Compressive Sensing

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Compressive Sensing


After discussing the progress in researching compressive sensing, the report provides the principles behind CS, the framework being used, and the five different types of algorithms used. It then outlines two case studies, one for biological systems and one for physical systems. Further uses for CS are then provided.

Research Progress

  • Despite the speed of the development of computational power, the capturing and processing of signals in many areas continue to pose a significant challenge. Therefore, practical solutions for these computational and storage challenges with high-dimensional data often rely on compression. Compression aims at finding the best algorithm that can achieve an acceptable distortion that is capable of being accurately restored after transmitted.
  • The Nyquist Sampling Theorem states that "A bandlimited continuous-time signal can be sampled and perfectly reconstructed from its samples if the waveform is sampled over twice as fast as its highest frequency component."
  • Research quickly asked, "Why are we transmitting all this data if we are not going to use it?" The search for a way to compress the data before it is collected began. Looking for a way to achieve sensing and comprehension in a single step, the research shifted the Compressive Sensing(CS) framework from the load of compression being on the sampling acquisition process to the reconstruction procedure. If the originating signal is guaranteed to be sparse, the new method can provide a high reconstruction accuracy.


  • Sparsity is the first important principle in the CS framework. Sparsity in the CS framework means that much of the coefficient vector s is zero. Continuous time-signals such as sound or image can be compressed and stored by omitting zero or small projection coefficients. The signal is compressible if numerous projection coefficients can be ignored.
  • The second principle, mutual coherence, refers to the reconstruction performance. The reconstruction accuracy can be raised by preserving low mutual coherence. Low mutual coherence shows that the sensing matrix and the dictionary are uncorrelated. In the current CS framework, the reconstruction process is "successful as long as the mutual coherence between the sensing matrix and the dictionary is low."

Compressive Sensing Framework

  • The CS framework enables the reconstruction of either naturally sparse or transformed sparse signals by using a smaller number of sampling data compared to the Nyquist theorem.
  • The CS framework includes three major sections. The first is the sensing matrix design, which characterizes the system and satisfies the mutual coherence properties.
  • The second section is the dictionary settings, which determine the appropriate sparse signal that represents the original signal or image.
  • Finally, the reconstruction algorithm is used to reconstruct the sparse representation signal.


A recent research study identified five different classifications of algorithms.

General CS-Based Reconstruction Algorithms

Reconstruction Algorithms based on Sparsifying Transforms

  • If the correct domain transform is used to sparsify a signal, the reconstruction provides improved results for a large compression ratio.

Reconstruction Algorithms based on Compression Ratio

  • Compressive Sensing is used to reduce the number of samples during either the first or third section of the framework. Research studies have used CS and achieved good reconstruction results for large compression ratios of up to 80%.

CS Reconstruction of 3D Ultrasound

  • As of April 2019, little research had been done in the field of 3D UltraSound (US) compressive sensing. Research work in this field has mainly two directions, 3D US image CS acquisition, and 3D US image reconstruction.

CS-Based Deep Learning Novel Methods.

  • The architecture of deep neural networks (DNNs) has used artificial intelligence to interpret ultrasounds. It has been used for US imaging in liver classification, locating standard plane in fetal US imaging, and classification of breast lesions. Research work using DNN and CS for US image reconstruction is being researched.

CS Case Studies

There are two categories in which CS is being used. The first is biological systems, and the second is physical systems.

Biological Systems

  • Biological systems are used to monitor human health related to the diagnosis of diseases, monitoring of daily activities, or vital signs. Compressive streaming is currently being used in MRIs and Ultrasounds.

Case Study — General Electric

Physical Systems

  • Physical systems health monitoring refers to checking physical systems. Structural health is related to checking cracks, vibration, or fatigues of materials in structures like buildings, pipelines, and aircraft.

Case Study — Nokia

Other Examples of CS Use

Compressed sensing is being used in greenhouses for the remote control of CO2, soil moisture, light, and temperature. It is also used in digital cameras for picture and video recording, the oil and gas industry in monitoring pipe corrosion, CCTV (Closed Circuit TV) for better surveillance, and monitoring of infrastructures like buildings, bridges, and roads.


Two trends in the use of CS are electromyography, where multiple data readings from various places on the body are sent as a compressed signal to the applications; and smart cities, which are using different type of IoTs to send data for tracking programs.


  • Wearable devices are being used to monitor biosignals in real-time as well as providing continuous monitoring. One of the many signals that provide critical information about human body status is electromyography (EMG). EMG signal is useful in monitoring muscle functionality and activity during sport, fitness, or daily life.
  • Surface electromyography (sEMG) is a suitable technique in several health monitoring applications. Its non-invasiveness and ease of use make it a popular choice. When it is used, however, there are usually multiple recording EMG signals from different parts of the body. This configuration results in "a large amount of data that increases the power consumption of wireless transmission, thus reducing the sensor lifetime."
  • Compressed sensing (CS) is seen as a promising solution that uses the signal to significantly reduce the number of samples required to reconstruct the signal.
  • A large variety of algorithms have been developed in recent years with this technique, using low-power wireless body area networks (WBANs) for sEMG monitoring.

Smart Cities

Compressed sensing is an integral part of the flourishing of smart cities. IESE Business School in Barcelona publishes an annual list of the top ten smart cities trending around the globe

Hong Kong

  • Hong Kong has a "high innovation index, almost 100% of its population have mobile telephones, and it has a high number of wireless access points globally." As part of its smart city strategy, Hong Kong has implemented an electronic ID (e-ID) system.