On The Multi-scale Geometric Analysis Based Image Compressive Sensing Techniques

As a new signal processing theory, the compressed sensing is utilized to get over the Nyquist sampling rate limit. The compressed sensing (CS) technology realizes the sampling and compression of the signal simultaneously via the measurement process. Being capable of approximating the original signal from a small amount of measurement by using effective reconstruction algorithm, CS greatly reduces the bandwidth requirements in the wireless transmission system. Sparse representation is a critical issue in CS framework. Wavelet transform (WT) is widely utilized in the conventional image signal analysis in the transform domain. Althrough, the WT can capture sparse representation characteristics of the image signal along horizontal, vertical and diagonal direction, it fails in characterizing the complicated geometric characteristics of image signal beyond the aforementioned three directions, therefore the WT will be far from optimal in terms of sparse representation of imge signal.In order to overcome the inefficiency of the WT, multi-scale geometric analysis is investigated in this paper to unveil its ability of sparse representation in the framework of compressed sensing. Firstly, it is shown that WT is not good at dealing with handing the high-dimensional line singularity, while Ridgelet transfomr (RT) offers better nonlinear approximation for a two-dimensional or higher order dimensional function with "line singularity". The comparison against the WT validates that the Ridgelet based compressing sensing framework can achieve much better image restoration quality for image signal with line ingularity. On the other hand, it is unveiled that RT is not as good as WT based compressive sensing image with rich texture.Secondly, in order to solve the detail missing of reconstructed image by using RT and improve the effectiveness of the sparse representation of image signal with rich texture characteristics, a layered compressed sensing processing technique is proposed to combine the WT and finite RT by taking into account of caputuring capabilities of different transform domain analysis method in the sparse representation of the image signal. It is validated that the proposed layered scheme can avoid the missing detail in the reconstructed image with "line singularity" and the "scratch phenomena" in the smooth region by only using the finite Ridgelet based CS framework, at the same time, it is suitable for processing images when there is rich texture property. Finally, considering that Ridgelet does not have a good performance in terms of sparse approximation for the curve (non-linear) singularity in image signal, it is proposed to employ the Bandelet transform (BT) based CS for image signal. This dissertation shows that BT based CS scheme provides effective adaptive tracking capability for geometrical regularity direction of the image, and it is capable of achieving a better sparse decomposition of the image adaptively. The comparison analysis is presented to validate that the BT based CS framework outperforms the layered CS scheme which combines both WT and finite RT in terms of capturing the edge (profile) and the texture characteristics in the context of the CS-based image signal processing technique.The analysis in this dissertation unveils the following facts:Multiscale geometric analysis (RT and BT) can effectively improve the sparse representation of the signal or image with prevalent high-dimensional line singularity. Combines both WT and finite RT is capable of achieving a better sparse decomposition of the image or signal when there is rich texture property, and BT based CS framework provides effective adaptive tracking capability for geometrical regularity direction of the signal or image, it outperforms combine the WT and finite RT based image processing. It has reference value in this dissertation that multiscale geometric analysis improves the sparse representation of the signal or image.