Optical Inverse Problem Research Articles

Extractor-attention-predictor network for quantitative photoacoustic tomography

Quantitative photoacoustic tomography (qPAT) holds great potential in estimating chromophore concentrations, whereas the involved optical inverse problem, aiming to recover absorption coefficient distributions from photoacoustic images, remains challenging. To address this problem, we propose an extractor-attention-predictor network architecture (EAPNet), which employs a contracting–expanding structure to capture contextual information alongside a multilayer perceptron to enhance nonlinear modeling capability. A spatial attention module is introduced to facilitate the utilization of important information. We also use a balanced loss function to prevent network parameter updates from being biased towards specific regions. Our method obtains satisfactory quantitative metrics in simulated and real-world validations. Moreover, it demonstrates superior robustness to target properties and yields reliable results for targets with small size, deep location, or relatively low absorption intensity, indicating its broader applicability. The EAPNet, compared to the conventional UNet, exhibits improved efficiency, which significantly enhances performance while maintaining similar network size and computational complexity.

Removal of algorithmic stagnation by augmented iterative phase retrieval

Retrieving the phase of an optical field using intensity measurements is one of the most widespread and studied inverse problems in classical optics. However, common iterative approaches such as the Gerchberg-Saxton algorithm and its derivatives suffer from the twin-image problem – the iterative minimisation stagnates and the recovered field contains features from both the target field and its point-reflection. We present a technique that leverages mathematical properties of the stagnated field, to constrain the problem and remove the twin image artefacts. This improvement in reconstruction robustness has implications in a range of fields, including applications in adaptive optics, holography and optical communications.

Freeform surface topology prediction for prescribed illumination via semi-supervised learning.

Despite significant advances in the field of freeform optical design, there still remain various unsolved problems. One of these is the design of smooth, shallow freeform topologies, consisting of multiple convex, concave and saddle shaped regions, in order to generate a prescribed illumination pattern. Such freeform topologies are relevant in the context of glare-free illumination and thin, refractive beam shaping elements. Machine learning techniques already proved to be extremely valuable in solving complex inverse problems in optics and photonics, but their application to freeform optical design is mostly limited to imaging optics. This paper presents a rapid, standalone framework for the prediction of freeform surface topologies that generate a prescribed irradiance distribution, from a predefined light source. The framework employs a 2D convolutional neural network to model the relationship between the prescribed target irradiance and required freeform topology. This network is trained on the loss between the obtained irradiance and input irradiance, using a second network that replaces Monte-Carlo raytracing from source to target. This semi-supervised learning approach proves to be superior compared to a supervised learning approach using ground truth freeform topology/irradiance pairs; a fact that is connected to the observation that multiple freeform topologies can yield similar irradiance patterns. The resulting network is able to rapidly predict smooth freeform topologies that generate arbitrary irradiance patterns, and could serve as an inspiration for applying machine learning to other open problems in freeform illumination design.

Mathematical and numerical challenges in diffuse optical tomography inverse problems

Mathematical and numerical challenges in diffuse optical tomography inverse problems

Quantitative photoacoustic tomography: modeling and inverse problems.

Quantitative photoacoustic tomography (QPAT) exploits the photoacoustic effect with the aim of estimating images of clinically relevant quantities related to the tissue's optical absorption. The technique has two aspects: an acoustic part, where the initial acoustic pressure distribution is estimated from measured photoacoustic time-series, and an optical part, where the distributions of the optical parameters are estimated from the initial pressure. Our study is focused on the optical part. In particular, computational modeling of light propagation (forward problem) and numerical solution methodologies of the image reconstruction (inverse problem) are discussed. The commonly used mathematical models of how light and sound propagate in biological tissue are reviewed. A short overview of how the acoustic inverse problem is usually treated is given. The optical inverse problem and methods for its solution are reviewed. In addition, some limitations of real-life measurements and their effect on the inverse problems are discussed. An overview of QPAT with a focus on the optical part was given. Computational modeling and inverse problems of QPAT were addressed, and some key challenges were discussed. Furthermore, the developments for tackling these problems were reviewed. Although modeling of light transport is well-understood and there is a well-developed framework of inverse mathematics for approaching the inverse problem of QPAT, there are still challenges in taking these methodologies to practice. Modeling and inverse problems of QPAT together were discussed. The scope was limited to the optical part, and the acoustic aspects were discussed only to the extent that they relate to the optical aspect.

Dense speed-of-sound shift imaging for ultrasonic thermometry

Objective. Develop a dense algorithm for calculating the speed-of-sound shift between consecutive acoustic acquisitions as a noninvasive means to evaluating temperature change during thermal ablation. Methods. An algorithm for dense speed-of-sound shift imaging (DSI) was developed to simultaneously incorporate information from the entire field of view using a combination of dense optical flow and inverse problem regularization, thus speeding up the calculation and introducing spatial agreement between pixels natively. Thermal ablation monitoring consisted of two main steps: pixel shift tracking using Farneback optical flow, and mathematical modeling of the relationship between the pixel displacement and temperature change as an inverse problem to find the speed-of-sound shift. A calibration constant translates from speed-of-sound shift to temperature change. The method performance was tested in ex vivo samples and compared to standard thermal strain imaging (TSI) methods. Main results. Thermal ablation at a frequency of 2 MHz was applied to an agarose phantom that created a speed-of-sound shift measured by an L12-5 imaging transducer. A focal spot was reconstructed by solving the inverse problem. Next, a thermocouple measured the temperature rise during thermal ablation of ex vivo chicken breast to calibrate the setup. Temperature changes between 3 °C and 15 °C was measured with high thermometry precision of less than 2 °C error for temperature changes as low as 8 °C. The DSI method outperformed standard TSI in both spatial coherence and runtime in high-intensity focused ultrasound-induced hyperthermia. Significance. Dense ultrasonic speed-of-sound shift imaging can successfully monitor the speed-of-sound shift introduced by thermal ablation. This technique is faster and more robust than current methods, and therefore can be used as a noninvasive, real time and cost-effective thermometry method, with high clinical applicability.

End-to-End Diverse Metasurface Design and Evaluation Using an Invertible Neural Network.

Employing deep learning models to design high-performance metasurfaces has garnered significant attention due to its potential benefits in terms of accuracy and efficiency. A deep learning-based metasurface design framework typically comprises a forward prediction path for predicting optical responses and a backward retrieval path for generating geometrical configurations. In the forward design path, a specific geometrical configuration corresponds to a unique optical response. However, in the inverse design path, a single performance metric can correspond to multiple potential designs. This one-to-many mapping poses a significant challenge for deep learning models and can potentially impede their performance. Although representing the inverse path as a probabilistic distribution is a widely adopted method for tackling this problem, accurately capturing the posterior distribution to encompass all potential solutions remains an ongoing challenge. Furthermore, in most pioneering works, the forward and backward paths are captured using separate models. However, the knowledge acquired from the forward path does not contribute to the training of the backward model. This separation of models adds complexity to the system and can hinder the overall efficiency and effectiveness of the design framework. Here, we utilized an invertible neural network (INN) to simultaneously model both the forward and inverse process. Unlike other frameworks, INN focuses on the forward process and implicitly captures a probabilistic model for the inverse process. Given a specific optical response, the INN enables the recovery of the complete posterior over the parameter space. This capability allows for the generation of novel designs that are not present in the training data. Through the integration of the INN with the angular spectrum method, we have developed an efficient and automated end-to-end metasurface design and evaluation framework. This novel approach eliminates the need for human intervention and significantly speeds up the design process. Utilizing this advanced framework, we have effectively designed high-efficiency metalenses and dual-polarization metasurface holograms. This approach extends beyond dielectric metasurface design, serving as a general method for modeling optical inverse design problems in diverse optical fields.

Some inverse problems of Fourier optics

We consider a general formulation of the problem of signal reconstruction from incomplete a priori information about it and measurements of the intensity of its Fourier transform. Some special cases are studied when a priori information is the even or odd part of the signal, as well as the real or imaginary part of the signal. Exact solutions in quadratures are constructed. An algorithm for solving the problem is proposed when only the signal and image intensities are known.

Analysis of relative error in perturbation Monte Carlo simulations of radiative transport.

Perturbation and differential Monte Carlo (pMC/dMC) methods, used in conjunction with nonlinear optimization methods, have been successfully applied to solve inverse problems in diffuse optics. Application of pMC to systems over a large range of optical properties requires optimal "placement" of baseline conventional Monte Carlo (cMC) simulations to minimize the pMC variance. The inability to predict the growth in pMC solution uncertainty with perturbation size limits the application of pMC, especially for multispectral datasets where the variation of optical properties can be substantial. We aim to predict the variation of pMC variance with perturbation size without explicit computation of perturbed photon weights. Our proposed method can be used to determine the range of optical properties over which pMC predictions provide sufficient accuracy. This method can be used to specify the optical properties for the reference cMC simulations that pMC utilizes to provide accurate predictions over a desired optical property range. We utilize a conventional error propagation methodology to calculate changes in pMC relative error for Monte Carlo simulations. We demonstrate this methodology for spatially resolved diffuse reflectance measurements with ±20% scattering perturbations. We examine the performance of our method for reference simulations spanning a broad range of optical properties relevant for diffuse optical imaging of biological tissues. Our predictions are computed using the variance, covariance, and skewness of the photon weight, path length, and collision distributions generated by the reference simulation. We find that our methodology performs best when used in conjunction with reference cMC simulations that utilize Russian Roulette (RR) method. Specifically, we demonstrate that for a proximal detector placed immediately adjacent to the source, we can estimate the pMC relative error within 5% of the true value for scattering perturbations in the range of . For a distal detector placed at transport mean free paths relative to the source, our method provides relative error estimates within 20% for scattering perturbations in the range of . Moreover, reference simulations performed at lower values showed better performance for both proximal and distal detectors. These findings indicate that reference simulations utilizing continuous absorption weighting (CAW) with the Russian Roulette method and executed using optical properties with a low ratio spanning the desired range of values, are highly advantageous for the deployment of pMC to obtain radiative transport estimates over a wide range of optical properties.

The Optical Inverse Problem in Quantitative Photoacoustic Tomography: A Review

Photoacoustic tomography is a fast-growing biomedical imaging modality that combines rich optical contrast with a high acoustic resolution, at depths in tissues. Building upon the foundation of this technique, novel quantitative photoacoustic tomography fully leverages its advantages while further delivering improved quantification capabilities to produce high-accuracy concentration estimates, which has attracted substantial research interest in recent years. The kernel challenge associated with quantitative photoacoustic tomography is an optical inverse problem aiming to recover the absorption coefficient distribution from the conventional photoacoustic image. Although the crucial importance of the optical inversion has been widely acknowledged, achieving it has remained a persistent challenge due to the inherent non-linearity and non-uniqueness. In the past decade, numerous methods were proposed and have made noticeable progress in addressing this concern. Nevertheless, a review has been conspicuously absent for a long time. Aiming to bridge this gap, the present study comprehensively investigates the recent research in this field, and methods identified with significant value are introduced in this paper. Moreover, all included methods are systematically classified based on their underlying principles. Finally, we summarize each category and highlight its remaining challenges and potential future research directions.

Level set‐based shape optimization approach for the inverse optical tomography problem

Level set‐based shape optimization approach for the inverse optical tomography problem

100 years of Emil Wolf: introduction.

The groundbreaking research and ideas introduced by Emil Wolf continue to inspire researchers and motivate ongoing research in the wave properties of light. This special issue commemorates the legacy of Emil Wolf with research in physical optics, with specific focus on those areas where Wolf was active, such as optical coherence theory, inverse problems, singular optics, imaging, and polarization, and the intersection of these fields of study. Here we discuss the life of Emil Wolf and his influence on optical science and the optics community.

Implementing neural network approach to create carbon-based optical nanosensor of heavy metal ions in liquid media

The present study is devoted to the creation of multifunctional optical carbon dots-based nanosensor to simultaneously measure concentrations of metal ions (Cu2+, Ni2+, Cr3+) and NO3– anion in liquid media. Such nanosensor operates on the basis of its fluorescence (FL) change under the influence of ions in the medium. However, the absence of analytical model, describing CD FL mechanism, the superposition of various luminescence quenching mechanisms during the interaction of carbon dots (CD) with cations, hampers the usage of classical approaches to solve this inverse multiparametric spectroscopic problem. To solve it neural networks were used that analyzed complex fluorescence signal from CD aqueous suspensions comprising Cu2+, Ni2+, Cr3+, NO3– ions in the concentration range from 0 to 4.95 mM. The following neural network architectures ensured optical spectroscopy inverse problem solution: multilayer perceptrons, 1D and 2D convolutional neural networks.The developed sensor enables simultaneous determination of the concentrations of heavy metal ions Cu2+, Ni2+, Cr3+ with a root mean squared error of 0.28 mM, 0.79 mM, 0.24 mM respectively. Based on the data given in the literature we can assert that the accuracy of the studied nanosensor satisfies the needs of monitoring the composition of waste and technological water. The developed nanosensor has a unique multimodality: with the simplicity of the synthesis protocol the sensor enables simultaneous determination of three heavy metal ions concentrations, while analogues are being developed mainly to measure the concentration of one (in rare cases two) heavy metal ions.

Method for reconstruction of axisymmetric capillary wave surface topography using inverse ray-tracing of refracted laser sheet

Capillary waves can be used to measure the fundamental fluid properties such as surface tension as well as, potentially, the viscosity of Newtonian fluids. This requires the measurement of various wave parameters, mainly wavelength, amplitude, and decay coefficient. However, the different scales of magnitudes make it a challenging task. Optical methods are well suited to analyze such problems due to their non-intrusive nature and high dynamic measurement resolution in both space and time. These methods are further categorized as point methods for a single probe measurement and space–time methods for transient measurement of the complete surface. Dynamic space–time methods are preferred despite the associated complex post-processing since they enable reconstruction of the wave surface. Some existing methods are discussed, and an improved method is then proposed to actually solve the associated inverse optics problem. In the method, an axisymmetric wave surface is re-constructed by analyzing the refracted laser sheet. The assumptions, simplifications, and constraints are taken to be compatible with experimental aspects for future validation. It is derived using the fundamental concepts in physics and the only major assumption of the axisymmetric nature of wave surface. The method exploits the underlying symmetry in the topography, making it more versatile, and suited for linear and non-linear capillary waves and waves with planar wavefront. The impact of parameters on the final result is determined through numerical simulations. Very low error (average and maximum) values are observed between reference and reconstructed topography for damped and undamped wave surfaces with a wide range of curvatures. Optimum values of critical parameters and associated reasoning are presented.

Photoacoustic image synthesis with generative adversarial networks

Photoacoustic tomography (PAT) has the potential to recover morphological and functional tissue properties with high spatial resolution. However, previous attempts to solve the optical inverse problem with supervised machine learning were hampered by the absence of labeled reference data. While this bottleneck has been tackled by simulating training data, the domain gap between real and simulated images remains an unsolved challenge. We propose a novel approach to PAT image synthesis that involves subdividing the challenge of generating plausible simulations into two disjoint problems: (1) Probabilistic generation of realistic tissue morphology, and (2) pixel-wise assignment of corresponding optical and acoustic properties. The former is achieved with Generative Adversarial Networks (GANs) trained on semantically annotated medical imaging data. According to a validation study on a downstream task our approach yields more realistic synthetic images than the traditional model-based approach and could therefore become a fundamental step for deep learning-based quantitative PAT (qPAT).

Iterative phase retrieval with a sensor mask.

As an important inverse imaging problem in diffraction optics, Fourier phase retrieval aims at estimating the latent image of the target object only from the magnitude of its Fourier measurement. Although in real applications alternating methods are widely-used for Fourier phase retrieval considering the constraints in the object and Fourier domains, they need a lot of initial guesses and iterations to achieve reasonable results. In this paper, we show that a proper sensor mask directly attached to the Fourier magnitude can improve the efficiency of the iterative phase retrieval algorithms, such as alternating direction method of multipliers (ADMM). Furthermore, we refer to the learning-based method to determine the sensor mask according to the Fourier measurement, and unrolled ADMM is used for phase retrieval. Numerical results show that our method outperforms other existing methods for the Fourier phase retrieval problem.

Novel regularization method for diffuse optical tomography inverse problem

Diffuse optical tomography (DOT) is an optical imaging technique using near-infrared spectroscopy (NIRS). Because the NIR light is highly scattered in biological tissue, the photon density drops rapidly with increasing depth. This makes the measurement accuracy of DOT in deep tissue significantly lower than in superficial tissue. Classical Regularized Least Square (RLS) based regularization methods with different penalty term (l1-norm, l2-norm, LASSO, Ridge, and Elastic Net etc.) come up short in finding the inclusions when the inclusion number is increased or one of the inclusion located on the upper surface and the other one located on the deeper surface of the tissue. This is the depth problem encountered in DOT. In this paper, a new least square based regularization method is proposed. Within the scope of the proposed method, both the weight matrix and penalty terms are modified in order to solve this problem. In this process, firstly, the A matrix is changed using the diagonal value of AAT based on DCA (Depth Compensation Algorithm). Then, a new penalty term using modified A matrix is added. It has been observed that when the noise level increases, proposed method give higher accuracy and more robust according to other RLS Method on synthetic data. Experimental results on synthetic data show that the new method improves inversion accuracy, reduces errors, and the lowers inversion instability compared to the state-of-the-art methods.

Information-statistical approach to inverse optical problem solution for 3D disperse systems with nano- and micro particles

Multiparameter analysis of simultaneous optical data for 3D disperse systems (consisted from nano- and/or microparticles of different nature) by information-statistical methods can help to estimate the share of different types of particles in mixtures. At the solution of inverse optical problem for unknown poly-component 3D DS, the comparison of measured parameters with the known ones from the set of mono-component 3D DS can help to identify the component content of the system under study. The approach was tested on the biomineral water mixtures of kaolin clay and bacterium coli bacillus with the help of the program based on the information-statistical theory. To solve the impurity optical recognition tasks, the Base of optical data for 3D DS is needed.

Study of optical and cooperative parameters of fibrous quartz heat shielding on experimental data on its reflection spectra

The paper presents the results of a study of a number of key optical characteristics of some highly porous thermal protection materials based on amorphous SiO2 fibers in the near and mid IR ranges. In the optical part of the statistical simulation model of such materials, based on the rigorous Mie theory, its application requires the introduction of cooperative corrections to the results of interaction with electromagnetic radiation of individual fragments. These corrections are introduced into the model in the form of an additional multiplier (Cs) of the scattering and absorption Mie efficiencies, which was usually used as a model tuning parameter when interpreting the experimental results of thermophysical studies. In this work, using the example of TZM-23M materials (Russia), made of a relatively thick (~9 μm) silica fiber PS-23 (Belarus), for the first time, data were obtained that make it possible to study the Cs parameter as a spectral quantity. The study is based on the results of experimental determination of the spectral hemispherical reflectivity of layers of material of two or more optical thicknesses in the wavelength range 0.83-16.65 μm. These data and Zege's asymptotic formulas were used to estimate the values of the absorption and scattering coefficients. The spectral parameter Cs is determined from the value of the scattering coefficient in the process of solving the inverse optical problem. The data obtained are compared with the results calculated on the basis of classical approximations of the optical constants. The results of this work can be useful for specialists in the field of radiation heat transfer and the interaction of electromagnetic radiation with complex partially transparent media with high spectral albedo scattering values. Keywords: Silica fibrous thermal protection, reflectivity, experiment, absorption and scattering spectrum, optical model, Mie theory, cooperative corrections.

Исследование оптических и кооперативных параметров волокнистой кварцевой теплозащиты по экспериментальным данным о ее спектрах отражения

The paper presents the results of a study of a number of key optical characteristics of some highly porous thermal protection materials based on amorphous SiO2 fibers in the near and mid IR ranges. In the optical part of the statistical simulation model of such materials, based on the rigorous Mie theory, its application requires the introduction of cooperative corrections to the results of interaction with electromagnetic radiation of individual fragments. These corrections are introduced into the model in the form of an additional multiplier Cs of the scattering and absorption Mie efficiencies, which was usually used as a model tuning parameter when interpreting the experimental results of thermophysical studies. In this work, using the example of TZM-23M materials (Russia), made of a relatively thick (~ 9 μm) silica fiber PS-23 (Belarus), for the first time, data were obtained that make it possible to study the Cs parameter as a spectral quantity. The study is based on the results of experimental determination of the spectral hemispherical reflectivity of layers of material of two or more optical thicknesses in the wavelength range 0.83 - 16.65 μm. These data and Zege's asymptotic formulas were used to estimate the values of the absorption and scattering coefficients. The spectral parameter Cs is determined from the value of the scattering coefficient in the process of solving the inverse optical problem. The data obtained are compared with the results calculated on the basis of classical approximations of the optical constants. The results of this work can be useful for specialists in the field of radiation heat transfer and the interaction of electromagnetic radiation with complex partially transparent media with high spectral albedo scattering values.