Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3592_Библиотеки_им_академика_М_И_Перельмана
.pdf
IOP Publishing
ℓ
https://t.me/medicina_free
Vascular and Intravascular Imaging Trends, Analysis, and
Challenges, Volume 1
Stent applications
Petia Radeva and Jasjit S Suri
Chapter 7
Blind inpainting and outlier detection using
logarithmic transformation and total variation
Manya V Afonso and J Miguel Sanches
In this work, we address the problem of image reconstruction with missing pixels or
outliers, when the locations of the corrupted pixels are not known and when the
noise is multiplicative, which is the algebraic model for ultrasound and synthetic
aperture radar imaging. A logarithmic transformation is applied to convert the
multiplication between the image, the binary mask and the Rayleigh distributed
speckle noise into an additive problem. The image and mask terms are then
estimated iteratively with total variation regularization applied on the image, and
regularization on the mask term, which imposes sparseness on the support set of
0
the missing pixels. The resulting alternating minimization scheme simultaneously
estimates the image and mask, in the same iterative process. Experimental results
show that the proposed method can deal with a larger fraction of missing pixels than
two-phase methods, which first estimate the mask and then reconstruct the image.
Furthermore, it was experimentally observed that applying the algorithm on radio
frequency (RF) images of the carotid artery or intravascular ultrasound images led
to an outlier map which clearly delineates the lumen, and can therefore be applicable
for segmentation.
7.1 Introduction
Faulty imaging sensors or bit errors during transmission can cause some pixels in an
image to be lost or corrupted by impulse noise [14, 38]. In the case of missing pixel
values, the corrupted pixels are assumed to have a value equal to zero, and the
problem of estimating the complete image is called the inpainting problem [11, 42, 53].
When the image is corrupted by impulse noise, the corrupted pixels have values
different from zero, which are drawn from a binary distribution (salt-and-pepper
noise), or from a uniform distribution (random valued impulse noise).
doi:10.1088/2053-2563/ab01fach7 7-1 ª IOP Publishing Ltd 2019
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
Ultrasound images and those from some other coherent imaging modalities are
corrupted by a type of noise called speckle noise, which results from interference
patterns. Although it is a random process, some works argue that it contains useful
information about the microstructure of the tissue and the properties of the incident
pressure field [43, 49]. It has been shown that the observed ultrasound RF image can
be well approximated as an element-wise multiplication between the image pixels
which represent the morphological structure of the organ being imaged, and the
speckle field [43]. The statistics of the speckle noise in ultrasound RF images has
been assumed to follow the Rayleigh distribution [1, 30, 58]. Other non-Rayleigh
distributions such as the gamma distribution [55, 57] and Rayleigh mixture models
[52] have also been considered in the literature.
Despeckling methods seek to reduce the speckle noise and make the morphological structures more clearly visible. Denoising methods developed for this
modality take into account the multiplicative algebraic relation and the Rayleigh
statistics of the noise [4, 13, 50, 64]. Some methods characterize the spiky
components of the speckle as outliers which do not fit in the statistical distribution
assumed. In [43], an outlier shrinkage step is applied on the log-transformed image
to eliminate these outliers. An adaptive window method to compute local statistics
and to discard local extrema and replace them by average values based on the local
statistics on the B-mode image was proposed in [56]. It was reported in [65] that
outliers in ultrasound images resulted from tissue shifting and made image registration difficult.
Image reconstruction and inpainting methods for multiplicative noise generally
require the pixel locations of the outliers to be known and therefore cannot be
applied to estimate the pixel values at the locations of the outliers. In [5], a new
method for blind image inpainting was proposed, which estimates the values of
pixels which are missing or corrupted with impulse noise, when their locations are
unknown. This method was also extended to non-additive and non-Gaussian noises.
In this chapter, we review this method for the case of Rayleigh multiplicative speckle
noise, and show that applying it on an ultrasound image can be used as a
segmentation method.
The image to be estimated has n pixels and is represented as a vector, say in
lexicographic ordering,
∈
n
. Let
be the number of observed pixels or
<mn
pixels free from impulse noise. The loss of pixels or observing a partial set of m pixels
out of n can be represented as an element-wise multiplication of the image with a
binary mask in which all but m pixels are zero. In our representation, this
observation process is represented as a multiplication of the vector
n × n identity matrix
missing pixels set to zero.
−nm
)
For multiplicative noise, the mapping from
with the respective diagonal elements corresponding to the
to the partially observed image
with a size
given by
is
η=·yAx(),
S
7-2
(7.1)
η
η
ℓ
1
ℓ
1
ℓ
ℓ
1
ℓ
ℓ
1
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
where the speckle noise term
is Rayleigh or Gamma distributed, and the
S
multiplication is element-wise.
When the image is corrupted by impulse noise, those pixels of
the zeros on the diagonal of the observation matrix
to the noise field
Our goal is to estimate the image
estimate the matrix
,
I
ηη=·+−yAx I A()( ).
G
n
from the partial observations
which indicates which pixels are outliers.
have a value that corresponds
I
corresponding to
(7.2)
and also
7.1.1 Related work
For standard inpainting, i.e. when the index set of the observed pixels or the
observation mask
is known, the reconstruction problem can be solved by one of
several existing methods for image reconstruction from a sparse set of observations.
Many of these methods, such as [2, 9, 10, 22, 32, 37, 41], were developed for the
additive and Gaussian noise model and often in the context of compressed sensing
[16, 26] reconstruction. However, it must be noted that in compressed sensing the
observation operator needs to satisfy conditions to lead to incoherent observations
[17]. For the removal of impulse noise, a common approach is to estimate the
support set of the noisy pixels using an outlier detection method, to obtain an
estimate of
and then apply the reconstruction method.
For the additive and Gaussian noise model, there exist several methods for
reconstructing an image with missing data and removing impulse noise. Early
approaches for estimating the missing values used median filtering, which discards
outliers [38]. The adaptive median filter (AMF) [38] and adaptive center weighted
median filter [21] were developed to detect the positions of noisy pixels with,
respectively, salt-and-pepper and random valued impulse noise. Median filters rely
on the accuracy of the neighboring pixels and inherently cannot deal with a large
percentage of outliers. In [25] an improved outlier detector based on thresholding, a
measure called the rank-ordered logarithmic difference with edge preserving
regularization (ROLD-EPR) was proposed, for random valued impulse noise.
Two-phase methods for estimating the image involve a mask estimation step, in
which the observation mask
is estimated using outlier detection, and a reconstruction step in which a standard convex optimization procedure is used with the
estimate of
impulse noise is a sum of an
regularizers, leading to an
using either the
[15, 19]. Another commonly used formulation for denoising with
-norm data fidelity term and total variation (TV) [20, 48]
-TV optimization problem [15]. Sparse regularization
or
norm regularizers on the impulse noise term or support set of
0
the outliers was used in [15, 24, 60] and [62]. An iteratively reweighted least squares
(IRLS) based method for mixed impulse and Gaussian noise removal was proposed in
[46, 47]withan
data fidelity term corresponding to the observed pixels, an
2
term
corresponding to the noisy pixels and TV regularization on the image.
Several of the above methods are based on alternating minimization, where two
or more variables are iteratively estimated through a Gauss–Seidel method [44].
7-3
ℓ
ℓ
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
A related approach is the augmented Lagrangian (AL)/alternating direction method
of multipliers (ADMM) framework [29], which has been used extensively in recent
work on image reconstruction/restoration because of its mathematical elegance and
computational speed [37, 63].
For the Poisson noise model, there exist methods for image reconstruction from a
partial set of pixels with the observation matrix known, such as [33, 34, 40]. Another
method proposed in [8] filled in missing data through the minimization of the image
gradient and an approximate solution of the mean curvature flow equation. Patchbased dictionary learning methods for the Poisson model were proposed in [35] for
image denoising and in [36] for inpainting.
For the Rayleigh speckle noise model, existing methods such as the classical
interpolation methods for ultrasound [54], as well as those based on TV regularization after logarithmic compression [6, 49], all require the sampling matrix to be
known. A denoising method for ultrasound with an outlier shrinkage step applied on
the log-transformed image was proposed in [43].
7.1.2 Contributions and organization
In [5], we proposed a method to estimate the image
and the observation mask
simultaneously, for the additive and Gaussian noise model. We formulated the
masking operation as a summation after logarithmic compression, and applied a TV
regularizer on the term corresponding to the logarithm of the image, and an
-norm
0
regularizer on the term corresponding to the mask. The TV regularizer encourages
the estimate of
to be piece-wise smooth, while the
-norm regularizer encourages
0
the mask term to be sparse. The problem was solved iteratively using a Gauss–Seidel
alternating minimization scheme. This method was extended to the multiplicative
and Rayleigh distributed speckle noise, and Poisson noise models. The data fidelity
terms corresponding to these statistical models allow the relation between the image
and observation mask to remain additive after logarithmic transformation.
In this chapter, we review the blind inpainting method for multiplicative and
Rayleigh distributed noise, and present results for outlier detection in ultrasound
images. We show that applying the blind inpainting algorithm on ultrasound images
of the carotid artery, without loss of pixels, produced outlier maps which corresponded roughly to the lumen and can be useful for segmenting the images.
We formulate the estimation problem and the blind inpainting algorithm in
section 7.2.2. In section 7.3, we present experimental results on inpainting. Subsection 7.3.2 presents results for the application of blind inpainting for segmentation
of the lumen.
,
7.2 Blind inpainting
We begin with the method for blind inpainting for the additive and Gaussian noise
model, which is mathematically simpler, and then elaborate the method for multiplicative and Rayleigh distributed noise. In section 7.2.1, the observation model is
7-4
η
λ
ℓ
λ
1
1
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
where
is an additive Gaussian noise term.
G
η=+yAx(),
G
(7.3)
7.2.1 Blind inpainting for additive noise
For standard inpainting with TV regularization on the image and with the
observation mask
known, the problem of estimating
as formulated and solved
in [ 2, 22]is
λ
2
(),
is the regularization
> 0
(7.4)
where
ˆ
=∥−∥+TVxAxyxarg min
is the isotropic total variation function and
TV (.)
1
x
22
parameter. Note that this problem assumes the additive and Gaussian noise model
and has an
When we need to estimate both the image
applied on
where we now have two regularization parameters,
-norm data fidelity term.
2
, leading to
ˆ
ˆ
=∥−∥++TVxA Ax y x A(, ) argmin
1
xA,
22()2
and the mask
λλ
12
2
λ >,0
12
, a regularizer
( ), (7.5)
ϕ
ϕ(.)
, for each of the two
regularizer terms. The problem (7.5) is difficult to solve because it is not separable
for our variables
xA
,)
.
In [5], a logarithmic transform was used to convert the masking problem into an
additive and separable one. Since
is a diagonal matrix
=
adiag( )
and the
masking operation is element-wise multiplication, the variable of optimization for
the observation matrix is the vector of diagonal elements
with index i is observed, the corresponding mask element
lost,
. Thus, a pixel k in vector
=a 0
i
is defined as the scalar product,
=×yxa.
ii
i
∈
=a
i
n
{0, 1}
. When a pixel
, and when pixel i is
(7.6)
is
It is not known a priori if a given pixel ykcorresponds to an observed one (
not (
). Rather than have
=a 0
k
defined to be a small value in the order of
that depends on the dynamic range of the image, typically greater than or equal to 3.
Defining
= alog( )
ii
Assuming that
on equation (7.6) converts it into an additive model,
when the pixel is not observed, where aiis
=a 0
i
−0K
or smaller, K being a positive integer
,
⎧
0, if is observed
⎨
v =
i
−iK
,otherwise
⎩
and
are always positive, applying a logarithmic transformation
=×yxalog log( ),
i
ii
. (7.7)
7-5
)or
=a
k
(7.8)
g
ℓ
ℓ
0
λ
λ
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
where
=
ii
and
xlog
=+yx alog log log ,
i
i
i
ylog( )
i
ii
v=+gu ,
ii
. A small positive bias term
δ=+
δ > 0
is added to
(7.9)
(7.10)
to guarantee positivity. The base of the logarithm used is 10, but we could use the
natural logarithm with the only difference being an additive constant term. The problem
is now estimating the vectors
We assume that our image
piece-wise smooth. We apply a TV regularizer on the log-transformed image
-norm regularizer on the log-transformed mask
0
correspond to the non-observed pixels and those elements of
0 correspond to the observed pixels. Since the
elements irrespective of their sign, minimizing
and
, given the log-transformed observationg.
and therefore its logarithmic transformation
. The negative elements of
therefore
which are equal to
-norm indicates the number of non-zero
0
minimizes the number of
∥v
are
,andthe
non-observed pixels.
The problem therefore becomes
λλ
12
2
TV( )
2
2
, (7.11)
0
where
ˆˆ
λ >,0
12
=∥−−∥++∥∥uv guv u v(, ) argmin
1
uv,
22
are the respective regularization parameters.
Since equation (7.11) is a separable problem, we can apply an iterative alternating
method as in [12]. We apply an iterative alternating minimization to solve equation
(7.11), by isolating the terms in each variable keeping the other fixed, leading to a
Gauss–Seidel scheme. Solving for
at iteration t,
tt() ()
ˆ
=∥−−∥+u guv uarg min
1
u
22
λ
1
2
TV( ).
2
(7.12)
This is a TV regularized denoising problem, the solution of which can be computed
efficiently using an algorithm such as Chambolle’s algorithm [18].
Similarly, for
at iteration t we have
tt() ()
ˆ
=∥−−∥+∥∥vguvvarg min
1
u
22
λ
2
2
2
.
0
(7.13)
This problem although non-convex, has a solution given by the hard threshold [27]
where
tt()
ˆ
Hvgu( ), (7.14)
=−
λ
2
2
is the hard threshold operator and is defined element-wise as
(.)
λ
2
⎧
0, if g u 2 ,
⎪
t
()
⎨
v
=
i
⎪
()
⎩
t
()
−
gu
i
i
,otherwise.
()
t
()
i
i
()
−⩽
λ
2
(7.15)
The steps (7.12) and (7.13) are run alternatingly until the stopping criterion is
satisfied. Continuation schemes can be used on the regularization parameters
λ,
12
in which they are multiplied by a factor greater than one, until they reach a certain
7-6
,
ℓ
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
maximum value, as was done in [59]. The estimates of the image and mask are
computed by inverting the logarithmic transformation,
ˆ
u
ˆ
10
=
and
ˆ
ˆ
v
. The
=
10
conditions for convergence [29, 34] do not require equation (7.12) to be solved
exactly, as long as the error sequence decreases and the parameter μ is positive.
7.2.2 Blind inpainting for Rayleigh multiplicative noise
We now extend the method described in the previous sub-section to the multiplicative and Rayleigh distributed noise model. An intuitive way would be to apply
the logarithmic transform to convert the observation model (7.1) into an additive
one. Then, we could apply the method described in the previous section. However,
this approach does not take into account the statistical model of the noise and the
appropriate data fidelity term for Rayleigh speckle noise.
We therefore extend the blind inpainting method to Rayleigh distributed multiplicative
speckle noise by using the appropriate data fidelity term. For multiplicative noise,
multiplying a pixel whose value is 0 will always lead to the corresponding observed pixel
being equal to 0 as well. Therefore, we interchange the order of the noisy observation and
masking so that our observation
is the result of observing the masked imagexunder
the noise model. The observation model changes from equation (7.1)to,
η=·yAx() .
S
(7.16)
For the Rayleigh multiplicative noise, the likelihood function is
2
y
yAx() exp
∣= −
p
∏
=
in1
ax
ii
i
⎛
⎜
⎜
⎝
y
i
ax
2( )
ii
⎞
⎟
. (7.17)
⎟
⎠
It is straightforward to show that (see [51] for more details) the associated data
fidelity term between
In equation (7.18), we see that there appears a term with the product
andxis
J
r
2
⎛
y
i
⎜
=+
∑
⎜
ax
2( )
⎝
=
in1
log( ) . (7.18)
ii
axyAx(, )
ii
⎞
⎟
⎟
⎠
ax
ii
and a
)
term involving its logarithm. Therefore, as before, we can work with the logtransformed variables
=
J
r
and
log
=++
∑
=
in1
=
2
⎛
y
i
⎜
⎜
2
⎝
. Thus equation (7.18) changes to
alog
−+
u
()
ii
euyuv(, , )
v
⎞
⎟
v
. (7.19)
ii
⎟
⎠
We now formulate our optimization problem, once again with TV regularization
on
and
regularization on
0
. The data fidelity term
is changed accordingly.
(.)
The problem (7.11) for the additive Gaussian noise model changes to the more
general problem
λλ
ˆˆ
=++∥∥JTVuv yuv u v(, ) argmin (, , )
uv,
12
()
2
. (7.20)
0
2
7-7
ℓ
μ
ℓ
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
Since equation (7.19) involves the sum of a linear term and an exponential term, it
is non-separable for
and
. Therefore, we need to use variable splitting [23]tobe
able to use the augmented Lagrangian/alternating direction method of multipliers
(AL/ADMM) to solve equation (7.20). We therefore introduce two auxiliary
variables
andwto act as the arguments of the TV and
regularizer terms,
0
respectively, leading to the constrained problem
λλ
12
JTVyuv z w
min ( , , )
uvzw,,,
subject to , .
++∥∥
==
uzvw
()
2
2
0
(7.21)
Using the augmented Lagrangian [39, 45], this problem can be shown to be
equivalent to the minimization problem
λλ
12
++∥∥
zw
()
2
2
2
2
2
d,
zw
0
2
,
2
are the so-called Bregman
(7.22)
where
μ ⩾,0
12
JTVyuv z w
min ( , , )
uvzw,,,
μμ
1
+∥−−∥+∥−−∥
uzd vwd
22
are the penalty parameters, and
update vectors [37]. This problem is split into four problems at each iteration by
gathering all the terms in each variable, and solving for each by keeping the others
fixed. Thus, the AL algorithm iterates between minimizing the objective function in
equation (7.22) with respect to
and
, leading to a Gauss–Seidel process (for more
details, see [2, 3, 31] and references therein) which at iteration t is summarized as
μ
+
ttt
(1) ()
=+−−
+
ttt
(1) ()
=+−−
+
tt
(1)
=−−+
+
tt
(1)
wvwdwarg min
=−−+∥∥
Ju yuv u z darg min ( , , )
u
Jvyuvvwdarg min ( , , )
v
μ
1
()
z
22
μ
2
()
w
22
t
(1) ()
+
ddz u
z
t
(1) ()
+
ddw v,.
w
t
=+ −
z
t
=+ −
w
As in the case of Gaussian noise, the
1
()
2
μ
2
()
2
2
λ
t
()
z
()
w
tt
(1) (1)
++
tt
(1) (1)
++
-TV denoising problem from equation
2
1
2
2
t
λ
2
(7.25) is solved using a few iterations of Chambolle’s algorithm and the
2
t
()
z
2
2
t
()
w
2
TVzuzdzarg min
()
2
0
(7.23)
(7.24)
(7.25)
(7.26)
−ℓℓ
20
regularized denoising problem from equation (7.26) is solved using the hard
threshold. The problems involving
, equations (7.23) and (7.24) can be solved
(.)
approximately using a few iterations of Newton’s method [44], after plugging in
equation (7.19).
7-8
λ
μ
λ
1
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
The proposed method for blind inpainting with Rayleigh multiplicative noise is
summarized in the following algorithm.
Algorithm. Blind inpainting—non-Gaussian noise
1. Input
2. Initialize parameters
.
(0) (0) (0) (0)
vzwdd,,, ,,
δ > 0
(0) (0)
.
zw
,
>K 0
,
,
λ,
12
, initial estimates
μ,
12
3. Set t = 0.
4. Repeat.
+
5. Compute
6. Compute
7. Compute
8. Compute
t
(1) ()
+
9.
z
t
(1) ()
+
10.
w
11. Update values of
12.
←+tt
t(1)
using Newton’s method to solve (7.23).
+
t(1)
using Newton’s method to solve (7.24).
+
t(1)
using Chambolle’s method to solve (7.25).
+
tt
(1)
←−
Hwvd
λμ
/
()
2
t
tt
z
t
w
(1) (1)
(1) (1)
←+ −
dz u
←+ −
dw v
2
++
tt
++
.
λ,
12
()
t
()
.
w
.
.
13. Until the stopping criterion is satisfied.
14. Set estimates
u
ˆ
= e
v
ˆ
,
= e
.
7.3 Experimental results
In this section we compare our proposed method for blind inpainting, with
inpainting using the additive model after logarithmic transformation. In the
synthetic experiments with the Lena and Cameraman images, we have the noise
free image for reference and use the normalized mean absolute error (NMAE) [28],
which is defined as
−ˆ∥∥∥xx x/
of the mask is measured in terms of the number of incorrectly estimated mask pixels,
which is obtained by the binary exclusive or XOR operation between the estimated
mask and the reference. A logical value equal to 1 is obtained at the mask pixels
estimated incorrectly, and zero otherwise. Hence, the sum over all the pixels of the
logical XOR operation is a measure of the errors in the estimate of the mask. All
experiments were performed on MATLAB on an Ubuntu Linux based server with
64 GB of RAM.
Results for the additive and Gaussian noise and Poisson noise models can be
found in [5].
7.3.1 Blind inpainting
To test our proposed method, we generate a random binary mask with a fraction of
its elements equal to zero and multiply it element-wise to our image corrupted with
multiplicative Rayleigh noise. The criteria used to evaluate the accuracy of
estimation are the NMAE, the structural similarity index measure (SSIM) [61]
and the fraction of incorrectly estimated mask pixels.
, the figure of merit. The accuracy of the estimation
11
7-9
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
We normalize the Lena image by dividing by the maximum pixel value. To
compare our method, we use a logarithmic transformation followed by inpainting
using a method for inpainting with additive noise. In this case, we use fast two-phase
deblurring [15], since AOP [62], although faster, also divides the observed image by
255, and in the case of Rayleigh speckle, the image is already normalized. We
summarize our results for the Lena and cameraman images for different fractions of
missing pixels, in table 7.1. We can see that taking into account the statistical model
offers an improvement in terms of the NMAE. The existing methods for blind
inpainting do not always work well for large fractions of missing pixels, above
0%
or take a long time, over 10 min.
For the Lena image, a cropped region from the noisy image with
of the pixels
0%
missing is shown in figure 7.1(b) and the respective estimates using the proposed
Table 7.1. Inpainting with Rayleigh noise. κ indicates the fraction of missing pixels. (*) The additive model is
used after logarithmic transformation [5] (© 2015 IEEE).
Lena Cameraman
κ Method Time (s) NMAE SSIM Mask
err. (%)
0.973 0.664 15.5 1.23 × 10
0.958 0.000 19 17.3 1.53 × 10
0.972 0.317 24.4 1.26 × 10
0.958 4.9 19.1 1.53 × 10
0.972 0.0835 156 1.27 × 10
0.25 Proposed 41 3.12 × 10
Additive(*) 11.3 3.81 × 10
0.5 Proposed 66.8 3.16 × 10
Additive(*) 75.8 3.82 × 10
0.7 Proposed 116 3.17 × 10
−06
−06
−06
−06
−06
Time (s) NMAE SSIM Mask
−05
−05
−05
−05
−05
err. (%)
0.975 11.6
0.961 0.001 14
0.974 2.13
0.961 4.29
0.973 0.484
,
Figure 7.1. Blind inpainting with Rayleigh noise with the Lena image: (a) original image (cropped),
(b) observed image with Rayleigh noise and
method; (d) estimate using inpainting with the additive model after logarithmic transformation; (e) observed
image with Rayleigh noise and
[
5] (© 2015 IEEE).
of its pixels missing; (c) estimate from (b) using the proposed
50%
of its pixels missing; and (f) estimate from (e) using the proposed method
70%
7-10
Соседние файлы в папке Библиотека им академика М.И. Перельмана