Noise-Enhanced and Human Visual System-Driven Image Processing: Algorithms and Performance Limits

This dissertation investigates the problem of image processing based on stochastic resonance (SR) noise and human visual system (HVS) properties, where several novel frameworks and algorithms for object detection in images, image enhancement and image segmentation as well as the method to estimate the performance limit of image segmentation algorithms are developed. Object detection in images is a fundamental problem whose goal is to make a decision if the object of interest is present or absent in a given image. We develop a framework and algorithm to enhance the detection performance of suboptimal detectors using SR noise, where we add a suitable dose of noise into the original image data and obtain the performance improvement. Micro-calcification detection is employed in this dissertation as an illustrative example. The comparative experiments with a large number of images verify the efficiency of the presented approach. Image enhancement plays an important role and is widely used in various vision tasks. We develop two image enhancement approaches. One is based on SR noise, HVS-driven image quality evaluation metrics and the constrained multi-objective optimization (MOO) technique, which aims at refining the existing suboptimal image enhancement methods. Another is based on the selective enhancement framework, under which we develop several image enhancement algorithms. The two approaches are applied to many low quality images, and they outperform many existing enhancement algorithms. Image segmentation is critical to image analysis. We present two segmentation algorithms driven by HVS properties, where we incorporate the human visual perception factors into the segmentation procedure and encode the prior expectation on the segmentation results into the objective functions through Markov random fields (MRF). Our experimental results show that the presented algorithms achieve higher segmentation accuracy than many representative segmentation and clustering algorithms available in the literature. Performance limit, or performance bound, is very useful to evaluate different image segmentation algorithms and to analyze the segmentability of the given image content. We formulate image segmentation as a parameter estimation problem and derive a lower bound on the segmentation error, i.e., the mean square error (MSE) of the pixel labels considered in our work, using a modified Cramér-Rao bound (CRB). The derivation is based on the biased estimator assumption, whose reasonability is verified in this dissertation. Experimental results demonstrate the validity of the derived bound

Review : Deep learning in electron microscopy

Deep learning is transforming most areas of science and technology, including electron microscopy. This review paper offers a practical perspective aimed at developers with limited familiarity. For context, we review popular applications of deep learning in electron microscopy. Following, we discuss hardware and software needed to get started with deep learning and interface with electron microscopes. We then review neural network components, popular architectures, and their optimization. Finally, we discuss future directions of deep learning in electron microscopy

A comparative evaluation for liver segmentation from spir images and a novel level set method using signed pressure force function

Thesis (Doctoral)--Izmir Institute of Technology, Electronics and Communication Engineering, Izmir, 2013Includes bibliographical references (leaves: 118-135)Text in English; Abstract: Turkish and Englishxv, 145 leavesDeveloping a robust method for liver segmentation from magnetic resonance images is a challenging task due to similar intensity values between adjacent organs, geometrically complex liver structure and injection of contrast media, which causes all tissues to have different gray level values. Several artifacts of pulsation and motion, and partial volume effects also increase difficulties for automatic liver segmentation from magnetic resonance images. In this thesis, we present an overview about liver segmentation methods in magnetic resonance images and show comparative results of seven different liver segmentation approaches chosen from deterministic (K-means based), probabilistic (Gaussian model based), supervised neural network (multilayer perceptron based) and deformable model based (level set) segmentation methods. The results of qualitative and quantitative analysis using sensitivity, specificity and accuracy metrics show that the multilayer perceptron based approach and a level set based approach which uses a distance regularization term and signed pressure force function are reasonable methods for liver segmentation from spectral pre-saturation inversion recovery images. However, the multilayer perceptron based segmentation method requires a higher computational cost. The distance regularization term based automatic level set method is very sensitive to chosen variance of Gaussian function. Our proposed level set based method that uses a novel signed pressure force function, which can control the direction and velocity of the evolving active contour, is faster and solves several problems of other applied methods such as sensitivity to initial contour or variance parameter of the Gaussian kernel in edge stopping functions without using any regularization term

Feature based estimation of myocardial motion from tagged MR images

In the past few years we witnessed an increase in mortality due to cancer relative to mortality due to cardiovascular diseases. In 2008, the Netherlands Statistics Agency reports that 33.900 people died of cancer against 33.100 deaths due to cardiovascular diseases, making cancer the number one cause of death in the Netherlands [33]. Even if the rate of people affected by heart diseases is continually rising, they "simply don’t die of it", according to the research director Prof. Mat Daemen of research institute CARIM of the University of Maastricht [50]. The reason for this is the early diagnosis, and the treatment of people with identified risk factors for diseases like ischemic heart disease, hypertrophic cardiomyopathy, thoracic aortic disease, pericardial (sac around the heart) disease, cardiac tumors, pulmonary artery disease, valvular disease, and congenital heart disease before and after surgical repair. Cardiac imaging plays a crucial role in the early diagnosis, since it allows the accurate investigation of a large amount of imaging data in a small amount of time. Moreover, cardiac imaging reduces costs of inpatient care, as has been shown in recent studies [77]. With this in mind, in this work we have provided several tools with the aim to help the investigation of the cardiac motion. In chapters 2 and 3 we have explored a novel variational optic flow methodology based on multi-scale feature points to extract cardiac motion from tagged MR images. Compared to constant brightness methods, this new approach exhibits several advantages. Although the intensity of critical points is also influenced by fading, critical points do retain their characteristic even in the presence of intensity changes, such as in MR imaging. In an experiment in section 5.4 we have applied this optic flow approach directly on tagged MR images. A visual inspection confirmed that the extracted motion fields realistically depicted the cardiac wall motion. The method exploits also the advantages from the multiscale framework. Because sparse velocity formulas 2.9, 3.7, 6.21, and 7.5 provide a number of equations equal to the number of unknowns, the method does not suffer from the aperture problem in retrieving velocities associated to the critical points. In chapters 2 and 3 we have moreover introduced a smoothness component of the optic flow equation described by means of covariant derivatives. This is a novelty in the optic flow literature. Many variational optic flow methods present a smoothness component that penalizes for changes from global assumptions such as isotropic or anisotropic smoothness. In the smoothness term proposed deviations from a predefined motion model are penalized. Moreover, the proposed optic flow equation has been decomposed in rotation-free and divergence-free components. This decomposition allows independent tuning of the two components during the vector field reconstruction. The experiments and the Table of errors provided in 3.8 showed that the combination of the smoothness term, influenced by a predefined motion model, and the Helmholtz decomposition in the optic flow equation reduces the average angular error substantially (20%-25%) with respect to a similar technique that employs only standard derivatives in the smoothness term. In section 5.3 we extracted the motion field of a phantom of which we know the ground truth of and compared the performance of this optic flow method with the performance of other optic flow methods well known in the literature, such as the Horn and Schunck [76] approach, the Lucas and Kanade [111] technique and the tuple image multi-scale optic flow constraint equation of Van Assen et al. [163]. Tests showed that the proposed optic flow methodology provides the smallest average angular error (AAE = 3.84 degrees) and L2 norm = 0.1. In this work we employed the Helmholtz decomposition also to study the cardiac behavior, since the vector field decomposition allows to investigate cardiac contraction and cardiac rotation independently. In chapter 4 we carried out an analysis of cardiac motion of ten volunteers and one patient where we estimated the kinetic energy for the different components. This decomposition is useful since it allows to visualize and quantify the contributions of each single vector field component to the heart beat. Local measurements of the kinetic energy have also been used to detect areas of the cardiac walls with little movement. Experiments on a patient and a comparison between a late enhancement cardiac image and an illustration of the cardiac kinetic energy on a bull’s eye plot illustrated that a correspondence between an infarcted area and an area with very small kinetic energy exists. With the aim to extend in the future the proposed optic flow equation to a 3D approach, in chapter 6 we investigated the 3D winding number approach as a tool to locate critical points in volume images. We simplified the mathematics involved with respect to a previous work [150] and we provided several examples and applications such as cardiac motion estimation from 3-dimensional tagged images, follicle and neuronal cell counting. Finally in chapter 7 we continued our investigation on volume tagged MR images, by retrieving the cardiac motion field using a 3-dimensional and simple version of the proposed optic flow equation based on standard derivatives. We showed that the retrieved motion fields display the contracting and rotating behavior of the cardiac muscle. We moreover extracted the through-plane component, which provides a realistic illustration of the vector field and is missed by 2-dimensional approaches

Understanding Physiological and Degenerative Natural Vision Mechanisms to Define Contrast and Contour Operators

BACKGROUND:Dynamical systems like neural networks based on lateral inhibition have a large field of applications in image processing, robotics and morphogenesis modeling. In this paper, we will propose some examples of dynamical flows used in image contrasting and contouring. METHODOLOGY:First we present the physiological basis of the retina function by showing the role of the lateral inhibition in the optical illusions and pathologic processes generation. Then, based on these biological considerations about the real vision mechanisms, we study an enhancement method for contrasting medical images, using either a discrete neural network approach, or its continuous version, i.e. a non-isotropic diffusion reaction partial differential system. Following this, we introduce other continuous operators based on similar biomimetic approaches: a chemotactic contrasting method, a viability contouring algorithm and an attentional focus operator. Then, we introduce the new notion of mixed potential Hamiltonian flows; we compare it with the watershed method and we use it for contouring. CONCLUSIONS:We conclude by showing the utility of these biomimetic methods with some examples of application in medical imaging and computed assisted surgery

Restauration d'images en IRM anatomique pour l'étude préclinique des marqueurs du vieillissement cérébral

Les maladies neurovasculaires et neurodégénératives liées à l'âge sont en forte augmentation. Alors que ces changements pathologiques montrent des effets sur le cerveau avant l'apparition de symptômes cliniques, une meilleure compréhension du processus de vieillissement normal du cerveau aidera à distinguer l'impact des pathologies connues sur la structure régionale du cerveau. En outre, la connaissance des schémas de rétrécissement du cerveau dans le vieillissement normal pourrait conduire à une meilleure compréhension de ses causes et peut-être à des interventions réduisant la perte de fonctions cérébrales associée à l'atrophie cérébrale. Par conséquent, ce projet de thèse vise à détecter les biomarqueurs du vieillissement normal et pathologique du cerveau dans un modèle de primate non humain, le singe marmouset (Callithrix Jacchus), qui possède des caractéristiques anatomiques plus proches de celles des humains que de celles des rongeurs. Cependant, les changements structurels (par exemple, de volumes, d'épaisseur corticale) qui peuvent se produire au cours de leur vie adulte peuvent être minimes à l'échelle de l'observation. Dans ce contexte, il est essentiel de disposer de techniques d'observation offrant un contraste et une résolution spatiale suffisamment élevés et permettant des évaluations détaillées des changements morphométriques du cerveau associé au vieillissement. Cependant, l'imagerie de petits cerveaux dans une plateforme IRM 3T dédiée à l'homme est une tâche difficile car la résolution spatiale et le contraste obtenus sont insuffisants par rapport à la taille des structures anatomiques observées et à l'échelle des modifications attendues. Cette thèse vise à développer des méthodes de restauration d'image pour les images IRM précliniques qui amélioreront la robustesse des algorithmes de segmentation. L'amélioration de la résolution spatiale des images à un rapport signal/bruit constant limitera les effets de volume partiel dans les voxels situés à la frontière entre deux structures et permettra une meilleure segmentation tout en augmentant la reproductibilité des résultats. Cette étape d'imagerie computationnelle est cruciale pour une analyse morphométrique longitudinale fiable basée sur les voxels et l'identification de marqueurs anatomiques du vieillissement cérébral en suivant les changements de volume dans la matière grise, la matière blanche et le liquide cérébral.Age-related neurovascular and neurodegenerative diseases are increasing significantly. While such pathological changes show effects on the brain before clinical symptoms appear, a better understanding of the normal aging brain process will help distinguish known pathologies' impact on regional brain structure. Furthermore, knowledge of the patterns of brain shrinkage in normal aging could lead to a better understanding of its causes and perhaps to interventions reducing the loss of brain functions. Therefore, this thesis project aims to detect normal and pathological brain aging biomarkers in a non-human primate model, the marmoset monkey (Callithrix Jacchus) which possesses anatomical characteristics more similar to humans than rodents. However, structural changes (e.g., volumes, cortical thickness) that may occur during their adult life may be minimal with respect to the scale of observation. In this context, it is essential to have observation techniques that offer sufficiently high contrast and spatial resolution and allow detailed assessments of the morphometric brain changes associated with aging. However, imaging small brains in a 3T MRI platform dedicated to humans is a challenging task because the spatial resolution and the contrast obtained are insufficient compared to the size of the anatomical structures observed and the scale of the xpected changes with age. This thesis aims to develop image restoration methods for preclinical MR images that will improve the robustness of the segmentation algorithms. Improving the resolution of the images at a constant signal-to-noise ratio will limit the effects of partial volume in voxels located at the border between two structures and allow a better segmentation while increasing the results' reproducibility. This computational imaging step is crucial for a reliable longitudinal voxel-based morphometric analysis and for the identification of anatomical markers of brain aging by following the volume changes in gray matter, white matter and cerebrospinal fluid

Characterizing functional and structural brain alterations driven by chronic alcohol drinking: a resting-state fMRI connectivity and voxel-based morphometry analysis

El balance del cerebro se altera a nivel estructural y funcional por el consumo de alcohol y puede causar trastornos por consumo de alcohol (TCA). El objetivo de esta Tesis Doctoral fue investigar los efectos del consumo crónico y excesivo de alcohol en el cerebro desde una perspectiva funcional y estructural, mediante análisis de imágenes multimodales de resonancia magnética (RM). Realizamos tres estudios con objetivos específicos: i) Para entender cómo las neuroadaptaciones desencadenadas por el consumo de alcohol se ven reflejadas en la conectividad cerebral funcional entre redes cerebrales, así como en la actividad cerebral, realizamos estudios en ratas msP en condiciones de control y tras un mes con acceso a alcohol. Para cada sujeto se obtuvieron las señales específicas de sus redes cerebrales tras aplicar análisis probabilístico de componentes independientes y regresión espacial a las imágenes funcionales de RM en estado de reposo (RMf-er). Después, estimamos la conectividad cerebral en estado de reposo mediante correlación parcial regularizada. Para una lectura de la actividad neuronal realizamos un experimento con imágenes de RM realzadas con manganeso. En la condición de alcohol encontramos hipoconectividades entre la red visual y las redes estriatal y sensorial; todas con incrementos en actividad. Por el contrario, hubo hiperconectividades entre tres pares de redes cerebrales: 1) red prefrontal cingulada media y red estriatal, 2) red sensorial y red parietal de asociación y 3) red motora-retroesplenial y red sensorial, siendo la red parietal de asociación la única red sin incremento de actividad. Estos resultados indican que las redes cerebrales ya se alteran desde una fase temprana de consumo continuo y prolongado de alcohol, disminuyendo el control ejecutivo y la flexibilidad comportamental. ii) Para comparar el volumen de materia gris (MG) cortical entre 34 controles sanos y 35 pacientes con dependencia al alcohol, desintoxicados y en abstinencia de 1 a 5 semanas, realizamos un análisis de morfometría basado en vóxel. Las principales estructuras cuyo volumen de MG disminuyó en los sujetos en abstinencia fueron el giro precentral (GPreC), el giro postcentral (GPostC), la corteza motora suplementaria (CMS), el giro frontal medio (GFM), el precúneo (PCUN) y el lóbulo parietal superior (LPS). Disminuciones de MG en el volumen de esas áreas pueden dar lugar a cambios en el control de los movimientos (GPreC y CMS), en el procesamiento de información táctil y propioceptiva (GPostC), personalidad, previsión (GFM), reconocimiento sensorial, entendimiento del lenguaje, orientación (PCUN) y reconocimiento de objetos a través de su forma (LPS). iii) Caracterizar estados cerebrales dinámicos en señales de RMf mediante una metodología basada en un modelo oculto de Markov (HMM en inglés)-Gaussiano en un paradigma con diseño de bloques, junto con distintas señales temporales de múltiples redes: componentes independientes y modos funcionales probabilísticos (PFMs en inglés) en 14 sujetos sanos. Cuatro condiciones experimentales formaron el paradigma de bloques: reposo, visual, motora y visual-motora. Mediante la aplicación de HMM-Gaussiano a los PFMs pudimos caracterizar cuatro estados cerebrales a partir de la actividad media de cada PFM. Los cuatro mapas espaciales obtenidos fueron llamados HMM-reposo, HMM-visual, HMM-motor y HMM-RND (red neuronal por defecto). HMM-RND apareció una vez el estado de tarea se había estabilizado. En un futuro cercano se espera obtener estados cerebrales en nuestros datos de RMf-er en ratas, para comparar dinámicamente el comportamiento de las redes cerebrales como un biomarcador de TCA. En conclusión, las técnicas de neuroimagen aplicadas en imagen de RM multimodal para estimar la conectividad cerebral en estado de reposo, la actividad cerebral y el volumen de materia gris han permitido avanzar en el entendimiento de los mecanismos homeostáticoLa ingesta d'alcohol altera el balanç del cervell a nivell estructural i funcional i pot causar trastorns per consum d' alcohol (TCA). L'objectiu d'aquesta Tesi Doctoral fou estudiar els efectes en el cervell del consum crònic i excessiu d'alcohol, des d'un punt de vista funcional i estructural i per mitjà d'anàlisi d'imatges de ressonància magnètica (RM). Vam realitzar tres anàlisis amb objectius específics: i) Per a entendre com les neuroadaptacions desencadenades pel consum d'alcohol es veuen reflectides en la connectivitat cerebral funcional entre xarxes cerebrals, així com en l'activitat cerebral, vam realitzar estudis en rates msP en les condicions de control i després d'un mes amb accés a alcohol. Per a cada subjecte vam obtindre els senyals de les xarxes cerebrals tras aplicar a les imatges funcionals de RM en estat de repòs una anàlisi probabilística de components independents i regressió espacial. Després, estimàrem la connectivitat cerebral en estat de repòs per mitjà de correlació parcial regularitzada. Per a una lectura de l'activitat cerebral vam adquirir imatges de RM realçades amb manganés. En la condició d'alcohol vam trobar hipoconnectivitats entre la xarxa visual i les xarxes estriatal i sensorial, totes amb increments en activitat. Al contrari, va haver-hi hiperconnectivitats entre tres parells de xarxes cerebrals: 1) xarxa prefrontal cingulada mitja i xarxa estriatal, 2) xarxa sensorial i xarxa parietal d'associació i 3) xarxa motora-retroesplenial i xarxa sensorial, sent la xarxa parietal d'associació l'única xarxa sense increment d'activitat. Aquests resultats indiquen que les xarxes cerebrals ja s'alteren des d'una fase primerenca caracteritzada per consum continu i prolongat d'alcohol, disminuint el control executiu i la flexibilitat comportamental. ii) Per a comparar el volum de MG cortical entre 34 controls sans i 35 pacients amb dependència a l'alcohol, desintoxicats i en abstinència de 1 a 5 setmanes vam emprar anàlisi de morfometria basada en vòxel. Les principals estructures on el volum de MG va disminuir en els subjectes en abstinència van ser el gir precentral (GPreC), el gir postcentral (GPostC), la corteça motora suplementària (CMS), el gir frontal mig (GFM), el precuni (PCUN) i el lòbul parietal superior (LPS). Les disminucions de MG en eixes àrees poden donar lloc a canvis en el control dels moviments (GPreC i CMS), en el processament d'informació tàctil i propioceptiva (GPostC), personalitat, previsió (GFM), reconeixement sensorial, enteniment del llenguatge, orientació (PCUN) i reconeixement d'objectes a través de la seua forma (LPS). iii) Caracterització de les dinàmiques temporals del cervell com a diferents estats cerebrals, en senyals de RMf mitjançant una metodologia basada en un model ocult de Markov (HMM en anglès)-Gaussià en imatges de RMf, junt amb dos tipus de senyals temporals de múltiples xarxes cerebrals: components independents i modes funcionals probabilístics (PFMs en anglès) en 14 subjectes sans. Quatre condicions experimentals van formar el paradigma de blocs: repòs, visual, motora i visual-motora. HMM-Gaussià aplicat als PFMs (senyals de RM funcional de xarxes cerebrals) va permetre la millor caracterització dels quatre estats cerebrals a partir de l'activitat mitjana de cada PFM. Els quatre mapes espacials obtinguts van ser anomenats HMM-repòs, HMM-visual, HMM-motor i HMM-XND (xarxa neuronal per defecte). HMM-XND va aparèixer una vegada una tasca estava estabilitzada. En un futur pròxim s'espera obtindre estats cerebrals en les nostres dades de RMf-er en rates, per a comparar dinàmicament el comportament de les xarxes cerebrals com a biomarcador de TCA. En conclusió, s'han aplicat tècniques de neuroimatge per a estimar la connectivitat cerebral en estat de repòs, l'activitat cerebral i el volum de MG, aplicades a imatges multimodals de RM i s'han obtés resultats que han permés avançar en l'enteniment dels mAlcohol intake alters brain balance, affecting its structure and function, and it may cause Alcohol Use Disorders (AUDs). We aimed to study the effects of chronic, excessive alcohol consumption on the brain from a functional and structural point of view, via analysis of multimodal magnetic resonance (MR) images. We conducted three studies with specific aims: i) To understand how the neuroadaptations triggered by alcohol intake are reflected in between-network resting-state functional connectivity (rs-FC) and brain activity in the onset of alcohol dependence, we performed studies in msP rats in control and alcohol conditions. Group probabilistic independent component analysis (group-PICA) and spatial regression were applied to resting-state functional magnetic resonance imaging (rs-fMRI) images to obtain subject-specific time courses of seven resting-state networks (RSNs). Then, we estimated rs-FC via L2-regularized partial correlation. We performed a manganese-enhanced (MEMRI) experiment as a readout of neuronal activity. In alcohol condition, we found hypoconnectivities between the visual network (VN), and striatal (StrN) and sensory-cortex (SCN) networks, all with increased brain activity. On the contrary, hyperconnectivities were found between three pairs of RSNs: 1) medial prefrontal-cingulate (mPRN) and StrN, 2) SCN and parietal association (PAN) and 3) motor-retrosplenial (MRN) and SCN networks, being PAN the only network without brain activity rise. Interestingly, the hypoconnectivities could be explained as control to alcohol transitions from direct to indirect connectivity, whereas the hyperconnectivities reflected an indirect to an even more indirect connection. These findings indicate that RSNs are early altered by prolonged and moderate alcohol exposure, diminishing the executive control and behavioral flexibility. ii) To compare cortical gray matter (GM) volume between 34 healthy controls and 35 alcohol-dependent patients who were detoxified and remained abstinent for 1-5 weeks before MRI acquisition, we performed a voxel-based morphometry analysis. The main structures whose GM volume decreased in abstinent subjects compared to controls were precentral gyrus (PreCG), postcentral gyrus (PostCG), supplementary motor cortex (SMC), middle frontal gyrus (MFG), precuneus (PCUN) and superior parietal lobule (SPL). Decreases in GM volume in these areas may lead to changes in control of movement (PreCG and SMC), in processing tactile and proprioceptive information (PostCG), personality, insight, prevision (MFG), sensory appreciation, language understanding, orientation (PCUN) and the recognition of objects by touch and shapes (SPL). iii) To characterize dynamic brain states in functional MRI (fMRI) signals by means of an approach based on the Hidden Markov model (HMM). Several parameter configurations of HMM-Gaussian in a block-design paradigm were considered, together with different time series: independent components (ICs) and probabilistic functional modes (PFMs) on 14 healthy subjects. The block-design fMRI paradigm consisted of four experimental conditions: rest, visual, motor and visual-motor. Characterizing brain states' dynamics in fMRI data was possible applying the HMM-Gaussian approach to PFMs, with mean activity driving the states. The four spatial maps obtained were named HMM-rest, HMM-visual, HMM-motor and HMM-DMN (default mode network). HMM-DMN appeared once a task state had stabilized. The ultimate goal will be to obtain brain states in our rs-fMRI rat data, to dynamically compare the behavior of brain RSNs as a biomarker of AUD. In conclusion, neuroimaging techniques to estimate rs-FC, brain activity and GM volume can be successfully applied to multimodal MRI in the advance of the understanding of brain homeostasis in AUDs. These functional and structural alterations are a biomarker of chronic alcoholism to explain impairments in executive control, reward evaluation and visuospatial processing.Pérez Ramírez, MÚ. (2018). Characterizing functional and structural brain alterations driven by chronic alcohol drinking: a resting-state fMRI connectivity and voxel-based morphometry analysis [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/11316

Mathematics and Algorithms in Tomography

This was the ninth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, emission tomography, X-ray CT, and vector tomography along with a wide range of mathematical analysis