Noise Reduction in MR brain image
via various transform domain schemes
Bhawna Goyal^{1*},
Sunil Agrawal^{1}, B.S. Sohi^{2}, Ayush
Dogra^{1}
^{1}Department of Electronics and Communications, UIET, Panjab University, Chandigarh
^{2}Vice Chancellor, Chandigarh University, Chandigarh
*Corresponding Author Email: bhawnagoyal28@gmail.com
ABSTRACT:
Despite the phenomenal progress in the field of image denoising it continues to be an active area of research and
still holds margin in improving the standard of the denoising
techniques. Image denoising has emerged as a significant
tool in medical imaging specifically. In this article we have compared and evaluated
three transform domain techniques on an
KEYWORDS: Curvelet, Shearlet, Tetrolet, Magnetic Resonance
image, Denoising, thresholding.
INTRODUCTION:
As the
number of pixels per unit area keeps on increasing, the modern image acquisition
devices are becoming increasing sensitive to noise [1]. Therefore there grows a
huge dependability on image denoising algorithms to reduce
the effect of noise and artefacts in the resultant image.
Noise can be perceived as a random fluctuation in the colour
information or the intensity value of the image pixels. Image denoising is a well studied field by researchers. A large sum
of image denoising techniques has been presented so far
[2]. Removal of noise is a fundamental operation in image processing and its applications
range from the direct i.e. photographic enhancement to the technical i.e. as a sub
problem in image reconstruction algorithms. It serves as a pre processing as well
as a post processing technique in image registration [3, 4] and image enhancement.
The degradation model of an image contaminated
with noise can be described as:
_{} 1
where h and f are column vectors and represent the true
and noisy signal respectively and n is the added noise[5].
During image acquisition two predominant sources
for noise are the detectors which posses the stochastic nature of photon count and
electronic fluctuations of the electronic devices. When there is enough illumination
which is usually the case the second source of noise gives rise to Additive White
Gaussian Noise (AWGN). These leads to the assumption of Gaussian noise models as
a common practise. Under the conditions with lower power
light source, short exposure time and phototoxity the
source of noise is signal dependant and it forms the reasonable basis to model the
output of the detectors as a Poisson distributed random vector. Other commonly known
noise is impulse noise [6].
Image denoising finds
application in remote sensing, surveillance, medical imaging etc. Image denoising can be considered as an area of vital and critical
importance in medical imaging as noise can be intercepted as an image deformity
and it can lead to false diagnosis. The basic fundamental of image denoising is to reconstruct a plausible estimate of the original
image from the distorted image and maintain a trade off
between removal of noise and feature preservation. The image denoising techniques have been forwarding parallel in three
domains i.e. spatial filters, transform domain methods and dictionary learning methods.
The underlying property of sparse representation
in image transforms i.e. an image can be represented as a sum of few high value
coefficients makes transform domain methods like Directionlet,
Shearlet, Curvelet, Tetrolet, Ridgelet, Ripplet ,Wedgelet very popular for
image denoising [2, 7, 8, 9]. In this article three popular
transform domain methods i.e. Curvelet, Shearlet and Tetrolet with a selected
thresholding have been implemented and contrast is withdrawn
amongst them. The transforms are modified Wavelet systems with more directional
sensitivity. Shearlet was able to overcome the limitations
possessed by Curvelet transform i.e. not to resolve 2D
singularities. Some of the other image transforms like Bandlets,
Wedgelets, Grouplets and Tetrolet were coined to work on image structures based on averaging
in the adaptive data neighbourhood point. In this category
Tetrolet is the most recent transform is the most recent
transform and has been discussed in context of denoising
in this article.
This paper is organised
as follows: Section 2 discusses the basic and related work in the field of denoising. Section 3 gives a brief insight into the Curvelet, Shearlet and Tetrolet transform. Section 4 discusses the experimental setup
and objective evaluation metrics. Results and discussions are presented in section
5. Section 6 finally concludes the article and gives the future scope.
Related
work:
There exists a plethora of denoising methods originating from various great disciplines
such as partial differential equations, spectral and multiresolution
analysis probability theory and statistics etc [10]. All the image denoising methods rely on some implicit and explicit assumptions
in order to separate the true image from the noise. In the field of denoising one can exploit the spatial domain filters, can work
with filters in frequency domain and can also rely on dictionary learning methods.
Spatial domain methods include local and non local filters which exploit the similarities
between either pixels or patches in an image [11].
Some of the local filters include Gaussian filter,
Weiner filter trained filter, bilateral filter, steering kernel regression method
(SKR), MSKR, KSPR [12, 13, 14]. A filter is said to be local if it exploits the
similarity between the pixels restricted by spatial distance. On the other the non
local filters exploit the similarity between the pixels in the entire image and
unrestricted by the spatial distance. However in the presence of high noise, the
similarity between pixels adjacent to each other gets corrupted due to which these
techniques are not able to perform well. Gaussian filter was designed to provide
image denoising. Since then its several improvisations
have been proposed to provide better edges preservation. Anisotropic filter removes
the blurring effect caused by the Gaussian filter. Total variation minimization
technique was designed to denoise the homogenous regions
in the image but the edges. Then the SUSAN filter was proposed for further better
edge preservation. This filter can average the pixels in the neighbourhood those are same spatial distance as the centre
of the pixel. Apart from these parametric methods SKR exploits the kernel regression
fact to denoise an image and remove artifacts. The motivation
behind SKR is pixels on the same side of the edge have a higher impact on the filtering
of the pixel then those on the other side of the edge. Trained filter: the basis
of this filter is a non parametric process in which weight shave been obtained by
the offline training of the large set of images [2].
In non local filters the weights are only marked
by the similarity. Since the proposal of the basic non local filter many further
improvements have been presented based on acceleration in the implementation of
the technique and improvement in qualitative and quantitative results. An effort
was made by M.Mahamoudi [15] by computing mean and gradient
values to fasten the non local manner. Further SVD and local variance were introduced
to eradicate dissimilar pixels. Also when FFT was used to calculate weights it made
the algorithm 50 times faster than the innate one [16]. Now taking into consideration
the improvement of qualitative and quantitative results Seven et al. proposed to
increase the number of weight values for the target patches by designing a rotationally
invariant block matching for non local filters. NLM was the first filter designed
taking into consideration the similarity of the entire image. The basic idea behind
non local filters can be mapped as the modification of the bilateral filter by calculating
the Euclidean distance between two patches instead of two pixels. In 2008 Goosen et al. proposed improved non local filters in various
aspects like the variance in noise is recorded at every location and the extra noise
was removed as a post processing technique. The
Image transforms consider transforming the images
into other domains in which the similarities of the transformed coefficients are
employed. These methods are particularly based on the assumption that the true signal
can be well approximated by a linear combination of few basis elements. Hence the
basic fundamental is to preserve few high magnitude transform coefficients that
contain the true signal intensity and thresholding rest
of the coefficients that contain noise so as to calculate the total estimated signal.
Transform domain has been researched in context of image denoising
for decades. A large number of variants have been proposed in this category like
DCT, Wavelets, BM3D ,Directionlet Curvelet,
Coutourlet, Ridgelet, Tetrolet, Shearlet and corresponding
thresholding techniques like sure shrink Visu shrink and Bayes shrink [7,17]. Wavelet is a multi resolution analysis
tool. As far as the coining of wavelet transform is concerned there are primitively
fifteen variants of wavelet namely Haar wavelet, called
the mother wavelet, Symlet, Mexican Hat Wavelet, Morlet, Mayerlet, Daubechies, Symlet, Coiefflet, Spline Wavelet, Battle
Lemarie Wavelet, DWT,
Image
Transforms
(a).
Curvelet Transform:
To overcome the limitations of Wavelet transform,
the Ridgelet and Curvelet transforms
were developed with an attempt to put a halt
on the inherent limit of wavelet denoising of images.
Ridgelet transform was based on radon transform and was
able to provide information regarding orientation of linear edges. However Ridgelet was not able to resolve the discontinuities in the
two dimensions. So in Candes and Donoho
proposed the second generation Curvelet Transform [19].
Curvelet transform with a purpose to achieve anisotropic
directionality uses a parabolic scaling law. Curvelet
is also a special case of Ripplet Type 1 Transform with
support c=1 and degree d=2. Combining multiscale Ridgelets with a spatial bandpass
filtering leads to the formation of Curvelet transform
with a purpose to isolate different scales. Curvelets
are able to work at all scales, locations and orientations [20]. The mathematical
background has been developed by Candes in [19] clearly.
Curvelet decomposition consists of four steps: subband filtering that is the object under consideration is
decomposed into subbands. Then each subband is smoothly windowed into squares of an approximate
scale. Each resulting square is normalized to unit scale. Then finally hard thresholding rule is followed for estimating the unknown Curvelet coefficients. According to [21] Curvelet reconstructions are able to display a higher sensitivity
then the wavelet based reconstruction also all the structures which are visually
detectable in the noisy image were clearly seen in the Curvelet
reconstructed image.
Shearlet Transform:
Multivariate functions like images that are
governed typically by anisotropic features such as edges are not optimally presented
by classical wavelets. Since then various directional representations systems have
been proposed like Curvelets, Contourlets
and Shearlets [22]. However only Shearlet
systems were able to provide a unified treatment for the continuum and digital world
along with giving optimally sparse approximations. Shearlet
lead to improvements in many image processing applications such as denoising, image enhancement [23, 24] and image fusion. The
discrete wavelet transform is a special type of composite wavelet transforms [25].
There are basically two types of Shearlet systems i.e.
band limited Shearlet and compactly supported Shearlet transform [22,25]. The representation of the discrete
Shearlet transform can be summarized as a cascade of three
steps: the first step includes transformation to classical Fourier transform and
changing of variables to pseudo polar coordinates which is followed by weighting
by a radical density compensation factor. Then it is decomposed into rectangular
tiles and inverse transform is applied to each of tiles after using the Bayes thresholding method. The main
reason Shearlet is able to perform well then Curvelet and Counterlet is the fact
that the Shearlets are affine systems that leads to parameterization
of directions by slopes instead of angles which helps in digital setting along extensive
theoretical framework. The mathematical equations are fully explained by Gitta. K in [22,25].
Tetrolet Transform:
the construction of Tetrolet
transform is similar to the idea of digital Wedgelets
[26] which works on image structures. Tetrolet is based
on adaptive Haar wavelet transform. Tetrolet was designed visualising
the non redundancy of the basis functions for sparse image
approximation. Tetrolets have four equal sized squares
which are called tetrimonies. The algorithm of Tetrolet transform includes the division of the image into 4*4
blocks. Then out of them the sparsest Tetrolet representation
is chosen. Then the rearranging of the high and the low pass coefficients of each
block into 2*2 blocks the various coefficients are finally stored. The output image
obtained is a decomposed image and this is an adaptive decomposition algorithm.
There are a number variants of Tetrolet transform available
namely Tetrolet 16, Tetrolet
Rel, Tetrolet Edge and Tetrolet 16 Rel Edge in context of
denoising along with Tensor Biorthogonal
9/7 wavelet filter bank. In this article we have worked with basic form of Tetrolet Transform.
Objective
Evaluation Metrics
The objective evaluation metrics calculated
for the quality of results so obtained in the context of this article are PSNR (peak
signal to noise ratio) and MSE (mean square error). PSNR is measured in decibels.
It is used as an objective image quality metrics between the source image and the
denoised image. Higher is the PSNR value better is the
quality of the denoised image so obtained. MSE is the
difference or cumulative error between the original and the denoised
image. Lower is the calculated MSE value better is the result of the denoised image [27].
(b)
Experimental setup
A
The all three algorithms
have been tested from low noise values i.e. σ = 5 and 10 to high range noise values i.e. σ = 20,
25 and 40. The flow chart for the implemented methodology is given in Fig.1
(b)
Experimental setup
A
The all three algorithms
have been tested from low noise values i.e. σ = 5 and 10 to high range noise values i.e. σ = 20,
25 and 40. The flow chart for the implemented methodology is given in Fig.1
The basics steps
in image denoising (as shown in Fig. 1)
are:
·
Preprocess
the
·
Add Gaussian
noise standard deviation at σ= 5,
10, 20, 25 and 40.
·
The image is transformed using Curvelet,
Shearlet and Tetrolet representation
·
Apply various
thresholding schemes to modify the images coefficients
(i.e., Soft/hard/ Bayes Shrink thresholding).
·
Apply the
inverse image transform is applied to reconstruct the synthesised
image
·
Denoised image is obtained at the output.
RESULTS AND DISCUSSIONS
To evaluate the performance of the state of
art denoising techniques they are applied to the synthetic
data i.e. a
Table1: PSNR and MSE values with three schemes
Standard deviation σ= 
Curvelet transform 
Shearlet transform 
Tetrolet transform 

PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 

5 
35.7284 
17.3875 
37.0407 
12.8532 
35.4546 
18.5191 
10 
32.3670 
37.7030 
33.8621 
26.7223 
32.8823 
33.4853 
15 
30.2738 
61.0521 
41.6073 
31.9391 
30.0181 
64.7544 
20 
28.7254 
87.20142 
30.4882 
58.1119 
27.3797 
118.8812 
40 
24.5603 
227.5354 
27.2656 
122.0465 
19.2797 
767.5630 
As the value of noise
level increases from σ= 5, 10, 20, 25 and 40 the PSNR values keep on
decreasing while the MSE value keeps on increasing after employing all three denoising techniques. Higher the noise level in the image lower
will be the restoration ability of the denoising technique.
At high noise levels the techniques tends to smoothen away the fine details along
with the elimination of noise, there is deterioration in the image details. In all
cases Shearlet transform outperforms Curvelet and Tetrolet transform in
terms of PSNR and MSE at least for moderate and large values of the noise levels.
In addition the Shearlet transform output images are visually more pleasant.
From the visual inspection and as well as from table1 it can be seen that Tetrolet is able to work at moderate noise levels, whereas its
performance degrades at high noise levels. As far as Curvelet
denosing scheme is concerned it yields better results
than Tetrolet based denoising
scheme. This fact can be clearly validated from the subjective and objective (Table1)
evaluation. So it can be stated that Shearlet transform is able to work well on denoising a
CONCLUSION:
The two directionally sensitive image transforms
that is Curvelet and Shearlet
transform are able to work well on the denoising of the
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DOI: 10.5958/0974360X.2016.00176.1