A Mathematical Algorithm for Improving the Medical Image
ANANTACHAI PADCHAROEN, DUANGKAMON KITKUAN
Department of Mathematics, Faculty of Science and Technology,
Rambhai Barni Rajabhat University,
Chanthaburi 22000,
THAILAND
*Corresponding author
Abstract: -In this paper, we present a hybrid model based on total generalized variation (TGV) and shearlet with
non-quadratic fidelity data terms for blurred images corrupted by impulsive and Poisson noises. Numerical ex-
periments demonstrate that the proposed can reduce the staircase effect while preserving edges and outperform
classical TV-based models in the peak signal-to-noise ratio (PSNR).
Key-Words: Total generalized variation (TGV), image restoration, Mathematical algorithm, Medical
Image, non-quadratic fidelity
Received: August 19, 2023. Revised: February 6, 2024. Accepted: March 3, 2024. Published: April 25, 2024.
1 Introduction
Image restoration is an inverse problem where the
objective is to recover a sharp image from a blurry
and noisy observation. Mathematically, a linear shift-
invariant imaging of the system is modeled as
(1)
where is the observed image, is the unknown im-
age, matrix is a linear transformation representing
convolution operation and is the noise. The goal of
image restoration is to recover from .
The structured matrix has many singular values
of different orders of magnitude close to the origin.
In particular, is severely ill-conditioned and may
be singular. This makes the solution of (1) be very
sensitive to the noise in the right-hand side In gen-
eral, a regularization method can be employed to com-
pute the approximate solutions that are less sensitive
to noise than the naive solution. Probably one of the
most popular regularization methods is Tikhonov reg-
ularization, [1], which replaces (1) by the minimiza-
tion problem
(2)
where denotes the Euclidean norm and is the
regularization parameter that controls the balance be-
tween the two terms for minimization.
In fact, Tikhonov regularization estimate is similar
to low pass filtering, therefore, it produces a smooth-
ing effect on the restored image, i.e., it penalizes
edges, which is not a good approximation of the origi-
nal image if it contains edges. To overcome this short-
coming, [2], proposed a total variation (TV) based
regularization technique, which preserved the edge
information in the restored image. In the case of TV
regularization, the estimated solution is obtained by
minimizing the objective function (the ROF model)
(3)
and, [3], studied -TV denoising model:
(4)
where and is the discrete TV
regularization term. Several efforts have been made
to improve the TV output, [4], [5], [6], [7], [8], [9].
2 Second order TGV (TGV) and
algorithm
Since optimization solution with total variational fil-
ter is very effective for preserving sharp edges, cor-
ners and other fine details of an image. However,
it also has some disadvantages, most notably the so-
called staircase effect, which is the unwanted occur-
rence of edges.
Recently, [10], developed the total generalized
variation (TGV) regularizer, which is assumed to be
the generalization of the total variational filter. To-
tal generalized variation consists and balances higher-
order derivatives of We are introducing some prop-
erties of the second order TGV which is given by
div
div (5)
where is the set of all symmetric matrices and
denotes the space of compactly sup-
ported. The divergences div
and
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
192
Volume 21, 2024
div
are defined by
div
and div
In order to simplify the computation, we give the
discretization of
Firstly, we denote
and
According to [27], [29], the discretized
is ap-
proximatively rewritten as the following minimiza-
tion:
(6)
where
and represent two first-order forward finite differ-
ence operators along the horizontal and vertical direc-
tions, respectively. Here, the operations
and are written as
and
3 Shearlet Transform
The shearlet transformis a very effective tool for tack-
ling the piecewise smooth images containing corners,
edges and spikes etc. The shearlets transform can
completely approximate the piecewise smooth im-
ages’ singular structures. Such property of shear-
lets is suitable particularly in image processing task
since irregular structures and singularities carry im-
portant details in an observed image. We can propose
our high order deblurring model for impulsive noise
which is denoted by
(7)
where is the th subband of the shear-
let transform of For numerical computation, we
adopt the fast finite shearlet transform (FFST), [11], in
which the construction is based on the Meyer scaling
and wavelet functions. Moreover, all the band wise
discrete shearlet transforms can be computed fastly
using the fast Fourier transform(FFT) and the dis-
crete inverse Fourier transform(IFT). For notational
simplicity, we use to interchange-
ably represent the continuous and the discrete shearlet
transform of continuous and discrete respectively.
Let be the FFT of the discrete 2D scaling function,
and be those of the discrete shearlets. Let
and be
the vectorizing and the matricizing operators. Then
we have
.
where and .denote convolution and componen-
twise multiplication. The above equation in vector
form is given by
diag
where diag, and diag is de-
fined as
diag and diag
where if and
We begin with a short review of ADMM, which
solves the model in the form of
subject to (8)
The Lagrangian is
, where is the scaled Lagrange multiplier
and is a positive parameter. The ADMM algorithm,
[12], [13], starts from and and iterates
(9)
We introduce one auxiliary variable and one quadratic
penalty term for each term. More specifically, we
introduce auxiliary variables
and
such that (7) is equivalent to
subject to
(10)
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
193
Volume 21, 2024
After applying the ADMM, we arrive at the following
algorithm:
(11)
The first four sub-problems are solved by shrink-
age explicitly. The -subproblem and -subproblem
can be solved by
shrink
shrink
(12)
where shrinksgn.
Since the -subproblem is componentwise separa-
ble, the solution to the -subproblem reads as
shrink
(13)
where the component of located at is
denoted by and the shrinkage operator
shrinkcan be formulated as follows
shrink
if 0
if 0
(14)
Similarly, solution for the -subproblem is formulated
as
shrink
(15)
where is the component of cor-
responding to the pixel and
shrink
if 0
if 0
(16)
where 0is a square null matrix and the Frobenius
norm of a matrix is denoted by
To solve the -subproblem, we obtain the
first-order necessary conditions for optimality as fol-
lows:
(17)
After grouping the like terms in (17), we obtain the
following linear system
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
194
Volume 21, 2024
where the block matrices are defined as
(18)
and
(19)
Next we multiply a preconditioner matrix from the
left to the linear system such that the coefficient ma-
trix is blockwise diagonal
This operation can also be equivalently performed
by multiplying each equation in (18) from the left
with By denoting )
diagand
diag
conjdiagwe have
.
.
.
.
.
.
.
.
.
(20)
Similarly to the scalar case, and can
be obtained by applying Cramer’s rule. Hence
and have the following closed forms
.
.
.
(21)
where the division is componentwise. Here is
defined to be
.. ..
.. ..
.. ..
where .is componentwise multiplication and
In conclusion, summing up the statements above,
this yields the resulting alternating minimization
method generalized in Algorithm 1.
Algorithm 1 TGV-ADMM.
Input: - linear transformation representing convo-
lution operation.
- observed image.
Choose:
Initialize:
For do the following computations
is determined by (12)
is determined by (12)
is determined by (13)
is determined by (15)
are determined by (21)
If the halting criteria are satisfied, it returns and
stops.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
195
Volume 21, 2024
4 Convergence analysis
The convergence follows directly from that of the
classic ADMM because the problem is convex and
the variables can be grouped into two
blocks and For fixed values of
, the updates of are independent of
one another. Because of this, the above iteration is
a direct application of ADMM.
Theorem 4.1. For and
then ADMM (11) converges.
Proof. By letting
in (8), and the
Lagrangian function be of the form
the convergence analysis of the classical results from
current ADMM, [10], [12], [13], [14], yields the fol-
lowing result.
5 Numerical experiments
In this section, we show some numerical examples us-
ing the Algorithm 1 in image restoration compared
with TV-ADMM and ADMM.
Our hybrid model is implemented via the alternat-
ing minimization method with the equivalent param-
eters and
Moreover, the regularization coefficients are firmly
chosen as and
The Peak signal to noise ratio (PSNR) in decibel (dB)
as follows:
where is an original image and is an estimated
image at iteration respectively.
The stopping criteria of the algorithm is
All codes were written
in Matlab 2017b and run on Dell i-5 Core laptop.
Experimental results are shown in Figure 1, Figure 2,
Figure 3 and Figure 4 in the Appendix.
6 Conclusions
We propose an algorithm to perform second order to-
tal generalized variation using shearlet regularization.
We compared its results with those of TV-ADMM and
ADMM on four images. On the contrary, the TGV-
ADMM had a better than two methods .
Acknowledgments
This work was supported by (i) Rambhai Barni Ra-
jabhat University (RBRU), (ii) Thailand Science Re-
search and Innovation (TSRI), and (iii) National
Science, Research and Innovation Fund (NSRF)
4369065 (Grant number 1121/2566).
References:
[1] A. Tikhonov and V. Arsenin, Solution of ill-
Posed Problems, Winston, Washington, DC,
1977.
[2] L. Rudin, S. Osher and E. Fatemi, Nonlin-
ear total variation based noise removal algo-
rithms, Physica D: Nonlinear Phenomena, Vol.
60, No.1-4, 1992, pp. 259-268.
[3] J. E. Aujol, G. Gilboa, T. Chan and S. Os-
her, Structure-texture image decomposition-
modeling, algorithms, and parameter selection,
Int. J. Comput. Vision, Vol.67, 2006, pp. 111-
136.
[4] F. Knoll, K. Bredies, T. Pock and R. Stoll-
berger , Second order total generalized vari-
ation (TGV) for MRI, Magn. Reson. Med.,
Vol.65, No.2, 2011, pp. 480-491 .
[5] W. Guo, J. Qin and W. Yin, A new detail-
preserving regularization scheme, SIAM J.
Imaging Sci., Vol.7, No.2, 2014, pp. 1309-1334
.
[6] T. Jia, Y. Shi, Y. Zhu and L. Wang, An image
restoration model combining mixed fi-
delity terms, J. Vis. Commun. Image R., Vol.38,
2016, pp. 461-473.
[7] X. Liu, Augmented Lagrangian method for to-
tal generalized variation based Poissonian im-
age restoration, Comput. Math. Appl., Vol.71,
No.8, 2016, pp. 1694-1705.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
196
Volume 21, 2024
[8] W. Wang and Q. Song, Color image restoration
based on saturation-value total variation plus
L1 fidelity, Inverse Problems, Vol.38, No.8,
2022, pp. 085009.
[9] D. V. Thong, S. Reich, P. Cholamjiak and L. D.
Long, Iterative methods for solving monotone
variational inclusions without prior knowledge
of the Lipschitz constant of the single-valued
operator, Numerical Algorithms, 2024.
[10] K. Bredies , K. Kunisch and T. Pock, Total gen-
eralized variation, SIAM J. Imaging Sci., Vol.3,
No.3, 2010, pp. 492-526.
[11] S. H user, Fast Finite Shearlet Transform,
preprint, arXiv:1202.1773, 2012.
[12] D. Gabay and B. Mercier, A dual algorithm
for the solution of nonlinear variational prob-
lems via finite element approximations, Com-
put. Math. Appl., Vol.2, No.1, 1976, pp. 17-40.
[13] R. Glowinski and A. Marroco, Sur
l’approximation, par éléments finis et la
résolution par pénalisation-dualité d’une
classe de problémes de Dirichlet non linéaires,
Revue française d’automatique, informatique,
recherche opérationnelle. Analyse numérique,
Vol. 9, No. R2, 1975, pp. 41-76.
[14] C. Wu and X. C. Tai, Augmented Lagrangian
method, dual methods, and split Bregman it-
eration for ROF, vectorial TV, and high or-
der models, SIAM J. Imaging Sci., Vol.3, No.3,
2010, pp. 300-339.
Authors’ contributions
These authors contributed equally to this work.
[Anantachai Padcharoen (anantachai.p@rbru.ac.th)&
Duangkamon Kitkuan (duangkamon.k@rbru.ac.th)]
Conflicts of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
197
Volume 21, 2024
APPENDIX
(a) Original image
(b) ADMM (PSNR=25.6174)
(c) TV-ADMM (PSNR=25.8667)
(d) Algorithm 1 (PSNR=26.3065)
Figure 1: Figure (a) shows the Original image, figure
(b) shows the restored image by ADMM, figure (c)
shows the restored image by TV-ADMM, figure (d)
shows the restored image by Algorithm 1.
(a) Original image
(b) ADMM (PSNR=26.4018)
(c) TV-ADMM (PSNR=26.6161)
(d) Algorithm 1 (PSNR=29.4125)
Figure 2: Figure (a) shows the Original image, figure
(b) shows the restored image by ADMM, figure (c)
shows the restored image by TV-ADMM, figure (d)
shows the restored image by Algorithm 1.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
198
Volume 21, 2024
(a) Original image
(b) ADMM (PSNR=21.2982)
(c) TV-ADMM (PSNR=22.1277)
(d) Algorithm 1 (PSNR=23.4826)
Figure 3: Figure (a) shows the original image, figure
(b) shows the restored image by ADMM, figure (c)
shows the restored image by TV-ADMM, figure (d)
shows the restored image by Algorithm 1.
(a) Original image
(b) ADMM (PSNR=23.0395)
(c) TV-ADMM (PSNR=23.7816)
(d) Algorithm 1 (PSNR=25.6356)
Figure 4: Figure (a) shows the original image, figure
(c) shows the restored image by TV-ADMM, figure
(d) shows the restored image by Algorithm 1.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.20
Anantachai Padcharoen, Duangkamon Kitkuan
E-ISSN: 2224-2902
199
Volume 21, 2024
