Self-Portrait of the Focusing Process in Speckle: II. Gouy Phase Shift for Defocus Correction and Pixel Depth Reassignment

This is the second article in a series of three dealing with the exploitation of speckle for aberration correction and reverberation compensation in reflection imaging. When probing heterogeneous media with waves, we have to cope with multi-scale fluctuations of the wave velocity. On the one hand, short-scale heterogeneities induce back-scattered echoes whose random interference generate a speckle pattern on the beamformed image. On the other hand, large-scale fluctuations of the wave-velocity can distort the focused wave-fronts, resulting in aberrations on the same image. In this paper, we show how the self-portrait of the wave evolves as a function of the speed-of-sound model. Strikingly, a Gouy phase shift is observed when the speed-of-sound model is optimal. This particularly sensitive feature enables: (i) an optimization of the speed-of-sound model for each pixel of the image; (ii) a local and fine compensation of defocus across the field-of-view, thereby compensating for most aberrations in the image. Experiment in a tissue-mimicking phantom and numerical simulations are first presented to validate our method. It is then applied to in-vivo liver data of a difficult-to-image patient. The speed-of-sound optimization allows an axial compensation of aberrations and a depth-reassignment of each singly-scattered echo to the actual position of the associated scatterer. As distance measurement is often critical for diagnosis, such a wave speed optimization can be crucial for ultrasound but also for any other imaging methods based on the principle of echo-location.

💡 Research Summary

This paper addresses a fundamental challenge in echo‑based imaging modalities such as ultrasound: the simultaneous presence of speckle generated by short‑scale heterogeneities and large‑scale wave‑speed variations that cause defocus and depth mis‑registration. Conventional real‑time scanners assume a homogeneous speed of sound (c₀) and perform delay‑and‑sum beamforming based on this single value. In reality, biological tissues exhibit speed variations from roughly 1400 m s⁻¹ (fat) to 1650 m s⁻¹ (muscle, skin). When c₀ differs from the true local speed, the focal plane, the isochronous plane, and the imaging plane no longer coincide, leading to axial shifts, loss of resolution, and erroneous depth information—issues that are especially problematic for diagnoses that rely on precise distance measurements.

The authors propose a novel method that exploits the “self‑portrait” of the coherent wave field, i.e., the focused reflection matrix, to infer the optimal local speed of sound for each pixel. The workflow begins with the acquisition of a full reflection matrix R(u_out, θ_in, t) using plane‑wave insonifications at multiple angles. From this matrix, a focused reflection matrix Rₓₓ(t, c₀) is constructed by decoupling input and output focal points, effectively representing the response between virtual transducers located at the same depth. By re‑parameterising the matrix in terms of the lateral offset Δx = x_out − x_in, the authors obtain a “de‑scan” matrix R_D(Δx, r) whose columns correspond to the reflection‑point spread function (RPSF) centered on each input focal point.

Crucially, the lateral width of each RPSF reaches a minimum when the assumed speed c₀ matches the true local speed c. However, individual RPSFs are corrupted by the random speckle pattern. To suppress this randomness, the authors apply a spatial window P(r − r_p) and compute a locally averaged, incoherent RPSF: RPSF_inc(Δ, r_p) = ⟨|R_L(Δ, r, r_p)|²⟩_r. The Δ‑profile of RPSF_inc exhibits a clear, sharp minimum as a function of c₀, which directly yields the optimal speed for the pixel centred at r_p. This minimum is accompanied by a Gouy phase shift—a rapid phase transition characteristic of a well‑focused wave—providing an additional, highly sensitive indicator of optimal focusing.

The method is validated in three stages. First, numerical simulations on synthetic media with known speed maps demonstrate accurate recovery of the ground‑truth speed distribution and confirm the presence of the Gouy phase shift at the optimum. Second, experiments on a tissue‑mimicking phantom (c ≈ 1542 m s⁻¹) show that using an incorrect c₀ (e.g., 1800 m s⁻¹) produces obvious axial stretching and resolution loss, whereas the optimal c₀ restores a focused image and reveals the Gouy phase signature. Third, in‑vivo liver data from a patient with hepatic steatosis are processed. The algorithm identifies distinct speed zones corresponding to subcutaneous fat, muscle, and liver tissue; the liver region exhibits a reduced speed (~1540 m s⁻¹) consistent with fatty infiltration. Applying the locally optimized speed map aligns the focal and isochronous planes, thereby correcting axial aberrations and enabling depth reassignment of each singly‑scattered echo to its true physical location.

Compared with existing approaches such as Computed Ultrasound Tomography in Echo mode (CUTE), which reconstruct a global speed map from transmission data, the present technique operates purely on reflection data and yields pixel‑wise speed estimates without requiring a separate transmission acquisition. Moreover, by leveraging speckle statistics, the method inherently filters multiple scattering and electronic noise, making it robust in low‑signal‑to‑noise scenarios typical of deep abdominal imaging.

Limitations are acknowledged. In regions where speckle is weak (e.g., dominated by strong specular reflectors) the RPSF width may not provide a reliable minimum, and the method’s computational cost—dominated by singular‑value decompositions of large reflection matrices—currently restricts it to offline processing. The authors suggest future work on GPU acceleration, multi‑frequency fusion to improve robustness against broadband speed variations, and extension to three‑dimensional imaging.

In summary, the paper introduces a speckle‑based, phase‑sensitive metric (the Gouy phase shift) to locally optimise the speed‑of‑sound model, thereby correcting defocus, improving resolution, and enabling accurate depth reassignment in echo‑based imaging. This advancement holds promise not only for clinical ultrasound but also for any modality that relies on echo‑location, such as radar and optical coherence tomography.