Planar Homography Estimation from Traffic Streams via Energy Functional Minimization

The 3x3 homography matrix specifies the mapping between two images of the same plane as viewed by a pinhole camera. Knowledge of the matrix allows one to remove the perspective distortion and apply any similarity transform, effectively making possible the measurement of distances and angles on the image. A rectified road scene for instance, where vehicles can be segmented and tracked, gives rise to ready estimates of their velocities and spacing or categorization of their type. Typical road scenes render the classical approach to homography estimation difficult. The Direct Linear Transform is highly susceptible to noise and usually requires refining via an further nonlinear penalty minimization. Additionally, the penalty is a function of the displacement between measured and calibrated coordinates, a quantity unavailable in a scene for which we have no knowledge of the road coordinates. We propose instead to achieve metric rectification via the minimization of an energy that measures the violation of two constraints: the divergence-free nature of the traffic flow and the orthogonality of the flow and transverse directions under the true transform. Given that an homography is only determined up to scale, the minimization is performed on the Lie group $SL(3)$, for which we develop a gradient descent algorithm. While easily expressed in the world frame, the energy must be computed from measurements made in the image and thus must be pulled back using standard differential geometric machinery to the image frame. We develop an enhancement to the algorithm by incorporating optical flow ideas and apply it to both a noiseless test case and a suite of real-world video streams to demonstrate its efficacy and convergence. Finally, we discuss the extension to a 3D-to-planar mapping for vehicle height inference and an homography that is allowed to vary over the image, invoking a minimization on Diff$(SL(3))$

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