This thesis reviews the method of Mafra, Schlotterer, and Stieberger (2011) for computing the full colour ordered N-point open superstring amplitude using the Pure Spinor formalism. We introduce relevant elements of super Yang-Mills theory and examine the basics of the Pure Spinor formalism, with a focus on tools for amplitude computation. We then define a series of objects with increasingly useful BRST transformation properties, which greatly simplify the calculations, and show how these properties can be determined using a diagrammatic method. Finally, we use the explicit four- and five-point amplitude computations as stepping stones to compute the general N-point amplitude, which factors into a set of kinematic integrals multiplying SYM subamplitudes.

Stereo camera systems can be utilized for different applications such as position estimation,distance measuring, and 3d modelling. However, this requires the cameras to be calibrated.This paper proposes a traditional calibration solution with Aruco-markers mounted on avehicle to estimate the pose of a stereo camera system in a tolling environment. Our method isbased on Perspective N Point which presumes the intrinsic matrix to be already known. Thegoal is to find each camera’s pose by identifying the marker corners in pixel coordinates aswell as in world coordinates. Our tests show a worst-case error of 21.5 cm and a potential forcentimetre accuracy. It also verifies validity by testing the obtained pose estimation live in thecamera system. The paper concludes that the method has potential for higher accuracy notobtained in our experiment due to several factors. Further work would focus on enlarging themarkers and widening the distance between the markers.

Lens distortions appear in almost all digital images and cause straight lines to appear curved in the image. This can contribute to errors in position estimations and 3D reconstruction and it is therefore of interest to correct for the distortion. If the camera is available, the distortion parameters can be obtained when calibrating the camera. However, when the camera is unavailable the distortion parameters can not be found with the standard camera calibration technique and other approaches must be used. Recently, variants of Perspective-n-Point (PnP) extended with lens distortionand focal length parameters have been proposed. Given a set of 2D-3D point correspondences, the PnP-based methods can estimate distortion parameters without the camera being available or with modified settings. In this thesis, the performance of PnP-based methods is compared to Zhang’s camera calibration method. The methods are compared both quantitatively, using the errors in reprojectionand distortion parameters, and qualitatively by comparing images before and after lens distortion correction. A test set for the comparison was obtained from a camera and a 3D laser scanner of an indoor scene.The results indicate that one of the PnP-based models can achieve a similar reprojection error as the baseline method for one of the cameras. It could also be seen that two PnP-based models could reduce lens distortion when visually comparing the test images to the baseline. Moreover, it was noted that a model can have a small reprojection error even though the distortion coefficient error is large and the lens distortion is not completely removed. This indicates that it is important to include both quantitative measures, such as reprojection error and distortion coefficient errors, as well as qualitative results when comparing lens distortion correction methods. It could also be seen that PnP-based models with more parameters in the estimation are more sensitive to noise.

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The Fast Multipole Method of Greengard and Rokhlin does the seemingly impossible: it approximates the quadratic scaling N-body problem in linear time. The key is to avoid explicitly computing the interactions between all pairs of N points. Instead, by organizing the data in a space-partitioning tree, distant interactions are quickly and efficiently approximated. Similarly, dual-tree algorithms, which approximate or eliminate parts of a computation using distance bounds, are the fastest algorithms for several fundamental problems in statistics and machine learning -- including all nearest neighbors, kernel density estimation, and Euclidean minimum spanning tree construction. We show that this overarching principle -- that by organizing points spatially, we can solve a seemingly quadratic problem in linear time -- can be generalized to problems involving interactions between sets of three or more points and can provide orders-of-magnitude speedups and guarantee runtimes that are asymptotically better than existing algorithms. We describe a family of algorithms, multi-tree algorithms, which can be viewed as generalizations of dual-tree algorithms. We support this thesis by developing and implementing multi-tree algorithms for two fundamental scientific applications: n-point correlation function estimation and Hartree-Fock theory. First, we demonstrate multi-tree algorithms for n-point correlation function estimation. The n-point correlation functions are a family of fundamental spatial statistics and are widely used for understanding large-scale astronomical surveys, characterizing the properties of new materials at the microscopic level, and for segmenting and processing images. We present three new algorithms which will reduce the dependence of the computation on the size of the data, increase the resolution in the result without additional time, and allow probabilistic estimates independent of the problem size through sampling. We provide both empirical evidence to support our claim of massive speedups and a theoretical analysis showing linear scaling in the fundamental computational task. We demonstrate the impact of a carefully optimized base case on this computation and describe our distributed, scalable, open-source implementation of our algorithms. Second, we explore multi-tree algorithms as a framework for understanding the bottleneck computation in Hartree-Fock theory, a fundamental model in computational chemistry. We analyze existing fast algorithms for this problem, and show how they fit in our multi-tree framework. We also show new multi-tree methods, demonstrate that they are competitive with existing methods, and provide the first rigorous guarantees for the runtimes of all of these methods. Our algorithms will appear as part of the PSI4 computational chemistry library.

This thesis attempts to create an application to determine head rotation angle in video recorded from a security camera. The application consists of three parts: face detection, facial landmarks detection and determination of person's head rotation. The face detection has been implemented using Viola-Jones and HOG algorithms. Facial landmarks detection has been done using algorithm based on active shape model. Two methods to calculate the head rotation angles have been used: the first method works with anthropometric head features. The second method uses Perspective-n-Point algorithm to find the right rotation angles. Finally, all algorithms implemented have been tested and the proper parameters have been determined.