Y.-M.R. Huang et al. (Eds.): PCM 2008, LNCS 5353, pp. 923–926, 2008.
© Springer-Verlag Berlin Heidelberg 2008
A Robust Denoising Filter with Adaptive Edge Preservation
Li-Cheng Chiu and Chiou-Shann Fuh
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan 10617
{d93010,fuh}@csie.ntu.edu.tw
Abstract. To preserve edge information during noise removal, bilateral filtering is designed as a powerful denoising filter using Gaussian function with geometric and photometric information. Unfortunately, peak noise will be regarded as effective high-frequency information and degrade the performance of bilateral filter. We design a new robust denoising filter employing image pre- processing and high-frequency edge detection. This filter can preserve details and remove noise simultaneously and it performs better than bilateral filter.
Keywords: Noise removal, bilateral filtering, peak noise, robust denoising filter.
1 Introduction
Bilateral filter [1] is an adaptive denoising filter using Gaussian kernel with distance and intensity information of pixels. Although bilateral filter has satisfactory results in most scenarios, its capability of noise removal will be reduced when peak noise is introduced. Besides bilateral filter, many edge-preserving algorithms [2-6] are proposed to remove noise. Anisotropic diffusion [2-4] uses heat flow to detect image edges during noise removal. Vector directional filters (VDFs) [5, 6] separate the processing of vector-valued signals into directional and magnitude processing.
In this paper, we propose a robust edge preservation and noise removal model. Our proposed method is non-iterative and takes both spatial closeness and photometric similarity into consideration.
2 Review of Bilateral Filter
Bilateral filter is proposed by Tomasi and Manduchi [1] and defined as ( ) = -1( ) ( ) ( , ) ( ( ), ( )) d , h x k x ∞ ∞ f ξ ξc x s f ξ f x ξ
−∞ −∞
∫ ∫
(1)where the normalization term is
-
( ) = ( , ) ( ( ), ( )) d . k x ∞ ∞c ξ x s f ξ f x ξ
∫ ∫
−∞ ∞ (2)924 L.-C. Chiu and C.-S. Fuh
An image f(x) is convolved by the termsc( , )ξ x ands f( ( ), ( ))ξ f x . Geometric closeness ( , )
cξ x has a form as
1 ( , ) 2
- ( )
( , ) = 2 d ,
d x
c x e
ξ
ξ σ (3)
where
( , ) = -
d ξ x ξ x (4)
is the Euclidean distance betweenξ and x. Photometric similarity s f( ( ), ( ))ξ f x is defined as
1 ( ( ), ( ))2
- ( )
( ( ), ( )) = 2 r ,
f f x
s f f x e
δ ξ
ξ σ (5)
where
( ( ), ( )) = f f x f( ) - ( )f x
δ ξ ξ (6)
is a measure of distance between the two intensity values f( )ξ andf x( ). Bilateral filter removes noise based on geometric spreadσdand photometric spreadσr. Each pixel will be influenced mainly by a pixel that has spatial closeness and photometric similarity. Therefore σd andσr decide the capability of noise removal and edge preservation for bilateral filter. Unfortunately, σd andσr are hardly chosen for removing peak noise and preserving edges simultaneously.
3 Proposed Denoising Filter
Inspired by the problem of bilateral filter, we propose a robust denoising filter to improve the capability of noise removal. Our model takes the high-frequency component h(x) of image f(x) convolved with Gaussian low-pass filter g(x) as the preserved index for the kernel filter. High-frequency component h(x) is defined as
{ } { }
( ) ( ) ( ) ( ) ( ) ,
h x = ∇ g x ∗f x = ∇g x ∗f x (7)
where∇g x( )denotes the first derivative of Gaussian function and can be described as
1 2
- ( ) 2
( ) 2 t .
x
t
g x x e σ
∇ = −σ (8)
The geometric spreadσtdecides the strength of noise suppression on original image.
Hence our model performs low-pass filtering according to this high-frequency index h(x) and has a form as
( ) = x k-1( )x ∞ ∞ f( ) ( , )d ,ξ ξ x ξ
−∞ −∞
Ω
∫ ∫
Φ (9)A Robust Denoising Filter with Adaptive Edge Preservation 925
where the normalization term is
- -
( ) = ( , )d .
k x ∞ ∞ ξ x ξ
∞ ∞Φ
∫ ∫
(10)The kernel filterΦ( , )ξ x is an adaptive Gaussian function described as
1 ( ) 2
- ( )
( , ) = 2 ,
h x
x e
ξ
ξ δ⋅
Φ (11)
whereδ is geometric spread of kernel filter to determine the smoothness.
Fig. 1 illustrates the simulation results of bilateral filter and our method. The noisy curve in Fig. 1 (a) is smoothed by a Gaussian low-pass filter with geometric spreadσt. The first derivative curves of noisy and smoothed curves in Fig. 1 (b) are obtained from Fig. 1 (a). It is observed that the zero-crossing of peak noise in the first derivative of smoothed curve is relatively lower than that of noisy curve in Fig. 1 (b).
Fig. 1 (c) shows the results of our method and bilateral filter applied on the same noisy curve. Both geometric and photometric spreads of bilateral filter are chosen much large to remove peak noise but the edge is smoothed relatively. On the contrary, our method can achieve similar peak noise removal and preserve the edge better than bilateral filter.
noisy curve sm oothed curve
1st derivative of noisy curve
1st derivative of sm oothed curve noisy curve
result from proposed method result from bilateral filter
(a) (b) (c)
Fig. 1.(a) A Gaussian low-pass filter is applied on a noisy curve. (b) The first derivative curves are obtained from noisy and smoothed curves of (a). (c) Denoised curves are obtained after applying our method and bilateral filter on the same noisy curve.
4 Experimental Results
To quantitatively evaluate the performance of our method, we compare the results of 3×3 median, bilateral filter, and our approach on Lena image which is corrupted by Gaussian noise with amplitudeσ = 5, 10, 15, and 20. Geometric spreadσd and photometric spread σr of bilateral filter are chosen as 1.0 and 2.0.
Parametersσtandδ of our method are 2.5 and 3.0. The results are compared using peak signal-to-noise ratio (PSNR) and signal-to-noise ratio (SNR) as the objective evaluations. Tables 1 shows the experimental data after applying three different methods. It is observed that our method performs the best in the three methods.
926 L.-C. Chiu and C.-S. Fuh
Table 1. PSNR and SNR are obtained by various denoising filters are applied on Lena image
σ = 5 σ = 10 σ = 15 σ = 20
PSNR SNR PSNR SNR PSNR SNR PSNR SNR 3×3 median 19.93 6.70 18.80 5.57 17.88 4.65 17.08 3.85 Bilateral filter 20.36 7.14 19.32 6.10 18.22 4.99 17.94 4.72 Our method 21.07 7.85 19.76 6.54 18.99 5.76 18.59 5.36
5 Conclusion
In conclusion, we propose a robust low-pass filter to obtain excellent edge preservation during noise removal. The pre-processing will take the effective high- frequency components of original images passing through a low-pass filter. The pre- filtering can be adjusted to efficiently recognize fluctuations on original images. The kernel filter will remove noise adaptively based on the extracted high-frequency information. Therefore our method can remove noise and preserve edges excellently.
References
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3. Sapiro, G., Ringach, D.L.: Anisotropic diffusion of multivalued images with application to color filtering. IEEE Trans. Image Processing 5(11), 1582–1586 (1996)
4. Tang, B., Sapiro, G., Caselles, V.: Diffusion of general data on non-flat manifolds via harmonic maps theory: the direction diffusion case. International Journal of Computer Vision 36(2), 149–161 (2000)
5. Trahanias, P.E., Venetsanopoulos, A.N.: Vector directional filters – a new class of multichannel image processing filters. IEEE Trans. Image Processing 2(4), 528–534 (1993) 6. Plataniotis, K.N., Androutsos, D., Venetsanopoulos, A.N.: Vector directional filters: An
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