The Theory Of Compressed Sensing And Its Application In Image Texture Segmentation

As we known, the Nyquist - Shannon sampling theorem points out the conditions needed for the signal exact reconstruction without distortion is that the sampling rate of the signal must be at least twice as the highest frequency. As for sparse signals, the compressed data of the original signal is only part of the original signal data. The mode of first sampling then compressing will cause waste of resources. The theory of Compressed Sensing as an emerging theory, offers a joint compression and sensing process for the sparse or compressible signal. The signal can be exactly reconstructed though the measurements which are much less than conventional sampling data. As for the Compressed Sensing theory, the sparsity of the signal is required. A revolutionary breakthrough has been brought to the traditional signal processing from the Compressed Sensing theory. The main work we have down is as follows:1. If the redundant dictionary is fixed, how to design a suitable measurement matrix so that the compressed sensing matrix composed by the measurement matrix and redundant dictionary has the optimal performance. We firstly investigate the newest academic achievements and progress at home and abroad in this direction. We improved the optimal projection proposed by M. Elad, making the compressed sensing matrix has optimal performance. The optimal performance is that, for the signals with the same sparsity, it can reconstruct the signal using the least measurements. In the other words, with the same measurements, the compressed sensing matrix can reconstruct the signal with the smallest error.2. The piecewise linear regularized solution paths algorithm proposed by S. Rosset and J. Zhu in 2007 is valid based on the premise that the matrix is full rank. Inspired by the method that Ch. Ong, S. Shao and J. Yang have used for modifying the Entire Regularization path algorithm for the Support Vector Machine proposed by T. Hastie, S. Rosset and J. Zhu. We improved this algorithm by solving the problem of the inversion of singular matrix. The new algorithm is verified through experiment.3. The Compressed Sensing theory is introduced into the texture segmentation problem. We firstly consider the texture image as a cycle-extension of small texture atom. We cast the texture recognition problem as one of classifying among multiple linear regression models. Secondly, the texture image is decomposed as a linear combination of the overcomplete dictionary by 1-minimization in the Compressed Sensing theory and the coefficients of the image is considered as the feature vector. Then, this texture class will be recognized by some criteria on the feature vector. Afterwards, as the texture class has been recognized, we are able to use the texture image's structure feature, consistency, to segment all the parts of the texture image belonging to this class. The edges will be segmented by the fine segmentation.