Optimal Folding of Data Flow Graphs based on Finite Projective Geometry using Lattice Embedding

A number of computations exist, especially in area of error-control coding and matrix computations, whose underlying data flow graphs are based on finite projective-geometry(PG) based balanced bipartite graphs. Many of these applications are actively being researched upon. Almost all these applications need bipartite graphs of the order of tens of thousands in practice, whose nodes represent parallel computations. To reduce its implementation cost, reducing amount of system/hardware resources during design is an important engineering objective. In this context, we present a scheme to reduce resource utilization when performing computations derived from PG-based graphs. In a fully parallel design based on PG concepts, the number of processing units is equal to the number of vertices, each performing an atomic computation. To reduce the number of processing units used for implementation, we present an easy way of partitioning the vertex set. Each block of partition is then assigned to a processing unit. A processing unit performs the computations corresponding to the vertices in the block assigned to it in a sequential fashion, thus creating the effect of folding the overall computation. These blocks have certain symmetric properties that enable us to develop a conflict-free schedule. The scheme achieves the best possible throughput, in lack of any overhead of shuffling data across memories while scheduling another computation on the same processing unit. This paper reports two folding schemes, which are based on same lattice embedding approach, based on partitioning. We first provide a scheme for a projective space of dimension five, and the corresponding schedules. Both the folding schemes that we present have been verified by both simulation and hardware prototyping for different applications. We later generalize this scheme to arbitrary projective spaces.Comment: 31 pages, to be submitted to some discrete mathematics journa

Geometric uncertainty models for correspondence problems in digital image processing

Many recent advances in technology rely heavily on the correct interpretation of an enormous amount of visual information. All available sources of visual data (e.g. cameras in surveillance networks, smartphones, game consoles) must be adequately processed to retrieve the most interesting user information. Therefore, computer vision and image processing techniques gain significant interest at the moment, and will do so in the near future. Most commonly applied image processing algorithms require a reliable solution for correspondence problems. The solution involves, first, the localization of corresponding points -visualizing the same 3D point in the observed scene- in the different images of distinct sources, and second, the computation of consistent geometric transformations relating correspondences on scene objects. This PhD presents a theoretical framework for solving correspondence problems with geometric features (such as points and straight lines) representing rigid objects in image sequences of complex scenes with static and dynamic cameras. The research focuses on localization uncertainty due to errors in feature detection and measurement, and its effect on each step in the solution of a correspondence problem. Whereas most other recent methods apply statistical-based models for spatial localization uncertainty, this work considers a novel geometric approach. Localization uncertainty is modeled as a convex polygonal region in the image space. This model can be efficiently propagated throughout the correspondence finding procedure. It allows for an easy extension toward transformation uncertainty models, and to infer confidence measures to verify the reliability of the outcome in the correspondence framework. Our procedure aims at finding reliable consistent transformations in sets of few and ill-localized features, possibly containing a large fraction of false candidate correspondences. The evaluation of the proposed procedure in practical correspondence problems shows that correct consistent correspondence sets are returned in over 95% of the experiments for small sets of 10-40 features contaminated with up to 400% of false positives and 40% of false negatives. The presented techniques prove to be beneficial in typical image processing applications, such as image registration and rigid object tracking

Design of a Real-time Image-based Distance Sensing System by Stereo Vision on FPGA

A stereo vision system is a robust method to sense the distance information in a scene. This research explores the stereo vision system from the fundamentals of stereo vision and the computer stereo vision algorithm to the final implementation of the system on a FPGA chip. In a stereo vision system, images are captured by a pair of stereo image sensors. The distance information can be derived from the disparities between the stereo image pair, based on the theory of binocular geometry. With the increasing focus on 3D vision, stereo vision is becoming a hot topic in the areas of computer games, robot vision and medical applications. Particularly, most stereo vision systems are expected to be used in real-time applications. In this thesis, several stereo correspondence algorithms that determine the disparities between stereo image pair are examined. The algorithms can be categorized into global stereo algorithms and local stereo algorithms depending on the optimization techniques. The global algorithms examined are the Dynamic Time Warp (DTW) algorithm and the DTW with quantization algorithm, while the local algorithms examined are the window based Sum of Squared Differences (SSD), Sum of Absolute Differences (SAD) and Census transform correlation algorithms. With analysis among them, the window based SAD correlation algorithm is proposed for implementation on a FPGA platform. The proposed algorithm is implemented onto an Altera DE2 board featuring an Altera Cyclone II 2C35 FPGA. The implemented module of the algorithm is simulated using ModelSim-Altera to verify the correctness of its functionality. Along with a pair of stere image sensors and a LCD monitor, a stereo vision system is built. The entire system realizes a real-time video frame rate of 16.83 frames per second with an image resolution of 640 by 480 and produces disparity maps in which the objects are clearly distinguished by their relative distance information

Kinematics and Robot Design I, KaRD2018

This volume collects the papers published on the Special Issue “Kinematics and Robot Design I, KaRD2018” (https://www.mdpi.com/journal/robotics/special_issues/KARD), which is the first issue of the KaRD Special Issue series, hosted by the open access journal “MDPI Robotics”. The KaRD series aims at creating an open environment where researchers can present their works and discuss all the topics focused on the many aspects that involve kinematics in the design of robotic/automatic systems. Kinematics is so intimately related to the design of robotic/automatic systems that the admitted topics of the KaRD series practically cover all the subjects normally present in well-established international conferences on “mechanisms and robotics”. KaRD2018 received 22 papers and, after the peer-review process, accepted only 14 papers. The accepted papers cover some theoretical and many design/applicative aspects

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics

Applications of monodromy in solving polynomial systems

Polynomial systems of equations that occur in applications frequently have a special structure. Part of that structure can be captured by an associated Galois/monodromy group. This makes numerical homotopy continuation methods that exploit this monodromy action an attractive choice for solving these systems; by contrast, other symbolic-numeric techniques do not generally see this structure. Naturally, there are trade-offs when monodromy is chosen over other methods. Nevertheless, there is a growing literature demonstrating that the trade can be worthwhile in practice. In this thesis, we consider a framework for efficient monodromy computation which rivals the state-of-the-art in homotopy continuation methods. We show how its implementation in the package MonodromySolver can be used to efficiently solve challenging systems of polynomial equations. Among many applications, we apply monodromy to computer vision---specifically, the study and classification of minimal problems used in RANSAC-based 3D reconstruction pipelines. As a byproduct of numerically computing their Galois/monodromy groups, we observe that several of these problems have a decomposition into algebraic subproblems. Although precise knowledge of such a decomposition is hard to obtain in general, we determine it in some novel cases.Ph.D