Ray Transformation Research Articles

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.

On the Light Ray Transform of Wave Equation Solutions

We study the light ray transform on Minkowski space-time and its small metric perturbations acting on scalar functions which are solutions to wave equations. We show that the light ray transform uniquely determines the function in a stable way. The problem is of particular interest because of its connection to inverse problems of the Sachs–Wolfe effect in cosmology.

On a Generalization of Exponential Ray Transform in Tomography

We consider the operator of generalized exponential ray transform whose values are the initial data in the emission computer tomography problem. We establish connections between exponential ray transforms of different orders and obtain the differential equations satisfied by the exponential ray transforms. We prove the uniqueness theorems for boundary value and initial-boundary value problems for the corresponding equations We also discuss close connections of generalized exponential ray transforms with integral geometry of tensor fields and tomography problems.

An Investigation of Radial Basis Function Method for Strain Reconstruction by Energy-Resolved Neutron Imaging

The main objective of the current work is to determine meshless methods using the radial basis function (rbf) approach to estimate the elastic strain field from energy-resolved neutron imaging. To this end, we first discretize the longitudinal ray transformation with rbf methods to give us an unconstrained optimization problem. This discretization is then transformed into a constrained optimization problem by adding equilibrium conditions to ensure uniqueness. The efficiency and accuracy of this approach are investigated for the situation of 2d plane stress. In addition, comparisons are made between the results obtained with rbf collocation, finite-element (fem) and analytical solution methods for test problems. The method is then applied to experimentally measured continuous and discontinuous strain fields using steel samples for an offset ring-and-plug and crushed ring, respectively.

Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken Ray Transform Data

The single-scatter approximation is fundamental in many tomographic imaging problems including x-ray scatter imaging and optical scatter imaging for certain media. In all cases, noisy measurements are affected by both local scatter events and nonlocal attenuation. Prior works focus on reconstructing one of two images: scatter density or total attenuation. However, both images are media specific and useful for object identification. Nonlocal effects of the attenuation image on the data are summarized by the broken ray transform (BRT). While analytic inversion formulas exist, poor conditioning of the inverse problem is only exacerbated by noisy measurements and sampling errors. This has motivated interest in the related star transforms incorporating BRT measurements from multiple source-detector pairs. However, all analytic methods operate on the log of the data. For media comprising regions with no scatter a new approach is required. We are the first to present a joint estimation algorithm based on Poisson data models for a single-scatter measurement geometry. Monotonic reduction of the log-likelihood function is guaranteed for our iterative algorithm while alternating image updates. We also present a fast algorithm for computing the discrete BRT forward operator. Our generalized approach can incorporate both transmission and scatter measurements from multiple source-detector pairs. Transmission measurements resolve low-frequency ambiguity in the joint image estimation problem, while multiple scatter measurements resolve the attenuation image. The benefits of joint estimation, over single-image estimation, vary with problem scaling. Our results quantify these benefits and should inform design of future acquisition systems.

The Light Ray Transform in Stationary and Static Lorentzian Geometries

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.

Realistic rendering of scenes with anisotropic media

We describe a method of tracing a backward (from camera) ray in a scene that contains birefrigent (uniaxial) media. The physics of scattering of an electromagnetic wave by a boundary between two media is well known and is a base for ray tracing methods; but processing of a backward ray differs from scattering of a “natural” forward ray. Say, when a backward ray refracts by a boundary, besides the energy transfer coefficient like for a forward ray, one must account for the radiance change due to beam divergence. We calculate this factor and prove it must be evaluated only for the first and the last media along the ray path while the contributions from the intermediate media mutually cancel. We present a closed numerical method that allows one to perform transformation of a backward ray on a boundary between two media either of which can be birefrigent. We hope it is more convenient and ready for usage in ray tracing engines than known publications. Calculation utilizes Helmholtz reciprocity to calculate directions of scattered rays and their polarization (i.e., Mueller matrices), which is advantageous over a straightforward “reverse” of forward ray transformation. The algorithm was integrated in the lighting simulation system Lumicept and allowed for an efficient calculation of images of scenes with crystal elements.

The Attenuated Geodesic Ray Transform on Tensors: Generic Injectivity and Stability

We consider the attenuated geodesic ray transform defined on pairs of symmetric 2-tensors and 1-forms on a simple Riemannian manifold. We prove injectivity and stability results for a class of generic simple metrics and attenuations containing real analytic ones. In fact, methods used in this paper can be modified to generalize our results for a class of non-simple manifolds similar to Stefanov and Uhlmann Am J Math 130(1):239–268 (2008).

Evolution and development of the fish jaw skeleton.

The evolution of the jaw represents a key innovation in driving the diversification of vertebrate body plans and behavior. The pharyngeal apparatus originated as gill bars separated by slits in chordate ancestors to vertebrates. Later, with the acquisition of neural crest, pharyngeal arches gave rise to branchial basket cartilages in jawless vertebrates (agnathans), and later bone and cartilage of the jaw, jaw support, and gills of jawed vertebrates (gnathostomes). Major events in the evolution of jaw structure from agnathans to gnathostomes include axial regionalization of pharyngeal elements and formation of a jaw joint. Hox genes specify the anterior-posterior identity of arches, and edn1, dlx, hand2, Jag1b-Notch2 signaling, and Nr2f factors specify dorsal-ventral identity. The formation of a jaw joint, an important step in the transition from an un-jointed pharynx in agnathans to a hinged jaw in gnathostomes involves interaction between nkx3.2, hand2, and barx1 factors. Major events in jaw patterning between fishes and reptiles include changes to elements of the second pharyngeal arch, including a loss of opercular and branchiostegal ray bones and transformation of the hyomandibula into the stapes. Further changes occurred between reptiles and mammals, including the transformation of the articular and quadrate elements of the jaw joint into the malleus and incus of the middle ear. Fossils of transitional jaw phenotypes can be analyzed from a developmental perspective, and there exists potential to use genetic manipulation techniques in extant taxa to test hypotheses about the evolution of jaw patterning in ancient vertebrates. This article is categorized under: Comparative Development and Evolution > Evolutionary Novelties Early Embryonic Development > Development to the Basic Body Plan Comparative Development and Evolution > Body Plan Evolution.

Determination of Discontinuities of a Function in a Domain with Refraction from Its Attenuated Ray Transform

Some results of numerical investigations are presented for the problem of determination of discontinuities of an unknown function that has the meaning of the internal source distribution and is given in a domain with absorption and refraction, on using the attenuated ray transform of the function. The refraction and the absorption coefficient are assumed to be given. The behavior of the available and newly constructed discontinuity indicator operators is investigated in some numerical tests. Some modification of discontinuity indicators was carried out for the purpose of applying them in the model of refractive tomography with absorption. Numerical methods are applied to investigate the possibility of using the operators of this kind for solving the problem of finding the discontinuities of a function from its attenuated ray transform; the degree is investigated of the influence on the recovery quality of such factors as the level of the noise introduced into the generated data, the parameters of metrics, the magnitude and variation of the absorption coefficient.

Means for Giving Flexibility to a Laser Beam Tool

Background. Existing opinion of laser ray, as universal instrument, so justly, as far as judgment is faithful that a metal-cutting instrument is universal. Universality is confirmed by instrument functionality, and the compared objects in this sense are practically equal. A difference is in that a metal-cutting instrument is created beforehand, including its special kinds, while a laser ray becomes an instrument in the moment of its use. The latter both complicates and simplifies work of technologist – his fantasy is unlimited, modernization and size modification of chart and construction of instrument are possible, even during a technological operation. It can be attained because of unique property of laser ray – his flexibility.Objective. Knowledge of this ray property does not guarantee its successful application, because it is not enough to use the known optical elements and their charts for transformation of ray to the effective and various instrument. Herein there is the aim of the paper – by unconventional and sometimes unobvious technological and designer decisions to show hidden possibilities of laser ray, as a flexible instrument.Methods. Principal incentive reason of diverse instrument creation is a necessity of the technological providing of new or set of the known flow sheets of zones of irradiation forming for the different types of treatment: from superficial (engraving, creation of regular surface relief) to the size (precision).Results. For realization of treatment flow sheets range the results of development and use of traditional and original means of laser ray transformation into an instrument (optical elements with changeable parameters, combined systems for the complex affecting of ray purveyance, for expansion of his influence zone, for medicine) are shown.Conclusions. Flexibility of laser ray at converting it into a processing instrument is less as inherent to its property, and more as chance for a technologist to simplify implementation of technological operations and diversify their results.

The Local Magnetic Ray Transform of Tensor Fields

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnet...

Tomographic reconstruction of residual strain in axisymmetric systems from Bragg-edge neutron imaging

In this paper, we address the tomographic reconstruction of elastic strain fields from Bragg-edge measurements. The non-trivial null space of the Longitudinal Ray Transform lies at the heart of this problem, precluding direction inversion from measured data. We have approached this problem by considering the physical constraints of equilibrium through the minimisation of strain energy. Through this approach, a method is developed for axisymmetric systems that is capable of reconstructing residual strain fields. This algorithm has been demonstrated for two such fields, and is capable of rejecting simulated Gaussian measurement noise of the magnitude expected in experimental data. In contrast to previous algorithms for this class of system, a clear path to general application (i.e. arbitrary geometry) exists.

Bragg-edge neutron transmission strain tomography for in situ loadings

An approach for tomographic reconstruction of three-dimensional strain distributions from Bragg-edge neutron transmission strain images is outlined and investigated. This algorithm is based on the link between Bragg-edge strain measurements and the Longitudinal Ray Transform, which has been shown to be sensitive only to boundary displacement. By exploiting this observation we provide a method for reconstructing boundary displacement from sets of Bragg-edge strain images. In the case where these displacements are strictly the result of externally applied tractions, corresponding internal strain fields can then be found through traditional linear-static finite element methods. This approach is tested on synthetic data in two-dimensions, where the rate of convergence in the presence of measurement noise and beam attenuation is examined.

Numerical Inversion of a Broken Ray Transform Arising in Single Scattering Optical Tomography

The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed, and robustness of our method in the case of both noise-free and noisy data.

A small look at the ear recognition process using a hybrid approach

The purpose of this document is to offer a combined approach in biometric analysis field, integrating some of the most known techniques using ears to recognize people. This study uses Hausdorff distance as a pre-processing stage adding sturdiness to increase the performance filtering for the subjects to use it in the testing process. Also includes the Image Ray Transform (IRT) and the Haar based classifier for the detection step. Then, the system computes Speeded Up Robust Features (SURF) and Linear Discriminant Analysis (LDA) as an input of two neural networks to recognize a person by the patterns of its ear. To show the applied theory experimental results, the above algorithms have been implemented using Microsoft C#. The investigation results showed robustness improving the ear recognition process.

Guest editorial: online multimedia signal and image processing

In recent years, multimedia has become an essential part of networking and online communication environments. As people connect to the Internet or to one another multimedia content such as photos, video and music, takes the lion’s share of the overall artwork traffic. Multimedia-based applications use a wide range of signal and image processing techniques to enhance user experience. Semantic multi-media retrieval has been a major research focus for the last decade. Recently, a large number of cutting-edge algorithms have been developed performing audio, image and video content extraction and object recognition. Artificial intelligence and machine learning techniques have been also extensively used to connect contextual and conceptual information to multimedia data. The importance of metadata has been boosted by the emergence of high-end devices such as tablets, smartphones, and other hand-held devices. Sematic-aware metadata can further improve intelligence of online multimedia environments. This special issue targets online applications ranging from traditional signal and imageprocessing methods to evolving semantic-based multimedia, audio, image and video technologies. Also, it aims to demonstrate novel, flexible and easy-to-use applications developed for the next generation of devices. This special issue received many submissions from researchers and practitioners working on multimedia signal and image processing. After two rounds of review, eventually seven high quality papers were chosen. The paper titled “Extending the Image Ray Transform for Shape Detection and Extraction” presents a novel approach to image analysis performing feature extraction at low level and complements it with high-level feature extraction to determine structure, exploiting the Hough transform. Authors analyze performance with images from the Caltech-256 dataset and describe how thir approach can select chosen shapes.

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and, therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but it is still ill-posed if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.

Extending the image ray transform for shape detection and extraction

A conventional approach to image analysis is to separately perform feature extraction at a low level (such as edge detection) and follow this with high level feature extraction to determine structure (e.g. by collecting edge points) using the Hough transform. The original image Ray Transform (IRT) demonstrated capability to extract structures at a low level. Here we extend the IRT to add shape specificity that makes it select specific shapes rather than just edges; the new capability is achieved by addition of a single parameter that controls which shape is selected by the extended IRT. The extended approach can then perform low-and high-level feature extraction simultaneously. We show how the IRT process can be extended to focus on chosen shapes such as lines and circles. Histogram patterns, which are an extension to this new capability, can describe extracted features showing that the extracted patterns are robust to change in orientation, position and scale. We confirm the new capability by using conventional methods for exact shape location, such as the Hough transform. We analyse performance with images from the Caltech-256 dataset and show that the new approach can indeed select chosen shapes. Further research will aim to capitalise on the new extraction ability to extend descriptive capability.