Compressive Sensing Based On Double Sparse Dictionary

Natural scenery images in people’s eyes are generally continuous simulation images,but modern computers can only efciently process digital signals. So the premise condi-tion of modern digital image processing is image sampling and quantization. The work ofShannon and Nyquist laid the foundation of digital sampling. Their work suggests that wewill be able to reconstruct the original signal without distortion from the discrete signalwhen sampling rate is not less than twice the highest frequency of sampling.Inspired by sparse coding technology, D. Donoho, e. Candes and t. Tao put forwardcompressive sensing theory. Compressed sensing combine the sampling process withcompression process and the sampling rate is far lower than Nyquist rate. Compressivesensing theory is very popular in the field of signal processing.Traditionally, the compressive signal is represented sparsely by orthogonal base. Werequire that the observation matrix is in incoherent with sparse base so that we can re-construct the original high-dimensional data from the low dimensional observation data.In fact, signals are usually sparser in redundant dictionary than that in orthogonal base.And if we reconstruct the original signal directly rather than reconstruct sparse recoverycoefcient, we just need to require the observation matrix meet some conditions withcoherent and redundant dictionary.In this paper, image will be divided into two parts–cartoon part and texture part,and we represent this two parts by two diferent Dictionaries. We theoretically prove thatit is not necessary to consider whether the redundant dictionary is incoherent with theobservation matrix, and we can reconstruct the signal exactly only with some restrictionson observation matrix.Several numerical simulation results are provided.