Study Of Ridgelet-based Image Sparse Decomposition

Image compression is one of the important parts of image processing.The traditional image compression technology generally has a low compression ratio when compressing a image, but when the image compression is in a high compression ratio, the recovered images from these compressed images especially the iamges having obvious texture image restoration are not very good, so we need a better image compression method to solve this problem. This paper focuses on the application of the compressive sensing theory in image compression, and presents a method based on image sparse decomposition using ridge wave as the sparse basis, hoping that the recovered images will have higher picture quality when the compressed image are in high compression ratio.First of all it introduces the basic theory of image compression and the compressive sensing.First,it introduces the represen-tation of the image and describes the classification of image compression including lossy compression (distortion compression) and lossless compression (no distortion compression).Then it introduces the image compression standard generally including still image compression and moving image compression.JPEG2000 is commonly used in still image compression standard. The results of this paper are also compared with that of JPEG2000.Then it describes the evaluation of image compression, mainly fidelity criteria which includes the principle of subjective evaluation and objective evaluation criteria. The common used objective evaluation criteria is the signal to noise ratio.The evaluation criteria used in the paper combines the two evaluation criteria to judge the quality of the reconstructed images. According to the specific circumstances of this thesis, it can be more realistic to compare the quality of the images reconstructed in the two different method when using the combined evaluation criteria compares the results of the experimental simulation and JPEG2000.The second part introduces the theory of compressive sensing.The compressive sensing theory and the conventional Nyquist sampling theorem are quite different.Assuming that the signal to be processed is sparse,you can transform it to the transform domain though a appropriate transformation method for sparse change, thenThen multiply the observation base that isn't associated with the sparse base and the sparse matrix that has been already sparsed.Finally,using a suitable reconstruction algorithm to recover the original image, that is, solving a optimal solution.It is a theory that can sample while achieving the purpose of compression, the compression process is as follows:First, if the signal X is compressible in an orthogonal basis,find the coefficient of variation which is the expression of the sparse signal.Second,design a stable M×N-dimensional observation matrixΦthat is not related to the exchange baseψ, then obtain the observational set by observing the sparse matrix (?),the process can also be expressed as a signal non-adaptive observed through a matrix,that is, find the appropriate observation matrix. Finally, design a fast reconstruction algorithms using optimization problem solving in the sense of 0-norm. Then it introduces the theory of the wavelet transform and ridgelet transform. It analyzes the excellent time-frequency localized of the wavelet transform. After the wavelet transform of the image, if you want low frequency resolution and high temporal resolution,you can get that from the high-frequency part of it, on the contrary, if you want high frequency resolution and low time resolution, you can get that from the low frequency part in it. Then it introduces the ridge transform, and analyzes the advantage of the ridge wavelet transform bacause it is easier than wavelet transform to describ the differences between curves, so the wave ridge is even more suitable to describe the mutation characteristics on the edge of the image.It is based on the compressed sensing theory,using ridge wave to sparse decompose the image signal,using the Gaussian random matrix as observation matrix and using the matching pursuit algorithm to solve optimization problems to reconstruct the image. The first step is sparse decomposition which is the focus of this paper. The first is sparse decomposition based on ridge wave, using the ridge wave as a sparse basis, and transform the airspace of the image from the airspace to the Slant Stank domain, and then transform it to the Ridgelet domain to complete sparse decomposition.Then observe the compressed stream with a random Gaussian matrix,and reconstruct the image using the matching pursuit method. The experiment uses the standard 256 * 256 Lena image, Cameraman image and Rice images, and respectively analyzes them at the compression ratio 10,55,80 and contrast the reconstructed image with the reconstructed image under JPEG2000 at the same compression ratio.Then you will found that at low compression ratio, the reconstructed image obtained by this method and the reconstructed image obtained by JPEG2000 are similar, but at the high compression ratio, using this method can reconstruct higher quality image, especially for those with significant texture feature, the effect of the recovered image is more evident. But the details are not too good. Then on account of an improved method, it proposed a new hierarchical encoding method by combining both of the performance advantages of wavelet and ridge wave.It smooothes the image to get the smooth image and texture image.The smooth layer uses SPIHT algorithm based on wavelet transfor.The texture layer uses a modified orthogonal finite ridgelet, and makes the corresponding improvement of SPIHT algorithm to achieve the texture layer sparse decomposition. Then compare the image reconstructed by the improved method with that reconstructed by JPEG2000 through experiments.Experiment uses a 256 * 256 Barbara image, and obtains better restoration results of the image.This is because the improved method makes best use of the image coding algorithm based on wavelet which has better performation in compressing the smoothed image,a smooth image compression for better performance, but also can avoid introducing "pseudo-marks" to the smoothed area in the reconstructed image. Experimental results show that the algorithm can not only improve the compression efficiency, while can also preserving most of the information of the details of the image.However, the proposed method has some shortcomings, it isn't perfect in the details. Although details of the image are reserved in the improved method, but the results are not very satisfactory.At this point ,it need further improvement to be perfect. First of all, Radon formula requires line integral on the graphics in all directions, but in the specific study, it is impossible to do line integral in all directions, it can only use limited data to restore the image, so there are some errors. when Radon formula is applied,it needs fast algorithm to compute.But the algorithm is not very easy to get, and an algorithm can not always abtain a good restoration results for all practical applications. So this is also a difficult problem needing to be to solved. The second is the algorithm which is used (matching pursuit) in the calculation uses the approximation instead of exact solution. The third that the observation matrix uses a random Gaussian matrix which has already been formed. We can considere structure a more appropriate observation matrix by ourselves, and the recovery effect may be better.These existent deficiencies can be improved and perfected through continuous researches and innovations.