Research On Image Reconstruction Of Compressive Sensing

Sampling is a transition which make signal from the analog source to the digital information. In the conventional sampling process, in order to avoid signal distortions, according to Nyquist-Shannon sampling theorem, a bandlimited analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency in the original signal. However, with the rapid and continuous progress of science and technology of information domain, people need to deal with more and more data with which current hardware capacity has been difficult to cope. Therefore, to find a new way of data sampling and processing has become an inevitable trend. Besides, the Nyquist–Shannon sampling theorem provides a sufficient condition, but not a necessary one, for perfect reconstruction. Recently, an emerging theory of signal acquirement named Compressive Sensing (CS) proposed by Candes and Donoho become one of the most hottest topic of signal sampling and image processing. According to the characteristics of being sparse represented within some orthogonal basis or the framework of compact support of original signal, compressive sensing system first employ non-adaptive linear projections preserve the structure of the signal, then obtain the measurements, in which measurement process is in accordance with much less requirements of sampling of the traditional Nyquist-Shannon sampling theorem. With solving the optimization problem to realize and implement the reconstruction in sparse domain, ultimately the purpose of reconstruction of the original signal or image will be carried out progressively. Simply, by given a sparse signal in a high dimensional space, compressive sensing system which combines with sampling and compression, can reconstruct that signal accurately and efficiently from fewer linear measurements much less than its actual dimension using sparse-prior characteristic of signal. Therefore, the theory will help people easing the pressure of hardware facility from the requirements of the huge amount in information processing. Compressive sensing has broad applications such as image denoising, image compression, medical imaging, synthetic aperture radar image (SAR), pattern recognition, feature extraction, etc.Supported by the National Science Foundation of China, this thesis concentrates on the compressive sensing theory and applications such image reconstruction. We are focus on the exploratory study of the sparse-prior characteristic of signal and the method of reconstruction in the applications of compressive sensing. The thesis describes the three main areas of compressive sensing theory: sparse representation, measurement matrix and reconstruction algorithm. According to the different sparse-prior characteristic of signal such as DCT transform domain, wavelet transform domain, multi-wavelet transform domain and contourlet transform domain, we analyze and compare the subjective quality evaluation and objective quality evaluation of the image reconstruction, respectively.Because the wavelet only has single scaling function and can not simultaneously satisfy the orthogonality, high vanishing moments, compact support, symmetry characteristic and regularity, compressive sensing of image reconstruction using multi-wavelet transforms is proposed. We choose Gaussian random matrix and Bernoulli random matrix as the measurement matrix in order to get the measurement of the original image by a linear projection. Then, we use multi-wavelet transform as the sparse-prior characteristic in the process of reconstruction. Finally, Complete the image reconstruction by using orthogonal matching pursuit algorithm to find the most valuable and matchable atoms of the atoms dictionary in each pursuit.Based on the inadequacy of direction of traditional orthogonal wavelet, can not be good at the sparse representation of image which contains the characteristic of piecewise smoothness and line singularity, a new compressive sensing of image reconstruction using sparse-prior characteristic of contourlet transforms is proposed. We also choose Gaussian random matrix as the measurement matrix to get the linear measurement of original image. In order to achieve the reconstruction of the original image, we use iterative threshold shrinkage algorithm to solve optimization problem.