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Concept# Tomographic reconstruction

Résumé

Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security.
This article applies in general to reconstruction methods for all kinds of tomography, but some of the terms and physical descriptions refer directly to the reconstruction of X-ray computed tomography.
Introducing formula
The projection of an object, resulting from the tomographic measurement process at a given angle \theta, is made up of a set of line i

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Study of advanced image processing; mathematical imaging. Development of image-processing software and prototyping in JAVA; application to real-world examples in industrial vision and biomedical imaging.

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Tomodensitométrie

La tomodensitométrie (TDM), dite aussi scanographie, tomographie axiale calculée par ordinateur (TACO), CT-scan (CT : computed tomography), CAT-scan (CAT : computer-assisted tomography), ou simpleme

Tomographie

vignette|Principe de base de la tomographie par projections : les coupes tomographiques transversales S1 et S2 sont superposées et comparées à l’image projetée P.
La tomographie est une technique d’,

Théorème de Radon

Le théorème de projection de Radon établit la possibilité de reconstituer une fonction réelle à deux variables (assimilable à une image) à l'aide de la totalité de ses projections selon des droites

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Ahmed Bassam Sayed Ayoub Mohamed Emam

In this thesis, we study the 3 challenges described above. First, we study different reconstruction techniques and assess the fidelity of each reconstruction results by means of structured illumination and phase conjugation. By reconstructing the 3D refractive index of the sample using different algorithms (i.e. Born, Rytov, and Radon) and then perform a numerical back-propagation of experimentally measured structured illumination pattern we are able to assess the fidelity of each reconstruction algorithms without prior information about the 3D RI distribution of the sample.The second part of the thesis is concerned with the 3D reconstruction of samples using intensity-only measurements which the need to holographically acquire them. We show that using intensity-only measurements, we could still be able to reconstruct the 3D volume of the sample with edge-enhanced effects which was proven useful for drug delivery applications in which nano-particles were identified on the cell membrane of immune T-cells in a drug delivery studies. Such reconstruction technique would result in more robust imaging system where the commercial imaging microscope systems can be incorporated with LEDs for high-quality speckle noise-free imaging systems. In addition, we show that under certain conditions, we can be able to reconstruct the 3D refractive index distribution of different samples.The third part of the thesis is contributing to high-speed complex wave-front shaping using DMDs. In that part, new modulation technique is demonstrated that can boost the speed of the current time-multiplexing techniques by a factor of 32. The modulation technique is based on amplitude modulation where an amplitude modulator is synchronized withvthe DMD to modulate the intensity of each bit-plane of an 8-bit image and then all the modulated bit-planes are linearly added on the detector. Such modulation technique can be used not only for structured illumination microscopy but also for high-speed 3D printing applications as well as projectors.The last part is concerned with using deep learning approaches to solve the missing cone problem usually accompanied with optical imaging due to the limited numerical aperture of the imaging system. Two techniques are discussed; the first is based on using a physical model to enhance the quality of the 3D RI reconstruction and the second is based on using deep neural network to solve the missing cone problem.

The recent advances in fluorescent molecular probes, photon detection instrumentation, and photon propagation models in tissue, have facilitated the emergence of innovative molecular imaging technologies such as Fluorescence Diffuse Optical Tomography (FDOT). FDOT has already successfully been applied on its own to resolve specific molecular targets and pathways in vivo. Recent attempts have tried to integrate FDOT with structural imaging technology into a multi-modal system. This thesis is amongst the first academic-industrial partnerships to develop a hybrid X-Ray Computed Tomography/FDOT platform for small animal imaging. In this study, we focus on image reconstruction algorithms for FDOT, and consider three main research directions. First, we investigate innovative discretization techniques for FDOT, in the context of variational reconstruction methods. Our goal is to treat in a consistent manner the propagation equation, and the regularization, needed in the reconstruction algorithm. This study uses the diffusion approximation for modeling the propagation of photons in tissue. We solve the diffusion equation with a novel Finite Volumes Method that is specifically designed to be compatible with the sparsity-inducing regularization developed in the second part of this work. Our approach results in a computationally efficient implementation of the forward model. Next, we study a novel regularization approach for FDOT which is based on sparsity-promoting regularization. This approach is expected to enhance the reconstruction when the fluorescence distribution presents a sparsity pattern in a suitable representation, or after linear transformation. We propose a variational framework based on ℓp-norm penalization in order to incorporate the sparsity a priori in the inversion algorithm. We demonstrate this approach on experimental data, using computationally efficient algorithms relying on convex optimization principles. Finally, we focus on hybrid imaging systems that combine FDOT with another imaging technique. The latter is employed to acquire a high-resolution image of the anatomy of the specimen under investigation, which provides valuable information for the FDOT reconstruction. We propose a new regularization method that allows the incorporation of the anatomical information in the FDOT reconstruction. Our method is based on the concept of group sparsity, and implemented in practice using ℓ2,1-norm penalization. We extend our sparsity-inducing algorithms to this case, and study the performance of the resulting implementation on experimental data.

Many PET scanners nowadays have the possibility to record event-by-event information, known as list mode data. This has the advantage of keeping the data in the highest possible resolution (both temporal and spatial). In most cases, list mode data are then binned into sinogram format before reconstruction. In this paper, we discuss at which stage normalisation factors should be introduced. It is shown that noise is greatly reduced by performing the normalisation after the binning. We illustrate this with acquired and simulated data for the quad HiDAC camera.

2008