ARTICLE INFO


ABSTRACT

Original
Research Article
Received:
7 October 2021
Accepted:
24 October 2021
KEYWORDS
PDE,
Image
denoising,
Fractional
derivative,
EulerLagrange
equation,
Grunwald
approximation,


This
paper generalizes the computational applicability of noninteger ordered
derivative based PDE (Partial Differential Equation) in image restoration
process. The fractional derivative of RiemannLiouville form gives the
several choices of fractional order in image denoising. The fractional form
of Euler Lagrange equation is the noninteger ordered generalization of
gradient type equations which can be used in minimization problems. The
Grunwald approximation discretizes the fractional PDE used in image
restoration, which shows some satisfactory results.

Introduction
The image processing
using partial differential equation (PDE) has given a new direction in image
restoration such as denoising and enhancement [18]. Image restoration is often
inevitable as a preprocessing of images especially for image segmentation and image
compression. The computational applicability of appropriate numerical
methodologies for the PDE models is a salient component of PDEbased image
restoration. In image restoration, the integer ordered derivative is used for
minimization of the total variation in the image. Many image restoration
methodologies use the minimization of Lagrangian [9].
In this paper we have
used the noninteger ordered derivative in image restoration. The noninteger
order (fractional) derivative was theorized long back in the correspondence of
Leibniz with L’Hospital in 1695 [10]. The fractional derivative was developed
as theoretical field of mathematics [1114]. Research studies in scaling
phenomena [1517] and in classical mechanics [1821] fractional calculus shows
many interesting applications. The use of noninteger ordered derivative in
diffusion process brings the dynamic nature of the nonlinearity of the image
filter. The use of fractional derivative in image restoration shows some
encouraging results. The chaotic dynamics of the image processing can be
optimized by the introduction of fractional Lagrangian to replace the
noninteger order of the fractional derivative.
Section II of this paper
gives the preliminaries idea of image restoration using Euler Lagrange (EL)
equation. Fractional derivative and fractional EL equation have been discussed
in section III, while Section IV deals with the concept of Grunwald
approximation for discretization of fractional EL equation. In section V, the
experimental results are shown and discussion is carried out. Finally, the conclusion
is drawn in section VI.
II. Preliminaries
For a given image
, the corrupted observed image,
can be viewed as
, (1)
where k is a blurring operator and e can be represented as the additive
Gaussian noise. The denoising of the image can be theorized as the
minimization of a functional of gradient, represented as
, where
(2)
Here
, can be defined as Lagrange constant and
represents the
gradient operator defined as
.
represents the
distance of norm 2. If p=1, the first term in
is described as the total variation. The minimization problem
(2) can be transformed into the differential form called the EulerLagrange
(EL) equation, which is the minimization problem in 2D of the form
(3)
The EL equation states
(4)
where
i.e.
(5)
When the EL equation (4)
is applied to equation (2), we get
(6)
Iterating the descent
direction by the time step t, the
resulting evolutionary EL equation can be represented as
(7)
where
is the constrained
parameter.
III. Denoising Using Fractional Derivative
The fractional framework
in partial differential equation has been used in a wide variety of
computational problems like turbulence, chaotic dynamics, quantization etc.
Riemann fractional derivative [11] is defined as
(8)
where equation is defined
as
ordered left Riemann
fractional derivative. Similarly, the
ordered right Riemann
fractional derivative is defined as,
(9)
Here n is an integer such that
and
is the gamma function
defined as [12],
(10)
In (3), if we consider L = 0, the fractional derivative is
called the RiemannLiouville form, and for L =
it
is the Liouville definition of fractional derivative [11,12]. Other definition
of fractional derivative also exists. In this paper RiemannLiouville form is
considered without loss of generality. It is easy to see that these derivatives
are nonlocal operator of convolution type.
The classical functional
gradient
is defined as
(11)
which brings the essence
of the x and y directional derivative. Now the fractional directional derivative
operator can be viewed as
, which can be defined as,
(12)
where
.Now if we define the Lagrangian fractional functional as
(13)
then we can state the following lemma,
Lemma
1:
Let L be an admissible Lagrangian
function and
, defined as (13), for
, be the associated fractional functional and differentiable
for all
, then for all h,
the differential is given by
, where
.
Proof:
Since
is differentiable for
,
for
. (14)
Again, RiemannLiouville
derivative is linear operator [11,12], which implies that,
(15)
As a consequence we
obtain
(16)
Using Taylor expansion,
(17)
For two functions
we can state that
[12],
(18)
(19)
Using (17) and (19)
which proves the above Lemma.
Now let define the
fractional divergence operator as
, and we can state the following corollary,
Corollary:
If the minimization of fractional 2D problem be
(20)
The fractional EL
equation states that,
(21)
Proof: Using the Lemma 1
and classical Du Bois Reymond Lemma,
obtain the result (21).
Using the concept of
fractional Lagrangian the denoising technique can be represented as
,
(22)
and when fractional Euler
Lagrangian (EL) equation is applied to (22), we get
(23)
Iterating the descent
direction by the time step t, we get,
(23)
where
is the constrained
parameter.
Since the numerator
can not be zero in
computation, the term can be regularized as
(24)
where
is very small.
IV. Numerical Approximation
The continuous domain
representation of the fractional PDE requires discretization for computational
feasibility. Right shifted Grunwald approximation can be used to discretize the
equation (6) similar to finite difference approximation . The right shifted
Grunwald formula can be represented as,
(25)
In image processing,
is bounded and
, hence equation (25) can be approximated as
(26)
where
and
are integers. Here
, where
is the size of image.
The two dimensional fractional PDE (6) used in processing of MRI can be
discretized using equation as follows,
(27)
V. Results
For numerical
experiments, we choose grey scale images. The image is perturbed with random
valued additive Gaussian noise (zero mean and have a certain variance). The
restoration procedure has been implemented Matlab R14. As shown in figure 1, we
present the experimental results utilizing the grayscale Zelda image in
pixels. The original image is shown in 1(a) which is perturbed with Gaussian noise of
zero mean and 0.01 variance, as shown in, figure 1(b). The perturbed image is
denoised by discretized fractional EL equation considering
,
and
. In this example, the denoising is carried out for several
choices of fractional order of the derivative after 20 iterations. The strong
relationship between the PSNR of the denoised image and the fractional ordered
shows that there
exists some maximum for which the PSNR of the denoised image is maximum in this
process. In 1(c) the output image is for the fractional order
, which is the denoised image derived from classical Euler
Lagrange equation. For this image the PSNR is 21.13 dB. In 1(d) the image is
denoised for the fractional order
, for which the PSNR is 32.60 dB, which is maximum PSNR
throughout the process, while the PSNR of the noisy image 1(b) is 11.96 dB.
(a)
(b)
(c)
(d)
Fig. 1. (a) Original Zelda
image (
pixels) (b) Perturbed image with Gaussian noise of zero mean
and 0.01 variance (c) restored image for fractiona order
and (d) restored image
with fractional order
In figure 2, the curves
show the relationship between the PSNR and the fractional order
for different iterations (10 iterations and 20 iterations).
The curve shows that there exists some point where the PSNR of the denoised
image is maximum. When the fractional order
is about 1, i.e.
, the equation (23) can be viewed as diffusion process [1],
while for
, the same equation (23) can be defined as super diffusive
process. For
, the equation (23) follows the wave equation.
Fig. 2: The PSNR (dB) vs
fractional order for different iterations
VI. Conclusion
The noninteger
ordered partial differential equation (PDE) is used in this paper for the
restoration of noisy image. The variation minimization problem is solved
through the introduction of fractional derivative in classical Euler Lagrange
equation. The noninteger ordered Euler Lagrange equation gives the scope of
choices for several fractional ordered used in fractional derivative. The
fractional variation is used to define the fractional generalization of
gradient type equation for denoising the image. The Grunwald approximation is
used to discretize the fractional EulerLagrange equation and the numerical
scheme is tested using different images which show some satisfactory results.
The choice of fractional order affects the denoising effect in the