![6
International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET)
IMAGE ENHANCEMENT AIDED DENOISING USING DUAL TREE COMPLEX WAVELET
TRANSFORM
K. Mounika1
, S. Rehana Banu2
.
1 Research Scholar,Department of Electronics AndCommunication Engineering, Chiranjeevi Reddy Institute of Engineering and Technology
Anantapur, A. P, India.
2 Assistant Professor,Department of Electronics AndCommunication Engineering, Chiranjeevi Reddy Institute of Engineering and Technology
Anantapur, A. P, India.
*Corresponding Author:
K. Mounika,
Research Scholar, Department of Electronics AndCom-
munication Engineering, Chiranjeevi Reddy Institute of
Engineering and Technology,Anantapur, A. P, India.
Email: mounika.rishi@gmail.com.
Year of publication: 2016
Review Type: peer reviewed
Volume: I, Issue : I
Citation: K. Mounika, Research Scholar"Image Enhance-
ment Aided Denoising Using Dual Tree Complex Wavelet
Transform" International Journal of Research and Inno-
vation on Science, Engineering and Technology (IJRISET)
(2016) 06-09
INTRODUCTION
The central idea to wavelets is to analyze (a signal) ac-
cording to scale. Imagine a function that oscillates like a
wave in a limited portion of time or space and vanishes
outside of it. The wavelets are such functions: wave-like
but localized. One chooses a particular wavelet, stretches
it ( to meet a given scale) and shifts it, while looking into
its correlations with the analyzed signal. This analysis is
similar to observing the displayed signal (e.g., printed or
shown on the screen) from various distances. The signal
correlations with wavelets stretched to large scales reveal
gross (“rude”) features, while at small scales fine signal
structures are discovered. It is therefore often said that
the wavelet analysis is to see both the forest and the trees.
In such a scanning through a signal, the scale and the
position can vary continuously or in discrete steps. The
latter case is of practical interest in this thesis. From an
engineering point of view, the discrete wavelet analysis is
a two channel digital filter bank (composed of the lowpass
and the highpass filters), iterated on the lowpass output.
The lowpass filtering yields an approximation of a sig-
nal (at a given scale), while the highpass (more precisely,
bandpass) filtering yields the details that constitute the
difference between the two successive approximations. A
family of wavelets is then associated with the bandpass,
and a family of scaling functions with the lowpass filters.
This concept is explained in Section. At the moment, we
address only the general aspects.
several wavelets are shown that are obtained from the
mother wavelet ψ(x) = (1 − 2x2)e−x2; this wavelet is the
second derivative of a Gaussian function and is called the
Mexican hat. Its first use was in computer vision, for mul-
tiscale edge detection.
The origins of the wavelet analysis can be traced to the
1909 Haar wavelet (that was not called by that name
then) and various “atomic decompositions” in the history
of mathematics. For a comprehensive review, see [Mey-
er93, p.13-31]. The current use of the name “wavelet” is
due to Grosman’s and Morlet’s work on geophysical signal
processing, which led to the formalization of the continu-
ous wavelet transform [Grosman84]. In the development
of wavelets, the ideas from
A few wavelets obtained from the Mexican hat mother wavelet
ψ(x) = (1 − 2x2
)e−x2
.
Abstract
This paper presents a novel way to reduce noise introduced or exacerbated by image enhancement methods, in par-
ticular algorithms based on the random spray sampling technique, but not only. According to the nature of sprays,
output images of spray-based methods tend to exhibit noise with unknown statistical distribution. To avoid inappropri-
ate assumptions on the statistical characteristics of noise, a different one is made. In fact, the non-enhanced image is
considered to be either free of noise or affected by non-perceivable levels of noise. Taking advantage of the higher sensi-
tivity of the human visual system to changes in brightness, the analysis can be limited to the luma channel of both the
non-enhanced and enhanced image. Also, given the importance of directional content in human vision, the analysis is
performed through the dual-tree complex wavelet transform , lanczos interpolator and edge preserving smoothing filters.
Unlike the discrete wavelet transform, the DTWCT allows for distinction of data directionality in the transform space.
For each level of the transform, the standard deviation of the non-enhanced image coefficients is computed across the
six orientations of the DTWCT, then it is normalized.
Keywords: dual-tree complex wavelet transform (DTWCT), lanczos interpolator, edge preserving smoothing filters.
International Journal of Research and Innovation in
Electronics and Communication Engineering (IJRIECE)](https://image.slidesharecdn.com/ijri-ece-01-02imageenhancementaideddenoisingusingdualtreecomplexwavelettransform-161122075118/85/Ijri-ece-01-02-image-enhancement-aided-denoising-using-dual-tree-complex-wavelet-transform-1-320.jpg)
![7
International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET)
LITERATURE
Although the field of image enhancement has been active
since before digital imagery achieved a consumer status,
it has never stopped evolving. The present work introduc-
es a novel multi-resolution denoising method, tailored to
address a specific image quality problem that arises when
using image enhancement algorithms based on random
spray sampling. While inspired by the peculiar problem
of such methods, the proposed approach also works for
other image enhancement methods that either introduce
or exacerbate noise. This work builds and expands on a
previous article by Fierro et al. [1].
Due to the peaked nature of sprays, a common sideeffect
of image enhancement methods that utilize spray sam-
pling is the introduction of undesired noise in the output
images. The magnitude and statistical characteristics of
said noise are not known a-priori, instead they depend on
several factors, like image content, spray properties and
algorithm parameters.
Among image denoising algorithms, multi-resolution
methods have a long history.
Independently of the specific transform used, the general
assumption in multi-resolution shrinkage is that image
data gives rise to sparse coefficients in the transform
space. Thus, denoising can be achieved by compress-
ing (shrinking) those coefficients that compromise data
sparsity. Such process is usually improved by an elabo-
rate statistical analysis of the dependencies between co-
efficients at different scales [17-20]. Yet, while effective,
traditional multi-resolution methods are designed to only
remove one particular type of noise (e.g. Gaussian noise).
Furthermore, only the input image is assumed to be giv-
en. Due to the unknown statistical properties of the noise
introduced by the use of sprays, traditional approaches
do not find the expected conditions, and thus their action
becomes much less effective.
The Dual-Tree Complex Wavelet Transform
The dual-tree difficult wavelet transform (CWT) is a some-
what existing improvement to the discrete wavelet trans-
form (DWT), with dynamic additional things: It is almost
shifting invariant and directionally selective in two and
progressive dimensions. It scopes this with a idleness
factor of only 2d for d-dimensional signals, which is sig-
nificantly lessor than the undecimated DWT. The mul-
tidimensional (M-D) dual-tree CWT is non discrete how-
ever is created on a computationally operative, divisible
filter bank (FB). This tutorial debates the principle behind
schedule the dual-tree transform, shows how difficult
wavelets with decent things can be planned, and displays
a kind of taxes in signal and image processing. We usage
the difficult number symbol C in CWT to avoid mispercep-
tion with the often-used acronym CWT for the (different)
continuous wavelet transform.
THE WAVELET TRANSFORM AND MULTISCALE ANAL-
YSIS
Since its progress 20 years ago, the wavelet variation has
been misused with unlimited achievement through the
scope of signal dealing out habits, in the development,
regularly redefining the state-of-the-art arrangement [10],
[11]. In a outer layer, the DWT changes the really vacillat-
ing sinusoidal root drives of the Fourier transform using
a static of in the vicinity fluctuating basis meanings called
wavelets. In the standard site, the wavelets are pressed
and moved types of a vital, real-valued band pass wavelet
!(t ). When foolishly carefully chosen and combined with
changes of a real-valued low-pass scaling function "(t ),
they system an orthonormal beginning enlargement for
one-dimensional (1-D) real-valued continuous-time sig-
nals [20]. That is, any determinate drive analog signal
x(t ) can be disintegrated in terms of wavelets and scaling
functions via
The scaling coefficients c(n) and wavelet coefficients d( j,
n) are considered via the middle products.
They arrange for a time-frequency analysis of the signal
by computing its frequency relaxed (organised by the
scale factor j) at dissimilar times (organised by the time
shift n). There occurs a same effective, right time difficulty
algorithm to compute the coefficients c(n) and d( j, n) from
a fine-scale demonstration of the signal (repeatedly purely
N examples) and vice versa created on dual octave-band,
discrete-time FBs that recursively placed on a discrete-
time low-pass filter h0(n), a greater-pass filter h1(n), and
up sample and down sample processes. Grasp “Real-Val-
ued Detached Wavelet Transform and Filter Banks” for
further contextual on wavelets, FBs, and their plan.
EXPERIMENTAL RESULTS
Proposed Block diagram
Input Low contrast noisy image.](https://image.slidesharecdn.com/ijri-ece-01-02imageenhancementaideddenoisingusingdualtreecomplexwavelettransform-161122075118/85/Ijri-ece-01-02-image-enhancement-aided-denoising-using-dual-tree-complex-wavelet-transform-2-320.jpg)
![8
International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET)
Contrast enhanced noisy image
Noise removal output with dtcwt
Bilateral filter output
Final contrast enhanced output
Input Low contrast noisy image
Bilateral filter output
Final contrast enhanced output
CONCLUSION
This project presents a noise reduction method based
on Dual Tree Complex Wavelet Transform coefficients
shrinkage. The main point of novelty is represented by its
application in post-processing on the output of an image
enhancement method (both the non enhanced image and
the enhanced one are required) and the lack of assump-
tions on the statistical distribution of noise. On the other
hand, the non-enhanced image is supposed to be noise-
free or affected by non perceivable noise.
REFERENCES
[1] M. Fierro, W.-J. Kyung, and Y.-H. Ha, “Dual-tree com-
plex wavelet transform based denoising for random spray
image enahcement methods,” in Proc. 6th Eur. Conf. Col-
our Graph., Imag. Vis., 2012, pp. 194–199.
[2] E. Provenzi, M. Fierro, A. Rizzi, L. De Carli, D. Ga-
dia, and D. Marini, “Random spray retinex: A new retinex
implementation to investigate the local properties of the
model,” vol. 16, no. 1, pp. 162–171, Jan. 2007.
[3] E. Provenzi, C. Gatta, M. Fierro, and A. Rizzi, “A spa-
tially variant whitepatch and gray-world method for color
image enhancement driven by local contrast,” vol. 30, no.
10, pp. 1757–1770, 2008.
[4] Ø. Kolås, I. Farup, and A. Rizzi, “Spatio-temporal
retinex-inspired envelope with stochastic sampling: A
framework for spatial color algorithms,” J. Imag. Sci.
Technol., vol. 55, no. 4, pp. 1–10, 2011.
[5] H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch,
“Adaptive bayesian wavelet shrinkage,” J. Amer. Stat. As-
soc., vol. 92, no. 440, pp. 1413–1421, 1997.
[6] A. Chambolle, R. De Vore, N.-Y. Lee, and B. Lucier,
“Nonlinear wavelet image Processing: Variational prob-
lems, compression, and noise removal through wavelet
shrinkage,” IEEE Trans. Image Process., vol. 7, no. 3, pp.](https://image.slidesharecdn.com/ijri-ece-01-02imageenhancementaideddenoisingusingdualtreecomplexwavelettransform-161122075118/85/Ijri-ece-01-02-image-enhancement-aided-denoising-using-dual-tree-complex-wavelet-transform-3-320.jpg)
![9
International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET)
[7] D. Cho, T. D. Bui, and G. Chen, “Image denoising
based on wavelet shrinkage using neighbor and level de-
pendency,” Int. J. Wavelets, Multiresolution Inf. Process.,
vol. 7, no. 3, pp. 299–311, May 2009.
[8] E. P. Simoncelli and W. T. Freeman, “The steerable
pyramid: A flexible architecture for multi-scale derivative
computation,” in Proc. 2nd Annu. Int. Conf. Image Pro-
cess., Oct. 1995, pp. 444–447.
[9] F. Rooms, W. Philips, and P. Van Oostveldt, “Inte-
grated approach for estimation and restoration of photon-
limited images based on steerable pyramids,” in Proc. 4th
EURASIP Conf. Focused Video/Image Process. Multime-
dia Commun., vol. 1. Jul. 2003, pp. 131–136.
[10] H. Rabbani, “Image denoising in steerable pyramid
domain based on a local laplace prior,” Pattern Recognit.,
vol. 42, no. 9, pp. 2181–2193, Sep. 2009.
[11] S. Foucher, G. Farage, and G. Benie, “Sar image fil-
tering based on the stationary contourlet transform,” in
Proc. IEEE Int. Geosci. Remote Sens. Symp., Jul.–Aug.
2006, pp. 4021–4024.
[12] W. Ni, B. Guo, Y. Yan, and L. Yang, “Speckle suppres-
sion for sar images based on adaptive shrinkage in con-
tourlet domain,” in Proc. 8th World Congr. Intell. Control
Autom., vol. 2. 2006, pp. 10017–10021.
[13] K. Li, J. Gao, and W. Wang, “Adaptive shrinkage for
image denoising based on contourlet transform,” in Proc.
2nd Int. Symp. Intell. Inf. Technol. Appl., vol. 2. Dec.
2008, pp. 995–999.
[14] Q. Guo, S. Yu, X. Chen, C. Liu, and W. Wei, “Shear-
let-based image denoising using bivariate shrinkage with
intra-band and opposite orientation dependencies,” in
Proc. Int. Joint Conf. Comput. Sci. Optim., vol. 1. Apr.
2009, pp. 863–866.
[15] X. Chen, C. Deng, and S. Wang, “Shearlet-based
adaptive shrinkage threshold for image denoising,” in
Proc. Int. Conf. E-Bus. E-Government, Nanchang, China,
May 2010, pp. 1616–1619.
[16] J. Zhao, L. Lu, and H. Sun, “Multi-threshold image
denoising based on shearlet transform,” Appl. Mech. Ma-
ter., vols. 29–32, pp. 2251–2255 , Aug. 2010.
[17] N. G. Kingsbury, “The dual-tree complex wavelet
transform: A new technique for shift invariance and di-
rectional filters,” in Proc. 8th IEEE Digit. Signal Process.
Workshop, Aug. 1998, no. 86, pp. 1 – 4.
[18] D. L. Donoho and I. M. Johnstone, “Threshold selec-
tion for wavelet shrinkage of noisy data,” in Proc. 16th
Annu. Int. Conf. IEEE Eng. Adv., New Opportunities Bi-
omed. Eng. Eng. Med. Biol. Soc., vol. 1. Nov. 1994, pp.
A24–A25.
[19] W. Freeman and E. Adelson, “The design and use of
steerable filters,” IEEE Trans. Pattern Anal. Mach. Intell.,
vol. 13, no. 9, pp. 891–906 , Sep. 1991.
[20] N. G. Kingsbury, “Image Processing with complex
wavelets,” Philos. Trans. Math. Phys. Eng. Sci., vol. 357,
no. 1760, pp. 2543–2560, 1999.
AUTHOR
K. Mounika,
Research Scholar,
Department of Electronics AndCommunication Engineering,
Chiranjeevi Reddy Institute of Engineering and Technology,
Anantapur, A. P, India.
S. Rehana Banu,
Assistant Professor,
Department of Electronics AndCommunication Engineering,
Chiranjeevi Reddy Institute of Engineering and Technology,
Anantapur, A. P, India.](https://image.slidesharecdn.com/ijri-ece-01-02imageenhancementaideddenoisingusingdualtreecomplexwavelettransform-161122075118/85/Ijri-ece-01-02-image-enhancement-aided-denoising-using-dual-tree-complex-wavelet-transform-4-320.jpg)
Ijri ece-01-02 image enhancement aided denoising using dual tree complex wavelet transform
- 1. 6 International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET) IMAGE ENHANCEMENT AIDED DENOISING USING DUAL TREE COMPLEX WAVELET TRANSFORM K. Mounika1 , S. Rehana Banu2 . 1 Research Scholar,Department of Electronics AndCommunication Engineering, Chiranjeevi Reddy Institute of Engineering and Technology Anantapur, A. P, India. 2 Assistant Professor,Department of Electronics AndCommunication Engineering, Chiranjeevi Reddy Institute of Engineering and Technology Anantapur, A. P, India. *Corresponding Author: K. Mounika, Research Scholar, Department of Electronics AndCom- munication Engineering, Chiranjeevi Reddy Institute of Engineering and Technology,Anantapur, A. P, India. Email: [email protected]. Year of publication: 2016 Review Type: peer reviewed Volume: I, Issue : I Citation: K. Mounika, Research Scholar"Image Enhance- ment Aided Denoising Using Dual Tree Complex Wavelet Transform" International Journal of Research and Inno- vation on Science, Engineering and Technology (IJRISET) (2016) 06-09 INTRODUCTION The central idea to wavelets is to analyze (a signal) ac- cording to scale. Imagine a function that oscillates like a wave in a limited portion of time or space and vanishes outside of it. The wavelets are such functions: wave-like but localized. One chooses a particular wavelet, stretches it ( to meet a given scale) and shifts it, while looking into its correlations with the analyzed signal. This analysis is similar to observing the displayed signal (e.g., printed or shown on the screen) from various distances. The signal correlations with wavelets stretched to large scales reveal gross (“rude”) features, while at small scales fine signal structures are discovered. It is therefore often said that the wavelet analysis is to see both the forest and the trees. In such a scanning through a signal, the scale and the position can vary continuously or in discrete steps. The latter case is of practical interest in this thesis. From an engineering point of view, the discrete wavelet analysis is a two channel digital filter bank (composed of the lowpass and the highpass filters), iterated on the lowpass output. The lowpass filtering yields an approximation of a sig- nal (at a given scale), while the highpass (more precisely, bandpass) filtering yields the details that constitute the difference between the two successive approximations. A family of wavelets is then associated with the bandpass, and a family of scaling functions with the lowpass filters. This concept is explained in Section. At the moment, we address only the general aspects. several wavelets are shown that are obtained from the mother wavelet ψ(x) = (1 − 2x2)e−x2; this wavelet is the second derivative of a Gaussian function and is called the Mexican hat. Its first use was in computer vision, for mul- tiscale edge detection. The origins of the wavelet analysis can be traced to the 1909 Haar wavelet (that was not called by that name then) and various “atomic decompositions” in the history of mathematics. For a comprehensive review, see [Mey- er93, p.13-31]. The current use of the name “wavelet” is due to Grosman’s and Morlet’s work on geophysical signal processing, which led to the formalization of the continu- ous wavelet transform [Grosman84]. In the development of wavelets, the ideas from A few wavelets obtained from the Mexican hat mother wavelet ψ(x) = (1 − 2x2 )e−x2 . Abstract This paper presents a novel way to reduce noise introduced or exacerbated by image enhancement methods, in par- ticular algorithms based on the random spray sampling technique, but not only. According to the nature of sprays, output images of spray-based methods tend to exhibit noise with unknown statistical distribution. To avoid inappropri- ate assumptions on the statistical characteristics of noise, a different one is made. In fact, the non-enhanced image is considered to be either free of noise or affected by non-perceivable levels of noise. Taking advantage of the higher sensi- tivity of the human visual system to changes in brightness, the analysis can be limited to the luma channel of both the non-enhanced and enhanced image. Also, given the importance of directional content in human vision, the analysis is performed through the dual-tree complex wavelet transform , lanczos interpolator and edge preserving smoothing filters. Unlike the discrete wavelet transform, the DTWCT allows for distinction of data directionality in the transform space. For each level of the transform, the standard deviation of the non-enhanced image coefficients is computed across the six orientations of the DTWCT, then it is normalized. Keywords: dual-tree complex wavelet transform (DTWCT), lanczos interpolator, edge preserving smoothing filters. International Journal of Research and Innovation in Electronics and Communication Engineering (IJRIECE)
- 2. 7 International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET) LITERATURE Although the field of image enhancement has been active since before digital imagery achieved a consumer status, it has never stopped evolving. The present work introduc- es a novel multi-resolution denoising method, tailored to address a specific image quality problem that arises when using image enhancement algorithms based on random spray sampling. While inspired by the peculiar problem of such methods, the proposed approach also works for other image enhancement methods that either introduce or exacerbate noise. This work builds and expands on a previous article by Fierro et al. [1]. Due to the peaked nature of sprays, a common sideeffect of image enhancement methods that utilize spray sam- pling is the introduction of undesired noise in the output images. The magnitude and statistical characteristics of said noise are not known a-priori, instead they depend on several factors, like image content, spray properties and algorithm parameters. Among image denoising algorithms, multi-resolution methods have a long history. Independently of the specific transform used, the general assumption in multi-resolution shrinkage is that image data gives rise to sparse coefficients in the transform space. Thus, denoising can be achieved by compress- ing (shrinking) those coefficients that compromise data sparsity. Such process is usually improved by an elabo- rate statistical analysis of the dependencies between co- efficients at different scales [17-20]. Yet, while effective, traditional multi-resolution methods are designed to only remove one particular type of noise (e.g. Gaussian noise). Furthermore, only the input image is assumed to be giv- en. Due to the unknown statistical properties of the noise introduced by the use of sprays, traditional approaches do not find the expected conditions, and thus their action becomes much less effective. The Dual-Tree Complex Wavelet Transform The dual-tree difficult wavelet transform (CWT) is a some- what existing improvement to the discrete wavelet trans- form (DWT), with dynamic additional things: It is almost shifting invariant and directionally selective in two and progressive dimensions. It scopes this with a idleness factor of only 2d for d-dimensional signals, which is sig- nificantly lessor than the undecimated DWT. The mul- tidimensional (M-D) dual-tree CWT is non discrete how- ever is created on a computationally operative, divisible filter bank (FB). This tutorial debates the principle behind schedule the dual-tree transform, shows how difficult wavelets with decent things can be planned, and displays a kind of taxes in signal and image processing. We usage the difficult number symbol C in CWT to avoid mispercep- tion with the often-used acronym CWT for the (different) continuous wavelet transform. THE WAVELET TRANSFORM AND MULTISCALE ANAL- YSIS Since its progress 20 years ago, the wavelet variation has been misused with unlimited achievement through the scope of signal dealing out habits, in the development, regularly redefining the state-of-the-art arrangement [10], [11]. In a outer layer, the DWT changes the really vacillat- ing sinusoidal root drives of the Fourier transform using a static of in the vicinity fluctuating basis meanings called wavelets. In the standard site, the wavelets are pressed and moved types of a vital, real-valued band pass wavelet !(t ). When foolishly carefully chosen and combined with changes of a real-valued low-pass scaling function "(t ), they system an orthonormal beginning enlargement for one-dimensional (1-D) real-valued continuous-time sig- nals [20]. That is, any determinate drive analog signal x(t ) can be disintegrated in terms of wavelets and scaling functions via The scaling coefficients c(n) and wavelet coefficients d( j, n) are considered via the middle products. They arrange for a time-frequency analysis of the signal by computing its frequency relaxed (organised by the scale factor j) at dissimilar times (organised by the time shift n). There occurs a same effective, right time difficulty algorithm to compute the coefficients c(n) and d( j, n) from a fine-scale demonstration of the signal (repeatedly purely N examples) and vice versa created on dual octave-band, discrete-time FBs that recursively placed on a discrete- time low-pass filter h0(n), a greater-pass filter h1(n), and up sample and down sample processes. Grasp “Real-Val- ued Detached Wavelet Transform and Filter Banks” for further contextual on wavelets, FBs, and their plan. EXPERIMENTAL RESULTS Proposed Block diagram Input Low contrast noisy image.
- 3. 8 International Journal of Research and Innovation on Science, Engineering and Technology (IJRISET) Contrast enhanced noisy image Noise removal output with dtcwt Bilateral filter output Final contrast enhanced output Input Low contrast noisy image Bilateral filter output Final contrast enhanced output CONCLUSION This project presents a noise reduction method based on Dual Tree Complex Wavelet Transform coefficients shrinkage. The main point of novelty is represented by its application in post-processing on the output of an image enhancement method (both the non enhanced image and the enhanced one are required) and the lack of assump- tions on the statistical distribution of noise. On the other hand, the non-enhanced image is supposed to be noise- free or affected by non perceivable noise. REFERENCES [1] M. Fierro, W.-J. Kyung, and Y.-H. Ha, “Dual-tree com- plex wavelet transform based denoising for random spray image enahcement methods,” in Proc. 6th Eur. Conf. Col- our Graph., Imag. Vis., 2012, pp. 194–199. [2] E. Provenzi, M. Fierro, A. Rizzi, L. De Carli, D. Ga- dia, and D. Marini, “Random spray retinex: A new retinex implementation to investigate the local properties of the model,” vol. 16, no. 1, pp. 162–171, Jan. 2007. [3] E. Provenzi, C. Gatta, M. Fierro, and A. Rizzi, “A spa- tially variant whitepatch and gray-world method for color image enhancement driven by local contrast,” vol. 30, no. 10, pp. 1757–1770, 2008. [4] Ø. Kolås, I. Farup, and A. Rizzi, “Spatio-temporal retinex-inspired envelope with stochastic sampling: A framework for spatial color algorithms,” J. Imag. Sci. Technol., vol. 55, no. 4, pp. 1–10, 2011. [5] H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, “Adaptive bayesian wavelet shrinkage,” J. Amer. Stat. As- soc., vol. 92, no. 440, pp. 1413–1421, 1997. [6] A. Chambolle, R. De Vore, N.-Y. Lee, and B. Lucier, “Nonlinear wavelet image Processing: Variational prob- lems, compression, and noise removal through wavelet shrinkage,” IEEE Trans. Image Process., vol. 7, no. 3, pp.
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