11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Compressed sensing | 5/6 | https://en.wikipedia.org/wiki/Compressed_sensing | reference | science, encyclopedia | 2026-05-05T14:40:18.609631+00:00 | kb-cron |
The Augmented Lagrangian method for the orientation field,
min
X
‖
∇
X
∙
d
‖
1
+
λ
2
‖
Y
−
Φ
X
‖
2
2
{\displaystyle \min _{\mathrm {X} }\lVert \nabla \mathrm {X} \bullet d\rVert _{1}+{\frac {\lambda }{2}}\ \lVert Y-\Phi \mathrm {X} \rVert _{2}^{2}}
, involves initializing
d
h
,
d
v
,
H
,
V
{\displaystyle d_{h},d_{v},H,V}
and then finding the approximate minimizer of
L
1
{\displaystyle L_{1}}
with respect to these variables. The Lagrangian multipliers are then updated and the iterative process is stopped when convergence is achieved. For the iterative directional total variation refinement model, the augmented lagrangian method involves initializing
X
,
P
,
Q
,
λ
P
,
λ
Q
{\displaystyle \mathrm {X} ,P,Q,\lambda _{P},\lambda _{Q}}
. Here,
H
,
V
,
P
,
Q
{\displaystyle H,V,P,Q}
are newly introduced variables where
H
{\displaystyle H}
=
∇
d
h
{\displaystyle \nabla d_{h}}
,
V
{\displaystyle V}
=
∇
d
v
{\displaystyle \nabla d_{v}}
,
P
{\displaystyle P}
=
∇
X
{\displaystyle \nabla \mathrm {X} }
, and
Q
{\displaystyle Q}
=
P
∙
d
{\displaystyle P\bullet d}
.
λ
H
,
λ
V
,
λ
P
,
λ
Q
{\displaystyle \lambda _{H},\lambda _{V},\lambda _{P},\lambda _{Q}}
are the Lagrangian multipliers for
H
,
V
,
P
,
Q
{\displaystyle H,V,P,Q}
. For each iteration, the approximate minimizer of
L
2
{\displaystyle L_{2}}
with respect to variables (
X
,
P
,
Q
{\displaystyle \mathrm {X} ,P,Q}
) is calculated. And as in the field refinement model, the lagrangian multipliers are updated and the iterative process is stopped when convergence is achieved. For the orientation field refinement model, the Lagrangian multipliers are updated in the iterative process as follows:
(
λ
H
)
k
=
(
λ
H
)
k
−
1
+
γ
H
(
H
k
−
∇
(
d
h
)
k
)
{\displaystyle (\lambda _{H})^{k}=(\lambda _{H})^{k-1}+\gamma _{H}(H^{k}-\nabla (d_{h})^{k})}
(
λ
V
)
k
=
(
λ
V
)
k
−
1
+
γ
V
(
V
k
−
∇
(
d
v
)
k
)
{\displaystyle (\lambda _{V})^{k}=(\lambda _{V})^{k-1}+\gamma _{V}(V^{k}-\nabla (d_{v})^{k})}
For the iterative directional total variation refinement model, the Lagrangian multipliers are updated as follows:
(
λ
P
)
k
=
(
λ
P
)
k
−
1
+
γ
P
P
k
−
∇
(
X
)
k
)
{\displaystyle (\lambda _{P})^{k}=(\lambda _{P})^{k-1}+\gamma _{P}P^{k}-\nabla (\mathrm {X} )^{k})}
(
λ
Q
)
k
=
(
λ
Q
)
k
−
1
+
γ
Q
(
Q
k
−
P
k
∙
d
)
{\displaystyle (\lambda _{Q})^{k}=(\lambda _{Q})^{k-1}+\gamma _{Q}(Q^{k}-P^{k}\bullet d)}
Here,
γ
H
,
γ
V
,
γ
P
,
γ
Q
{\displaystyle \gamma _{H},\gamma _{V},\gamma _{P},\gamma _{Q}}
are positive constants.
===== Advantages and disadvantages ===== Based on peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) metrics and known ground-truth images for testing performance, it is concluded that iterative directional total variation has a better reconstructed performance than the non-iterative methods in preserving edge and texture areas. The orientation field refinement model plays a major role in this improvement in performance as it increases the number of directionless pixels in the flat area while enhancing the orientation field consistency in the regions with edges.
== Applications == The field of compressive sensing is related to several topics in signal processing and computational mathematics, such as underdetermined linear systems, group testing, heavy hitters, sparse coding, multiplexing, sparse sampling, and finite rate of innovation. Its broad scope and generality has enabled several innovative CS-enhanced approaches in signal processing and compression, solution of inverse problems, design of radiating systems, radar and through-the-wall imaging, and antenna characterization. Imaging techniques having a strong affinity with compressive sensing include coded aperture and computational photography. Conventional CS reconstruction uses sparse signals (usually sampled at a rate less than the Nyquist sampling rate) for reconstruction through constrained
l
1
{\displaystyle l_{1}}
minimization. One of the earliest applications of such an approach was in reflection seismology which used sparse reflected signals from band-limited data for tracking changes between sub-surface layers. When the LASSO model came into prominence in the 1990s as a statistical method for selection of sparse models, this method was further used in computational harmonic analysis for sparse signal representation from over-complete dictionaries. Some of the other applications include incoherent sampling of radar pulses. The work by Boyd et al. has applied the LASSO model- for selection of sparse models- towards analog to digital converters (the current ones use a sampling rate higher than the Nyquist rate along with the quantized Shannon representation). This would involve a parallel architecture in which the polarity of the analog signal changes at a high rate followed by digitizing the integral at the end of each time-interval to obtain the converted digital signal.
=== Photography === Compressed sensing has been used in an experimental mobile phone camera sensor. The approach allows a reduction in image acquisition energy per image by as much as a factor of 15 at the cost of complex decompression algorithms; the computation may require an off-device implementation. Compressed sensing is used in single-pixel cameras from Rice University. Bell Labs employed the technique in a lensless single-pixel camera that takes stills using repeated snapshots of randomly chosen apertures from a grid. Image quality improves with the number of snapshots, and generally requires a small fraction of the data of conventional imaging, while eliminating lens/focus-related aberrations.
=== Holography === Compressed sensing can be used to improve image reconstruction in holography by increasing the number of voxels one can infer from a single hologram. It is also used for image retrieval from undersampled measurements in optical and millimeter-wave holography.
=== Facial recognition === Compressed sensing has been used in facial recognition applications.
=== Magnetic resonance imaging === Compressed sensing has been used to shorten magnetic resonance imaging scanning sessions on conventional hardware. Reconstruction methods include