Efficient Localization of Point Scatterers via Sparse Optimization on Measures

Kernkonzepte

The authors propose a "linearize and locally optimize" approach to efficiently recover the locations and intensities of point scatterers from far-field measurements of time-harmonic acoustic waves.

Zusammenfassung

The authors consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities, modeled by the Foldy-Lax system. They propose a two-step approach to recover the locations and intensities of the point scatterers from far-field measurements:

Linear step: Solve a convex program in the space of measures (Beurling LASSO) to obtain an initial estimate, leveraging results on the sparse spikes problem.
Nonlinear step: Locally optimize a non-convex functional involving the nonlinear forward operator, using the output of the linear step as initialization.

The authors provide theoretical guarantees for the linear step, showing that the output is close to the true measure when the scatterers have small intensities and are sufficiently separated. They also provide numerical evidence that the second step allows for accurate recovery in more involved settings.
The key contributions are:

Connecting the inverse scattering problem to the sparse spikes problem and leveraging existing results.
Deriving quantitative recovery guarantees for the linear step, accounting for the linearization error.
Proposing a "linearize and locally optimize" approach that combines the benefits of the linear and nonlinear steps.
Developing a JAX-based Python package that can run efficiently on CPUs, GPUs and TPUs.

Statistiken

The authors provide the following key figures and metrics:

The Foldy-Lax model for the scattering of acoustic waves by point-like inhomogeneities (Eq. 4, 5).
Bounds on the linearization error between the Foldy-Lax and Born approximation models (Propositions 4.1, 4.2).
Theoretical recovery guarantees in terms of the minimal separation between scatterers, number of measurements, and noise level (Theorems 4.3, 4.4).

Zitate

"The inverse scattering problem can be recast as the recovery of a discrete measure, which encodes the locations and the intensities of the scatterers, from nonlinear measurements."
"We propose to investigate how the guarantees and reconstruction methods developed for the sparse spikes problem can still be leveraged."
"We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated."

Wichtige Erkenntnisse aus

Localization of point scatterers via sparse optimization on measures

by Giovanni S. ... um arxiv.org 04-24-2024

https://arxiv.org/pdf/2402.00737.pdf

Tiefere Fragen

How can the regularization parameters λb and λf be automatically selected in practice, without relying on a grid search

To automatically select the regularization parameters λb and λf without relying on a grid search, one approach could be to use techniques such as cross-validation or model selection algorithms.

Cross-validation: Split the dataset into training and validation sets. Train the model with different values of λb and λf on the training set and evaluate their performance on the validation set. Choose the parameters that give the best performance on the validation set.

Model selection algorithms: Algorithms like Bayesian optimization or sequential model-based optimization can be used to search for the optimal values of λb and λf. These algorithms iteratively explore the parameter space based on the performance of the model and adjust the parameters accordingly.

Regularization path: Another approach is to use a regularization path method, where a range of λ values are tested simultaneously, and the optimal value is selected based on the performance metrics.

By implementing these techniques, the regularization parameters can be automatically selected based on the data and model performance, without the need for manual grid search.

Can the proposed approach be extended to handle more general types of inhomogeneities, beyond the point scatterer model

The proposed approach can be extended to handle more general types of inhomogeneities beyond the point scatterer model by adapting the forward operators and the optimization framework to accommodate the specific characteristics of the new model. Here are some ways to extend the approach:

Different types of inhomogeneities: Modify the forward operators to account for different types of inhomogeneities, such as extended scatterers or continuous distributions of scatterers. This may involve changing the form of the Green's functions and the modeling of the scatterers.

Nonlinear forward operators: Extend the optimization framework to handle nonlinear forward operators that arise in more complex scattering models. This may require more sophisticated optimization algorithms and regularization techniques.

Additional constraints: Incorporate additional constraints or prior information about the inhomogeneities into the reconstruction algorithm. This could include constraints on the shape, size, or intensity of the scatterers.

Experimental validation: Validate the extended approach using simulations or experimental data to ensure its effectiveness in real-world scenarios.

By adapting the approach to handle more general types of inhomogeneities, it can be applied to a wider range of inverse scattering problems in various fields.

What are the potential applications of the efficient localization of point scatterers in fields such as material science, medical imaging, or optics

The efficient localization of point scatterers has several potential applications in various fields:

Material Science: In material science, the ability to accurately localize point scatterers can help in detecting defects, impurities, or structural anomalies in materials. This information is crucial for quality control and material characterization.

Medical Imaging: In medical imaging, the localization of point scatterers can be used to detect and characterize abnormalities such as tumors, calcifications, or foreign bodies in tissues. This can aid in early diagnosis and treatment planning.

Optics: In optics, the localization of point scatterers is valuable for studying the interaction of light with nanoparticles, microstructures, or biological cells. This information is essential for designing optical devices, sensors, and imaging systems.

Acoustic Imaging: Beyond the mentioned fields, the localization of point scatterers can also be applied in acoustic imaging for detecting anomalies in underwater environments, structural health monitoring, and seismic analysis.

Overall, the efficient localization of point scatterers has diverse applications across different disciplines, contributing to advancements in imaging, diagnostics, and material analysis.