When light, sound or any kind of wave travels through a complex medium like fog, murky water, or biological tissue, it scatters in many directions. Each particle or irregularity in the medium changes the path of the waves, scrambling them and blurring the resulting image. This is why doctors struggle to image deep inside tissue using ultrasound, why optical microscopes can’t see through thick samples, and why radar and sonar sometimes miss objects hidden behind clutter.
Scientists have long looked for ways to focus waves through such disordered environments — and while many have tried to compensate for scattering, their success has been limited when the medium becomes very opaque.
A team led by Alexandre Aubry at ESPCI Paris and collaborators from Vienna and Aix-en-Provence wanted to turn this problem around. Instead of correcting or undoing the scattering, they wondered if something in the wave patterns remains stable even in the middle of all that complexity. That is, could they identify and locate a target based on the part of the signal that still carries its unique ‘fingerprint’?
Their new study, published in Nature Physics, introduces a mathematical tool called the fingerprint operator that allows exactly this. This operator can detect, locate, and even characterise an object hidden inside a strongly scattering medium by comparing the reflected light to a reference pattern recorded in simpler conditions. The method can work for sound, light, radar, and other kinds of waves.
At the heart of the technique is the reflection matrix, a large dataset recording how each source in an array of sources sends a wave into the medium and how every receiver picks up the returning echoes. Each element of this matrix contains information about how waves bounce off of different points, so together they capture the complete response of the system.
To find a target within this sea of signals, the researchers introduced the fingerprint operator, written as Γ = R × R₀†, where R is the measured reflection matrix from the complex medium and R₀ is a reference matrix measured for the same target in clear, homogenous conditions. The dagger (†) indicates a mathematical conjugate that makes the comparison sensitive to how well the two patterns match. By calculating how strongly the two matrices correlate, the team obtained a likelihood index, which indicates how likely it is that a target with certain properties — e.g. position, size or shape — is present at a given spot.
Effectively the team has developed a way to image hidden objects using scattered light.
The researchers tested this concept with ultrasound. They used arrays containing up to 1,024 transducers (devices that convert energy from one form to another) to send and receive sound waves. First, they embedded small metal spheres inside a suspension of glass beads mixed with water, making for a strongly scattering environment.
In the granular suspension, conventional ultrasound couldn’t see the buried metal spheres at all. The multiple scattering caused an exponential loss of contrast with depth, making the target signals roughly a 100x weaker than the background noise. Yet when the fingerprint operator was applied, the two spheres appeared sharply on the reconstructed likelihood map, each represented by a bright peak at its correct location. The contrast improvement reached factors of several hundred, strong enough to rule out false positive signals with a probability of error smaller than 1 in a hundred million.
This success came from the fingerprint operator’s ability to filter out diffuse, randomly scattered waves and isolate those faint waves that behave as if the medium were transparent. In simple terms, the operator is a mathematical tool that can use the complexity of the target’s own echo to cancel the complexity of the medium.
The same approach worked inside a foam that mimicked human tissue. A normal ultrasound image was dominated by speckle (random bright and dark spots caused by small scattering events), rendering a small pre-inserted marker nearly invisible. But when the fingerprint operator was applied to the data, the marker was revealed clearly and precisely.
To its credit, the fingerprint operator doesn’t require scientists to fully known the medium, only the ability to record a reflection matrix and a reference response. It can then use these resources to find patterns that survive scattering and extract meaningful information.
For medicine, this could improve ultrasound detection of small implants, needles, and markers that currently get lost in tissue noise. It could also help map the internal fibre structure of muscles or hearts, providing new diagnostic insights into diseases like cardiomyopathy and fibrosis. In materials science, it could reveal the orientation of grains in metals or composites. In military settings, it could locate targets hidden behind foliage or turbulent water.
The approach is also computationally efficient: according to the researchers’ paper, generating the likelihood map takes about the same time as developing a standard ultrasound image and can be adapted for moving targets by incorporating motion parameters into the fingerprint.
Finally, the idea animating the study also challenges a long-standing view that multiple scattering is purely a nuisance, incapable of being useful. The study overturns this view by extracting information from the multiple scattering data, using the fingerprint operator to account for how a target’s own echoes evolve through scattering, and leveraging those distortions to detect it more confidently.
Featured image credit: Rafael Peier/Unsplash.