Discretization schemes and numerical approximations of PDE impainting models and a comparative evaluation on novel real world MRI reconstruction applications

While various PDE models are in discussion since the last ten years and are widely applied nowadays in image processing and computer vision tasks, including restoration, filtering, segmentation and object tracking, the perspective adopted in the majority of the relevant reports is the view of applied mathematician, attempting to prove the existence theorems and devise exact numerical methods for solving them. Unfortunately, such solutions are exact for the continuous PDEs but due to the discrete approximations involved in image processing, the results yielded might be quite unsatisfactory. The major contribution of This work is, therefore, to present, from an engineering perspective, the application of PDE models in image processing analysis, from the algorithmic point of view, the discretization and numerical approximation schemes used for solving them. It is of course impossible to tackle all PDE models applied in image processing in this report from the computational point of view. It is, therefore, focused on image impainting PDE models, that is on PDEs, including anisotropic diffusion PDEs, higher order non-linear PDEs, variational PDEs and other constrained/regularized and unconstrained models, applied to image interpolation/ reconstruction. Apart from this novel computational critical overview and presentation of the PDE image impainting models numerical analysis, the second major contribution of This work is to evaluate, especially the anisotropic diffusion PDEs, in novel real world image impainting applications related to MRI

A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance

For strongly directed anisotropic processes such as coherence-enhancing diffusion filtering it is crucial to use numerical schemes with highly accurate directional behavior. We show that this is not possible in a satisfactory way when discretizations are limited to 3 x 3 stencils. As a consequence, we investigate a novel algorithm based on 5 x 5 stencils. It utilizes recently discovered differentiation filters with optimized rotation invariance. By juxtaposing it with several common algorithms we demonstrate its superior behavior with respect to the following properties: rotation invariance, avoidance of blurring artifacts (dissipativity), and accuracy. The latter one is evaluated by deriving an analytical solution for coherence-enhancing diffusion filtering of images with circular symmetry. Furthermore, we show that the new scheme is 3 to 4 times more efficient than explicit schemes on 3 x 3 stencils. It does not require to solve linear systems of equations, and it can be easily implemented in any dimension

PDE Based Enhancement of Color Images in RGB Space

International audienceA novel method for color image enhancement is proposed as an extension of scalar diffusion-shock filter coupling model, where noisy and blurred images are denoised and sharpened. The proposed model is based on using single vectors of the gradient magnitude and the second derivatives as a technique to relate different color components of the image. This model can be viewed as a generalization of Bettahar-Stambouli filter to multi-valued images. The proposed algorithm is more efficient than the mentioned filter and some previous works on color image denoising and deblurring without creating false colors

Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains

Statistical Computing on Non-Linear Spaces for Computational Anatomy

International audienceComputational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings

Upper airways segmentation using principal curvatures

Esta tesis propone una nueva técnica para segmentar las vías aéreas superiores. Esta propuesta permite la extracción de estructuras curvilíneas usando curvaturas principales. La propuesta permite la extracción de éstas estructuras en imágenes 2D y 3D. Entre las principales novedades se encuentra la propuesta de un nuevo criterio de parada en la propagación del algoritmo de realce de contraste (operador multi-escala de tipo sombrero alto). De la misma forma, el criterio de parada propuesto es usado para detener los algoritmos de difusión anisotrópica. Además, un nuevo criterio es propuesto para seleccionar las curvaturas principales que conforman las estructuras curvilíneas, que se basa en los criterios propuestos por Steger, Deng et. al. y Armande et. al. Además, se propone un nuevo algoritmo para realizar la supresión de nomáximos que permite reducir la presencia de discontinuidades en el borde de las estructuras curvilíneas. Para extraer los bordes de las estructuras curvilíneas, se utiliza un algoritmo de enlace que incluye un nuevo criterio de distancia para reducir la aparición de agujeros en la estructura final. Finalmente, con base en los resultados obtenidos, se utiliza un algoritmo morfológico para cerrar los agujeros y se aplica un algoritmo de crecimiento de regiones para obtener la segmentación final de las vías respiratorias superiores.This dissertation proposes a new approach to segment the upper airways. This proposal allows the extraction of curvilinear structures based on the principal curvatures. The proposal allows extracting these structures from 2D and 3D images. Among the main novelties is the proposal of a new stopping criterion to stop the propagation of the contrast enhancement algorithm (multiscale top-hat morphological operator). In the same way, the proposed stopping criterion is used to stop the anisotropic diffusion algorithms. In addition, a new criterion is proposed to select the principal curvatures that make up the curvilinear structures, which is based on the criteria proposed by Steger, Deng et. al. and Armande et. al. Furthermore, a new algorithm to perform the non-maximum suppression that allows reducing the presence of discontinuities in the border of curvilinear structures is proposed. To extract the edges of the curvilinear structures, a linking algorithm is used that includes a new distance criterion to reduce the appearance of gaps in the final structure. Finally, based on the obtained results, a morphological algorithm is used to close the gaps and a region growing algorithm to obtain the final upper airways segmentation is applied.Doctor en IngenieríaDoctorad

Nonlocal smoothing and adaptive morphology for scalar- and matrix-valued images

In this work we deal with two classic degradation processes in image analysis, namely noise contamination and incomplete data. Standard greyscale and colour photographs as well as matrix-valued images, e.g. diffusion-tensor magnetic resonance imaging, may be corrupted by Gaussian or impulse noise, and may suffer from missing data. In this thesis we develop novel reconstruction approaches to image smoothing and image completion that are applicable to both scalar- and matrix-valued images. For the image smoothing problem, we propose discrete variational methods consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures. We obtain edge-preserving filters by the joint use of such measures rich in texture content together with robust non-convex penalisers. For the image completion problem, we introduce adaptive, anisotropic morphological partial differential equations modelling the dilation and erosion processes. They adjust themselves to the local geometry to adaptively fill in missing data, complete broken directional structures and even enhance flow-like patterns in an anisotropic manner. The excellent reconstruction capabilities of the proposed techniques are tested on various synthetic and real-world data sets.In dieser Arbeit beschäftigen wir uns mit zwei klassischen Störungsquellen in der Bildanalyse, nämlich mit Rauschen und unvollständigen Daten. Klassische Grauwert- und Farb-Fotografien wie auch matrixwertige Bilder, zum Beispiel Diffusionstensor-Magnetresonanz-Aufnahmen, können durch Gauß- oder Impulsrauschen gestört werden, oder können durch fehlende Daten gestört sein. In dieser Arbeit entwickeln wir neue Rekonstruktionsverfahren zum zur Bildglättung und zur Bildvervollständigung, die sowohl auf skalar- als auch auf matrixwertige Bilddaten anwendbar sind. Zur Lösung des Bildglättungsproblems schlagen wir diskrete Variationsverfahren vor, die aus nichtlokalen Daten- und Glattheitstermen bestehen und allgemeine auf Bildausschnitten definierte Unähnlichkeitsmaße bestrafen. Kantenerhaltende Filter werden durch die gemeinsame Verwendung solcher Maße in stark texturierten Regionen zusammen mit robusten nichtkonvexen Straffunktionen möglich. Für das Problem der Datenvervollständigung führen wir adaptive anisotrope morphologische partielle Differentialgleichungen ein, die Dilatations- und Erosionsprozesse modellieren. Diese passen sich der lokalen Geometrie an, um adaptiv fehlende Daten aufzufüllen, unterbrochene gerichtet Strukturen zu schließen und sogar flussartige Strukturen anisotrop zu verstärken. Die ausgezeichneten Rekonstruktionseigenschaften der vorgestellten Techniken werden anhand verschiedener synthetischer und realer Datensätze demonstriert