$$

These match if

$$\omega = 2\sigma^2$$

. So Nakagami's scale parameter is exactly twice the variance of the Gaussian I/Q components — the same physical quantity, written in a different convention.

The offset issue from Part 1 carries through: at m = 1, the Nakagami mean is still

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

in the underlying units, and the envelope-to-scatterer-strength mapping still has that

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

factor sitting in the middle. Nakagami doesn't fix that — it wasn't designed to. What Nakagami fixes is the shape rigidity.

What Nakagami buys you, and what it doesn't

The right way to describe Nakagami is that it has no built-in assumptions about scatterer count. Both parameters, m and ω, are free — estimated from the local data — and the model adapts to whatever regime the tissue is in. This is different from a model that has a hand-tuned hyperparameter you have to pick a priori.

The standard estimator is method of moments:

$$\hat{m} \;=\; \frac{\big(\mathbb{E}[x^2]\big)^2}{\mathrm{Var}(x^2)}, \qquad \hat{\omega} \;=\; \mathbb{E}[x^2]$$

Both quantities come directly from the sample. There's a maximum-likelihood variant that's slightly more efficient in small windows, but for most practical purposes moment-based estimation is what the literature uses and what tools implement.

What Nakagami does not buy you:

• Pixel independence. The estimator above assumes samples inside the window are i.i.d. In ultrasound they aren't — the PSF spreads every scatterer's contribution across a neighborhood, and pixels are correlated by construction. Local m estimates carry a PSF-shaped bias that grows with the ratio of PSF width to window size. Part 4 is about this.

• A separation between distribution and display. Nakagami models the envelope amplitude. If your data has gone through log compression, gain adjustment, and gamma correction — which is what every commercial hospital scanner does before writing DICOM — the pixel values aren't envelope amplitudes anymore. You can still fit Nakagami, but the m you get is a pseudo-m on the display domain, not the true envelope-domain m. Part 3 is about this.

• A guarantee that m corresponds to a specific tissue property. m tracks the scatterer density in a statistical sense, but the mapping isn't one-to-one with any single clinical variable. Interpreting m maps as diagnostic requires calibration against ground truth for the specific tissue and scanner combination.

Below are the visuals that accompanied the original discussion — Nakagami PDFs at several m values, and a formulation diagram tying the model to the physical scattering setup.

Nakagami probability density function shown for a range of shape parameter m values.

Physical formulation tying the Nakagami model to the underlying scattering setup.

Next

Nakagami models the amplitude distribution at each pixel. What it doesn't model is the relationship between pixels — and that relationship is never zero in ultrasound, because the PSF couples them.

Before we get to correlation, though, there's a step in between: what happens when you estimate m locally across the image, sliding a window and producing a spatial map of the shape parameter. That's Part 3 — where Nakagami stops being a noise model and becomes a tissue characterization tool.

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References

• My paper on Bayesian restoration for B-mode ultrasound, using a system model that folds in both Nakagami-like envelope statistics and correlated noise: Ultrasonic Imaging, 2019
• ORCID (all publications): 0000-0003-2633-9371
• Previous in series: Part 1 — Rayleigh Distribution

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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.