8.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Compressed sensing | 4/6 | https://en.wikipedia.org/wiki/Compressed_sensing | reference | science, encyclopedia | 2026-05-05T14:40:18.609631+00:00 | kb-cron |
===== Iterative model using a directional orientation field and directional total variation ===== To prevent over-smoothing of edges and texture details and to obtain a reconstructed CS image which is accurate and robust to noise and artifacts, this method is used. First, an initial estimate of the noisy point-wise orientation field of the image
I
{\displaystyle I}
,
d
^
{\displaystyle {\hat {d}}}
, is obtained. This noisy orientation field is defined so that it can be refined at a later stage to reduce the noise influences in orientation field estimation. A coarse orientation field estimation is then introduced based on structure tensor, which is formulated as:
J
ρ
(
∇
I
σ
)
=
G
ρ
∗
(
∇
I
σ
⊗
∇
I
σ
)
=
(
J
11
J
12
J
12
J
22
)
{\displaystyle J_{\rho }(\nabla I_{\sigma })=G_{\rho }*(\nabla I_{\sigma }\otimes \nabla I_{\sigma })={\begin{pmatrix}J_{11}&J_{12}\\J_{12}&J_{22}\end{pmatrix}}}
. Here,
J
ρ
{\displaystyle J_{\rho }}
refers to the structure tensor related with the image pixel point (i,j) having standard deviation
ρ
{\displaystyle \rho }
.
G
{\displaystyle G}
refers to the Gaussian kernel
(
0
,
ρ
2
)
{\displaystyle (0,\rho ^{2})}
with standard deviation
ρ
{\displaystyle \rho }
.
σ
{\displaystyle \sigma }
refers to the manually defined parameter for the image
I
{\displaystyle I}
below which the edge detection is insensitive to noise.
∇
I
σ
{\displaystyle \nabla I_{\sigma }}
refers to the gradient of the image
I
{\displaystyle I}
and
(
∇
I
σ
⊗
∇
I
σ
)
{\displaystyle (\nabla I_{\sigma }\otimes \nabla I_{\sigma })}
refers to the tensor product obtained by using this gradient. The structure tensor obtained is convolved with a Gaussian kernel
G
{\displaystyle G}
to improve the accuracy of the orientation estimate with
σ
{\displaystyle \sigma }
being set to high values to account for the unknown noise levels. For every pixel (i,j) in the image, the structure tensor J is a symmetric and positive semi-definite matrix. Convolving all the pixels in the image with
G
{\displaystyle G}
, gives orthonormal eigen vectors ω and υ of the
J
{\displaystyle J}
matrix. ω points in the direction of the dominant orientation having the largest contrast and υ points in the direction of the structure orientation having the smallest contrast. The orientation field coarse initial estimation
d
^
{\displaystyle {\hat {d}}}
is defined as
d
^
{\displaystyle {\hat {d}}}
= υ. This estimate is accurate at strong edges. However, at weak edges or on regions with noise, its reliability decreases. To overcome this drawback, a refined orientation model is defined in which the data term reduces the effect of noise and improves accuracy while the second penalty term with the L2-norm is a fidelity term which ensures accuracy of initial coarse estimation. This orientation field is introduced into the directional total variation optimization model for CS reconstruction through the equation:
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}}
.
X
{\displaystyle \mathrm {X} }
is the objective signal which needs to be recovered. Y is the corresponding measurement vector, d is the iterative refined orientation field and
Φ
{\displaystyle \Phi }
is the CS measurement matrix. This method undergoes a few iterations ultimately leading to convergence.
d
^
{\displaystyle {\hat {d}}}
is the orientation field approximate estimation of the reconstructed image
X
k
−
1
{\displaystyle X^{k-1}}
from the previous iteration (in order to check for convergence and the subsequent optical performance, the previous iteration is used). For the two vector fields represented by
X
{\displaystyle \mathrm {X} }
and
d
{\displaystyle d}
,
X
∙
d
{\displaystyle \mathrm {X} \bullet d}
refers to the multiplication of respective horizontal and vertical vector elements of
X
{\displaystyle \mathrm {X} }
and
d
{\displaystyle d}
followed by their subsequent addition. These equations are reduced to a series of convex minimization problems which are then solved with a combination of variable splitting and augmented Lagrangian (FFT-based fast solver with a closed form solution) methods. It (Augmented Lagrangian) is considered equivalent to the split Bregman iteration which ensures convergence of this method. The orientation field, d is defined as being equal to
(
d
h
,
d
v
)
{\displaystyle (d_{h},d_{v})}
, where
d
h
,
d
v
{\displaystyle d_{h},d_{v}}
define the horizontal and vertical estimates of
d
{\displaystyle d}
.