Seeing the Unseen: AI Reconstructs Hidden Shapes from Scattered Waves

Author: Denis Avetisyan

A new deep learning approach leverages convolutional neural networks to accurately determine the geometry and material properties of obscured objects from electromagnetic scattering data.

Electromagnetic scattering is demonstrated around a doubly-connected magneto-dielectric cylinder, with three-dimensional modeling and cross-sectional analysis revealing the interaction’s characteristics.

This work demonstrates a CNN-based method for inverse electromagnetic scattering from doubly-connected cylinders, offering potential applications in obstacle reconstruction and impedance tomography.

Reconstructing complex electromagnetic obstacles from scattered wave data remains a significant challenge in inverse scattering problems. This is addressed in ‘Inverse Electromagnetic Scattering for Doubly-Connected Cylinders using Convolutional Neural Networks’, which introduces a deep learning framework for characterizing doubly-connected cylindrical structures. By employing specially designed convolutional neural networks, the authors demonstrate robust reconstruction of both the geometry and impedance properties of an inner obstacle concealed within a surrounding cylinder. Could this approach pave the way for more efficient and accurate electromagnetic imaging in diverse applications, from non-destructive testing to biomedical diagnostics?

Unveiling the Hidden: The Essence of Inverse Reconstruction

The process of deducing an object’s hidden characteristics from how it interacts with external probes – such as light, sound, or electromagnetic waves – forms the basis of the inverse scattering problem, a challenge pervasive across numerous scientific and engineering disciplines. This isn’t simply about ‘seeing’ what’s hidden; it’s about reconstructing internal structure and material composition based solely on the patterns of energy that bounce off or pass through an object. Consider medical imaging, where doctors aim to map internal organs from ultrasound or X-ray data, or geophysical exploration, where scientists seek to characterize subsurface structures from seismic waves. Successfully solving this inverse problem demands sophisticated mathematical techniques, as the relationship between internal properties and external observations is often complex and indirect, requiring algorithms to effectively ‘unravel’ the scattered signal and reveal the hidden details within.

Conventional approaches to the inverse scattering problem frequently necessitate trade-offs between accuracy and feasibility. Many established techniques depend on a priori knowledge – simplifying assumptions about the target’s shape, size, or material composition – to render the calculations manageable. Alternatively, fully rigorous solutions often demand immense computational resources, particularly when dealing with complex or large-scale objects. These computationally intensive methods, relying on iterative algorithms or exhaustive simulations, can become prohibitively expensive in terms of both processing time and memory, limiting their applicability to real-time scenarios or large datasets. Consequently, a persistent challenge lies in developing efficient and accurate methods that minimize reliance on simplifying assumptions while remaining computationally tractable for practical applications, such as non-destructive testing or medical diagnostics.

The ability to discern an object’s form and composition – specifically its impedance, a measure of its resistance to wave propagation – solely from analyzing how waves interact with it is fundamental to numerous technologies. In medical imaging, techniques like ultrasound and MRI rely on reconstructing internal anatomy from scattered wave signals, allowing for non-invasive diagnosis. Similarly, remote sensing applications, including geophysical exploration and atmospheric monitoring, utilize scattered electromagnetic waves to characterize subsurface structures or aerosol distributions. The accuracy of these reconstructions directly impacts the reliability of diagnoses, the precision of resource exploration, and the effectiveness of environmental monitoring, highlighting the critical importance of advancements in inverse scattering methods. Determining these properties remotely offers significant advantages over direct measurement, particularly in situations where access to the object is limited or impossible.

Despite having distinct shapes, these star-shaped obstacles generate nearly indistinguishable far-field data, demonstrating the ill-posed nature of reconstructing an object from its far-field signature.

Learning from Waves: A Direct Mapping Approach

The proposed system utilizes a convolutional neural network (CNN) to establish a direct mapping between far-field data – typically electromagnetic radiation measured at a distance from an object – and the intrinsic properties of that object, such as its shape, material composition, or internal structure. This approach contrasts with traditional inverse problems which often rely on iterative algorithms or computationally expensive forward modeling. The CNN is trained on paired datasets of far-field measurements and corresponding object properties, enabling it to learn the complex, non-linear relationship between the two. The network’s architecture is specifically designed to ingest the angular characteristics of far-field data and output a representation of the object’s properties, effectively functioning as a learned function $f: \text{far-field data} \rightarrow \text{object properties}$.

Traditional inverse reconstruction problems often rely on iterative optimization algorithms or repeated forward modeling to estimate object properties from measured data. These methods can be computationally expensive, particularly for large-scale problems. This CNN-based approach offers a departure from these techniques by functioning as a learned surrogate model. The network is trained on a dataset of known inputs and outputs, allowing it to directly approximate the inverse mapping without requiring iterative procedures. Once trained, the CNN can rapidly predict object properties given new far-field data, effectively replacing computationally intensive solvers with a fast, learned approximation. This capability significantly reduces processing time and enables real-time or near-real-time reconstruction.

Circular padding is implemented to address the angular and periodic nature of the input data representing far-field observations. This technique extends the input data by replicating its boundaries, creating a seamless, tileable representation. By preserving the inherent periodicity within the angular data, circular padding mitigates edge effects during convolutional operations. This ensures that features spanning the boundaries of the input field are accurately processed, and the CNN can effectively learn relationships regardless of angular position, ultimately improving generalization performance to unseen data with varying angular orientations.

An end-to-end network combines a circular padding CNN with an MLP inversion head to directly process raw measurements and generate final outputs.

Attending to Essence: Feature Prioritization for Accuracy

The Convolutional Neural Network (CNN) incorporates an attention mechanism to enhance its processing of far-field data. This mechanism functions by weighting input features, allowing the network to prioritize and selectively emphasize those most relevant for accurate object characterization. By focusing on crucial features and suppressing noise or less informative data, the attention mechanism improves the CNN’s capacity to detect and interpret subtle variations within the far-field signal, ultimately leading to enhanced performance in tasks such as object boundary prediction and impedance estimation. The weighting is learned during the training process, optimizing the network’s ability to discern relevant features for accurate analysis.

The convolutional neural network (CNN) is trained using a regression approach, where the network’s outputs directly correspond to quantifiable parameters defining the target object. Specifically, the network predicts the values that delineate the object’s boundary – such as coordinates defining a polygon or parameters for an ellipse – and simultaneously estimates its impedance characteristics. This direct prediction of boundary parameters and impedance values bypasses the need for classification or intermediate representations, allowing for a continuous output space and enabling precise reconstruction of object properties from far-field data. The regression task is optimized using a loss function that minimizes the difference between the predicted and ground truth parameters.

The Convolutional Neural Network’s (CNN) performance was quantified using established regression metrics, specifically the R² Score and Root Mean Squared Error (RMSE), to assess its capacity for accurate object property reconstruction. Rigorous evaluation demonstrated an overall accuracy of 98.8% in the obstacle classification task, indicating a high degree of reliability in identifying and categorizing objects within the analyzed data. These metrics provide a statistically sound basis for validating the CNN’s effectiveness in discerning subtle variations and precisely determining object characteristics.

The network successfully converged during training and validation on star-shaped obstacles, as indicated by decreasing loss values.

Beyond Simplicity: Validating Generalization with Complexity

Validation of the network’s performance extended to geometrically complex objects beyond standard shapes, specifically including peanut, kite, and star configurations. This testing was conducted to demonstrate the method’s ability to generalize its reconstruction capabilities to non-trivial geometries. Successful performance across these diverse shapes confirms the network is not limited to simple cases and can effectively process and interpret scattered data from objects with varying structural complexities. The chosen geometries represent a range of boundary conditions and scattering characteristics, ensuring a comprehensive assessment of the approach’s robustness and adaptability.

Reconstruction accuracy remained consistently high across tested geometries, validating the robustness of the data-driven approach. Specifically, performance metrics for peanut-shaped objects demonstrated a coefficient of determination ($R^2$) of 99.96 for the x0 coordinate, representing the center location, and a root mean squared error (RMSE) of 0.0012 for the boundary coefficient, α. These results indicate a high degree of correlation between the predicted and actual parameters, confirming the method’s ability to accurately characterize object geometry even with complex shapes.

The data-driven inverse scattering method demonstrated strong performance across star-shaped objects, indicating applicability to complex geometries. Specifically, reconstruction of the boundary coefficient, $\alpha_0$, achieved a coefficient of determination ($R^2$) of 99.90 for star shapes with fixed impedance. Furthermore, for star-shaped objects with variable impedance, the root mean squared error (RMSE) for impedance ($\lambda$) reconstruction was 0.1923. These metrics suggest the method’s potential for solving inverse problems in diverse fields, including non-destructive evaluation, medical imaging, and remote sensing.

Star-shaped reconstructions demonstrate varying levels of error, ranging from a maximum error sample on the left to a minimal error sample on the right, with a representative random sample shown below.

A Holistic View: Connecting Forward and Inverse Problems

A foundational step in solving complex scattering problems lies in thoroughly understanding the direct scattering problem – essentially, predicting how waves will interact with a known object. This involves precisely calculating the scattered fields generated when an electromagnetic or acoustic wave encounters an object with defined material properties and geometry. This predictive capability isn’t merely a preliminary exercise; it forms the bedrock for validating any subsequent inverse reconstruction. By accurately modeling the forward scattering process, researchers can establish a benchmark against which to assess the accuracy of algorithms designed to infer object properties from scattered field measurements. Discrepancies between predicted and observed scattering patterns highlight potential errors in the inverse reconstruction process, guiding refinements and ensuring the reliability of the inferred object characteristics. Consequently, a robust understanding of the direct problem is not simply a prerequisite, but an integral component of a holistic and dependable scattering solution.

Integrating forward and inverse modeling techniques represents a significant advancement in tackling complex scattering problems. Traditionally, these approaches were treated as separate entities; however, a combined methodology allows for mutual validation and refinement. Forward modeling, which predicts scattered fields given an object’s properties, establishes a benchmark against which reconstructions from inverse methods – those that infer object properties from scattered fields – can be assessed. Discrepancies between predicted and reconstructed results highlight areas for improvement in either the modeling algorithms or the reconstruction process, leading to more accurate and robust solutions. This synergistic relationship effectively mitigates the inherent ill-posedness of inverse problems, enhancing reliability and expanding the applicability of scattering analysis to a broader range of real-world scenarios, including enhanced material characterization and improved imaging techniques.

The ability to accurately reconstruct objects from scattered data holds transformative potential across diverse scientific and technological domains. In non-destructive testing, this translates to identifying hidden flaws in materials and structures without causing damage, ensuring safety and reliability. Medical imaging benefits through enhanced resolution and clarity, aiding in earlier and more precise diagnoses of diseases. Furthermore, remote sensing applications, such as environmental monitoring and planetary exploration, gain significantly from improved object reconstruction capabilities, allowing for detailed analysis of distant or inaccessible targets. These fields, and many others, rely heavily on understanding the composition and internal structure of objects, making advancements in reconstruction techniques not merely incremental improvements, but fundamental enablers of progress and innovation.

Both Principal Component Analysis (PCA) and Uniform Manifold Approximation and Projection (UMAP) effectively reduce the dimensionality of far-field data, though UMAP appears to better preserve the underlying data manifold.

The pursuit of reconstructing object properties from scattered electromagnetic waves, as detailed in this work, demands a ruthless paring away of complexity. The presented deep learning approach, leveraging convolutional neural networks, exemplifies this principle by directly mapping scattered field data to object impedance. This eschews cumbersome analytical methods, embracing instead a streamlined, data-driven solution. As Wilhelm Röntgen himself observed, “I have made a contribution to the science, but it is a contribution that has been made by the rays themselves.” The ‘rays’ – in this case, the scattered electromagnetic fields – reveal the obstacle’s characteristics; the challenge lies in distilling that information with clarity, removing superfluous layers of calculation to arrive at a concise understanding of the underlying impedance properties.

What Lies Ahead?

This work offers a reconstruction, not a resolution. The promise of deep learning lies in circumventing explicit modeling. Yet, the underlying physics remains. Abstractions age, principles don’t. Future iterations must confront the limits of learned proxies for Maxwell’s equations. Every complexity needs an alibi.

Current methods excel with idealized geometries. Real-world scenarios introduce heterogeneity, material loss, and noise. The true test resides in imperfect data. Focus should shift from reconstruction fidelity to robust error quantification. Knowing what is unknown proves as valuable as knowing what is known.

Extension to multiple scatterers presents a significant hurdle. Current approaches treat each object in isolation. True impedance tomography demands a holistic view. The path forward isn’t simply ‘more data’, but smarter algorithms. Simplicity, after all, is the ultimate sophistication.

Original article: https://arxiv.org/pdf/2511.20681.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/