Smoothing is an important part of the pre-processing step in Signal Processing. A signal, which is purified from noise as much as possible, is necessary to achieve our aim. There are many smoothing algorithms which give good result on a stationary data, but these smoothing algorithms don&rsquo / t give expected result in a non-stationary data. Studying Acceleration data is an effective method to see whether the smoothing is successful or not. The small part of the noise that takes place in the Displacement data will affect our Acceleration data, which are obtained by taking the second derivative of the Displacement data, severely. In this thesis, some linear and non-linear smoothing algorithms will be analyzed in a non-stationary dataset.

Smoothing is an important part of the pre-processing step in Signal Processing. A signal, which is purified from noise as much as possible, is necessary to achieve our aim. There are many smoothing algorithms which give good result on a stationary data, but these smoothing algorithms don&rsquo / t give expected result in a non-stationary data. Studying Acceleration data is an effective method to see whether the smoothing is successful or not. The small part of the noise that takes place in the Displacement data will affect our Acceleration data, which are obtained by taking the second derivative of the Displacement data, severely. In this thesis, some linear and non-linear smoothing algorithms will be analyzed in a non-stationary data set.

This thesis presents an analysis of several smoothness energies (also called smoothing energies) in geometry processing, and introduces new methods as well as a mathematical proof of correctness and convergence for a well-established method. Geometry processing deals with the acquisition, modification, and output (be it on a screen, in virtual reality, or via fabrication and manufacturing) of complex geometric objects and data. It is closely related to computer graphics, but is also used by many other fields that employ applied mathematics in the context of geometry. The popular Laplacian energy is a smoothness energy that quantifies smoothness and that is closely related to the biharmonic equation (which gives it desirable properties). Minimizers of the Laplacian energy solve the biharmonic equation. This thesis provides a proof of correctness and convergence for a very popular discretization method for the biharmonic equation with zero Dirichlet and Neumann boundary conditions, the piecewise linear Lagrangian mixed finite element method. The same approach also discretizes the Laplacian energy. Such a proof has existed for flat surfaces for a long time, but there exists no such proof for the curved surfaces that are needed to represent the complicated geometries used in geometry processing. This proof will improve the usefulness of this discretization for the Laplacian energy. In this thesis, the novel Hessian energy for curved surfaces is introduced, which also quantifies the smoothness of a functions, and whose minimizers solve the biharmonic equation. This Hessian energy has natural boundary conditions that allow the construction of functions that are not significantly biased by the geometry and presence of boundaries in the domain (unlike the Laplacian energy with zero Neumann boundary conditions), while still conforming to constraints informed by the application. This is useful in any situation where the boundary of the domain is not an integral part of the problem itself, but just an artifact of data representation---be it, because of artifacts created by an imprecise scan of the surface, because information is missing outside of a certain region, or because the application simply demands a result that should not depend on the geometry of the boundary. Novel discretizations of this energy are also introduced and analyzed. This thesis also presents the new developability energy, which quantifies a different kind of smoothness than the Laplacian and Hessian energies: how easy is it to unfold a surface so that it lies flat on the plane without any distortion (surfaces for which this is possible are called developable surfaces). Developable surfaces are interesting, as they can be easily constructed from cheap material such as paper and plywood, or manufactured with methods such as 5-axis CNC milling. A novel definition of developability for discrete triangle meshes, as well as a variety of discrete developability energies are also introduced and applied to problems such as approximation of a surface by a piecewise developable surface, and the design and fabrication of piecewise developable surfaces. This will enable users to more easily take advantages of these cheap and quick fabrication methods. The novel methods, algorithms and the mathematical proof introduced in this thesis will be useful in many applications and fields, including numerical analysis of elliptic partial differential equations, geometry processing of triangle meshes, character animation, data denoising, data smoothing, scattered data interpolation, fabrication from simple materials, computer-controlled fabrication, and more.

Smoothing filters have been extensively used for noise removal and image restoration. Directional filters are widely used in computer vision and image processing tasks such as motion analysis, edge detection, line parameter estimation and texture analysis. It is practically impossible to tune the filters to all possible positions and orientations in real time due to huge computation requirement. The efficient way is to design a few basis filters, and express the output of a directional filter as a weighted sum of the basis filter outputs. Directional filters having these properties are called "Steerable Filters." This thesis work emphasis is on the implementation of proposed computationally efficient separable and steerable Gaussian smoothers on a Xilinx VirtexII Pro FPGA platform. FPGAs are Field Programmable Gate Arrays which consist of a collection of logic blocks including lookup tables, flip flops and some amount of Random Access Memory. All blocks are wired together using an array of interconnects. The proposed technique [2] is implemented on a FPGA hardware taking the advantage of parallelism and pipelining.

Image processing plays a key role in vision systems. Its function is to extract and enhance pertinent information from raw data. In robotics, processing of real-time data is constrained by limited resources. Thus, it is important to understand and analyse image processing algorithms for accuracy, speed, and quality. The theme of this thesis is an implementation and comparative study of algorithms related to various image processing techniques like edge detection, corner detection and thinning. A re-interpretation of a standard technique, non-maxima suppression for corner detectors was attempted. In addition, a thinning filter, Hall-Guo, was modified to achieve better results. Generally, real time data is corrupted with noise. This thesis also incorporates few smoothing filters that help in noise reduction. Apart from comparing and analysing algorithms for these techniques, an attempt was made to implement correlation-based optic flow