Problem Of Optical Tomography Research Articles

Inverse problems of combined photoacoustic and optical coherence tomography.

Optical coherence tomography (OCT) and photoacoustic tomography are emerging non‐invasive biological and medical imaging techniques. It is a recent trend in experimental science to design experiments that perform photoacoustic tomography and OCT imaging at once. In this paper, we present a mathematical model describing the dual experiment. Because OCT is mathematically modelled by Maxwell's equations or some simplifications of it, whereas the light propagation in quantitative photoacoustics is modelled by (simplifications of) the radiative transfer equation, the first step in the derivation of a mathematical model of the dual experiment is to obtain a unified mathematical description, which in our case are Maxwell's equations. As a by‐product, we therefore derive a new mathematical model of photoacoustic tomography based on Maxwell's equations. It is well known by now that without additional assumptions on the medium, it is not possible to uniquely reconstruct all optical parameters from either one of these modalities alone. We show that in the combined approach, one has additional information, compared with a single modality, and the inverse problem of reconstruction of the optical parameters becomes feasible. © 2016 The Authors. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.

Radiative transport and optical tomography with large datasets.

We consider the inverse problem of optical tomography in the radiative transport regime. We report numerical tests of a direct reconstruction method that is suitable for use with large datasets. Reconstructions of experimental data obtained from a noncontact optical tomography system are also reported.

Pixel-based parametric source depth map for Cerenkov luminescence imaging

Optical tomography represents a challenging problem in optical imaging because of the intrinsically ill-posed inverse problem due to photon diffusion. Cerenkov luminescence tomography (CLT) for optical photons produced in tissues by several radionuclides (i.e.: 32P, 18F, 90Y), has been investigated using both 3D multispectral approach and multiviews methods. Difficult in convergence of 3D algorithms can discourage to use this technique to have information of depth and intensity of source. For these reasons, we developed a faster 2D corrected approach based on multispectral acquisitions, to obtain source depth and its intensity using a pixel-based fitting of source intensity. Monte Carlo simulations and experimental data were used to develop and validate the method to obtain the parametric map of source depth. With this approach we obtain parametric source depth maps with a precision between 3% and 7% for MC simulation and 5–6% for experimental data. Using this method we are able to obtain reliable information about the source depth of Cerenkov luminescence with a simple and flexible procedure.

Determination of space-dependent radiative properties in diffuse optical tomography using a wavelet multi-scale method

This paper deals with the estimation of radiative property distributions of participating media from a set of light sources and sensors located on the boundaries of a medium. This is the so-called diffuse optical tomography problem. Such a non-linear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is designed.

Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, theL2-inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.

Super-aperture metrology: overcoming a fundamental limit in imaging smooth highly curved surfaces.

The imaging of smooth, highly curved or tilted surfaces is widely recognized as one of the most challenging and unsolved problems in optical imaging and metrology today. The reason is that even when such surfaces are imaged using high aperture microscope objectives the steepness of the features causes the light to be reflected in such a way that it is not captured by the lens. This is true even in the limiting case of unity numerical aperture since the illuminating light may also be reflected in the forward direction. In order to overcome this fundamental problem we have developed a method whereby such specimens are covered with a readily removable organic fluorescent film thereby creating an isotropic scattering surface. We show that we are readily able to detect slopes with angles close 90° using a 0.75 NA objective--an 82% improvement over the theoretical aperture limit. Issues of variation in film thickness deposition are shown to be readily accommodated. This approach may be used with other fluorophore materials, organic or inorganic, since there is no need for biocompatibility in this application.

Regularization opportunities for the diffuse optical tomography problem

In optical tomography, the radiative properties of the medium under investigation are estimated from information contained in measurements provided by a set of light sources and sensors located on the frontier of the probed medium. Such a non-linear ill-posed inverse problem is usually solved through optimization with the help of gradient-type methods. Since it is well known that such inverse problems are ill-posed, regularization is required. This paper compares three distinct regularization strategies for two different optimization algorithms, namely the damped Gauss–Newton and BFGS algorithms, for the two-dimensional diffuse approximation to the radiative transfer equation in the frequency domain. More specifically, the mesh-based regularization is combined with the Tikhonov regularization in the damped Gauss–Newton algorithm. For the BFGS algorithm, the mesh-based regularization is combined with the Sobolev gradients method. Moreover, a space-dependent Sobolev gradients method is proposed for the first time. The performance of the proposed algorithms are compared by utilizing synthetic data. The deviation factor and correlation coefficient are used to quantitatively compare the final reconstructions. Also, three levels of noise are considered to characterize the behaviour of the proposed methods against measurement noises. Numerical results indicate that the BFGS algorithm outperforms the damped Gauss–Newton in many aspects.

Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography.

In optical tomography, there exist certain spatial frequency components that cannot be measured due to the limited projection angles imposed by the numerical aperture of objective lenses. This limitation, often called as the missing cone problem, causes the under-estimation of refractive index (RI) values in tomograms and results in severe elongations of RI distributions along the optical axis. To address this missing cone problem, several iterative reconstruction algorithms have been introduced exploiting prior knowledge such as positivity in RI differences or edges of samples. In this paper, various existing iterative reconstruction algorithms are systematically compared for mitigating the missing cone problem in optical diffraction tomography. In particular, three representative regularization schemes, edge preserving, total variation regularization, and the Gerchberg-Papoulis algorithm, were numerically and experimentally evaluated using spherical beads as well as real biological samples; human red blood cells and hepatocyte cells. Our work will provide important guidelines for choosing the appropriate regularization in ODT.

Improved regularization reconstruction from sparse angle data in optical diffraction tomography.

In this paper, we propose an improved deterministic regularization algorithm to handle the sparse angle data problem in optical diffraction tomography. Based on optical diffraction tomography and the deterministic regularization algorithm, the regularization iteration is performed in the space domain and the frequency domain simultaneously, which greatly reduces the computational cost. By applying piecewise-smoothness and positivity constraints as the penalty function, the missing frequency spectrum is effectively recovered and the internal refractive index distribution of the specimen is accurately reconstructed. Using simulated and experimental results, we show that the proposed regularization algorithm allows accurate refractive index reconstruction from very sparse angle data in optical diffraction tomography.

A wavelet multi-scale method for the inverse problem of diffuse optical tomography

This paper deals with the estimation of optical property distributions of participating media from a set of light sources and sensors located on the boundaries of the medium. This is the so-called diffuse optical tomography problem. Such a non-linear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is developed, implemented, and validated. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Numerical results show that the proposed method brings more stability with respect to the ordinary L-BFGS method and enhances the reconstructed images for most of initial guesses of optical properties.

Direct Sampling Method for Diffusive Optical Tomography

In this work, we are concerned with the diffusive optical tomography (DOT) problem in the case when only one or two pairs of Cauchy data are available. We propose a simple and efficient direct sampling method to locate inhomogeneities inside a homogeneous background and solve the DOT problem in both full and limited aperture cases. This new method is easy to implement and less expensive computationally. Numerical experiments demonstrate its effectiveness and robustness against noise in the data. This provides a new promising numerical strategy for the DOT problem.

The Toast++ software suite for forward and inverse modeling in optical tomography.

We present the Toast++ open-source software environment for solving the forward and inverse problems in diffuse optical tomography (DOT). The software suite consists of a set of libraries to simulate near-infrared light propagation in highly scattering media with complex boundaries and heterogeneous internal parameter distribution, based on a finite-element solver. Steady-state, time- and frequency-domain data acquisition systems can be modeled. The forward solver is implemented in C++ and supports performance acceleration with parallelization for shared and distributed memory architectures, as well as graphics processing computation. Building on the numerical forward solver, Toast++ contains model-based iterative inverse solvers for reconstructing the volume distribution of absorption and scattering parameters from boundary measurements of light transmission. A range of regularization methods are provided, including the possibility of incorporating prior knowledge of internal structure. The user can link to the Toast++ libraries either directly to compile application programs for DOT, or make use of the included MATLAB and PYTHON bindings to generate script-based solutions. This approach allows rapid prototyping and provides a rich toolset in both environments for debugging, testing, and visualization.

Near-Field Imaging of Infinite Rough Surfaces in Dielectric Media

This paper is concerned with an inverse surface scattering problem in near-field optical imaging, which is to reconstruct the scattering surface of a dielectric medium with a resolution beyond the diffraction limit. It is a nontrivial extension of the authors' work on near-field imaging of infinite rough surfaces from impenetrable to penetrable media [G. Bao and P. Li, SIAM J. Appl. Math., 73 (2013), pp. 2162--2187], where a more sophisticated transmission problem needs to be considered. The scattering surface is modeled as a small and smooth deformation of a plan surface. Based on a transformed field expansion, an analytic solution, which is given as a power series, is derived for the direct scattering problem. By neglecting high order terms in the power series, the original nonlinear inverse problem is linearized; explicit and unified reconstruction formulas are deduced for both reflection and transmission configurations. A spectral cut-off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components, which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near-field regime. The method requires only a single illumination at a fixed frequency and is realized efficiently by the fast Fourier transform. Numerical results show that the method is simple, stable, and effective in reconstructing scattering surfaces of dielectric media with subwavelength resolution.

Near-field imaging of perfectly conducting grating surfaces

A novel approach is presented to solving the inverse diffractive grating problem in near-field optical imaging, which is to reconstruct perfectly conducting grating surfaces with resolution beyond the diffraction limit. The grating surface is assumed to be a small and smooth deformation of a plane surface. An analytical solution of the direct grating problems is derived by using the method of transformed field expansion. Based on the analytic solution, an explicit reconstruction formula is deduced for the inverse grating problem. The method requires only a single incident field and is realized efficiently by using the fast Fourier transform. Numerical results show that the method is simple, stable, and effective in reconstructing grating surfaces with super-resolved resolution.

The study of sensitivity and precession of a single-source diffused tomography for detecting of target depth in biological phantom

Optical imaging can be used to study the cancerous stages of breast cancer; and this imaging is noninvasive and safe for healthy tissues. The key problem of optical imaging is the image reconstruction which depends on photon migration in biological tissues; because the study of photon migration in the biological tissues is a complicated problem. The diffusion equation is sometimes used to simulate the photon migration in the biological tissues. Due to limitation of diffusion equation and its approximated nature, we want to explore the accuracy and precision of this method. So in this study, we report the design of a single-source diffuse imaging system with simulating code based on finite element method (FEM) to detect the location of tumor in breast phantoms. The comparison between the reconstructed results and actual values can be considered as a criterion of accuracy of this diffused system. In this experimental setup, to reduce the expense of computational procedure, single source is applied, and the accuracy and precision of this single-source are investigated.

Computational Methodology of Optical Molecular Imaging

Molecular imaging emerging in the early 21st century has been rapidly developed. Over more than 10 years, optical molecular imaging has become an important tool in preclinical research and has been translated into the clinical for clinical diagnosis and therapy. Computational methodology has shown its indispensable position in developing molecular probes, building imaging instruments, researching reconstruction algorithms, and studying signal and image processing. Relevant research not only adapts existing computational methods but also develops novel strategies to tackle with challenging problems in optical molecular imaging. This special issue is devoted to the topic of computational algorithms and strategies in optical molecular imaging. Five exciting papers assembled in this special issue present a common theme, that is, improving the imaging and image processing performance in optical molecular imaging by using computational techniques. The papers address several problems in optical molecular imaging, including optical tomographic reconstruction for bioluminescence tomography and image processing such as image segmentation, 3D rendering, and cell morphological measurements. In detail, J. Yu et al. make use of a hybrid multilevel adaptive finite element reconstruction scheme and sparse regularization to acquire quantitative information of bioluminescent sources. The work of Q. Wu et al. is to take advantage of a high-order photon propagation model and Bregman optimization algorithm to improve bioluminescence tomography. The morphology information of living cells and 3D tissue images at the micrometer level can be obtained by using existing computational algorithms from the developed digital holographic microscopy and multichannel spectral imaging laser scanning confocal microscope in the research progress of Y. Wang et al. and Y. Zhang et al., respectively. J. Zheng et al. report that retinal vessels are also efficiently extracted with multiscale hessian-enhancement-based nonlocal mean filter.

Addressing the Inverse Problem of Far-Field Imaging: A Noniterative Exact Solution for Phase in Imaging

This talk describes an approach to a solution of inverse problems in far-field imaging with application to optical, electron/ion beam, x-ray and other imaging modalities. Two aspects of the inverse problem in optical imaging are phase and resolution. Whereas techniques such as STED, PALM, STORM etc address the latter, they lack phase information as does near-field optics. Furthermore, all of these techniques are serial. We introduce an approach based on a controllable nanoscopic far-field optical point source integrated into parallel imaging. In the first demonstration of this development we focus on the inverse problem of the phase of an image which has never been solved exactly and can only be approached through iterative methods with all their problems of nonconvergence, slow convergence, convergence to local minima, and stagnation. We now show that it is possible to obtain, with such a method, an exact solution to the inverse problem of phase both experimentally and theoretically. Our method is based on the breakthrough that crystallography experienced in phase retrieval for large molecular entities by Max Perutz's introduction of “heavy atoms” using the method of isomorphous replacement. Scanning probe microscopy and its full integration with optical microscopy allows us to apply these X-ray concepts to implement “heavy atom” restoration of phase in optical phase retrieval. In analogy to the heavy atom method, we acquire Fourier intensities in place of an X-ray diffraction pattern, and in place of the heavy atom, we utilize a nanometrically translatable point source of light, coherently related to the far-field illumination this leads to 3D optical imaging. The methodology has super-resolution potential, and thus, heavy atom restoration of phase with super-resolution (HARPS) has the potential to provide super-resolution 3D images in real time.

Near-Field Imaging of Infinite Rough Surfaces

This paper is concerned with an inverse rough surface scattering problem in near-field optical imaging, which is to reconstruct the scattering surface with a resolution beyond the diffraction limit. The surface is assumed to be a small and smooth deformation of a plane surface. Based on a transformed field expansion, the boundary value problem with complex scattering surface is converted into a successive sequence of two-point boundary value problems in the frequency domain, where an analytic solution for the direct scattering problem is derived from the method of integrated solution. By neglecting the high order terms in the power series expansion, the nonlinear inverse problem is linearized and an explicit inversion formula is obtained. A spectral cut-off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components, which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near-field regime. The method works for sound soft, sound hard, and impedance surfaces, requires only a single illumination at a fixed frequency, and is realized efficiently by the fast Fourier transform. Numerical results show that the method is simple, stable, and effective in reconstructing scattering surfaces with subwavelength resolution.

Trapping and Photoacoustic Detection of CTCs at the Single Cell per Milliliter Level with Magneto‐Optical Coupled Nanoparticles

Circulating tumor cells (CTCs) have been reported to correlate most closely with cancer development, and can serve as an important marker for metastatic malignancy, tumor recurrence, and prediction of prognosis and therapeutic effi cacy. Detecting and quantifying CTCs, however, have proven to be challenging due to their low abundance in blood. Based on magneto-optical coupled nanoprobes (made of gold nanorod and iron oxide nanoparticles) and photoacoustic (PA) imaging, we report the development of an enabling technology that can detect CTCs at single cell/mL level. Remarkably, at this low cell concentration, approximately 67% of circulating tumor cells can be captured and imaged with just one pass through the magnetic trapping zone. Compared to the conventional in vitro assays, this technology offers signifi cantly improved sensitivity, because it is inherently compatible with large sample volumes. Compared to more advanced in vivo CTC detection methods, this technology can solve the low throughput problem of optical imaging, and streamline the photoacoustic imaging process by combining magnetic enrichment and digital readout into a single step. Most cancer deaths are caused by metastasis, a process

Source localization approach for functional DOT using MUSIC and FDR control

In this paper, we formulate diffuse optical tomography (DOT) problems as a source localization problem and propose a MUltiple SIgnal Classification (MUSIC) algorithm for functional brain imaging application. By providing MUSIC spectra for major chromophores such as oxy-hemoglobin (HbO) and deoxy-hemoglobin (HbR), we are able to investigate the spatial distribution of brain activities. Moreover, the false discovery rate (FDR) algorithm can be applied to control the family-wise error in the MUSIC spectra. The minimum distance between the center of mass in DOT and the Montreal Neurological Institute (MNI) coordinates of target regions in experiments was between approximately 6 and 18 mm, and the displacement of the center of mass in DOT and fMRI ranged between 12 and 28 mm, which demonstrate the legitimacy of the DOT-based imaging. The proposed brain mapping method revealed its potential as an alternative algorithm to monitor the brain activation.