Super-resolution in disordered media using neural networks

TL;DR

The paper tackles high-resolution imaging in strongly scattering media by estimating the ambient Green's function sensing matrix ${\cal G}$ from large, diverse data sets. It presents a three-step method: (i) an initial unordered dictionary learning to approximate ${\cal G}$, (ii) a nonconvex optimization to refine the columns, and (iii) a connectivity-based multidimensional scaling to order the columns and reconstruct the image grid, enabling super-resolution via an effective aperture. It also introduces an unlabeled dictionary-learning approach using encoder-decoder neural networks trained on data without ground-truth labels, followed by DBSCAN clustering to form a robust dictionary and subsequent ordering by MDS. Numerical simulations in the C-band demonstrate cross-range super-resolution and accurate grid reconstruction, with results comparable to time-reversal focusing in random media. The dual paths—classical optimization and neural-network-based learning—highlight practical trade-offs: controllable accuracy versus initialization requirements and robustness to randomness, offering a viable route to super-resolved imaging in complex media.

Abstract

We propose a methodology that exploits large and diverse data sets to accurately estimate the ambient medium's Green's functions in strongly scattering media. Given these estimates, obtained with and without the use of neural networks, excellent imaging results are achieved, with a resolution that is better than that of a homogeneous medium. This phenomenon, also known as super-resolution, occurs because the ambient scattering medium effectively enhances the physical imaging aperture. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

Figures (7)

• Figure 1: Imaging schematic and illustration of the effective aperture. In the simulations, the cross-range resolution in homogeneous media is $\lambda L/a=8\lambda$, where $\lambda$ is the central wavelength, $a$ is the physical array aperture and $L$ is the range. The range resolution is $c_0/B$ where $c_0$ is the background propagation speed and $B$ is the bandwidth. In the scattering medium the resolution is about $2\lambda$ which translates to an effective aperture that is four times the physical aperture of the array, i.e.,$a_{eff} \approx 4 a = 120 \lambda$. The range resolution in the scattering medium does not change.
• Figure 2: Physical and effective aperture Left: Comparison of range and cross-range between random and homogeneous medium, with fixed array size and bandwidth. Super-resolution in cross-range is observed compared to the homogeneous medium, where resolution in cross-range is $\lambda L/a = 8\lambda$ and in range it is $c/B=5 \lambda$. Middle: Effect of bandwidth with fixed array size: No effect on resolution, but a large enough bandwidth guarantees statistical stability over different random media with similar statistics. Right: Effect of physical array size with fixed bandwidth. The array size does not affect the main peak in different random media but the noise level off the main peak decreases with array size.
• Figure 3: Grid reconstruction using multidimensional scaling. Left: the grid as reconstructed using the true Green's functions; Middle: grid obtained from the estimated Green's functions using MDS; Right: the grid obtained from MDS after rotation and scaling assuming the location of 3 "anchor" points is known, superimposed over true grid, plotted with red circles. This is the reconstruction that leads to the images shown in Figure \ref{['fig:image']}.
• Figure 4: Super-resolution vs. time reversal. Top left: physical time reversal focusing on the known grid using the true Green's functions. Top right: the proposed imaging algorithm, migration image using the estimated dictionary elements on the reconstructed grid from the ordered sensing matrix. We can see that we achieve similar resolution in both cases, which is able to resolve to closely located sources. Bottom left: migration image in a homogeneous medium. Bottom right: Homogeneous medium Green's function migration applied to random medium data. The resolution is inferior to that obtained by the estimated sensing matrix. Array size is $30 \lambda$ and the bandwidth is $1GHz$, with a $5GHz$ central frequency.
• Figure 5: Neural network architecture. We use a two hidden layer network for the encoder. The output of the encoder is fed into the linear decoder network. Modulus thresholding activation \ref{['e:leakyrelu']} is used in the red layers and LeakyReLU activation is used for the blue layers. Before activation functions are applied, the input is standardized using batch normalization, as described in Ioffe2015. The green layer is linear, without an activation function applied. The hidden dimensions of the encoder networks are $4096$, and $2048$, respectively. The decoder matrix has $1024$ columns. We train the networks to minimize the loss function $\mathcal{L}= \|I(y)-y\|^2_2+\mu \|\ E(y)\|_1.$