Double Bilateral Filtering for Image Noise Removal
Double Bilateral Filtering for Image Noise Removal Herng-Hua Chang?and Woei C.Chu?
Institute of Biomedical Engineering
National Yang-Ming University
Beitou112,Taipei,Taiwan
Abstract
Bilateral?ltering is a popular denoising technique that smooths images while preserving edges by means of a non-linear combination of adjacent pixel values.We propose a double bilateral?lter that extends the classical bilateral ?ltering for image noise removal.A new median-metric weighting function is introduced by incorporating a median ?lter into a second bilateral?lter.Balancing between two bilateral?lters,the double bilateral?lter is shown to per-form better than or at least equal to the traditional bilateral ?lter.A variety of images contaminated by various degrees of Gaussian,impulse,and mixed noise were used to assess the performance of this new?ltering method.It is indicated that the double bilateral?lter outperformed several existing methods in both visual image quality and restored signal quantity.
1.Introduction
Noise reduction and elimination is the process of remov-ing noise from a deteriorated image while keeping its fea-tures intact.It is one of the major concerns and fundamental operations in computer vision and image processing.Im-ages can pick up noise from a variety of sources:during acquisition and transmission,due to cameras’quality,il-lumination,resolution,and calibration.Noise presence is exhibited by displeasing information that is not related to the scene under study.For most typical applications,im-age noise can be adequately modeled with additive Gaus-sian noise and impulse noise.
Additive Gaussian noise is usually characterized by adding to each pixel a random value from a zero-mean
?This work was supported in part by the National Science Council under Research Grant No.NSC96-2811-E-010-001and NSC97-2811-B-010-028,E-mail:emwave313@https://www.wendangku.net/doc/c215076421.html,
?Professor of Biomedical Engineering,E-mail:wchu@https://www.wendangku.net/doc/c215076421.html,.tw Gaussian distribution,whose variance determines the inten-sity of the corrupting noise.This zero-mean property en-ables such noise to be removed by locally averaging pixel values.In particular,Gaussian low-pass?ltering computes a weighted average of pixel values in the neighborhood in such a way that the weights decrease with distance from the kernel center.Though the Gaussian?lter smoothes noise quite ef?ciently edges are blurred signi?cantly.
One of the nonlinear methods to?ltering noise while pre-serving the sharpness is the anisotropic diffusion technique [6].In their approach,pixel values are averaged from neigh-borhoods,whose size and shape depend on local image vari-ation that is measured at every point.The diffusion coef?-cient is chosen to be an appropriate function of the image gradient in such a way as to encourage smoothing within a region in preference to smoothing across the boundaries.It requires iterative steps to solve partial differential equations that may raise issues of stability depending on the compu-tational situation.
A noniterative scheme for edge preserving smoothing is the bilateral?ltering[8]and SUSAN[7]based on the orig-inal idea of Overton and Weymouth[4].Though their al-gorithms were developed independently,the essence is the same:combining both geometric closeness in the spatial domain and gray value similarity in the range as a nonlin-ear?lter for image denoising.It has been shown that the bilateral?lter performed effectively in Gaussian noise sup-pression and it bas been the object of further studies[3,5]. Zhang and Allebach[9]proposed an adaptive bilateral?lter for sharpness enhancement and noise removal with a pa-rameter training procedure.In addition,the relationship be-tween bilateral?ltering and anisotropic diffusion has been investigated by Barash[1].
A common type of impulse noise is the salt-and-pepper noise that is characterized by replacing a portion of pixel values with either the maximum or minimum value in the dynamic range.The median?lter was once the most pop-ular nonlinear?lter for removing impulse noise due to its amazing ef?cacy and computational ef?ciency.Recently,
2009 World Congress on Computer Science and Information Engineering
Garnett et al.[2]proposed a trilateral?lter algorithm that extended the bilateral?ltering for removing impulse noise of random values.
In this paper,we propose a new noise removal algorithm based on a double bilateral?lter framework that extends the classical bilateral?lter.The double bilateral?lter consists of two independent bilateral?lters that work simultaneously to constitute the restored result.A median-metric compo-nent is introduced to the weighting function of a new bilat-eral?lter by incorporating a median?lter into the nonlinear average of adjacent pixels.By balancing between two bi-lateral?lters,a?exible and robust?ltering framework is obtained that produces more satisfactory results.
2.Bilateral Filtering
The idea of the bilateral?lter[8]is to combine gray lev-els based on both the geometric closeness and photometric similarity that is in favor of near values to distant values in both domain and range.Two weighting functions regard-ing spatial and radiometric are designed to replace a pixel value with an average of similar and nearby pixel values in a(2N+1)×(2N+1)neighborhood.In theory,any shape of weighting functions can be used but it is usually a Gaus-sian function in terms of the Euclidean distance between the arguments.More speci?cally,let(θx,θy)be the location of the pixel under consideration and
Ψθ
x,θy ={(μx,μy):(μx,μy)∈[θx?N,θx+N]
×[θy?N,θy+N]}(1)
be the pixels in the neighborhood of(θx,θy).The weight-ing functions for the spatial and radiometric components are de?ned respectively as
W Sθ
x,θy (μx,μy)=exp
?
|(μx,μy)?(θx,θy)|2
2σ2
S
(2)
and
W Rθ
x,θy (μx,μy)=exp
?
|I(μx,μy)?I(θx,θy)|2
2σ2R
(3)
where I(·,·)is the intensity value at the given position. Then,the ensemble weight in the bilateral?lter is the prod-uct of(2)and(3):
Wθ
x,θy (μx,μy)=W Sθ
x,θy
(μx,μy)W Rθ
x,θy
(μx,μy).(4)
In practice,each pixel is?ltered using normalized weights
as
?I(θ
x,θy)=
(μx,μy)∈Ψ
Wθ
x,θy
(μx,μy)I(μx,μy)
(μx,μy)∈Ψ
Wθ
x,θy
(μx,μy)
(5)
(a)
(b)
(c)
Figure1.Sensitivity analyses in PSNR of pa-
rameterβin restoring images corrupted by
different degrees of additive Gaussian noise
using the double bilateral?lter.(a)Baboon
image corrupted withσ=10restored using
σS=5andσM=σR=15.(b)Lena image
corrupted withσ=60restored usingσS=3
andσM=σR=180.(c)Bird image cor-
rupted withσ=40restored usingσS=3and
σM=σR=160.
(a)
(b)
(c)
Figure 2.Restoration of Bird image corrupted by additive Gaussian noise using the double bilateral ?lter.(a)Original Bird image (512×512).(b)The image corrupted by Gaussian noise with σ=40.(c)The result after double bilateral ?ltering.
where ?I
(θx ,θy )is the ?ltered image at location (θx ,θy ).The parameters σS and σR are used to adjust the in?uence of W S and W R ,respectively.They can be treated as rough thresholds for identifying pixels suf?ciently close or similar to the pixel being ?ltered.
3.Double Bilateral Filtering
Our double bilateral ?lter is motivated by the bilateral ?lter,which combines range ?ltering with domain ?ltering for edge preserving smoothing.We introduce a new range ?lter that incorporates the median ?lter into the weighting function as
W M
θx ,θy (μx ,μy )=exp ?|I M (μx ,μy )?I M (θx ,θy )|2
2σ2M
(6)
where W M
θx ,θy represents the median-metric component and I M (·,·)represents the median intensity in a (2D +1)×(2D +1)neighborhood of the corresponding position (·,·).Accordingly,the ensemble weight of the second bilateral ?lter is de?ned as
W θx ,θy (μx ,μy )=W S θx ,θy (μx ,μy )W M
θx ,θy
(μx ,μy ).(7)
Table https://www.wendangku.net/doc/c215076421.html,parison of restoration results in PSNR on Bird and Baboon images corrupted by additive Gaussian noise with σ=10be-tween different ?ltering methods.
Method
Bird Baboon Median ?lter 3×325.3820.61Median ?lter 5×523.1319.55Gaussian ?lter 3×325.4627.56Anisotropic ?lter 27.4028.20Bilateral ?lter 27.9028.24Double bilateral ?lter
28.71
29.27
This new bilateral weighting function is used to ?lter the median intensity of each pixel through normalization as
?I M (θx ,θy )= (μx ,μy )∈ΨW
θx ,θy (μx ,μy )I M (μx ,μy ) (μx ,μy )∈ΨW
θx ,θy (μx ,μy )
(8)
By combining the bilateral ?lters in (5)and (8),we obtain the double bilateral ?ltering:
I (θx ,θy )=(1?β)?I
(θx ,θy )+β?I M (θx ,θy )(9)
where β,0≤β≤1,is a weight for balancing these two
bilateral ?lters.
4.Experimental Results
In order to study the performance of the double bilateral ?lter,a number of images have been contaminated with a variety of noise.We compared the denoising results with several existing methods in both visual image quality and restored signal quantity.In particular,we measured the peak signal-to-noise ratio (PSNR)de?ned as
P SNR =10·log 10 L i L j I 2
max
i j |I (i,j )?I (i,j )|2
=10·log 10 L i L j 255
2
i j |I (i,j )?I (i,j )|2
(10)where I represents the original L i ×L j image,I represents
the restored image,and I max represents the maximum pos-sible pixel value of the image,which is 255in our 8-bit grayscale case.
4.1.Sensitivity of parameters
We ?rst studied the sensitivity of parameter βto the de-noised results in PSNR while keeping σM =σR on im-ages corrupted by additive Gaussian noise with σ=10,60,
(a)
(b)
(c)
(d)
(e)(f)
Figure https://www.wendangku.net/doc/c215076421.html,parison of the restoration of Lena image corrupted by Gaussian noise.(a)Original image (512×512).(b)Corrupted with σ=30.(c)Restored with the Gaussian ?l-ter.(d)Restored with the anisotropic ?lter.(e)Restored with the bilateral ?lter.(f)Restored with the double bilateral ?lter.
and 40for Baboon,Lena,and Bird images,respectively.As shown in Fig.1,the values of PSNR increased as the contri-bution of βwas getting larger.For slightly degraded image,say σ=10in Fig.1(a),better PSNR results were obtained using smaller values of β.On the other hand,larger val-ues of βwere advantageous for moderately and seriously degraded images [see Figs.1(b)and (c)].Figure 2shows the restoration result of Bird image corrupted by Gaussian noise with σ=40using the double bilateral ?lter.It is clear that a fair amount of noise was eliminated while preserving edge boundaries and ?ne details.
Table https://www.wendangku.net/doc/c215076421.html,parison of restoration results on Lena image corrupted by Gaussian noise.Method
σ=10σ=20σ=30Median ?lter 3×331.0526.6723.98Median ?lter 5×528.7027.1026.06Gaussian ?lter 5×530.6125.0023.68Anisotropic ?lter 32.0325.8723.73Bilateral ?lter 31.8528.0526.69Double bilateral ?lter
32.58
28.82
27.76
4.2.Experiments with additive Gaussian
noise
To understand the ability of denoising additive Gaussian noise,we compared the double bilateral ?lter with the clas-sical median ?lter,Gaussian ?lter,anisotropic diffusion ?l-ter [6],and bilateral ?lter [8]on a variety of noisy images.Table 1summarizes the restoration results in PSNR on Bird and Baboon images corrupted by additive Gaussian noise with σ=10,which indicates that the double bilateral ?l-ter outperformed other methods.Further comparison was made by ?ltering Lena image corrupted by a series of addi-tive Gaussian noise with σranging from 10to 30.As pre-sented in Table 2,the double bilateral ?lter produced higher PSNR values in all degrees of Gaussian noise relative to other methods.Figure 3illustrates the restoration results of Lena image corrupted by additive Gaussian noise with σ=30using the Gaussian ?lter,anisotropic ?lter,bilat-eral ?lter,and double bilateral ?https://www.wendangku.net/doc/c215076421.html,paring with other methods,it is indicated that the double bilateral ?lter re-moved a large amount of noise while preserving edges and sharpness,e.g.,the hair.
4.3.Experiments with mixed noise
For completeness,we studied the ability of the double bi-lateral ?lter in restoring images contaminated by a series of mixed noise.Fig.4shows the restoration results of Bird im-age corrupted by a mix of Gaussian noise with σ=20and impulse noise with p =20%using the bilateral ?lter and double bilateral ?lter https://www.wendangku.net/doc/c215076421.html,paring with the origi-nal,it is observed that a certain amount of noise was elim-inated while preserving edge boundaries and tiny details.The double bilateral ?lter outperformed the bilateral ?lter regarding the PSNR value (21.25vs.17.38dB)and the vi-sual image quality was more https://www.wendangku.net/doc/c215076421.html,stly,we compared the double bilateral ?lter with other methods in denoising Bird image corrupted by mixed noise with different degrees of additive Gaussian noise (σ=10,30,50,and 80)and a constant impulse noise (p =20%).As illustrated in Fig.
(a)
(b)
(c)
Figure4.Restoration of Bird image corrupted
by mixed noise.(a)The image(360×360)
corrupted by a mix of Gaussian noise with
σ=20and impulse noise with p=20%.(b)
Restored with the bilateral?lter(PSNR=17.38
dB).(c)Restored with the double bilateral?l-
ter(PSNR=21.25
dB).
https://www.wendangku.net/doc/c215076421.html,parison of restoration results
in PSNR on Bird image corrupted by mixes of
different degrees of additive Gaussian noise
(σ=10,30,50,and80)and a constant impulse
noise(p=20%).
5,the PSNR values of using the double bilateral?lter were
quite higher than other methods in all experiments.
5.Conclusions
A new noise removal?lter based on the extension of
the bilateral?lter has been introduced.It is indicated that
the proposed method is always better than or at least equal
to the performance of the traditional bilateral?lter in any
denoising situation.We have studied the effect of the pa-
rameterβon the restoration results using a variety of noisy
images.Due to the incorporation of the median?lter into
the denoising framework,this double bilateral?lter outper-
formed several existing methods in both visual image qual-
ity and restored signal quantity.Though the double bilat-
eral?lter was developed for improving the performance of
?ltering Gaussian noise,we found that it was also capable
of handling impulse noise.In addition,our noise removal
algorithm had better performance in restoring images cor-
rupted by the combination of Gaussian and impulse noise.
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