Journal of X-Ray Science and Technology 31 (2023) 555–572
DOI 10.3233/XST-221326
IOS Press
555
Truncated total variation in fractional
B-spline wavelet transform for micro-CT
image denoising
Dongjiang Ji∗ , Xiying Xue and Chunyu Xu
School of Science, Tianjin University of Technology and Education, Tianjin, China
Received 18 October 2022
Revised 19 January 2023
Accepted 16 February 2023
Abstract.
BACKGROUND: In medical applications, computed tomography (CT) is widely used to evaluate various sample characteristics. However, image quality of CT reconstruction can be degraded due to artifacts.
OBJECTIVE: To propose and test a truncated total variation (truncation TV) model to solve the problem of large penalties
for the total variation (TV) model.
METHODS: In this study, a truncated TV image denoising model in the fractional B-spline wavelet domain is developed
to obtain the best solution. The method is validated by the analysis of CT reconstructed images of actual biological Pigeons
samples. For this purpose, several indices including the peak signal-to-noise ratio (PSNR), structural similarity index (SSIM)
and mean square error (MSE) are used to evaluate the quality of images.
RESULTS: Comparing to the conventional truncated TV model that yields 22.55, 0.688 and 361.17 in PSNR, SSIM and
MSE, respectively, using the proposed fractional B-spline-truncated TV model, the computed values of these evaluation
indices change to 24.24, 0.898 and 244.98, respectively, indicating substantial reduction of image noise with higher PSNR
and SSIM, and lower MSE.
CONCLUSIONS: Study results demonstrate that compared with many classic image denoising methods, the new denoising
algorithm proposed in this study can more effectively suppresses the reconstructed CT image artifacts while maintaining the
detailed image structure.
Keywords: Digital image processing, image denoising, fractional B-spline wavelet, truncated total variation (truncated TV)
model, Split-Bregman iteration
1. Introduction
Micro-computed tomography (micro-CT) is an important tool in biomedical research and preclinical applications, which can provide visual inspection and quantitative information of small animals
and biological samples [1]. Currently, high-quality micro-CT images need to acquire projection data
in many views, which limits the system throughput. In real application, especially the biomedical
applications, to reduce radiation dose while maintaining the diagnostic performance is a major chal∗
Corresponding author: Dongjiang Ji, School of Science, Tianjin University of Technology and Education, Tianjin, 300222,
China. E-mail: zjkjdj@tute.edu.cn.
ISSN 0895-3996/$35.00 © 2023 – IOS Press. All rights reserved.
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lenge. The approach to decrease radiation dose is to reduce the number of projections. However, the
term of deformation or damage to the image sample that arise on account of scarcity of projections
is usually called the aliasing distortions, which will create the visual noise in the reconstructed CT
images [2]. However, the noise of CT images can be controlled by three main denoising: projection
domain, iterative reconstruction and image denoising based on post-processing [3]. In this paper, we
use post-processing methods to reduce the noise of CT images.
A traditional post-processing method to reduce noise from image is spatial filtering. Spatial filters
reduce noise by applying filtering directly on original noisy CT image. In spatial domain, linear filters
were adopted to denoise the images, but it was not successful for preserving edge over the images.
Mean filtering was used to reduce the gaussian noise but for high noise it produces blurry images. To
overcome that, Some of methods was further used, such as non-local means [4], total variation [5],
and BM3D [7], while some use newer deep learning techniques [8]. The most investigated transform
in denoising is the wavelet transform [9]. The basic idea of wavelet transform is to use a cluster of
functions to represent or approximate a signal or function [10]. Different components in the image
correspond to different frequency components. The frequency corresponding to the smooth part of the
image is the low frequency part, and the frequency corresponding to the noise and detail part is the
high frequency part [11]. While removing noise, image denoising preserves image details as much
as possible, so it needs targeted selection when removing high frequency components [12]. The scale
factor in the wavelet transform reflects the frequency information. By changing the wavelet coefficients
at different scales, the high frequency is suppressed and the low frequency is retained, so as to achieve
the purpose of denoising [13, 14].
Spline function plays an important role in wavelet theory. Any scaling function (or wavelet) can
be written as a convolution of polynomial B-spline and singular distribution [15]. Compared with
other wavelet functions, spline wavelets have clearer mathematical expressions and a higher degree
of approximation. B-spline is the basis function that constitutes the spline function [17]. In 2000,
Unser proposed the fractional B-spline function [18, 19], which extended the original integer-order
B-spline to the fractional order, and constructed the fractional-order B-spline wavelet. Fractional Bspline wavelets have good properties such as decay, two-scale equations and fractional approximation.
In the literature [20], the meaning of fractional order is explained from the perspective of filter, and it
is proved that the fractional B-spline wavelet can well describe the features with fractional singularity
such as fine texture in the image [21].
As an edge-preserving smoothing filter, TV is widely used due to its simplicity and effectiveness in
removing noise-like structures [5]. However, it penalizes larger gradient amplitudes and may reduce
the contrast during the smoothing process. Therefore, a method called truncated TV is proposed, which
uses a high-sparse regularization method to establish its mathematical model [6]. Compared with TV,
truncated TV attempts to partially penalize the gradient of the image to ensure that only important
structures in the image are retained and uninteresting structures in the image are smoothed. In addition,
the algorithm can take full advantage of the existing fast TV solver without multiple modifications, so
that it has the same computational efficiency as TV [6].
Based on the characteristics of the fractional B-spline wavelet transform and the truncated TV model,
this paper proposes a truncated TV image denoising model in the fractional B-spline wavelet domain
(referred to as the fractional B-spline truncated TV model). The proposed truncated TV image denoising
model based on fractional B-spline wavelet transform possesses three significant characteristics: 1)
the fractional B-spline wavelet has the characteristics of characterizing the fractional singularity of the
sample; 2) the truncated total variation (truncation TV) model solves the problem of large penalties for
the total variation (TV) model; 3)Since the staircase often becomes the high-frequency components in
the fractional wavelet domain, the fractional wavelet shrinkage technique can suppress the staircase
caused by the truncated TV model to some extent [22, 23].
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
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The structure of the paper is organized as follows. The second section introduces the theory of fractional B-spline wavelet transform and truncated TV model. The third part presents the truncated total
variation in fractional B-spline wavelet transform for Micro-CT image denoising, and the fourth part
uses this method to conduct experimental analysis on the reconstructed image of the actual biological
sample pigeon. Finally, the fifth section gives some concluding remarks.
2. Methodology
2.1. Fractional B-spline wavelet transform
By analogy with the classical B-splines, we define the fractional causal B-splines by taking the
(α + 1)th fractional difference of the one-sided power function [18]:
α
(x) =
β+
+∞
1
1
α
(−1)k
α+1
+ x+ =
(α + 1)
(α + 1) k≥0
α+1
k
(x − k)α+
(1)
α
(α+1)
= (k+1)(α−k+1)
.
k
We also introduce the reversed versions of these functions: the anticausal B-splines of degree α [18],
where (α + 1) =
0
α −x
x e
dx, (x − k)α+ = (max (x − k, 0))α ,
α
(x) = α+1
β−
−
α
x−
(α + 1)
α
(−x)
= β+
(2)
α
where x−
= (−x)α+ and where α− denotes the fractional backward difference operator
α− f (x) =
k
(−1)
k≥0
α
k
f (x + k)
(3)
To symmetrize the construction and to be in the position to calculate fractional B-spline inner
products, the centered fractional B-splines of degree α are defined as [18]:
α−1
α−1
β∗α (x) = β+2 (x) ∗ β−2 (x)
(4)
where * denotes convolution operator and β∗α can be calculated as:
β∗α (x) =
α + 1
1
1
α∗ |x|α∗ =
(−1)k |x − k|α∗
k (α + 1)
(α + 1) k=Z
where
|x|α∗ =
⎧
|x|α
,α =
/ 2m(noteven)
⎨ −2 sin(πα/2)
⎩
x2m log|x|
, α = 2m(even)
(−1)1+m π
α
α
=
k
k + α2
Fractional B-spline function has decay, Riesz basis, approximation, and satisfies two-scale relation.
Therefore, the fractional B-spline function cluster can form a multi-resolution analysis, so that the
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
Fig. 1. Analysis and synthesis filterbank with fractional spline filters.
corresponding wavelet basis function can be constructed and the image can be transformed by fractional
B-spline wavelet.
In particular, the fractional B-spline function satisfies the two-scale relation
x
2
α
β+
=
α
(x − k)
hα+ (k) β+
k∈Z
This equation can be established by direct manipulation of the time domain formulae. The low-pass
filter corresponding to the fractional B-spline wavelet transform is obtained by performing Fourier
transform on the two-scale relationship [19]:
H+α
jω
e
√ 1 + e−jω
= 2
2
Orthogonalize H+α directly to get:
H⊥α
where Aα ejω =
n∈Z
jω
e
= H+α
e
jω
α+1
(5)
A2α+1 ejω
A2α+1 e2jω
(6)
α
α
(x) β+
(x + n) dx.
e−jω β+
The corresponding high-pass filter is [19]:
Gα⊥ ejω = e−jω H⊥α −e−jω .
(7)
The analysis and synthesis process of fractional B-spline wavelet transform is shown in Fig. 1.
2.2. Truncated Total Variation (Truncate TV)
We denote the input image by f and the smoothed result by u. Thus, the truncated TV is expressed
as [6]
U(x) =
(T (ux ) + T (uy )dxdy
T (u) =
|u||u| < ε
εotherwise
(8)
(9)
where ux , uy ) is the gradient of u. Truncated TV only penalizes gradients whose magnitudes are smaller
than the threshold ε, while for those whose magnitudes are greater than ε, the proposed penalty doesn’t
penalize them.
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
559
Truncated TV is non-convex and difficult to optimize due to its “truncated” shape. However, Equation
(9) can be re-expressed using L0 regularization. Taking ε as a parameter, T (u) defined in Equation (9)
is equivalent to
φ(u, l) = min ε|l|0 + |u − l|
(10)
l
where | · |0 is zero power operator, i.e. |l|0 = 1 if l =
/ 0, else |l|0 = 0.
3. Proposed method
Combining the fractional B-spline wavelet transform and truncated TV, the noise reduction model
in this paper is shown in Equation (11):
u = arg min WFBS f − WFBS u22 + α φ (WFBS ux , l1 ) + φ WFBS uy , l2 1
(11)
u,l1 ,l2
The first term on the right side of the above formula is the data fidelity term, and the second term is the
fractional B-spline-truncated TV term. Where f is the noise-containing image to be processed, u is the
image after noise reduction, WFBS is the fractional B-spline wavelet
transform, and α is the parameter
of the truncated TV regular term defined in the wavelet domain. φ (WFBS ux , l1 ) + φ WFBS uy , l2 1
defined as:
N−1
N−1
0 φ (WFBS ux , l1 ) + φ WFBS uy , l2 =
min
ε
l
+
W
u
−
l
1i,j
FBS xi,j
1i,j 1
i=0 j=0
(12)
0 + min ε l2i,j + WFBS uyi,j − l2i,j where φ(WFBS ux , l1 ) = min ε|l1 |0 + |WFBS ux − l1 |}, φ(WFBS uy , l2 ) = min ε|l2 |0 + |WFBS uy −
l2 |}.
Directly minimizing functional Equation (11) is difficult because it involves L1 and L2 penalty terms.
We adopt an alternating optimization strategy with split Bregman framework [24]. We introduce two
dual variables b1 and b2 , corresponding to WFBS ux − l1 and WFBS uy − l2 respectively, and re-express
the objective function as follows:
min
⎧
⎨N−1
N−1
⎩
WFBS fi,j − WFBS ui,j
2
+ αε
i=0 j=0
⎫
N−1
0 ⎬
N−1
0
l1i,j + l2ii,j + α
b1i,j + b2i,j ⎭
N−1
N−1
i=0 j=0
i=0 j=0
(13)
where b1 = WFBS ux − l1 , b2 = WFBS uy − l2 .
By using Lagrangian multipliers, Equation (13) can be converted to an unconstraint problem:
min
⎧
⎨N−1
N−1
⎩
WFBS fi,j − WFBS ui,j
i=0 j=0
+λ
2
+ αε
N−1
N−1
N−1
N−1
0 0
l1i,j + l2i,j + α
b1i,j + b2i,j i=0 j=0
N−1
N−1
i=0 j=0
2
b1i,j − WFBS uxi,j + l1i,j
+λ
i=0 j=0
N−1
N−1
i=0 j=0
2
b2i,j − WFBS uyi,j + l2i,j
⎫
⎬
⎭
(14)
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
By introducing Bregman distance, Equation (14) becomes:
min
⎧
⎨N−1
N−1
⎩
WFBS fi,j − WFBS ui,j
2
+ αε
i=0 j=0
+λ
N−1
N−1
N−1
N−1
0 0
l1i,j + l2i,j + α
b1i,j + b2i,j i=0 j=0
i=0 j=0
2
N−1
N−1
b1i,j − WFBS uxi,j + l1i,j − t1i,j
+λ
i=0 j=0
N−1
N−1
2
b2i,j − WFBS uyi,j + l2i,j − t2i,j
i=0 j=0
⎫
⎬
⎭
(15)
The above joint minimizing problem can be solved alternately by decoupling it into several subproblems, described as follows:
(1) Calculate WFBS u subproblem with fixed l1 , l2 , b1 , b1 , t1 , t2
WFBS ui+1 = min
⎧
⎨N−1
N−1
WFBS u ⎩
+λ
N−1
N−1
WFBS fi,j − WFBS ui,j
2
i=0 j=0
b1i i,j − WFBS uxi,j + l1i i,j − t1i i,j
2
+ b2i i,j − WFBS uyi,j + l2i i,j − t2i i,j
2
⎫
⎬
i=0 j=0
(16)
⎭
We derive the Euler-Lagrange equation as follows:
i
∂ b2i + l2i − t2i
∂ b1 + l1i − t1i
WFBS f − WFBS u + λ
+
− WFBS u = 0
∂x
∂y
(17)
where is the Laplace operator. Equation (17) can be efficient solved by using Gauss-Seidel iteration
algorithm or FFT operator.
(2) Calculate b1 and b2 with fixed WFBS u, l1 , l2 , t1 , t2
The unique minimizer of this subproblem can be obtained by applying the shrinkage operator:
b1i+1 = shrink
i
i α
WFBS ui+1
x − l1 + t1 ,
(18)
λ
i
i
b2i+1 = shrink WFBS ui+1
y − l2 + t2 ,
α
λ
(19)
where
x
max(x − α, 0)
|x|
(20)
i
i
i+1
t1i+1 = WFBS ui+1
x + t1 − l1 − b1
(21)
i
i
i+1
t2i+1 = WFBS ui+1
y + t2 − l2 − b2
(22)
shrink(x, α) =
(3) Update t1 , t2 with fixed WFBS u, l1 , l2 , b1 , b2
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
561
Table 1
The steps of the fractional B-spline-truncated TV image noise reduction algorithm
Algorithm1
Input: f , N, b1 = b2 = t1 = t2 = l1 = l2 = 0, αWFBS , τWFBS , εTTV , αTTV , λTTV ;
Perform the fractional B-spline wavelet transform on f to get WFBS f , WFBS u = WFBS f ;
For i from 1 to N:
Update WFBS u by solving (17);
Update b1 , b2 using (18) and (19);
Update t1 , t2 using (21) and (22);
Update l1 , l2 using (24) and (25):
End
Perform the inverse transform of fractional B-spline wavelet transform on WFBS u to obtain u.
Output: u
(4) Update l1 , l2 with fixed WFBS u, t1 , t2 , b1 , b2
⎧
N−1
N−1
0 0 N−1
arg min ⎨ N−1
2
i+1 i+1
i+1
i+1
l1 , l2 =
αε
+λ
l1i,j − WFBS ui+1
+
b
−
t
l1i,j + l2i,j x
1
1
i,j
i,j
i,j
l1 , l2 ⎩
i=0 j=0
i=0 j=0
+
i+1
i+1
l2i,j − WFBS ui+1
yi,j + b2i,j − t2i,j
2
(23)
The solution of this subproblem is:
l1i+1 =
l2i+1 =
⎧
⎨ t1i+1 − b1i+1 + WFBS ui+1 if t1i+1 − b1i+1 + WFBS ui+1 < αε
x
⎩
0
x
λ
otherwise
⎧
⎨ t2i+1 − b2i+1 + WFBS ui+1 if t2i+1 − b2i+1 + WFBS ui+1 < αε
y
⎩
0
y
otherwise
λ
(24)
(25)
(5) Finally, perform the inverse transform of the fractional B-spline wavelet transform on WFBS u to
obtain u.
The whole algorithm is summarized as Table 1. It can be seen from Table 1 that all sub-problems
have closed form solutions, so that the algorithm can be solved very quickly.
4. Experiments
4.1. Experimental data
Use MATLAB R2017a software to realize the real image noise reduction experiment on the computer
side (Intel(R) Core(TM)i3-5005U CPU @2.00 GHz,4.00GB). Two groups of real biological sample
pigeon CT reconstruction images are used for image denoising experiments.
The pigeon samples were scanned at a voltage of 100 kV and a current of 110 ␮A. The source to
rotation axis distance is 305 mm, the source to detector distance is 852.776 mm. As shown in Fig. 2,
the left is the fully projected reconstructed image (FDK reconstruction from full projections), and the
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
Fig. 2. Comparison of full projection reconstruction image and sparse projection reconstruction image. The grayscale images
were normalized to the range [0, 255], and the display window was [0, 255].
right is the sparse projection reconstructed image (FDK reconstruction from 200 projections). The size
of the two groups of pictures is 1024 × 1024 and 2048 × 2048.
As shown in the rectangular frame in the figure, after comparing the images and the partial enlargement of the image, due to the sparse projection CT reconstruction image contains obvious noise, the
internal structure is not clear, the detail texture is blurred, and there are obvious strip artifacts on the
periphery.
4.2. Evaluation index
Quantitative evaluation uses three indicators, peak signal-to-noise ratio (PSNR), mean square error
(MSE), and structural similarity index (SSIM), to evaluate the quality of images [25, 26].
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
563
Fig. 3. The PSNR value of the first denoised image varies with parameters.
Fig. 4. The PSNR value of the second denoised image varies with parameters.
Define two values, one is MSE, and the other is PSNR, the formula is as follows:
M N
1
MSE (x, y) =
xi,j − yi,j
M × N i=1 j=1
PSNR (x, y) = 10 log10
Peak2
MSE
= 20 log10
2
Peak
√
MSE
(26)
(27)
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
Table 2
PSNR, MSE, and SSIM between full projection reconstruction
image and sparse projection reconstruction image
PSNR (dB)
MSE
SSIM
22.32
18.61
380.88
895.19
0.5151
0.5026
Sample I
Sample II
Among them, x represents the reference image, y represents the noise reduction image, M × N is
the size of x and y; xi,j and yi,j represent the pixel intensity of x and y in a certain pixel (i, j), and
Peak represents the maximum pixel intensity in the normalized image (Usually the gray level of the
image, generally the value is 255). The general value range of PSNR:20–40, the larger the value, the
better the image quality. SSIM is a full-reference image quality evaluation index, which measures the
similarity of two images from three aspects: brightness, contrast, and structure. The range of SSIM is
0 to 1. The larger the SSIM value, the higher the similarity. Given two images x and y, the SSIM of
the two images can be calculated as follows:
SSIM(x, y) =
(2μx μy + c1 ) 2σxy + c2
μ2x + μ2y + c1
σx2 + σy2 + c2
(28)
where μx is the average of x, μy is the average of y, σx2 is the variance of x, σx2 is the variance of y, and σxy
is the covariance of x and y. c1 and c2 are constants used to maintain stability. The quantitative results
are shown in Table 2. The PSNR, MSE, and SSIM of the noisy reconstructed images to be processed
for the first biological sample are 22.3229 dB, 380.8818, and 0.6228. It can be clearly seen that the
fractional B-spline-truncated TV model noise reduction image has the highest PSNR, SSIM, and lower
MSE, and the noise reduction effect is better, and the image structure can be better maintained.
The quantitative results (PSNR, MSE and SSIM) between full projection reconstructed image and
sparse projection reconstructed image. are shown in Table 2.
5. Results and discussion
Apply fractional B-spline-truncated TV model and existing truncated total variation (truncated TV)
method [6], wavelet transform soft threshold noise reduction (WT) [9], weighted least squares (WLS)
method [27], and L0 smoothing method [28] to reduce the noise of the CT reconstructed image. In
the parameter selection of the fractional B-spline-truncated TV model algorithm, there are mainly five
parameters: αWFBS , τWFBS , εTTV , αTTV and λTTV . Where αWFBS is the degree of a fractional B spline and
τWFBS is dummy shift. εTTV , αTTV and λTTV are the same parameters as truncated TV. These parameters
have a great influence on the results of noise reduction, so it is very important to optimize their values.
In this article, we use a greedy strategy to determine the best parameter values one by one. As shown
in the figure, PSNR is used as the evaluation index. In order to determine the best value of a certain
parameter, while keeping other parameters unchanged, select different parameter values to obtain
different noise-reduction images, and then calculate the corresponding PSNR value, and determine the
optimal parameter value from the highest PSNR. The parameter values of other algorithms are also
determined in this way.
In actual experiments, we combined the three indicators of PSNR, MSE and SSIM to determine the
best parameter sets for different algorithms. In the truncated TV algorithm, ε is the threshold parameter
in the model, α is the parameter of the truncated TV regular term, and λ is the parameter introduced in the
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
565
Table 3
The best parameter values of different methods in the first biological sample
Truncated
TV
WT
WLS
L0
smoothing
fractional
B-spline-truncated
TV model
ε = 0.001;
α = 0.8;
λ=7
Threshold
vector
[26,14]
λ = 0.1;
α=1
λ = 0.0006;
κ=2
αWFBS = 5;
τWFBS = 0.05;
εTTV = 0.001;
αTTV = 0.1;
λTTV =0.03
iterative solution of each sub-problem in the model. In the wavelet transform soft threshold denoising
algorithm, the selected wavelet function coif3 is subjected to two-layer wavelet decomposition, and
the best threshold vector is obtained through experiments. λ is the balance parameter between the data
item and the smoothness in the WLS algorithm [27]. Increasing λ will produce a smoother image. And
α gives a degree of control over the affinities by non-linear scaling the gradients. Increasing α will
result in sharper preserved edges. λ is a smoothing parameter that controls the degree of smoothness
in the L0 algorithm, and κ is a parameter that controls the rate [28]. A smaller κ value results in more
iterations and sharper edges.
5.1. The first biological sample
Table 3 shows the best values of all the parameters of several noise reduction algorithms corresponding to the image to achieve their best noise reduction results. From Fig. 5, when the image to
be processed is degraded due to noise, several noise reduction methods can reduce the image noise,
but the image after the noise reduction method in this article looks the most clear, internal details and
external edges are more obvious. The frame in the figure represents the partial magnification area.
After partially magnifying the image, compare the processing effects of several methods on image
details and edges. As shown by the arrow in Fig. 6, the internal structure of the image is clearer after
the truncated TV denoising, and the image can retain the detailed texture after the wavelet threshold
denoising. After smoothing and WLS denoising, the image structure and details are smoothed, and
part of the image information is lost. The method in this paper can have the advantages of truncated
TV and fractional wavelet transform in terms of noise reduction. It can effectively reduce noise while
maintaining image structure and detailed texture.
As shown by the arrow in Fig. 7, among several methods, the image edge of the method proposed
in this article is the clearest, WLS is relatively blurred, and the remaining methods are relatively clear.
For the obvious streak artifacts at the periphery of the image, several methods have not been able to
effectively remove them. as shown by the elliptical frame in Fig. 7.
The quantitative results are shown in Table 4. The PSNR, MSE, and SSIM of the noisy reconstructed
images to be processed for the first biological sample are 22.3229 dB, 380.8818, and 0.6228. It can
be clearly seen that the fractional B-spline-truncated TV model noise reduction image has the highest
PSNR, SSIM, and lower MSE, and the noise reduction effect is better, and the image structure can be
better maintained.
Table 5 shows the time consumption of several algorithms. When processing the first biological
sample image, several methods take less time, of which WLS is relatively long.
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
Fig. 5. Comparison of noise reduction results for the first biological sample (a) original CT image; (b) truncated TV; (c) WT;
(d) WLS; (e) L0 smoothing; (f) fractional B-spline-truncated TV model. The grayscale images were normalized to the range
[0, 255], and the display window was [0, 255].
Fig. 6. Comparison of internal details for the first biological sample (a) original CT image; (b) truncated TV; (c) WT; (d)
WLS; (e) L0 smoothing; (f) fractional B-spline-truncated TV model. The grayscale images were normalized to the range [0,
255], and the display window was [0, 255].
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
567
Fig. 7. Comparison of external edges for the first biological sample (a) original CT image; (b) truncated TV; (c) WT; (d)
WLS; (e) L0 smoothing; (f) fractional B-spline-truncated TV model. The grayscale images were normalized to the range [0,
255], and the display window was [0, 255].
Table 4
PSNR, MSE, and SSIM of the first biological sample processed with several methods of
noise reduction
PSNR (dB)
MSE
SSIM
Truncated TV
WT
WLS
L0 smoothing
22.5537
361.1696
0.6881
22.5087
364.9335
0.6837
22.4519
369.7316
0.6822
22.4664
368.5037
0.6810
fractional
B-splinetruncated TV
model
26.3019
152.3685
0.7727
5.2. The second biological sample
Table 6 shows the best values of all parameters of several noise reduction algorithms. It can be seen
from Fig. 8 and the partially enlarged Figs. 9 and 10 that the proposed method enables the internal
structure and detailed texture of the image to be well maintained, as well as the edges are clearer. As
shown by the arrow and the elliptical frame in the figure.
The PSNR, MSE, and SSIM of the noisy reconstructed images to be processed for the second
biological sample are 18.6116 dB, 895.1922 and 0.5939, adjust all the parameters of several noise
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
Table 5
Comparison of calculation costs of different methods in the first biological sample
Algorithm
Image size
Time (seconds)
Truncated TV
WT
WLS
L0 smoothing
Fractional B-spline-truncated TV model
1024 × 1024
1024 × 1024
1024 × 1024
1024 × 1024
1024 × 1024
3.692015
2.721092
7.006485
5.637512
4.587047
Table 6
The best parameter values of different methods in the second biological sample
Truncated
TV
WT
WLS
L0
smoothing
fractional Bspline-truncated
TV model
ε = 0.0045;
α = 0.5;
λ=9
Threshold
vector
[20,20]
λ = 0.2;
α = 1.2
λ = 0.001;
κ=2
αWFBS = 8.5;
τWFBS = - 0.25;
εTTV =0.004;
αTTV =0.5;
λTTV =0.02
reduction method algorithms to obtain high PSNR, SSIM and low MSE. It can be seen from Table 7
that PSNR, MSE, and SSIM of the proposed method are better than other methods.
Table 8 shows the time-consuming for several algorithms. When processing the second biological
sample image, the WLS algorithm takes a long time compared to other algorithms due to the large
image size, while the method in this paper takes a mid-range amount of time.
6. Conclusion
In this article, we propose a truncated TV image denoising model based on the fractional B-splines
wavelet domain to suppress noise and maintain the structure of the image. Through the noise reduction processing of the noisy biological sample pigeon CT image, the experimental results show that,
compared with other methods, this new model can effectively suppress noise, and the processed image
effect is better. In addition, the model can maintain high PSNR, SSIM, and low MSE.
In our study, the truncated total variation in fractional B-spline wavelet transform domain is effective
for two groups of real biological sample compared other methods (e.g., truncated TV method, WT,
WLS, and L0 smoothing method). However, the shortcoming of the proposed algorithm is that its
parameters should be tuned for the best performance, and there is no proof that the new model is
true for all kinds of structures. In recent years, many wavelet constructions (e.g., contourlet, curvelet,
shearlet) have been developed and used for different image processing tasks. Take shearlet, for instance.
The shearlet is more effective in approximating piecewise smooth images containing rich geometric
information. We believe that shearlet combined with the truncated total variation can be applied for
samples with complex textures in image denoising.
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
569
Fig. 8. Comparison of noise reduction results for the second biological sample (a) original CT image; (b) truncated TV; (c)
WT; (d) WLS; (e) L0 smoothing; (f) fractional B-spline-truncated TV model. The grayscale images were normalized to the
range [0, 255], and the display window was [0, 255].
Fig. 9. Comparison of internal details for the second biological sample (a) original CT image; (b) truncated TV; (c) WT; (d)
WLS; (e) L0 smoothing; (f) fractional B-spline-truncated TV model. The grayscale images were normalized to the range [0,
255], and the display window was [0, 255].
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D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
Fig. 10. Comparison of external edges for the second biological sample (a) original CT image; (b) truncated TV; (c) WT; (d)
WLS; (e) L0 smoothing; (f) fractional B-spline-truncated TV model. The grayscale images were normalized to the range [0,
255], and the display window was [0, 255].
Table 7
PSNR, MSE, and SSIM of the second biological sample processed with several methods of
noise reduction
PSNR (dB)
MSE
SSIM
Truncated
TV
WT
WLS
L0 smoothing
18.7950
858.1855
0.7041
18.7303
871.0662
0.6839
18.3808
944.0603
0.6927
18.7390
869.3158
0.6873
fractional Bspline-truncated
TV model
24.2394
244.9836
0.8980
Table 8
Comparison of calculation costs of different methods in the second biological sample
Algorithm
Image size
Time (seconds)
Truncated TV
WT
WLS
L0 smoothing
fractional B-spline-truncated TV model
2048 × 2048
2048 × 2048
2048 × 2048
2048 × 2048
2048 × 2048
13.993614
3.356346
457.006485
21.684386
17.549406
D. Ji et al. / Truncated total variation in fractional B-spline wavelet transform
571
Acknowledgments
This work was supported by the Foundation of Tianjin University of Technology and Education
[Grant Nos. KJ12-01, KJ17-36].
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