CS (Compressed sensing)
Compressed Sensing (CS) is a signal processing technique that has emerged as a powerful tool for efficient data acquisition and reconstruction in various fields of science and engineering. CS is based on the idea that a signal or image that is sparse or compressible in a certain domain can be accurately reconstructed from a small number of linear projections or measurements, thus reducing the amount of data that needs to be acquired and processed.
In traditional signal processing, a signal is sampled at a fixed rate according to the Nyquist-Shannon sampling theorem, which requires that the sampling rate be at least twice the highest frequency present in the signal. This means that a high sampling rate is needed to capture all the details of a signal or image, which can result in large amounts of data that are expensive to store and process.
CS, on the other hand, exploits the fact that many signals and images have a sparse or compressible representation in some domain, such as the Fourier domain or wavelet domain. This means that the signal can be represented by a small number of basis functions or coefficients, which can be efficiently measured and reconstructed from a much smaller number of samples than required by traditional sampling techniques.
The basic idea behind CS can be illustrated using a simple example of reconstructing a sparse signal. Consider a signal x that has N samples and is K-sparse, meaning that it has only K non-zero coefficients in some basis or representation. To reconstruct this signal using traditional sampling, we would need to acquire at least N samples, which is proportional to the signal length. However, using CS, we can acquire only M << N measurements of the signal, where M is much smaller than N and K << M. The measurements can be taken in a random fashion, such as by selecting random rows from a sensing matrix A, which has M rows and N columns.
The measurements y can be represented as y = Ax, where A is the sensing matrix and x is the sparse signal. Since x is K-sparse, we can use a sparse signal recovery algorithm to find an estimate of the original signal from the measurements y. One popular algorithm for this purpose is the Basis Pursuit algorithm, which solves the following optimization problem:
minimize ||x||_1 subject to y = Ax
where ||x||_1 denotes the l1 norm of x, which is the sum of the absolute values of its coefficients. This problem can be solved efficiently using convex optimization techniques, such as linear programming or greedy algorithms.
The success of CS in sparse signal recovery depends on the properties of the sensing matrix A, which should be designed to satisfy the so-called Restricted Isometry Property (RIP). This property ensures that the sensing matrix preserves the distances between sparse signals, meaning that it does not distort or mix their coefficients. The RIP can be characterized by a constant delta, which measures the deviation of the sensing matrix from an orthonormal basis. If delta is small enough, then the sparse signal recovery problem can be solved with high probability using convex optimization techniques.
In practice, many signals and images are not exactly sparse, but rather compressible, meaning that they have a sparse representation in some basis or dictionary. CS can be extended to handle compressible signals by using a sparsifying dictionary or basis, such as the wavelet basis, instead of a fixed orthonormal basis. This leads to the so-called Compressed Sensing with Sparsity Constraints (CSSC) framework, which solves the following optimization problem:
minimize ||x||_1 subject to y = Ax, ||Dx||_0 <= K
where D is the sparsifying dictionary, ||x||_0 denotes the l0 pseudo-norm of x, which counts the number of non-zero coefficients, and K is the desired sparsity level. This problem can also be solved efficiently using convex optimization techniques, such as iterative thresholding or greedy algorithms.
CS has many applications in various fields, including signal processing, imaging, communications, and machine learning. In signal processing, CS can be used for compressed sensing of analog signals, such as audio and video, as well as for channel estimation in wireless communication systems. In imaging, CS can be used for fast MRI (magnetic resonance imaging) and CT (computed tomography) scans, as well as for image denoising and reconstruction from incomplete data. In machine learning, CS can be used for feature selection and dimensionality reduction, as well as for training sparse models.
One of the key advantages of CS is its ability to reduce the amount of data that needs to be acquired and processed, which can result in significant cost savings and faster processing times. For example, CS can reduce the number of MRI scans needed to produce an image, which can reduce the cost and discomfort for patients, as well as increase the throughput of the imaging system. CS can also reduce the amount of data that needs to be transmitted in wireless communication systems, which can increase the spectral efficiency and reduce the power consumption.
Another advantage of CS is its ability to handle noisy and incomplete data, which is common in real-world applications. CS algorithms can be designed to incorporate noise and measurement errors into the reconstruction process, leading to more robust and accurate estimates of the signal or image. CS can also handle missing or corrupted data, by exploiting the sparsity or compressibility of the signal to fill in the missing values or correct the errors.
In summary, Compressed Sensing is a powerful signal processing technique that has revolutionized the way we acquire and process data. By exploiting the sparsity or compressibility of signals and images, CS can significantly reduce the amount of data that needs to be acquired and processed, while maintaining high accuracy and robustness. CS has many applications in various fields and is a topic of active research and development, with many new algorithms and applications being proposed and tested.