Bearing failure is one of the foremost causes of breakdown in rotating machinery. Such failure can be catastrophic and can result in costly downtime. Bearing condition monitoring has thus played an important role in machine maintenance. In condition monitoring, the observed signal at a measurement point is often corrupted by extraneous noise during the transmission process. It is important to detect incipient faults in advance before catastrophic failure occurs. In condition monitoring, the early detection of incipient bearing signal is often made difficult due to its corruption by background vibration (noise). Numerous advanced signal processing techniques have been developed to detect defective bearing signals but with varying degree of success because they require a high Signal to Noise Ratio (SNR), and the fault components need to be larger than the background noise. Vibration analyses in the time and frequency domains are commonly used to detect machinery failure, but these methods require a relatively high SNR. Hence, it is essential to minimize the noise component in the observed signal before post processing is conducted. In this research, detection of failure in rolling element bearing faults by vibration analysis is investigated. The expected time intervals between the impacts of faulty bearing components signals are analysed using the blind deconvolution technique as a feature extraction technique to recover the source signal. Blind deconvolution refers to the process of learning the inverse of an unknown channel and applying it to the observed signal to recover the source signal of a damaged bearing. The estimation time period between the impacts is improved by using the technique and consequently provides a better approach to identify a damaged bearing. The procedure to obtain the optimum inverse equalizer filter is addressed to provide the filter parameters for the blind deconvolution process. The efficiency and robustness of the proposed algorithm is assessed initially using different kinds of corrupting noises. The result show that the proposed algorithm works well with simulated corrupting periodic noises. This research also shows that blind deconvolution behaves as a notch filter to remove the noise components. This research involves the application of blind deconvolution technique with optimum equalizer design for improving the SNR for the detection of damaged rolling element bearings. The filter length of the blind equalizer needs to be adjusted continuously due to different operating conditions, size and structure of the machines. To determine the optimum filter length a simulation test was conducted with a pre-recorded bearing signal (source) and corrupted with varying magnitude noise. From the output, the modified Crest Factor (CF) and Arithmetic Mean (AM) of the recovered signal can be plotted versus the filter length. The optimum filter length can be selected by observation when the plot converges close to the pre-determined source feature value. The filter length is selected based on the CF and AM plots, and these values are stored in a data training set for optimum determination of filter length using neural network. A pre-trained neural network is designed to train the behaviour of the system to target the optimum filter length. The performance of the blind deconvolution technique was assessed based on kurtosis values. The capability of blind deconvolution with optimum filter length developed from the simulation studies was further applied in a life bearing test rig. In this research, life time testing is also conducted to gauge the performance of the blind deconvolution technique in detecting a growing potential failure of a new bearing which is eventually run to failure. Results from unseeded new bearing tests are different, because seeded defects have certain defect characteristic frequencies which can be used to track a specific damaged frequency component. In this test, the test bearing was set to operate continuously until failures occurred. The proposed technique was then applied to monitor the condition of the test bearing and a trend of the bearing life was established. The results revealed the superiority of the technique in identifying the periodic components of the bearing before final break-down of the test bearing. The results show that the proposed technique with optimum filter length does improve the SNR of the deconvolved signal and can be used for automatic feature extraction and fault classification. This technique has potential for use in machine diagnostics.

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This thesis presents a new approach to blind deconvolution algorithms. The proposed method is a combination of a classical blind deconvolution subspace method and a marginalized particle filter. It is shown that the new method provides better performance than just a marginalized particle filter, and better robustness than the classical subspace method. The properties of the new method make it a candidate for further exploration of its potential application in acoustic blind dereverberation. / Thesis / Master of Applied Science (MASc)

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An iterative solution to the problem of blind image deconvolution is presented whereby a previous image estimate is explicitly used in the new image estimation process. The previous image is pre-filtered using an adaptive, non-parametric stabilizing function that is updated based on a current error estimate. This function is experimentally shown to dramatically benefit the convergence rate for the a priori restoration case. Noise propagation from one iteration to the next is reduced by the use of a second, regularizing operator, resulting in a hybrid iteration technique. Further, error terms are developed that shed new light on the error propagation properties of this method and others by quantifying the extent of noise and regularization error propagation. Optimal non-parametric, frequency adaptive stabilizing and regularization functions are then derived based on this error analysis. / Thesis / Master of Engineering (ME)

In this thesis, we conduct functional Magnetic Resonance Imaging (fMRI) data analysis with the aim of grouping the brain voxels depending on their responsiveness to a neural task. We mathematically treat the fMRI signals as the convolution of the neural stimulus with the hemodynamic response function (HRF). We first estimate a time series including HRFs for each of the observed fMRI signals from a given set and we cluster them in order to identify the groups of brain voxels. The HRF estimation problem is studied within the Bayesian framework through a blind deconvolution algorithm using MAP approach under completely unsupervised and model-free settings, i.e, stimulus is assumed to be unknown and also no particular shape is assumed for the HRF. Only using a given fMRI signal together with a weak Gaussian prior distribution imposed on HRF favoring &lsquo / smoothness&rsquo / , our method successfully estimates all the components of our framework: the HRF, the stimulus and the noise process. Then, we propose to use a modified version of Hausdorff distance to detect similarities within the space of HRFs, spectrally transform the data using Laplacian Eigenmaps and finally cluster them through EM clustering. According to our simulations, our method proves to be robust to lag, sampling jitter, quadratic drift and AWGN (Additive White Gaussian Noise). In particular, we obtained 100% sensitivity and specificity in terms of detecting active and passive voxels in our real data experiments. To conclude with, we propose a new framework for a mathematical treatment for voxel-based fMRI data analysis and our findings show that even when the HRF is unpredictable due to variability in cognitive processes, one can still obtain very high quality activation detection through the method proposed in this thesis.

A Bussgang based blind deconvolution algorithm called self-correcting multi-channel Bussgang (SCMB) blind deconvolution algorithm was proposed. Unlike the original Bussgang blind deconvolution algorithm where the probability density function (pdf) of the signal being recovered is assumed to be completely known, the proposed SCMB blind deconvolution algorithm relaxes this restriction by parameterized the pdf with a Gaussian mixture model and expectation maximization (EM) algorithm, an iterative maximum likelihood approach, is employed to estimate the parameter side by side with the estimation of the equalization filters of the original Bussgang blind deconvolution algorithm. A feedback loop is also designed to compensate the effect of the parameter estimation error on the estimation of the equalization filters. Application of the SCMB blind deconvolution framework for binary image restoration, multi-pass synthetic aperture radar (SAR) autofocus and inverse synthetic aperture radar (ISAR) autofocus are exploited with great results.

Low-dimensional signal structures naturally arise in a large set of applications in various fields such as medical imaging, machine learning, signal, and array processing. A ubiquitous low-dimensional structure in signals and images is sparsity, and a new sampling theory; namely, compressive sensing, proves that the sparse signals and images can be reconstructed from incomplete measurements. The signal recovery is achieved using efficient algorithms such as \ell_1-minimization. Recently, the research focus has spun-off to encompass other interesting low-dimensional signal structures such as group-sparsity and low-rank structure. This thesis considers low-rank matrix recovery (LRMR) from various structured-random measurement ensembles. These results are then employed for the in depth investigation of the classical blind-deconvolution problem from a new perspective, and for the development of a framework for the efficient sampling of correlated signals (the signals lying in a subspace). In the first part, we study the blind deconvolution; separation of two unknown signals by observing their convolution. We recast the deconvolution of discrete signals w and x as a rank-1 matrix wx* recovery problem from a structured random measurement ensemble. The convex relaxation of the problem leads to a tractable semidefinite program. We show, using some of the mathematical tools developed recently for LRMR, that if we assume the signals convolved with one another live in known subspaces, then this semidefinite relaxation is provably effective. In the second part, we design various efficient sampling architectures for signals acquired using large arrays. The sampling architectures exploit the correlation in the signals to acquire them at a sub-Nyquist rate. The sampling devices are designed using analog components with clear implementation potential. For each of the sampling scheme, we show that the signal reconstruction can be framed as an LRMR problem from a structured-random measurement ensemble. The signals can be reconstructed using the familiar nuclear-norm minimization. The sampling theorems derived for each of the sampling architecture show that the LRMR framework produces the Shannon-Nyquist performance for the sub-Nyquist acquisition of correlated signals. In the final part, we study low-rank matrix factorizations using randomized linear algebra. This specific method allows us to use a least-squares program for the reconstruction of the unknown low-rank matrix from the samples of its row and column space. Based on the principles of this method, we then design sampling architectures that not only acquire correlated signals efficiently but also require a simple least-squares program for the signal reconstruction. A theoretical analysis of all of the LRMR problems above is presented in this thesis, which provides the sufficient measurements required for the successful reconstruction of the unknown low-rank matrix, and the upper bound on the recovery error in both noiseless and noisy cases. For each of the LRMR problem, we also provide a discussion of a computationally feasible algorithm, which includes a least-squares-based algorithm, and some of the fastest algorithms for solving nuclear-norm minimization.

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"Blind" inversion of the effects of a given operator on a signal is an extremely difficult task that has no easy solutions. However,. Dr. Hany Farid has published several works that each individua:lly appear to achieve exactly this seemingly impossible result. In this work, we contribute a comprehensive overview of the published applications of blind process inversion, as well as provide the generalized form of the algorithms and requirements that are found in each of these applications, thereby formulating and explaining a general framework for blind process inversion using Farid's Algorithm. Additionally, we explain the knowledge required to derive the ROSA-based cost function on which Farid's Algorithm depends. As our primary contribution, we analyze the algorithmic complexity of this cost function based on the way it is currently, naively calculated, and derive a new algorithm to compute this cost function that has greatly reduced algorithmic complexity. Finally, we suggest an additional application of Farid's Algorithm to the problem of blindly estimating true camera response functions from a single image.