In Analog-to-Digital Converters(ADC), the signal is first sampled at a rate higher than or equal to the Nyquist, then quantized and encoded. In Analog-to-Informations (AIC), the sampling and compression are combined in one step, mathematically:
y=Φ⋅x
where y is compressed signal, Φ is the sensing matrix, and x is the input signal. My questions are:
* Simulating the above equation using MATLAB is downsampling, First, the input signal is read to obtain a vector of say N samples representing the signal and then multiplied by the matrix Φ.
Does this mean that we sample the signal to obtain the vector N? If so, where is the benefit? How sampling and compression are combined in one step? How can the sampling and compression be combined to perform compressive sensing in a practical way?
The reduction process is carried out by the measurement matrix (M*N) that is selected. The benefit is the process of taking the information , which represent the values that contain the main information large value of the data in the time domain or the process of taking the low frequency in frequency domain and at the same time removing the values that are less important and therefore this removal is considered a process of data compression
Actually, the matrix multiplication process is just a mathematical model of the actual AIC process. This measurement acquisition process can be done in the analog domain using analog multipliers and integrators to eventually provide you with exactly the same y vector, which was sampled below Nyquist rate. For example, see the figure below.
However, several applications also depend on sampling the signal first at or above Nyquist rate, then applying the measurement matrix. This is useful when the sample rate is not the issue, rather, you need a low-complexity compression method, for example, in a low-power sensing node that can't afford neither transmitting all acquired data nor a complex data compression mechanism. Therefore, we could just apply the sensing matrix, transmit the compressed data to another node which can afford the complicated recovery process.
The figure was taken from:
http://cs.signalprocessing.it/low-power.html
I would add more examples for some applications that current hardware limitations cannot overcome.
First is the limitation of the image sensor. The computation and power for readout N pixels. Pixels need enough light, readout requires power, etc so it prevents the very fast/high frame rate camera (thousands of frames per second) or very large image (billion pixels). So instead of reading them all, we collect the summation of frames/pixels follows the compressive sensing formula.
The application of low-complexity compression as Ahmad M. Rateb is not very practical for imaging due to the poor compression ratio and reconstruction quality. We can save the "compression power" but again waste the transmission and storage space. It still far bellow the conventional standard.
Many medical image applications will follow the compressive sensing formula due to their projection based measurements like MRI, electron tomography, etc.
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