A Mathematical Introdution To Compressive Sensing Answers

Compressive Sensing

Compressive sensing is a double-sided problem where one side is the sensory side, and the other side is the recovery of the sensed data.

From: Compressive Sensing in Healthcare , 2020

Neural signal compressive sensing

Denise Fonseca Resende , ... Carlos A. Duque , in Compressive Sensing in Healthcare, 2020

Abstract

Compressive sensing is a recent highly applicative approach. It enables efficient data sampling at a much lower rate than the requirements indicated by the Nyquist theorem. Compressive sensing possesses several advantages, such as the much smaller need for sensory devices, much less memory storage, higher data transmission rate, many times less power consumption. Due to all these advantages, compressive sensing has been used in a wide range of applications. An application field of compressive sensing in health care is in neuro-signal acquisition. This chapter reviews the different aspects of neuro-signal compressive sensing in the literature. It provides an intuitive explanation of the formulas of implementing compressive sensing in neuro-signal-based applications with descriptive figures and block diagrams.

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Recovery in compressive sensing: a review

Mahdi Khosravy , ... Carlos A. Duque , in Compressive Sensing in Healthcare, 2020

2.1 Introduction

In the era of the Internet of Things, and the increasing amount of information of all types, as well as the growing capability of modern devices in the processing of information of high quality, modern technology faces an increasing amount of information. These high levels of information quantity need: (i) to be initially sensed like signal acquisition, (ii) to be transmitted as in telecommunication systems, and (iii) to be processed. All these three tasks require data processing, storage memory, high transition bandwidth, and after all more power, especially for sensing. In the realm of signals and images, the compression was suggested and used wherein a signal/image was reshaped and saved in much lower size. The compressed signal/image was decompressed to the original size as it was needed for use. To avoid un-necessary sampling of the redundant information and bearing the side effect mentioned above, compressive sensing was suggested. It was pioneered by Candes, Romberg, and Tao and by Donoho [1–5]. Compressive sensing samples the signal by a much smaller number of samples than required by the Nyquist–Shannon theorem. It is based on an assumption of sparsity for the signal. Naturally, this assumption is true for most data forms of information in nature. Compressive sensing mainly is a challenge to (i) compressively measure a signal while its information content is kept preserved, (ii) to recover the original signal after compressive sensing. The compressive method has a great application potential and can be used in a wide range of applications, like location-based services [6], signal processing [7,8], smart environments [9], texture analysis [10], telecommunications [11–16], public transportation systems [17], acoustic OFDM [18], power line communications [19], power quality analysis [20–22], power system planning [23,24], human motion analysis [25], medical image processing [26–28], human–robot interaction [29], electrocardiogram processing [30–35], sentiment mining [36], text data processing [37], image enhancement [38,39], image adaptation [40], software intensive systems [41], agriculture machinery [42,43], and data mining [44,45]. This chapter gives a quick review of both aspects of compressive sensing.

2.1.1 Compressive sensing formulation

Compressive sensing is modeled as follows in its standard formulation. Consider x as a signal vector of length n belonging to the vector space of $R^{n}$ . Then x can be compressively sensed; in other words, it can be presented as a vector y of length m belonging to the vector space $R^{m}$ as follows:

(2.1) $y_{m \times 1} = A_{m \times n} x_{n \times 1} .$

Note that m is much shorter than n, $m < < n$ , which explains the term 'compressive sensing'. A is called a compressive sensing matrix. Fig. 2.1 shows the general schematic of compressive sensing wherein a long signal vector is sensed as a much shorter signal vector. As mentioned earlier two main questions of compressive sensing are as follows:

1.: What conditions must a compressive sensing matrix obey to maximally preserve the information content of the signal?
2.: Having the compressed sensed vector y , how can the signal vector x be recovered from y ?

This chapter briefly answers these two main questions of compressive sensing with main focus on the second question.

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Deterministic compressive sensing by chirp codes: a descriptive tutorial

Mahdi Khosravy , ... Noboru Babaguchi , in Compressive Sensing in Healthcare, 2020

Abstract

Compressive sensing is a quite recent signal/image acquisition achievement with the capability of sampling data via a much smaller number of sensory devices. It is even less than the theoretical requirement set by the Shannon–Nyquist sampling theorem. Compressive sensing foundations are established on the sparsity of the data with informative characteristics. While most of the compressive sensing techniques use the sensing matrices of random structure, there is a category of compressive techniques that use deterministic paradigms instead of random approaches. Deterministic compressive sensing is in both aspects of the structure of the sensing matrix as well as the recovery process wherein the compressive matrix structure is well known, and therefore the recovery structure is founded on this a priori knowledge. Deterministic compressive sensing is by using chirp codes with the compression capability of reducing the signal length by the factor of the square root of the signal length. Chirp codes compressive sensing theory and implementation are clarified in detail through a descriptive tutorial in this chapter.

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Compressive sensing theoretical foundations in a nutshell

Mahdi Khosravy , ... Noboru Babaguchi , in Compressive Sensing in Healthcare, 2020

Abstract

Compressive sensing is a well-established technique for signal/image acquisition with a considerably low sampling rate. It efficiently samples the data in a rate much lower than the classic requirement in uniform sampling by the Nyquist–Shannon sampling rate. Compressive sensing is based on a sparsity consideration of the information sources and it results in much lower requirement as regards the rate of data collection, sensory devices, required memory storage, and the power needed for sensory devices. This chapter briefly reviews the theoretical fundamental requirements for compressive data acquisition if it is to maintain the possibility of original data recovery. The connection of compressive sensing with sparseness of information and its confronting with the Nyquist sampling theorem is discussed. Compressive sensing comprises two main challenges: (i) How to design a compressive sensing matrix which senses a signal segment with a much smaller number of measurements than the signal segment length, ensuring that the information inside the signal is preserved. (ii) How to recover the signal from a segment of less shorter measurements. The chapter mainly explains the answer to the first question, as it clarifies the connection of compressive sensing with sparsity. The recovery methods in compressive sensing are out of this chapter's scope, and it mainly focuses on how to compressively sense data to keep the possibility of recovering the non-compressed form of data. The chapter explains the conditions required for a compressive sensing matrix. In a nutshell, the chapter aims to offer a clear understanding and overview to the reader of the theoretical fundamentals of compressive sensing.

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Carbon Nanotube-Based Infrared Camera Using Compressive Sensing

Bo Song , ... Liangliang Chen , in Nano Optoelectronic Sensors and Devices, 2012

14.2.5 Compressive Sensing Applications

Compressive sensing could minimize the measurements required in the hardware of the sensing component, so it has many practical applications. Richard Baraniuk and Kevin Kelly of Rice University developed an interesting implantation for compressive sensing [46, 47]. They use only one photodiode to sense and acquire images and then successfully perform reconstruction by compressive sensing. A digital micromirror device (DMD) was used in their "single-pixel camera" as a measurement matrix generator, and with the help of lens, the original image would be reflected and focused onto the signal photodiode. Thus what the photodiode measured is the linear projection from the original image into the measurement matrix generated by the DMD, not the entirety pixels in the original image. The measurement times are far the lower than the length of original signal (the number of pixels in the original image).

Another application for compressive sensing is the fast radar imaging system [48]. Because of the presence of a compressive sensing implant in the radar imaging system, the bandwidth of the sensing device is decreased. Also, instead of a complicated signal receiver device, a compressive sensing signal recovery system could obtain as a good performance as with the expensive hardware in a radar system. Compressive sensing can also be used in biology and medical areas such as DNA chips [49], fast imaging magnetic resonance system [50], and so forth. One of the significant advantages of compressive sensing is that it can decrease the hardware and power requirements in sensing devices, so a very popular research area is using compressive sensing to built up distribute sensor network. Researchers from Rice University successfully use the joint sparsity recovery method to decrease the noise and error level in sensor networks [51].

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A descriptive review to sparsity measures

Mahdi Khosravy , ... Noboru Babaguchi , in Compressive Sensing in Healthcare, 2020

Abstract

Compressive sensing is a recent data sampling technique with a variety of advantages over the classical Shannon–Nyquist based technique. The main theoretical approach to compressive sensing is based on the informative value of data according to sparsity where the higher sparsity indicates the higher information content. Therefore, while data samples are linearly mixed and sensed by a much smaller number of sensors and result in compressively sensed data of much less volume, the sparsity maximization is a strong approach to retrieving the original higher volume data from the compressed one. The sparsity analysis is the main approach to the idea of compressive sensing, and an efficient measure of sparsity has a key role in this regard. Although k-sparsity is the sparsity measure in use by compressive sensing techniques, it being well established in the theoretical analysis of compressive sensing, there are a variety of sparsity measures. This chapter reviews the sparsity measures from the k-sparsity already in use to be compared with other more complicated sparsity measures.

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Compressive sensing in practice and potential advancements

Ayan Banerjee , Sandeep K.S. Gupta , in Compressive Sensing in Healthcare, 2020

4.6 Conclusion

Compressive sensing has been a topic for extensive research in recent times. It has significant promise of high compression rates, however, in practice there are several problems. The biggest problem comes from the random sampling requirement. Reconstruction with random sampling assumption results in uniform distribution of error. Although this is not a problem as far as overall RMSE error is concerned, but it can cause high error rates on significant spatio-temporal characteristics of the data. As such several advancements are possible that perform selective sampling, blocked ROI-based compressive sensing and non-linear model driven CS. These possibilities are only tested in specific applications and have not been fully justified as consistent optimization potentials in the CS domain. Hence, a plethora of research topics are available to be explored by researchers in this domain.

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Deterministic compressive sensing by chirp codes: a MATLAB® tutorial

Mahdi Khosravy , ... Carlos A. Duque , in Compressive Sensing in Healthcare, 2020

7.1 Introduction

Compressive sensing [1–5] is a new road to signal/image data sampling. It is performed using by much less required samples than one stated by the Shannon–Nyquist theorem. It deploys the sparsity nature of the information where just some samples are associated with information, most of them not having any considerable information. Compressive sensing applies the sparsity in sensing the data, and by sensing the data in a compressed way, it needs much fewer sensors. In a wide range of applications, the compressive sensing can be applied, like data mining [6,7], text processing [8], signal processing [9,10], agriculture on-board data processing [11,12], image enhancement [13,14], acoustic OFDM [15], medical image processing [16–18], image adaptation [19] Electrocardiogram processing [20–26], telecommunications [27–32], power quality analysis [33–35], power line communications [36], etc.

This chapter is a step-by-step MATLAB® tutorial for deterministic compressive sensing by chirp codes [37].

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Compressive sensing of electroencephalogram: a review

Mateus M. de Oliveira , ... Carlos A. Duque , in Compressive Sensing in Healthcare, 2020

13.5.1 Block sparse Bayesian learning

The CS approach can only be successfully applied to signals that are sparse in some domain. However, the EEG signal does not comply with this requirement.

In order to overcome this issue, the authors of [76] applied Block Sparse Bayesian Learning (BSBL) to recover EEG signals acquired using the CS paradigm. The results showed that the BSBL approach presents better performance than other CS reconstruction algorithms, e.g., $l_{1}$ and CoSaMP.

In [77], the authors have also applied BSBL for EEG signals. Moreover, they applied a faster version of the BSBL algorithm and the results showed significant speed increase. Besides, the paper implements the algorithm in Field Programmable Gate Array (FPGA).

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Compressive sensing of electrocardiogram

Felipe Meneguitti Dias , ... Carlos A. Duque , in Compressive Sensing in Healthcare, 2020

9.4.6 Signal compression

CS involves different steps in the signal compression pipeline. It smartly samples the signal using a simple encoder and transfers all the mathematical complexity to the encoder. Besides, CS is not the state-of-the-art compression technique; therefore, it is usually not used just for compression. This subsection will show some applications where the main focus is signal compression using CS.

In [87], a compression pipeline is developed in a multi-channel electrocardiogram signal (MECG). First, a technique called Principal Component Analysis (PCA) is applied to the MECG signal. In order to reduce even more the information dimensionality, the components with maximum variance from PCA are projected over a sparse binary sensing matrix. In [88] , compressive sensing techniques are used as a lossy digital signal compression. The compression capabilities are compared to a conventional signal compression technique: Set Partitioning In Hierarchical Trees (SPIHT).

In order to estimate parameters from a compressed signal, e.g., QRS location, one has first to reconstruct the signal and then apply the appropriate technique to calculate the parameter. However, this approach is not feasible in a real-time application because signal reconstruction is a computationally complex task. The authors of [89] propose a framework to get information from a compressed ECG signal. They were able to detect the QRS complexes and estimate the heart rate in ECG signals that were compressed using CS.

Reference [90] proposes three methods of compression and compares them. It used CR, PRD, and the accuracy of the QRS detection as evaluation metrics. Initially, an Adaptive Linear Prediction method was proposed. A linear prediction was used to estimate the current sample of the signal $x [n]$ of the ECG signal from its previous samples. We write

(9.10) $\hat{x} [x] = \sum_{k = 1}^{m} h^{k} x [n - k]$

where $\hat{x} [n]$ is the estimate of $x [n]$ and $x [n]$ are the predictor coefficients.

Later, a technique known as Compressive Sampling Matching Pursuit was proposed. This method has four stages: sampling, redundancy removal, quantization, and Huffman coding. Fig. 9.6 schematically summarizes the method [90].

Finally, a third method based on decimation and characterized by losses is proposed. The authors concluded that the third method was the best one.

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