Compressive Sensing/Sampling (CS), a kind of new paradigm for simultaneous sampling and compression, has attracted considerable attention recently. Without being restricted to the constraint of Nyquist rate, compressive sensing can, in theory, perfectly reconstruct the original signal under the constraints that if only a few samples or measurements extracted from an original signal are available and the signal is sparse in certain domain. The unique characteristic of CS is that sampling and compression can be simultaneously achieved such that CS is suitably used for resource-limited digital devices and sensors.
Our study in CS particularly mainly focuses on theoretical analyses. Our goal is to close the gap between theoretical performance bound and practice. In fact, we find that, under noiseless or noisy environments, though several CS recovery algorithms have been presented, there exists a big gap between theoretical recovery bounds and practical results. Basically, this is due to the fact that theoretical analyses are based on some strict assumptions that are hard to be satisfied practically, leading to worst-case results. Since CS is under-determined, a unique solution can be obtained only if the original signal is sparse in certain domain. As a result, on one hand if the given samples are not enough, then a correct or sparest solution cannot be obtained and on the other hand samples will be wasted if they are more than required.
Inspired by the above hard problems, we are planning to derive the theoretical bounds for three CS recovery models. Our methodology will be based on conic geometry. In the future, we will, respectively, explore the problems, including multiple measurement vectors (MMVs) without noise interferences model、single measurement vector (SMV) with noise interferences model、and MMVs with noise interferences model. These indicate that, according to our methodology, we can achieve the best recovery results by the given parameters in advance. Most importantly, we are able to close the gap between the theoretical bounds and practical results.
So far, our current results on compressive sensing are summarized as follows.
1. We have proposed a sparse Fast Fourier Transform (sFFT) method that can be faster than MIT''s methods (also known as state-of-the-art) with better reconstructed results. The advantage also includes the easy selection of parameters and easy implementation in sFFT without needing to know sparsity of a signal.
2. For large-scale data, cost-efficient compressive sensing with fast reconstructed high-quality results is very challenging. we propose a new big media data compressive sensing method, composed of operator-based strategy in the context of fixed point continuation method and weighted LASSO with tree structure sparsity pattern. The main characteristic of our method is free from any assumptions and restrictions.
3. We have developed compressed sensing detector design for space shift keying for MIMO systems, compressed sensing-based clone identification method for sensor networks, and compressed sensing-based cooperative spectrum sensing in cognitive radio networks.
4. We have presented a distributed compressive video sensing (DCVS) method to simultaneously sensing and compressing videos.