VFAST Transactions on Mathematics
VFAST Transactions on 2021,
Mathematics
http://vfast.org/journals/index.php/VTM@
ISSN(e): 2309-0022, ISSN(p): 2411-6343
VFAST
Transactions
on
Mathematics
Volume 10, Number 1, January-June, 2022
pp: 01-13
Image Driven Isotropic Diffusivity
and Complementary
Regularization Approach for
Image Denoising Problem
Memoona Pirzada1* , Khuda Bux Amur1,2†‡ , Muzaffar Bashir Arain1,3†ğ , Rajab Ali
Malookani1,4†¶
1*,2,3,4
Department of Mathematics and Statistics, Quaid-e-Awam, University of
Engineering, Science and Technology, Nawabshah, Pakistan
Keywords: Image
Denoising, Optimization,
Partial Differential
Equation, Diffusion,
Regularization. Subject
Classification: Applied
Mathematics,
Mathematical Image
Processing.
Journal Info:
Submitted:
April 15, 2022
Accepted:
June 16, 2022
Published:
June 20, 2022
Abstract
We present the idea of image driven isotropic diffusivity along
with complementary regularization for image denoising problem. The method
is based on the optimization of a quadratic function in L2 norm. The minimization of the energy functional leads to the Partial Differential equation (PDE)-based
problem. We are looking for a steady state solution of equivalent time dependent problem. We discretize the problem with standard finite differences. The
steady-state numerical solution of the time dependent problem leads to the iterative procedure, which allow to compute a regularized version of the solution
as a denoised image. We have applied our designed model on synthetic as well
as real images. The numerous experiments have been conducted to analyse the
performance of the method for the different choices of scaling parameters. From
the quality of the obtained results and comparative study it is observed that the
proposed model performs well as compared to well existing methods.
*Correspondence Author Email Address:
umememoona35@gmail.com
This work is licensed under a Creative Commons Attribution 3.0 License.
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VFAST Transactions on Mathematics
1 Introduction
The Noise tells the unwanted information in digital images. Noise is the major issue while transferring
the images through all kinds of electronic communication. The images are often corrupted by the noise
during their acquisition and transmission [6]. The purpose of image denoising is to remove the noise from
the image. As it is not possible to completely remove the noise from image due to its random nature, but
the goal is to suppress the noise in the images as much as possible to maintain the main features of the
original image. Keeping in view the random nature of the noise in the image the denoising is a challenging
problem for the researchers because the restoration process also causes the blurring effects in images
and many other uncontrollable factors like illumination inconsistency, that affect the output results
[16]. To avoid such brightness variations in the images, the problem is still needs to be addresses with
novel strategies. To cope with such issues the Variational approach in imaging technologies have been
considered as one of the modern approaches where one must minimize an energy functional consisting
of Regularization and Data terms. The regularization term plays the role of filling-in effect and data term
reduces the deviations in original and noisy image. The optimization of such energy functional leads to
PDE’s (Partial Differential Equations) based image processing problems [3]. In this work the novel idea of
regulation has been proposed in the spirit of the complementary data regularization for the optical flow
computations [35] and choice of smoothing term in the spirt of disparity driven diffusivity approach for
stereo vision problem [34].
2 Related works
As a mathematical perspective, the image denoising problems are inverse in nature, therefore their solution as the removal of the noise from images is a highly demanded area of the research [2, 5, 6, 8, 9,
11, 16, 17, 22, 24, 25, 27, 29, 31]. As the unique solution of the denoising problems is not always possible
therefore the image practitioners are interested for new approaches to improve the denoising techniques
along with the suitable choice of mathematical models [1–4, 12–14, 18, 19, 21, 26, 28, 34, 35]. The authors
[7, 8, 17] have presented the various noise types, noise models and the classification of image denoising
techniques. The Variational approach in imaging technologies is one of the novel approaches, and variety
of methods have been proposed in the literature [12, 14, 18, 29]. Recently a novel approach [30], known
as flexible stroller regularization method which is based on the combination of fidelity term and STROLLR
regularizer. The STROLLR regularization has been claimed as better regularization agent for natural images. Author mainly considered image denoising, inpainting, as imaging applications. Another approach
was proposed in the literature as structure-constrained low rank approximation (SLR) method based on
wiener filtering regularization for image denoising [33]. The elliptic partial differential equations-based
approach was appeared in [32] where the performance of the optimization problem was investigated for
the appropriate choice of the regularization parameters.
According to the available literature, the modeling of the effective regularizers play the crucial part in the
image denoising and other ill-posed imaging optimization problems. Typically, such optimization problems lead to Partial differential Equations moreover these solution strategies have been considered as
modern approaches in image processing problems [19, 28].
In the view of the above discussion, we focus on Partial Differential Equation’s (PDE’s) based imaging approach for denoising problem. The variational approach for the denoising problem leads to optimize the
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VFAST Transactions on Mathematics
following energy functional.
E(u) =
Z
M(u, f ) +
| {z }
Data term
dxdy
V(∇u)
| {z }
(1)
Regularization term
The first term in E is called the data term measures the fidelity to the data f and reduces/minimizes
the oscillations in the noisy image. The second term is called the regularization term which performs the
smoothing effects [27].
Variational models have been proposed as fast and stable methods with high accuracy results [2, 9]. However, there is still a challenging task to maintain the good balance between the choice of regularization
term and data term.
It has been observed in the literature that these terms may contradict each other as still no one is confident
to say that which one of both terms contributes as major denoising agent during the denoising process.
To overcome such difficulty the Idea of complementary approach was successfully applied on optical flow
problem and was suggested to be applied of other imaging applications [35] which were further tested on
denoising problem [2]. In this work we propose the novel idea of modeling with the combined the complementary approach of [35] with the idea of the choice of diffusion term as the disparity driven diffusivity
approach for disparity map computations [34].
3 Proposed Denoising Method
The denoising is defined as the computation of the image u : Ω ⊂ R2 −→ R as diffused image from a
noisy image f (x, y) with additive noise η(x, y) as:
f (x, y) = g(x, y) + η(x, y)
(2)
here g(x, y) is original image. As it is not possible to directly remove the noise from the noisy image f (x, y)
therefore we propose in this work the energy optimization method. Keeping the good balance between the
regularization and data terms we combine the idea of the image driven diffusivity approach for disparity
map computations(stereo vision problem)[34], and the idea of complementary approach for optical flow
problem as given in [35]. The proposed energy minimization problem in L2 norm is given as (3)
E(u) = q
1
2
2
β
2
λ + |▽f |
(u – f )
L2 (Ω)
+ q
1
2
2
α
2
λ + |▽g|
|▽u|
(3)
L2 (Ω)
The terms α, β are the strictly positive smoothing parameters, λ is the small positive parameter used
to avoid the division by zero. These scaling parameters control the level of smoothness in regularized
solution of the ill-posed problem. ∥.∥2L(Ω) is a L2 (Ω) norm on image domain. In the integral form the above
equation can be written as
Z Z
Z Z
α
β
2
q
q
(4)
E[u(x, y)] =
|∇u|2 dxdy
|(u – f )| dxdy +
2
Ω
Ω λ + |∇f |2
λ + |∇g|
We recall the following direct result from the calculus of variation [18] to obtained from the minimum
of the above energy functional
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VFAST Transactions on Mathematics
Theorem 1. If a functional v[y(x)] having a variation achieves a maximum or a minimum at y = y0 (x), where
y(x) an interior point of the domain of the defination of the functional, then at y = y0 (x),
δv = 0
(5)
Applying the above direct method of calculus of variation, the variational optimization of the model (4)
yields the following Euler’s Lagrange equation;
α∇
β
(u)
=0
q
(u – f ) – div q
(6)
2
2
λ + |▽g|
λ + |▽f |
∂u
Where we consider the black boundary conditions for an image ∂u
∂x = 0, ∂y = 0 To solve the problem (6)
one is therefore interested to find the steady state solution of the following time-dependent problem:
α∇
∂u
β
(u)
(u – f ) – div q
=q
(7)
∂t
2
λ + |▽g|2
λ + |▽f |
With the boundary conditions for an image
4 Problem Discretization
Suppose tha Ω is the image domain, we discretize the image domain as Ωh as set of finite grid points
(pixels). Such a grid leads to the numerical solution of the given problem as uh : Ωh → R. We introduce
the following discrete solution space as
Uh = {uh : Ωh → R}
The discrete version {uh for the solution u(x, y) for the second order PDE. in this work is denoted as
ui,j = u(xi , yj )
We define the following standard discrete approximations for the partial derivatives. The uniform grid size
has been considered as h > 0 .
(ui+1,j – ui–1,j )
∂u
≈
∂x
2(△x)
(ui,j+1 – ui,j–1 )
∂u
≈
∂y
2(△y)
!
(ui+1,j – 2ui,j + ui–1,j )
∂2u
≈
∂x2
(△x)2
!
(ui,j+1 – 2ui,j + ui,j–1 )
∂2u
≈
2
2
∂y
△y
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VFAST Transactions on Mathematics
By substituting the above discrete operators in the problem (7) and after simplification, we obtained the
final numerical scheme in space-time domain
uki+1,j + uki–1,j – 4uki,j + uki,j+1 + uki,j–1
uki,j – fi,j
k+1
k
q
ui,j = ui,j + ∆t[α
]
–βq
λ + |∇g|2
λ + |∇f |2
(8)
we consider the following discrete reflexive boundary conditions u(i+1,j) = u(i–1,j) , u(i,j+1) = u(i,j–1) For 0 ≤ i ≤ P
and 0 ≤ j ≤ Q, where P and Q are number of rows and columns respectively and k is the number of
diffusion iterations. The Explicit Finite Difference Scheme(8) is simple but conditionally stable, therefore
the care has been taken in the choice of sensitive parameters effecting the stability criteria during the
computational process. Furthermore, the grid size is uniform, otherwise, serious numerical instability
problems may occur during computations. The above discrete model (8) has been implemented on three
benchmark images in this work. We prepare all the simulations in MATLAB.
5 Confidence Measures
To evaluate the quality of our mathematical model and compare the findings with well existing methods
some confidence measures from the existing literature [15, 29], have been proposed in this work. We
mainly focus in this work on the confidence measures like Peak-Signal-to-Noise-Ratio (PSNR), Structural
Similarity Index (SSIM) and Mean Square Error (MSE).
The Mean Square Error (MSE) is the average squared error between the filtered image and the original
image. Mathematically it is defined as
MSE =
1 X
[g(i, j) – u(i, j)]2
M∗N
(9)
Where M and N is the width and the height of the images, u(i, j) denote the diffused image and the original
image g(i, j). The i and j are the rows and columns of pixels of both the original and enhanced images. The
Peak-Signal-to-Noise-Ratio (PSNR) is the ratio between the maximum possible power of a signal and the
power of corrupting noise that affects the fidelity of its representation and is defined as
PSNR = 10log10
R2
MSE
(10)
Where R is the signal part and MSE (Mean Square Error) is the noise part. In our case the image is considered as a signal and an 8-bit image has a maximum intensity of 255 and so that is taken as the R value.
The smaller values of the PSNR worsen the quality of image, and the higher values of PSNR lead to good
quality images. The Structural Similarity Index (SSIM) is perceptual metric that quantifies the image
quality degradation caused by image processing like data compression or by losses in data transmission.
Structural Similarity Index (SSIM) actually measures the perceptual difference between two similar images.
SSIM is calculated by the following formula.
SSIM =
(2µu µf + c1 )(2σu f + c2 )
(µ2u + µ2f + c1 )(σu2 + σf2 + c2 )
(11)
Where µu , µf are the means and σu , σf are the variance of the signal respectively and σuf is the covariance
between u and f , and c1 , c2 are the constant values used to avoid instability. Smaller values reduce the
visual quality of the image.
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VFAST Transactions on Mathematics
6 Numerical Experiments
The numerical experiments have been performed on three different benchmark images as data sets . All
the computations have been performed in MATLAB. We consider the data sets as the "Lena Image”(1a,
“Peppers”2a and “Synthetic” 3a). The data images were downloaded from the [20].
The images have been degraded with Gaussian noise with a standard deviation σ = 30. The image
diffusion results from the conducted numerical experiments have been shown in the figures (1),(2),(3).
Further more the obtained quantitative results as numerical values including the error profiles, PSNR,
MSE and SSIM with different choices of scaling parameter α, β are placed in the tables (1),(2),(3).
The comparative study is the important part of this work therefore the comparison of the numerical
results have been shown in the tables (4),(5) where the results have been compared with other well
known denoising methods like, Charbonnier, Perona-Malik and Total variation models [10, 20, 23, 24, 29].
Furthermore comparison have been shown through interesting plots for for structural similarity index,
Peak-Signal-to-Noise-Ratio (PSNR) of images restored from different methods including the our method.
7 Results and Discussion
A novel image diffusion model based on the idea of the combined Image Driven anisotropic Diffusivity [35]
and the complementary regularization approach [1, 2] have been successfully implemented in this paper.
The Numerical experiments have been conducted on three benchmark images i.e. Lena image, Peppers
image and Synthetic images as discussed in previous section.
It is observed from the numerical experiments that the optimal choice of scaling regularization parameters
like α, and β reveal the significant effects on the quality of recovered images, keeping in view such smoothing effects the numerous experiments have been conducted with different choice of theses regularization
parameters as shown in the tables (1),(2) and (3) and figures (1),(2) and (3). It is observed from the simulation results that most suitable choice of the parameters α and β in this computational framework is 1.0
and β = 0.066. This choice is very interesting aspect of modeling the diffusion term using the data driven
approach in the sense that at this stage no further scaling is needed for Smoothing term. Furthermore
simulation results shows that the proposed method removes noise more efficiently and enhanced image
meaningfully see the figures (1),(2) and (3).
To evaluate the performance of the proposed model we compared the results of the proposed model with
some classical denoising models such as TV model in [24], Perona-Malik model in [23] and Charbonnier
model in [10]. The comparative study is given in the tables (4) and (5). The results show that the proposed
model is a novel idea of regularization and performs better than the well-rated existing methods.
Further the better performance of the method is also assured from the simulation plots (4), (5), and (6)
where the red, green, black and blue indicate the Charbonnier method, Perona-Malik, Total variation and
our method respectively which shows that blue line shows the peak values in the PSNR, SSIM and lowest
value in MSE as compared to other models. For more better visual quality image profiles after performing
denoising with proposed method and well existed methods is presented in figure (7). In table (1),(2),(3), we
calculate the results up to 350 iterations of our model by taking the different values of the parameters α
and β in the range 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1 but at the value α = 1 and β = 0.066 we obtain the outperforms
results as shown in tables (4) and (5).
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VFAST Transactions on Mathematics
(a) Original
(b) Noisy Image
(c) α = 0.1, β = 0.09
(d) α = 0.5, β = 0.09
(e) α = 0.9, β = 0.01
(f) α = 1, β = 0.06
Figure 1. Diffusion at different values of α and β
(a) Original
(b) Noisy Image
(c) α = 0.1, β = 0.01
(d) α = 0.1, β = 0.09
(e) α = 0.9, β = 0.09
(f) α = 1, β = 0.06
Figure 2. Diffusion at different values of α and β
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VFAST Transactions on Mathematics
(a) Original
(b) Noisy Image
(c) α = 0.9, β = 0.01
(d) α = 0.1, β = 0.09
(e) α = 0.9, β = 0.09
(f) α = 1, β = 0.06
Figure 3. Diffusion at different values of α and β
Figure 4. Structural Similarity Index (SSIM), Peak-Signal-to-Noise-Ratio (PSNR) and Mean Square Error (MSE) of a Lena
Image versus number of iterations
Figure 5. Structural Similarity Index (SSIM), Peak-Signal-to-Noise-Ratio (PSNR) and Mean Square Error (MSE) of a synthetic Image versus number of iterations
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VFAST Transactions on Mathematics
Figure 6. Structural Similarity Index (SSIM), Peak-Signal-to-Noise-Ratio (PSNR) and Mean Square Error (MSE) of a
peeper Image versus number of iterations
Figure 7. Image Profiles after denoising Lena, Synthetic and Peppers by Proposed Method
Table 1. PSNR, SSIM and MSE of our model on Lena Image (results taken after 350 Iterations)
α
β
PSNR
MSSIM
MSE
Iterations
0.1
0.1
0.1
0.5
0.5
0.5
0.9
0.9
0.5
1
1
1
1
0.01
0.05
0.09
0.01
0.05
0.09
0.01
0.05
0.09
0.01
0.05
0.09
0.06666
30.0301
27.2544
25.7535
27.9219
29.7328
29.3235
26.5371
29.3784
27.7309
26.2850
29.2461
29.7218
29.5516
0.8572
0.7587
0.7144
0.8616
0.8572
0.8290
0.8311
0.8711
0.8572
0.8242
0.8718
0.8609
0.8684
6.0615
8.7786
10.5174
6.7914
6.1767
6.7207
7.8615
6.1175
6.1775
8.0930
6.1550
6.1317
6.0915
350
350
350
350
350
350
350
350
350
350
350
350
350
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VFAST Transactions on Mathematics
Table 2. PSNR, SSIM and MSE of our model on Synthetic Image (results taken after 350 Iterations)
α
β
PSNR
MSSIM
MSE
Iterations
0.1
0.1
0.1
0.5
0.5
0.5
0.9
0.9
0.5
1
1
1
1
0.01
0.05
0.09
0.01
0.05
0.09
0.01
0.05
0.09
0.01
0.05
0.09
0.06666
32.9089
30.5216
28.9128
29.4532
32.6429
32.500
27.4289
31.6830
32.6395
27.051
31.6297
32.484
32.1199
0.9046
0.7023
0.5956
0.9383
0.9044
0.8523
0.9255
0.9317
0.9042
0.9225
0.9337
0.9064
0.9236
4.3862
6.0606
7.2967
5.7332
4.4656
4.7345
7.3281
4.6027
4.4677
7.6846
4.6833
4.5159
4.5510
350
350
350
350
350
350
350
350
350
350
350
350
350
Table 3. Table 3: PSNR, SSIM and MSE of our model on Pepper Image (results taken after 350 Iterations
α
β
PSNR
MSSIM
MSE
Iterations
0.1
0.1
0.1
0.5
0.5
0.5
0.9
0.9
0.5
1
1
1
1
0.01
0.05
0.09
0.01
0.05
0.09
0.01
0.05
0.09
0.01
0.05
0.09
0.06666
31.8734
28.3252
26.6785
30.3883
31.6862
30.8686
28.7624
31.6822
28.4562
28.4562
31.5917
31.7499
31.7615
0.8858
0.7621
0.7091
0.9060
0.8806
0.8456
0.8853
0.9017
0.8801
0.8801
0.9039
0.8855
0.8965
4.9327
7.7830
904717
4.9803
4.9682
5.6633
5.8412
4.6838
6.0393
6.0393
4.6768
4.8864
4.7341
350
350
350
350
350
350
350
350
350
350
350
350
350
Table 4. Peak-Signal-to-Noise-Ratio (PSNR) of images restored from various methods
Image
Charnonnier
PM
TV
Our Method
Lena
Synthetic
Peepers
24.31
24.36
26.31
25.47
29.04
27.55
29.07
32.06
30.86
29.55
32.11
31.74
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VFAST Transactions on Mathematics
Table 5. Structural Similarity Index (SSIM) of images restored from various methods
Image
Charnonnier
PM
TV
Our Method
Lena
Synthetic
Peepers
0.7556
0.8345
0.8261
0.7853
0.9181
0.8441
0.8645
0.8867
0.8856
0.8684
0.9236
0.8965
Author Contributions
Ms. Memoona as first author contributed in the optimization of the proposed problem by applying the
direct method of calculus of variations and discretization of the problem using Finite Difference Method.
Professor Dr. Khuda Bux amur as second author contributed the main idea for the mathematical formulation of the proposed model. Mr. Muzaffar Bashir as third author contributed in carrying out the
simulations. Dr. Rajab Ali Malookani as fourth author contributed his expertise in technical writing.
Compliance with Ethical Standards
It is declare that all authors don’t have any conflict of interest.
Funding Information
No Funding
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