Compressed Sensing: Matching Pursuit

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In summary, the Matching Pursuit algorithm efficiently solves the system Ax=b by finding the column of A that best matches b, removing the projection of b along that direction, and repeating this process until a termination criterion is met.
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EngWiPy
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Hi,

I need a geometric explanation for this algorithm: Suppose we have the system Ax=b, where A is and M-by-N matrix, x is N-dimensional s(<<N)-sparse vector, and b is M-dimensional vector. There is an algorithm called Matching Pursuit to solve the above system efficiently:

1- Find the column of A that best match b.
2- The projection of b along that direction is removed and an updated b is obtained.
3- Repeat 1 and 2 until some termination criterion is met.

How these 3 steps solve the above system of linear equations?

Thanks in advance
 
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What is compressed sensing?

Compressed sensing is a technique in signal processing that allows for the reconstruction of a signal from a small number of measurements, even if the signal is not sparse or compressible in its original form. It is based on the idea that a signal can be represented by a small number of significant coefficients in a suitable basis or dictionary, rather than its full representation.

What is matching pursuit?

Matching pursuit is a specific algorithm used in compressed sensing to efficiently find the sparsest representation of a signal. It works by iteratively selecting the basis or dictionary elements that best match the signal, and subtracting the contribution of these elements from the original signal. This process continues until a desired level of accuracy is achieved or a predetermined number of iterations is reached.

What are the advantages of compressed sensing and matching pursuit?

Compressed sensing and matching pursuit have several advantages, including the ability to accurately reconstruct signals from a small number of measurements, even if the signal is not sparse. This reduces the data storage and acquisition requirements, making it suitable for applications with limited resources. It also allows for fast and efficient signal processing, making it useful for real-time applications.

What are some applications of compressed sensing and matching pursuit?

Compressed sensing and matching pursuit have a wide range of applications, including medical imaging, wireless sensor networks, and data compression. They are also used in various fields such as astronomy, geophysics, and radar imaging, where large amounts of data need to be acquired and processed.

What are the limitations of compressed sensing and matching pursuit?

While compressed sensing and matching pursuit have many advantages, they also have some limitations. These techniques are computationally intensive and may not be suitable for real-time applications where speed is critical. They also require careful selection of the dictionary or basis elements, and the choice of these elements may affect the accuracy of the reconstruction. Additionally, they may not work well for signals that are not sparse or compressible in a suitable basis.

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