Self-Portrait of the Focusing Process in Speckle: III. Tailoring Complex Spatio-Temporal Focusing Laws To Overcome Reverberations in Reflection Imaging

This is the third article in a series of three dealing with the exploitation of speckle for imaging purposes. In complex media, a fundamental limit is the multiple scattering phenomenon that completely blurs the imaging process in depth. Matrix imaging can provide a relevant framework for solving this problem. As it proved to be an adequate tool for probing reverberations in speckle [E. Giraudat et al., Part I], we will show how it can be used to tailor complex spatio-temporal focusing laws to monitor the interference between the multiply-reflected paths and the ballistic component of the wave-field. To do so, we extend the distortion matrix concept to the frequency domain. An iterative phase reversal process operated from the space-time Fourier space is then used to compensate for reverberations and optimize both the axial and transverse resolution of the confocal image. Here, we first present an experimental proof-of-concept consisting in imaging a tissue-mimicking phantom through a reverberating plate before outlining the potential and the limits of this strategy for transcranial ultrasound and beyond.

💡 Research Summary

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This paper, the third in a series on exploiting speckle for imaging, tackles the persistent problem of reverberations (multiple reflections) that severely degrade ultrasound imaging in highly scattering media such as the human skull. Traditional adaptive focusing methods correct only for forward‑propagation aberrations using simple time‑delay or phase‑shift adjustments, and therefore cannot compensate for the complex spatio‑temporal distortions introduced by reverberating layers.

The authors extend the concept of the distortion matrix, previously defined in the spatial domain, to the frequency domain. By recording the full reflection matrix (R_{u\theta}(t)) (the response of each transmit‑receive pair as a function of time) and applying a space‑time Fourier transform, they obtain a frequency‑space representation that contains all the information about the medium’s scattering and reverberation properties. Within a selected speckle region that is statistically homogeneous, they compute a spectral correlation matrix

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