Non-interferometric methods of phase estimation for application in optical tomography

Abstract

We describe here two non-interferometric methods for the estimation of the phase of transmitted wavefronts through refracting objects. The phase of the wavefronts obtained is used to reconstruct either the refractive index distribution of the objects or their contours. Refraction corrected reconstructions are obtained by the application of an iterative loop incorporating digital ray tracing for forward propagation and a modified filtered back projection (FBP) for reconstruction. The FBP is modified to take into account non-straight path propagation of light through the object. When the iteration stagnates, the difference between the projection data and an estimate of it obtained by ray tracing through the final reconstruction is reconstructed using a diffraction tomography algorithm. The reconstruction so obtained, viewed as a correction term, is added to the estimate of the object from the loop to obtain an improved final refractive index reconstruction.     

Simultaneous determination of the refractive index and wedge angle of an optical wedge plate using a photorefractive holographic interferometer

Abstract

A simple and effective method for simultaneously determining the refractive index and the wedge angle of an optical wedge plate is described. The method is based on a real-time holographic interferometer which uses a photorefractive crystal as the recording and reconstruction medium. The wedge sample under test is inserted into a rectangular cell that is placed in the object light beam of the holographic interferometer. The interference patterns produced before and after a reference liquid is poured into the cell are received by a CCD camera and stored in a computer, respectively. The refractive index and the wedge angle of the wedge sample are determined by measuring the number of fringes falling inside a fixed aperture. The principle of the method is analysed and some experimental results with adequate accuracy are given.     

Single-shot full resolution region-of-interest (ROI) reconstruction in image plane digital holographic microscopy

We describe a numerical processing technique that allows single-shot region-of-interest (ROI) reconstruction in image plane digital holographic microscopy with full pixel resolution. The ROI reconstruction is modelled as an optimization problem where the cost function to be minimized consists of an L2-norm squared data fitting term and a modified Huber penalty term that are minimized alternately in an adaptive fashion. The technique can provide full pixel resolution complex-valued images of the selected ROI which is not possible to achieve with the commonly used Fourier transform method. The technique can facilitate holographic reconstruction of individual cells of interest from a large field-of-view digital holographic microscopy data. The complementary phase information in addition to the usual absorption information already available in the form of bright field microscopy can make the methodology attractive to the biomedical user community.     

Direct reconstruction of complex refractive index distribution from boundary measurement of intensity and normal derivative of intensity

We present an optical tomographic reconstruction method to recover the complex refractive index distribution from boundary measurements based on intensity, which are the logarithm of intensity and normal derivative of intensity. The method, which is iterative, repeatedly implements the forward propagation equation for light amplitude, the Helmholtz equation, and computes appropriate sensitivity matrices for these measurements. The sensitivity matrices are computed by solving the forward propagation equation for light and its adjoint. The results of numerical experiments show that the data types ln(I) and partial differential I/ partial differential n reconstructed, respectively, the imaginary and the real part of the object refractive index distribution. Moreover, the imaginary part of the refractive index reconstructed from partial differential I/ partial differential n and the real part from ln(I) failed to show the object's inhomogeneity. The value of the propagation constant, k, used in our simulations was 50, and this value resulted in smoothing of the reconstructed inhomogeneities. Thus we have shown that it is possible to reconstruct the complex refractive index distribution directly from the measured intensity without having to first find the light amplitude, as is done in most of the currently available reconstruction algorithms of diffraction tomography.     

Mart Algorithms for Circular and Helical Cone-Beam Tomography

The present work is concerned with the evaluation of the performance and the efficient implementation of multiplicative algebraic reconstruction technique (MART) to reconstruct three-dimensional (3D) objects for two different source/detector trajectories. Three types of MART algorithms are tested on a numerical phantom (Defrise), and they are implemented on a 3D X-ray system of Vikram Sarabhai Space Centre (VSSC). Circular and helical cone-beam trajectories are used. The results are compared with convolution backprojection (CBP) algorithm for each trajectory. It is found that iterative algorithms perform better than their counterpart, the transform-based CBP algorithm, whenever tomography systems are ill-conditioned due to limited views and/or noisy projection data.     

Deterministic versus stochastic level-set regularization in nonlinear phase contrast tomography

A new nonlinear level-set regularization method to reconstruct the complex refractive index distribution with in-line phase contrast tomography measurements is presented under the assumption that the index is piecewise constant. The nonlinear iterative approach is based on the Fréchet derivative of the intensity recorded at a single propagation distance and for several projection angles. The algorithm is successfully applied to a multi-material object for several noise levels. Better reconstruction results are achieved with a stochastic perturbation of the level-set function. This evolution corresponds to a stochastic evolution of the shape of the reconstructed regions. The reconstruction errors can be further decreased with topological derivatives. The different algorithms are tested on various multi-material objects.     

A modified algebraic reconstruction technique taking refraction into account with an application in terahertz tomography

Terahertz (THz) tomography is a rather novel technique for non-destructive testing that is particularly suited for the testing of plastics and ceramics. Previous publications showed a large variety of conventional algorithms adapted from computed tomography or ultrasound tomography which were directly applied to THz tomography. Conventional algorithms neglect the specific nature of THz radiation, i.e. refraction at interfaces, reflection losses and the beam profile (Gaussian beam), which results in poor reconstructions. The aim is the efficient reconstruction of the complex refractive index, since it indicates inhomogeneities in the material. A hybrid algorithm has been developed based on the algebraic reconstruction technique (ART). ART is adapted by including refraction (Snell’s law) and reflection losses (Fresnel equations). Our method uses a priori information about the interface and layer geometry of the sample. This results in the ‘Modified ART for THz tomography’, which reconstructs simultaneously the complex refractive index from transmission coefficient and travel time measurements.     

Determination of the refractive index of dehydrated cells by means of digital holographic microscopy

Spatial distributions of the integral refractive index in dehydrated cells of human oral cavity epithelium are obtained by means of digital holographic microscopy, and mean refractive index of the cells is determined. The statistical analysis of the data obtained is carried out, and absolute errors of the method are estimated for different experimental conditions.     

Automatic image-processing system and fast-reconstruction technique for holographic-interferometry computer tomography

A fast automatic processing system for holographic interferograms for use in optical tomography is discussed. A hot air flow field above a combustor with a rectangular opening is investigated by a fast reconstruction technique (the simultaneous algebraic reconstruction technique), and the temperature distribution of a section in the air flow field is reconstructed. The reconstructed temperatures are consistent with the temperatures measured by a thermocouple.     

Non‐iterative reconstruction with a prior for undersampled radial MRI data

This paper develops an FBP‐MAP (filtered backprojection, maximum a posteriori) algorithm to reconstruct MRI images from undersampled data. An objective function is first set up for the MRI reconstruction problem with a data fidelity term and a Bayesian term. The Bayesian term is a constraint in the temporal dimension. This objective function is minimized using the calculus of variations. The proposed algorithm is non‐iterative. Undersampled dynamic myocardial perfusion MRI data were used to test the feasibility of the proposed technique. It is shown that the non‐iterative Fourier–Bayesian reconstruction method effectively incorporates the temporal constraint and significantly reduces the angular aliasing artifacts caused by undersampling. A significant advantage of the proposed non‐iterative Fourier–Bayesian technique over the iterative techniques is its fast computation time and its ability to reach the optimal solution. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 53–58, 2013.     

Hologram reconstruction corrected for measurements through layers with different refractive indices in digital in-line holographic microscopy

Digital in-line holographic microscopy (DIHM) using point sources has been shown to be a versatile technique, especially for three-dimensional tracking of particles or microorganisms. However, the spherical source wave is altered when measurements are performed through layers with different refractive indices, such as water cuvettes. The situations where a layer of medium with a refractive index different than that of the predominant surrounding propagation medium (usually air) is situated behind or in front of the plane to be reconstructed are analyzed in detail, and a general approach for reconstruction under such circumstances is developed. The proposed refractive index correction is tested experimentally and compared to conventional reconstruction algorithms. Using 3D traces of swimming algal spores, the influence on the velocity calculation is also shown.     

Fourier-domain digital holographic optical coherence imaging of living tissue

Digital holographic optical coherence imaging is a full-frame coherence-gated imaging approach that uses a CCD camera to record and reconstruct digital holograms from living tissue. Recording digital holograms at the optical Fourier plane has advantages for diffuse targets compared with Fresnel off-axis digital holography. A digital hologram captured at the Fourier plane requires only a 2D fast Fourier transform for numerical reconstruction. We have applied this technique for the depth-resolved imaging of rat osteogenic tumor multicellular spheroids and acquired cross-section images of the anterior segment and the retinal region of a mouse eye. A penetration depth of 1.4 mm for the tumor spheroids was achieved.     

Iterative image reconstruction using prior knowledge

A method is proposed to reconstruct signals from incomplete data. The method, which can be interpreted both as a discrete implementation of the so-called prior discrete Fourier transform (PDFT) spectral estimation technique and as a variant of the algebraic reconstruction technique, allows one to incorporate prior information about the reconstructed signal to improve the resolution of the signal estimated. The context of diffraction tomography and image reconstruction from samples of the far-field scattering amplitude are used to explore the performance of the method. On the basis of numerical computations, the optimum choice of parameters is determined empirically by comparing image reconstructions of the noniterative PDFT algorithm and the proposed iterative scheme.     

Hybrid digital holographic imaging system for three-dimensional dense particle field measurement

To apply digital holography to the measurement of three-dimensional dense particle fields in large facilities, we have developed a hybrid digital holographic particle-imaging system. The technique combines the advantages of off-axis (side) scattering in suppressing speckle noise and on-axis (in-line) recording in lowering the digital sensor resolution requirement. A camera lens is attached to the digital sensor to compensate for the weak object wave from side scattering over a large recording distance. A simple numerical reconstruction algorithm is developed for holograms recorded with a lens without requiring complex and impractical mathematical corrections. We analyze the effect of image sensor resolution and off-axis angle on system performance and quantify the particle positioning accuracy of the system. The holographic system is successfully applied to the study of inertial particle clustering in isotropic turbulence.     

Quantitative phase and refractive index measurements with point-source digital in-line holographic microscopy

Point-source digital in-line holographic microscopy with numerical reconstruction is ideally suited for quantitative phase measurements to determine optical path lengths and to extract changes in refractive index within accuracy close to 0.001 on the submicrometer length scale. This is demonstrated with simulated holograms and with detailed measurements on a number of different micrometer-sized samples such as suspended drops, optical fibers, as well as organisms of biological interest such as E. coli bacteria, HeLa cells, and fibroblast cells.     

Orthographic double-beam holographic interferometry for limited-view optical tomography

A simultaneous algebraic reconstruction technique (SART) that was first applied to limited-view optical tomography reconstruction of a three-dimensional asymmetric refractive-index field or a temperature distribution is presented and numerically simulated. Within two orthographic-view directions and limited view ranges, the reconstruction speed and accuracy of the SART are adequate. A new orthographic double-beam holographic-interferometric system from which multidirectional interferometric data can be obtained is first built. An asymmetric heat air-flow field is experimentally investigated by the use of orthographic double-beam holographic interferometry with double exposures. The temperatures reconstructed with the SART are consistent with those measured with a thermocouple.     

Numerically robust minimal‐scan reconstruction algorithms for diffraction tomography via radon transform inversion

It is widely believed that measurements from a full angular range of 2π are generally required to exactly reconstruct a complex‐valued refractive index distribution in diffraction tomography (DT). In this work, we developed a new class of minimal‐scan reconstruction algorithms for DT that utilizes measurements only over the angular range 0 ≤ ? ≤ 3π/2 to perform an exact reconstruction. These algorithms, referred to as minimal‐scan estimate‐combination (MS‐E‐C) reconstruction algorithms, effectively operate by transforming the DT reconstruction problem into a conventional x‐ray CT reconstruction problem that requires inversion of the Radon transform. We performed computer simulations to compare the noise and numerical properties of the MS‐E‐C algorithms against existing filtered backpropagation‐based algorithms. © 2002 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 84–91, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10014     

Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens

We present an iterative tomographic algorithm to reconstruct refractive-index profiles for meridional planes of the lens of the spherical fish eye from measurements of deflection angles of refracted rays. Numerical simulations show that the algorithm allows accuracy up to the fourth decimal place, provided that the refractive index can be regarded as an analytical function of the radial coordinate and the experimental errors are neglected. An experimental demonstration is given by applying the algorithm to retrieve the refractive-index profile of a spherical fish lens. The method is conceptually simple and does not require matching of the index of the surrounding medium to that of the surface of the lens, and the related iterative algorithm rapidly converges.     

Angle-multiplexed holographic data storage with minimum cross talk noise

The cross talk noise-to-signal ratio (NSR) of an angle-multiplexed holographic data storage system is studied, and we propose a method to determine the optimized multiplexing spacing with which the cross talk noise can be less than the conventional method. In our method, the optimization location at the image plane can be chosen arbitrarily, so the multiplexing of asymmetrical image patterns can be optimized. In particular, we investigate the 90° scheme and the transmission scheme angle multiplexing. For the 90° scheme, a holographic medium with a higher refractive index is recommended for cross talk-limited multiplexing. For the transmission scheme, a holographic medium with a lower refractive index is recommended for angular range-limited multiplexing. In addition, for the transmission scheme, a larger angle between the object arm and the reference arm results in less cross talk noise, whereas the highest storage density is achieved at a 45° angle.     

Comparison of methods for determining the bias index of a dichromated gelatin hologram

Determination of the bias refractive index of a holographic emulsion before exposure and after development is an important factor in the design of holographic optical elements. Several experimental methods are discussed for determining the bias index of a volume hologram in dichromated gelatin, and the results for each technique are presented. It is shown experimentally that these measurement methods yield different results for the same hologram, and the cause of the differences is proposed to be a variation of the bias index with depth in the hologram. An index measurement technique is also presented that accounts for variation in the bias index and is shown to yield an accurate value for the bias index.