6.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Compressed sensing | 3/6 | https://en.wikipedia.org/wiki/Compressed_sensing | reference | science, encyclopedia | 2026-05-05T14:40:18.609631+00:00 | kb-cron |
In the CS reconstruction models using constrained
ℓ
1
{\displaystyle \ell _{1}}
minimization, larger coefficients are penalized heavily in the
ℓ
1
{\displaystyle \ell _{1}}
norm. It was proposed to have a weighted formulation of
ℓ
1
{\displaystyle \ell _{1}}
minimization designed to more democratically penalize nonzero coefficients. An iterative algorithm is used for constructing the appropriate weights. Each iteration requires solving one
ℓ
1
{\displaystyle \ell _{1}}
minimization problem by finding the local minimum of a concave penalty function that more closely resembles the
ℓ
0
{\displaystyle \ell _{0}}
norm. An additional parameter, usually to avoid any sharp transitions in the penalty function curve, is introduced into the iterative equation to ensure stability and so that a zero estimate in one iteration does not necessarily lead to a zero estimate in the next iteration. The method essentially involves using the current solution for computing the weights to be used in the next iteration.
====== Advantages and disadvantages ====== Early iterations may find inaccurate sample estimates, however this method will down-sample these at a later stage to give more weight to the smaller non-zero signal estimates. One of the disadvantages is the need for defining a valid starting point as a global minimum might not be obtained every time due to the concavity of the function. Another disadvantage is that this method tends to uniformly penalize the image gradient irrespective of the underlying image structures. This causes over-smoothing of edges, especially those of low contrast regions, subsequently leading to loss of low contrast information. The advantages of this method include: reduction of the sampling rate for sparse signals; reconstruction of the image while being robust to the removal of noise and other artifacts; and use of very few iterations. This can also help in recovering images with sparse gradients. In the figure shown below, P1 refers to the first-step of the iterative reconstruction process, of the projection matrix P of the fan-beam geometry, which is constrained by the data fidelity term. This may contain noise and artifacts as no regularization is performed. The minimization of P1 is solved through the conjugate gradient least squares method. P2 refers to the second step of the iterative reconstruction process wherein it utilizes the edge-preserving total variation regularization term to remove noise and artifacts, and thus improve the quality of the reconstructed image/signal. The minimization of P2 is done through a simple gradient descent method. Convergence is determined by testing, after each iteration, for image positivity, by checking if
f
k
−
1
=
0
{\displaystyle f^{k-1}=0}
for the case when
f
k
−
1
<
0
{\displaystyle f^{k-1}<0}
(Note that
f
{\displaystyle f}
refers to the different x-ray linear attenuation coefficients at different voxels of the patient image).
===== Edge-preserving total variation (TV)-based compressed sensing =====
This is an iterative CT reconstruction algorithm with edge-preserving TV regularization to reconstruct CT images from highly undersampled data obtained at low dose CT through low current levels (milliampere). In order to reduce the imaging dose, one of the approaches used is to reduce the number of x-ray projections acquired by the scanner detectors. However, this insufficient projection data which is used to reconstruct the CT image can cause streaking artifacts. Furthermore, using these insufficient projections in standard TV algorithms end up making the problem under-determined and thus leading to infinitely many possible solutions. In this method, an additional penalty weighted function is assigned to the original TV norm. This allows for easier detection of sharp discontinuities in intensity in the images and thereby adapt the weight to store the recovered edge information during the process of signal/image reconstruction. The parameter
σ
{\displaystyle \sigma }
controls the amount of smoothing applied to the pixels at the edges to differentiate them from the non-edge pixels. The value of
σ
{\displaystyle \sigma }
is changed adaptively based on the values of the histogram of the gradient magnitude so that a certain percentage of pixels have gradient values larger than
σ
{\displaystyle \sigma }
. The edge-preserving total variation term, thus, becomes sparser and this speeds up the implementation. A two-step iteration process known as forward–backward splitting algorithm is used. The optimization problem is split into two sub-problems which are then solved with the conjugate gradient least squares method and the simple gradient descent method respectively. The method is stopped when the desired convergence has been achieved or if the maximum number of iterations is reached.
===== Advantages and disadvantages ===== Some of the disadvantages of this method are the absence of smaller structures in the reconstructed image and degradation of image resolution. This edge preserving TV algorithm, however, requires fewer iterations than the conventional TV algorithm. Analyzing the horizontal and vertical intensity profiles of the reconstructed images, it can be seen that there are sharp jumps at edge points and negligible, minor fluctuation at non-edge points. Thus, this method leads to low relative error and higher correlation as compared to the TV method. It also effectively suppresses and removes any form of image noise and image artifacts such as streaking.