=Paper=
{{Paper
|id=Vol-1452/paper19
|storemode=property
|title=Adaptive Regularization Algorithm Paired with Image Segmentation
|pdfUrl=https://ceur-ws.org/Vol-1452/paper19.pdf
|volume=Vol-1452
|dblpUrl=https://dblp.org/rec/conf/aist/Serezhnikova15
}}
==Adaptive Regularization Algorithm Paired with Image Segmentation==
https://ceur-ws.org/Vol-1452/paper19.pdf
Adaptive Regularization Algorithm Paired with
Image Segmentation
Tatiana Serezhnikova1,2
1
Krasovsky Institute of Mathematics and Mechanics UB RAS
2
Ural Federal University, Ekaterinburg, Russia
sti@imm.uran.ru
Abstract. This paper presents the subsequent improvements of our
adaptive regularization algorithm. The current version of the algorithm
contains an additional stabilizer. This stabilizer includes the coefficient,
which utilizes prior information about segmentation of an image to be
processed. We demonstrate that our adaptive regularization algorithm
can exploit only the observed image segmentation for both the adaptive
coefficient calculation and successful restoration of the original image
segmentation.
Keywords: image restoration, Tikhonov regularization, adaptation, im-
age segmentation
1 Introduction
In the model experiments, see [1-4], we demonstrated, that our adaptive reg-
ularization algorithm can successfully restore and provide more deblurred and
denoisy intensity of the original model image. In this paper, we improve our
algorithm, which is based on the results from [5-7], by an additional stabilizer.
This stabilizer includes the coefficient, which utilizes some information about
segmentation of an image to be restored.
Most artificial objects are made of surfaces, resulting in images with several
subimages (or a single subimage) together with their boundary lines.
These facts motivate us to use ideas from image segmentation in order to
improve our algorithm. In the sequel, we describe the stabilizer coefficient con-
struction, using ideas from image segmentation.
Image segmentation algorithms are based on properties of intensity values:
discontinuity and similarity, see [8-11]. Starting with the observed object inten-
sity, setting β(x, y) ≡ 0 in our adaptive algorithm, we successfully apply the
original approach for both sub-images and their boundaries. In the next iter-
ations, the adaptive algorithm utilizes this initial result for calculation of the
coefficient β.
Thus, we consider image segmentation both as result and tool in every image
restoration.
We prove numerically that the proposed algorithm can reveal the latent image
segmentation of observed images.
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The paper is organized as follows. In Section 2, we describe regularizing
functionals and present our adaptive technique. In Section 3, we describe the
calculation process of our stabilizer coefficient. The last section concludes the
paper.
2 Mathematical Model for Image Deblurring
We consider the following two-dimensional Fredholm integral equation of the
first kind:
Z1 Z1
Au ≡ K(x − ξ, y − η)u(x, y)dx, dy = f (ξ, η). (1)
0 0
In image restoration, the estimation of u from the observation of f is referred
to as the two-dimensional image deblurring problem.
We construct and develop the novel technique for numeric solution of equa-
tion (1). Abstract methods with convergence analysis of regularization for this
problem are presented in [3-7].
The foundation of the regularization method is given by
Z
min ||Ah u − fδ ||2L2 + α (||u||2L2 + J(u)) : u ∈ U , J(u) = |∇u| dx, (2)
D
where ∇u denotes the gradient of smooth function u, (u ∈ W11 (D)), J(u) is
the total variation of the function u on D. The practical implementation of this
method requires minimization of the functional in (2).
The novelty was that we proposed to add the version of the iterative tech-
nique, containing additional parameters βi,j , βi,j ≥ 0 , see [1-4]:
X
uk = arg min {Φα
N (u) + βi,j (ui,j − uk−1 2 N
i,j ) : u ∈ R } . (3)
i,j
Here, N = n2 , Φα
N (u) is the discrete form of the functional in (2).
P
βi,j (ui,j − uk−1 2
i,j ) is a discrete form for the additional stabilizer in (3).
i,j
We use the iterative subgradient method in order to compute uk defined in
(3), see details in [1-4].
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3 Stabilizer coefficient selection
3.1 Image Segmentation: Motivation
The previous version of the algorithm (3) used the constant β in every mesh
point of discrete models. The low resulting accuracy of this two-dimensional
model led us to change the way of {βi,j } selection.
As we said above, most man-made objects are made of surfaces, resulting
in images with a single image or some sub-images. The quality of the images
restoration features the quality of sub-images restoration together with sub-
image boundary lines restoration.
In [1], we proposed the more general form of the stabilizer I β :
Z
I = β(x, y)[u(x, y) − uk (x, y)]2 dxdy,
β
(4)
Q
we used
β(x, y) = umax , for (x, y) ∈ Q,
(5)
β(x, y) = 0, for (x, y) ∈ Π/Q, where
umax = max{utrue (x, y) , (x, y) ∈ Π = [0, 1] × [0, 1]}, (6)
(x, y) ∈ Q ⇔ utrue (x, y) = umax . (7)
So, in [1] our regularization algorithm stabilizer includes the coefficient β,
which uses some information about true image segmentation, see (6) and (7).
This facts motivates us to exclude true image parameters. We approach this
purpose using image segmentation ideas.
3.2 Image Segmentation as Tool
In this paper we propose to begin calculations with the parameter β ≡ 0.
So, in this case, we use standard regularization method:
Z
min ||Ah u − fδ ||2L2 + α (||u||2L2 + J(u)) : u ∈ U , J(u) = |∇u| dx, (8)
D
where ∇u denotes the gradient of smooth function u, J(u) is the total
variation of the function u on D.
After obtaining the solution of problem (8), we get approximate ũ0 (x, y) , Q0 , ΓQ0 ,
and we use them for the βi,j construction:
βi,j ∈ {o; c}, c = max{ũ0 (x, y), (x, y) ∈ Q0 ∪ ΓQ0 }. (9)
168
3.3 Image Segmentation as Result
Now, using β calculated in (9), the algorithm computes the final image restora-
tion including its final image segmentation.
Numerical experiments have confirmed our ideas. Thus, we see in Figure 1:
(a) the observed image; (b) the restoration image;
(c) the central image segment restoration, Q0 ∪ ΓQ0 , calculated with β ≡ 0;
(b) the central image segment restoration, Q̃∪ Γ̃Q̃ , calculated with adapted β.
a) The observed image b) The recovered image
c) The model segment restoration d) The model segment restoration
for β(x, y) ≡ 0 for the adapted stabilizer coeffi-
cient β(x, y)
Fig. 1. Model experiments
It is clear, that we can write the final image segmentation of the recovered
image in the form:
Π = {Π/Q̃} ∪ Q̃, (10)
were Q̃ ∪ Γ̃Q̃ we see in Figure 1.(d), Π = [0, 1] × [0, 1].
169
3.4 Conclusion
In this paper, we have described and verified the image segmentation approach
for the construction of stabilizer coefficient β.
Starting with the observed object intensity and setting the stabilizer coef-
ficient β ≡ 0 in the regularization algorithm, we successfully apply the initial
approach for both the sub-image and the sub-image boundary, Q0 , ΓQ0 .
After that, we use calculated results for construction of the adaptive stabilizer
coefficient β. At this step the image segmentation is a tool.
Using the calculated coefficient β, the algorithm computes the final recovered
image. We obtain a good localization of segments, Q̃, ΓQ̃ , and we see a more
deblurred final image segmentation.
As a result, in this paper, we have described and verified that our adaptive
algorithm does not need a priori information. For successful image restoration
and restoration of image segmentation, our adaptive algorithm can utilize the
image of an observed object only.
We plan the following modifications and experiments:
– inclusion of the solution estimate for equation u(x, y, ΓQ (x, y)) = c to the
estimate list;
– validation of β k inclusion in (4), where k is the iteration number;
– processing of thin lines and narrow segments.
Acknowledgments The author expresses special gratitude to Prof. V.V.Vasin
from the Institute Mathematics and Mechanics of Ural Branch of RAS. The
author would like to thank the colleagues from AIST Program and Organizing
Committees for their helpful advice and guidance in the paper preparations.
This work was supported by the Program of UB RAS “Mathematical models,
algorithms, high-performance computational and information technologies and
applications”, project 15-7-1-13.
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