Three-dimensional image reconstruction in J-PET using Filtered Back Projection method

We present a method and preliminary results of the image reconstruction in the Jagiellonian PET tomograph. Using GATE (Geant4 Application for Tomographic Emission), interactions of the 511 keV photons with a cylindrical detector were generated. Pairs of such photons, flying back-to-back, originate from e+e- annihilations inside a 1-mm spherical source. Spatial and temporal coordinates of hits were smeared using experimental resolutions of the detector. We incorporated the algorithm of the 3D Filtered Back Projection, implemented in the STIR and TomoPy software packages, which differ in approximation methods. Consistent results for the Point Spread Functions of ~5/7,mm and ~9/20, mm were obtained, using STIR, for transverse and longitudinal directions, respectively, with no time of flight information included.Comment: Presented at the 2nd Jagiellonian Symposium on Fundamental and Applied Subatomic Physics, Krak\'ow, Poland, June 4-9, 2017. To be published in Acta Phys. Pol.

Elementary test for non-classicality based on measurements of position and momentum

We generalise a non-classicality test described by Kot et al. [Phys. Rev. Lett. 108, 233601 (2010)], which can be used to rule out any classical description of a physical system. The test is based on measurements of quadrature operators and works by proving a contradiction with the classical description in terms of a probability distribution in phase space. As opposed to the previous work, we generalise the test to include states without rotational symmetry in phase space. Furthermore, we compare the performance of the non-classicality test with classical tomography methods based on the inverse Radon transform, which can also be used to establish the quantum nature of a physical system. In particular, we consider a non-classicality test based on the so-called filtered back-projection formula. We show that the general non-classicality test is conceptually simpler, requires less assumptions on the system and is statistically more reliable than the tests based on the filtered back-projection formula. As a specific example, we derive the optimal test for a quadrature squeezed single photon state and show that the efficiency of the test does not change with the degree of squeezing

Statistical Image Reconstruction for High-Throughput Thermal Neutron Computed Tomography

Neutron Computed Tomography (CT) is an increasingly utilised non-destructive analysis tool in material science, palaeontology, and cultural heritage. With the development of new neutron imaging facilities (such as DINGO, ANSTO, Australia) new opportunities arise to maximise their performance through the implementation of statistically driven image reconstruction methods which have yet to see wide scale application in neutron transmission tomography. This work outlines the implementation of a convex algorithm statistical image reconstruction framework applicable to the geometry of most neutron tomography instruments with the aim of obtaining similar imaging quality to conventional ramp filtered back-projection via the inverse Radon transform, but using a lower number of measured projections to increase object throughput. Through comparison of the output of these two frameworks using a tomographic scan of a known 3 material cylindrical phantom obtain with the DINGO neutron radiography instrument (ANSTO, Australia), this work illustrates the advantages of statistical image reconstruction techniques over conventional filter back-projection. It was found that the statistical image reconstruction framework was capable of obtaining image estimates of similar quality with respect to filtered back-projection using only 12.5% the number of projections, potentially increasing object throughput at neutron imaging facilities such as DINGO eight-fold

Exact reconstruction formulas for a Radon transform over cones

Inversion of Radon transforms is the mathematical foundation of many modern tomographic imaging modalities. In this paper we study a conical Radon transform, which is important for computed tomography taking Compton scattering into account. The conical Radon transform we study integrates a function in $\R^d$ over all conical surfaces having vertices on a hyperplane and symmetry axis orthogonal to this plane. As the main result we derive exact reconstruction formulas of the filtered back-projection type for inverting this transform.Comment: 8 pages, 1 figur

Error analysis for filtered back projection reconstructions in Besov spaces

Filtered back projection (FBP) methods are the most widely used reconstruction algorithms in computerized tomography (CT). The ill-posedness of this inverse problem allows only an approximate reconstruction for given noisy data. Studying the resulting reconstruction error has been a most active field of research in the 1990s and has recently been revived in terms of optimal filter design and estimating the FBP approximation errors in general Sobolev spaces. However, the choice of Sobolev spaces is suboptimal for characterizing typical CT reconstructions. A widely used model are sums of characteristic functions, which are better modelled in terms of Besov spaces $\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)$. In particular $\mathrm{B}^{\alpha,1}_1(\mathbb{R}^2)$ with $\alpha \approx 1$ is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error $$\|f-f_L^\delta\| \le \|f-f_L\|+ \|f_L - f_L^\delta\|$$ splits into an approximation error and a data error, where $L$ serves as regularization parameter. In this paper, we study the approximation error of FBP reconstructions for target functions $f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)$ with positive $\alpha \not\in \mathbb{N}$ and $1 \leq p,q \leq \infty$. We prove that the $\mathrm{L}^p$-norm of the inherent FBP approximation error $f-f_L$ can be bounded above by \begin{equation*} \|f - f_L\|_{\mathrm{L}^p(\mathbb{R}^2)} \leq c_{\alpha,q,W} \, L^{-\alpha} \, |f|_{\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)} \end{equation*} under suitable assumptions on the utilized low-pass filter's window function $W$. This then extends by classical methods to estimates for the total reconstruction error.Comment: 32 pages, 8 figure

Comprehensive analysis of high-performance computing methods for filtered back-projection

This paper provides an extensive runtime, accuracy, and noise analysis of Computed To-mography (CT) reconstruction algorithms using various High-Performance Computing (HPC) frameworks such as: "conventional" multi-core, multi threaded CPUs, Compute Unified Device Architecture (CUDA), and DirectX or OpenGL graphics pipeline programming. The proposed algorithms exploit various built-in hardwired features of GPUs such as rasterization and texture filtering. We compare implementations of the Filtered Back-Projection (FBP) algorithm with fan-beam geometry for all frameworks. The accuracy of the reconstruction is validated using an ACR-accredited phantom, with the raw attenuation data acquired by a clinical CT scanner. Our analysis shows that a single GPU can run a FBP reconstruction 23 time faster than a 64-core multi-threaded CPU machine for an image of 1024 X 1024. Moreover, directly programming the graphics pipeline using DirectX or OpenGL can further increases the performance compared to a CUDA implementation