10 research outputs found
Recommended from our members
Geometric Sparsity in High Dimension
While typically complex and high-dimensional, modern data sets often have a concise underlying structure. This thesis explores the sparsity inherent in the geometric structure of many high-dimensional data sets.
Constructing an efficient parametrization of a large data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach, guided by geometry, consists in recovering a local parametrization (a chart) using the local tangent plane. In practice, the data are noisy and the estimation of a low-dimensional tangent plane in high dimension becomes ill posed. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise.
We present an approach that uses the geometry of the data to guide our definition of locality, discovering the optimal balance of this noise-curvature trade-off. Using eigenspace perturbation theory, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals the optimal scale for local tangent plane recovery. Additionally, we are able to accurately and efficiently estimate the curvature of the local neighborhood, and we introduce a geometric uncertainty principle quantifying the limits of noise-curvature perturbation for tangent plane recovery. An algorithm for partitioning a noisy data set is then studied, yielding an appropriate scale for practical tangent plane estimation.
Next, we study the interaction of sparsity, scale, and noise from a signal decomposition perspective. Empirical Mode Decomposition is a time-frequency analysis tool for nonstationary data that adaptively defines modes based on the intrinsic frequency scales of a signal. A novel understanding of the scales at which noise corrupts the otherwise sparse frequency decomposition is presented. The thesis concludes with a discussion of future work, including applications to image processing and the continued development of sparse representation from a geometric perspective
Acoustic emission source detection and wave generation on a working plate using piezoelectric actuators
In modern automated assembly lines, part feeders play an important role to separate and sort parts. Agility and flexibility are required to achieve a good yield for different types of parts. The common working principle is to vibrate a working plate driven by electromagnetic actuators in order to orient or separate parts on it. Then, a vision system will help a robotic arm to pick up the parts if they are correctly placed. However, the vibration can only create a random movement for the parts. Hence, a new design of working plate using piezoelectric actuators has been proposed to solve this problem. This thesis presents the modelling of the piezoelectric actuator driven working plate in order to provide a theoretical tool to explore the full potential of this new design. Despite of directly using eigenmodes to create non-homogeneous vibrations, it is also possible to use the piezoelectric actuator to detect the position of parts or to create a vibration at a specific position. An overview of possible technics which can achieve these tasks is presented. Among all solutions, the time-reversal method shows it is the best candidate. According to the time-reversal method, an excitation pulse response is the key element to establish the theoretical study. Firstly, the expression of vibration amplitude on a plate due to an excitation pulse has been given. A damping term is added to the result, which can be determined by impedance measurement thanks to the modelling of the electrical characteristics of the coupled piezoelectric-mechanical system. The second part deals with the detection. After the modelling of the interaction between the piezoelectric actuator and the plate, the complete time-reversal process modelling can be achieved. An example of its application on a beam is given, which shows the restored vibration amplitude has one main peak at the original position of the excitation and several smaller associated peaks elsewhere. This result demonstrates it is sufficient to detect a part dropping position with the time-reversal method. The third part focuses on the pulse generation. The associated peaks can be removed by a combination of several excitation pulse responses. The position of piezoelectric actuators should also meet conditions predicted by theoretical analysis. The damping term is equally taken into consideration to improve the pulse reconstruction. In parallel, the design of an electric drive for such piezoelectric system is also a challenge, due to the great capacitance of the piezoelectric actuator, high supply voltage and wide dynamic frequency range. A solution based on self-oscillating class-D amplifiers has been proposed because it allows reaching nearly the frequency limit of the switching components. The theoretical study gives a complete modelling of the system and it implies a stability condition for the choice of the parameters. The output of this drive is a non-periodic PWM (Pulse Width Modulation) signal. An original method is proposed to give a theoretical estimation of the frequency spectrum, which is useful to determine the cutoff frequency of the output low-pass filter. At the end, simulations and experiments are given to verify the theoretical results and to show the effectiveness to detect excitation pulse positions and the possibility to create a pulse vibration at a specific position