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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Compressed sensing | 2/6 | https://en.wikipedia.org/wiki/Compressed_sensing | reference | science, encyclopedia | 2026-05-05T14:40:18.609631+00:00 | kb-cron |
Compressed sensing takes advantage of the redundancy in many interesting signals—they are not pure noise. In particular, many signals are sparse, that is, they contain many coefficients close to or equal to zero, when represented in some domain. This is the same insight used in many forms of lossy compression. Compressed sensing typically starts with taking a weighted linear combination of samples also called compressive measurements in a basis different from the basis in which the signal is known to be sparse. The results found by Emmanuel Candès, Justin Romberg, Terence Tao, and David Donoho showed that the number of these compressive measurements can be small and still contain nearly all the useful information. Therefore, the task of converting the image back into the intended domain involves solving an underdetermined matrix equation since the number of compressive measurements taken is smaller than the number of pixels in the full image. However, adding the constraint that the initial signal is sparse enables one to solve this underdetermined system of linear equations. The least-squares solution to such problems is to minimize the
L
2
{\displaystyle L^{2}}
norm—that is, minimize the amount of energy in the system. This is usually simple mathematically (involving only a matrix multiplication by the pseudo-inverse of the basis sampled in). However, this leads to poor results for many practical applications, for which the unknown coefficients have nonzero energy. To enforce the sparsity constraint when solving for the underdetermined system of linear equations, one can minimize the number of nonzero components of the solution. The function counting the number of non-zero components of a vector was called the
L
0
{\displaystyle L^{0}}
"norm" by David Donoho. Candès et al. proved that for many problems it is probable that the
L
1
{\displaystyle L^{1}}
norm is equivalent to the
L
0
{\displaystyle L^{0}}
norm, in a technical sense: This equivalence result allows one to solve the
L
1
{\displaystyle L^{1}}
problem, which is easier than the
L
0
{\displaystyle L^{0}}
problem. Finding the candidate with the smallest
L
1
{\displaystyle L^{1}}
norm can be expressed relatively easily as a linear program, for which efficient solution methods already exist. When measurements may contain a finite amount of noise, basis pursuit denoising is preferred over linear programming, since it preserves sparsity in the face of noise and can be solved faster than an exact linear program. Typical-case reconstruction thresholds have also been studied with statistical-mechanical methods. Kabashima, Wadayama, and Tanaka used the replica method to analyze reconstruction limits for
L
p
{\displaystyle L_{p}}
-norm minimization with large random measurement matrices, and Takeda and Kabashima extended such analysis to correlated measurement matrices.
=== Total variation-based CS reconstruction ===
==== Motivation and applications ====
===== Role of TV regularization ===== Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). For signals, especially, total variation refers to the integral of the absolute gradient of the signal. In signal and image reconstruction, it is applied as total variation regularization where the underlying principle is that signals with excessive details have high total variation and that removing these details, while retaining important information such as edges, would reduce the total variation of the signal and make the signal subject closer to the original signal in the problem. For the purpose of signal and image reconstruction,
ℓ
1
{\displaystyle \ell _{1}}
minimization models are used. Other approaches also include the least-squares as has been discussed before in this article. These methods are extremely slow and return a not-so-perfect reconstruction of the signal. The current CS Regularization models attempt to address this problem by incorporating sparsity priors of the original image, one of which is the total variation (TV). Conventional TV approaches are designed to give piece-wise constant solutions. Some of these include (as discussed ahead) – constrained
ℓ
1
{\textstyle \ell _{1}}
-minimization which uses an iterative scheme. This method, though fast, subsequently leads to over-smoothing of edges resulting in blurred image edges. TV methods with iterative re-weighting have been implemented to reduce the influence of large gradient value magnitudes in the images. This has been used in computed tomography (CT) reconstruction as a method known as edge-preserving total variation. However, as gradient magnitudes are used for estimation of relative penalty weights between the data fidelity and regularization terms, this method is not robust to noise and artifacts and accurate enough for CS image/signal reconstruction and, therefore, fails to preserve smaller structures. Recent progress on this problem involves using an iteratively directional TV refinement for CS reconstruction. This method would have 2 stages: the first stage would estimate and refine the initial orientation field – which is defined as a noisy point-wise initial estimate, through edge-detection, of the given image. In the second stage, the CS reconstruction model is presented by utilizing directional TV regularizer. More details about these TV-based approaches – iteratively reweighted l1 minimization, edge-preserving TV and iterative model using directional orientation field and TV- are provided below.
==== Existing approaches ====
===== Iteratively reweighted ℓ1 minimization =====