International Conference on Advanced Language Processing and Web Information Technology
Speckle Reducing Anisotropic Diffusion based on Directions of Gradient Hye Suk Kim, Keon Hee Park, Hyo Sun Yoon and Guee Sang Lee Department of Computer Science, Chonnam National University, 300 Youngbong-dong, Buk-gu, Gwangju 500-757, Korea
[email protected] ,
[email protected]
rise to a speckle removal filter. The despeckle filter based on anisotropic diffusion was introduced by Perona and Malik[2] and has constituted since then a powerful tool for signal and image enhancement. Although the existing despeckle filters are termed as “edge preserving” and “feature preserving” there exist major limitations of the filtering approach. First, the filters are sensitive to the size and shape of the filter window. Given a filter window that is too large (compared to the scale of interest), over-smoothing will occur and edges will be blurred. A small window will decrease the smoothing capability of the filter and will leave speckle. In terms of window shape, a square window will lead to corner rounding of rectangular $ features that are not oriented at perfect 90 rotations, for example. Second, the existing filters do not enhance edges - they only inhibit smoothing near edges. When any portion of the filter window contains an edge, the coefficient of variation will be high and smoothing will be inhibited. Therefore, noise/speckle in the neighborhood of an edge (or in the neighborhood of a point feature with high contrast) will remain after filtering. Third, the despeckle filters are not directional. In the vicinity of an edge, all smoothing is precluded, instead of inhibiting smoothing in directions perpendicular to the edge and encouraging smoothing in directions parallel to the edge. In this paper, Prewitt mask is employed to detect the directions of gradient and anisotropic diffusion is processed based on directions of gradient. We first analyzed the anisotropic diffusion of Perona and Malik [2] and RAD [3] in Section 2. Section 3 presents the proposed anisotropic diffusion method based on directions of gradient. Section 4 shows the experimental results and conclusions are given in Section 5.
Abstract In this paper, we focus on the problem of speckle removal and edge preservation by means of anisotropic diffusion in ultrasound images. The conventional anisotropic diffusion is performed in four directions (North, South, East, West)without condition in the previous methods [2], [3], [6]. In this paper, a new anisotropic diffusion method based on directions of gradient is proposed. At first, K is calculated by using GHAD (Gradient HostogramBased Anisotropic Diffusion)[6] for anisotropic diffusion. Secondly, Prewitt mask is performed for the magnitude of gradient and directions of gradient. As a result, when the magnitude of gradient is large enough, the pixel is considered as the edge. In the case of edge, the weight of diffusion is selected adaptively according to the directions of gradient. Otherwise, the proposed diffusion is performed at 8-directions (North, South, East, West, North_Left, North_Right, South_Left, South_Right) which are not an edge. Experiments results show that the proposed method can eliminate noise/speckle and can remain contour of edge in ultrasound images with speckle noises.
1. Introduction Anisotropic Diffusion of Ultrasound Constrained by Speckle Noise Model Ultrasound is a low cost, noninvasive imaging modality that has proved popular for many medical applications. However, the coherent nature of ultrasound results in images that are corrupted by speckle noise which reduces the utility of ultrasound for less than highly trained users and also complicates image processing tasks such as feature segmentation. Recently, Yu and Acton[7] have proposed a novel filtering scheme based on the filter first described by Lee and Frost. The authors find a relation between the former and the anisotropic diffusion equation and give
978-0-7695-3273-8/08 $25.00 © 2008 IEEE DOI 10.1109/ALPIT.2008.88
2. Anisotropic Diffusion
198
The inverse proportion edge stopping function in equation (1) can remove the noise in the large area efficiently, because its diffusion is in inverse proportion to gradient. On the other hand, this function can’t preserve edge information. The exponent edge stopping function in equation (2) can remain edge. And the more the number of the diffusion, the more salient noises with high gradient are in equation (2) function. Black et al. [3] used the RAD(Robust Estimation Diffusion) theory to choose a better edge-stopping function.
Anisotropic diffusion algorithms remove noise from an image by modifying the image via a partial differential equation. Perona and Malik [2] replaced the classical isotropic diffusion equation with
w I (i , j , t ) wt
div [ g ( I ) I ]
where I is the gradient magnitude, and g I is an “edge-stopping” function. The expression for the divergence operator simplifies to I t div(c(i, j , t )I ) , c(i, j , t ) g ( I (i, j , t ) ) .
g ( I )
With the finite difference scheme and central differencing in spatial domain, the 2-D anisotropic diffusion equation can be expressed as
I (i, j, t 1)
If ( I ) 2 d K
I (i, j, t ) O [cN I N (i, j, t ) cE I E (i, j, t )]
where 0 d O d 0.25 controls the rate of diffusion and are the mnemonic subscripts for North, South, East ,West . The local image gradients are approximated by directional differences I N (i , j ) I (i 1, j ) I (i , j ) N , S , E ,W
I W (i , j ) I (i , j 1) I (i , j ) I E (i , j ) I (i , j 1) I (i , j ) and
g (| I S |),
CW
g (| IW |), CE
g (| I E |).
3. The proposed method
g ( I )
§ ( I ) · ¸ 1 ¨¨ ¸ © K ¹
e
The proposed method using directions of gradient is summarized as follows. Step 1) K is calculated by using GHAD[6] In this paper, GHAD [6] method uses the appropriate parameter K adopted, because GHAD [6] shown that GHAD outperforms the previous anisotropic diffusion [2], [3], [4], [5] in the restoration of a piecewise constant image corrupted by Gaussian noise. It establishes an explicit connection between the parameter K and the number Q of edge-elements of the filtered image. For example, a 5 x 5 image has 40 possible edgeelements (horizontal directions 1 ~ 20 , vertical
(1)
1 2
(( I / K )2
, the diffusion is carried out.
zero in the edge stopping function shown in equation (3).
This function is chosen to satisfy g ( x) o 0 when x o f so that the diffusion is “stopped” across edges. Perona and Malik [2] suggested two different edge stopping g () functions in their anisotropic diffusion equation (1) and (2). g ( I )
(3)
g (I ) o 0 , and if I K , then g (I ) o 1 . While the edge stopping g () functions of Perona and Malik [2] are performed, its influence does not descend all the way to zero. But RAD[3] can choose a more “robust” norm from the robust statistics literature which does descend to zero. In other words, if 2 ( I ) 2 is larger than K , the gradient descends to
I S (i , j ) I (i 1, j ) I (i , j )
g (| I N |), CS
2
2
Otherwise, the diffusing is not carried out, where K is an edge magnitude parameter. In the anisotropic diffusion method, the gradient magnitude is used to detect an image edge or boundary as a step discontinuity in intensity. If I ! K , then
cS I S (i, j, t ) cw I w (i, j, t )
CN
2 ª ( I )2 º ( I )2 d K ° «1 » K2 ¼ , ®¬ ° otherwise ¯0
(2)
199
The magnitude of gradient is given by equation (5).
directions 21 ~ 40 ), and each one of 40 gradient o magnitudes must appear once and only once in . Fig. 1 illustrates the possible edge-elements of 5 x 5 image.
I i , j 1 1 2 3 21 22 23 I i 1, j 5 6 7 26 27 28 9 10 11 31 32 33 13 14 15 36 37 38 17 18 19 Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
4
24 8 29 12 34 16 39 20
Q
25
Q
30
D
Q
Q
Q
35
Q
40
Q
Q
y
Gx
)u
180 pi
(6)
C category $ category D
Q
G
arctan(
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
(5)
The directions of gradient are given by equation (6).
I i, j
Q
G x2 G y2
M
45$
90$
Q
category A
Fig. 1. The possible edge-elements of 5 X 5 image
0
category B
45$
90$
category A
In GHAD, the user specifies the desired number
Q of edge-elements in the final filtered signal/image.
Fig. 2. Directions of gradient using Prewitt mask operations
Alternatively, the user can specify the proportion Y Q / n , where n is the number of possible edge-elements of the image. In each diffusion iteration, GHAD computes the absolute gradient values throughout the image. These gradient values are sorted in increasing order, generating the ordered
Directions of gradient shown in Fig. 2. are classified as four categories. Category A) The difference of the pixel values in the East direction and the West direction are very small, the difference of the pixel values in the North direction and the South direction is large. Category B) The difference of the pixel values in the North _ Right direction and the South_ Left direction are very small, the difference of the pixel values in the North_ Left direction and the South _ Right direction is large. Category C) The difference of the pixel values in the North direction and the South direction are very small, the difference of the pixel values in the East direction and the West direction is large. Category D) The difference of the pixel values in the North_ Left direction and the South _ Right direction are very small, the difference of the pixel values in the North _ Right direction and the South_ Left direction is large.
o
sequence ( 0 , ... n1 ) . In order to make the output image to have Q edge elements, GHAD set n 1 in every iteration. K
With this setting, the diffusion considers Q frontiers between neighboring pixels as edge-elements to be preserved, and n Q frontiers as homogeneous regions where the noise must be suppressed. All above criteria try to find a parameter K to appropriately separate edges from noises. However, we argue that the classification of a pair of neighboring pixels in “edge” or “noise” depends not only on the image I to be filtered, but also on the desired scale of the filtered image. If the user wants to obtain a coarsescale image with only the key edges, a large K should be specified. And if the user wants to obtain a finescale image with all the detailed edges, a small K should be chosen.
Step 3) Multiply edge stopping function by Weight which is decided adaptively based on directions of gradient. As a result of step 2, when the magnitude of gradient is large than threshold, the pixel is considered as the edge. In the case of edge, the 8 neighboring pixels of the edge belong to one of three kinds of direction patterns A, B and C. A pattern is that the pixels belong to the similar direction of the gradient of
Step 2) Directions of gradient is detected by using Prewitt mask operations Prewitt operators is organized as follows:
Gx
ª 1 0 1º « 1 0 1» G « » y «¬ 1 0 1»¼
ª1 1 1º «0 0 0» « » «¬ 1 1 1»¼
(4)
200
the edge. B pattern is that the pixels belong to the perpendicular direction of the gradient of the edge. C pattern is that the pixels belong to the other directions except for the similar direction and the perpendicular direction of the gradient of the edge. And as a result of step 2, when the magnitude of gradient is not large than threshold, the pixel is not an edge. Also, C pattern is used for diffusion in this case. According to the pattern, diffusion is performed adaptively to preserve edges and remove noises.
g (I )
1 § ( I 1 ¨¨ © K
)· ¸¸ ¹
2
u weight
diagonal edge
diagonal edge
(category B)
(category D)
vertical edge
vertical edge
( category C )
( category C )
horizontal edge (category A) (a) Original image
(7)
In case of A pattern: weight can be a value between 1.1 and 1.5 in proportion to the magnitude of gradient. In case of B pattern: weight is 0. In case of C pattern: weight is 1.
(b) Noisy image
(c) Proposed method Smudged edge
smudged edge
4. Experiments result
Impulse noise
Impulse noise
Fig. 3 depicts the restoration of a piecewise constant image corrupted by Gaussian noise. The original image Fig. 3(a) has background gray level 184, with edge with intensities 117 ~and 119. This image was converted to floating-point range [0, 1], and the zeromean Gaussian noise with standard deviation 0.06 was added, resulting in Fig. 3(b). The inverse proportion edge stopping function of Perona and Malik eliminates many important edges when iterated until convergence (Fig. 3 (d)). This happens because the grayscale intensity leaks continuously through smooth edges. RAD could neither completely eliminate noise nor preserve edges sharp. And impulse noise/speckle in the neighborhood of an edge (or in the neighborhood of a point feature with high contrast) remain after filtering (Fig. 3 (e)) Otherwise, filtering the noisy image Fig. 3(c) by proposed method with Y = 0.3% (number of iterations t 10 ), the Fig. 3(c) was obtained (PSNR=38.4726dB). The image quality of the proposed method is better than those of Fig. 3(d) and Fig. 3(e). The proposed method is the best one in remaining contour of edge and removing noise between anisotropic diffusions methods.
(d) Inverse proportion method
(e) RAD
Fig. 3. Restoration of a piecewise constant image corrupted by Gaussian noise Experiments results show that the image quality of the proposed method based on directions of gradient over conventional methods can be better up to 3.1509~8.9261(dB). Table 1. Comparison of PSNR Types of diffusions proposed method Inverse proportion RAD
Diffusion number 10
K ratio PSNR (dB) 38.4726 35.3217 29.5465
(Y ) 0.3
Fig. 4 shows the result of applying the lena image. Lena image was converted to floating-point range [0, 1], and the zero-mean Gaussian noise with standard deviation 0.1 was added, resulting in Fig. 4(b). Experiments results show that the PSNR of the proposed method based on directions of gradient over conventional methods can be better up to 0.4179 ~ 3.1069 (dB).
201
(a) Ultrasound image (b) Proposed method (c) Inverse proportion edge stopping function (d) RAD
5. Conclusion (a)
(b) smudged edge
(c)
The anisotropic diffusion filters do not enhance edges. When any portion of the filter window contains an edge, the coefficient of variation will be high and smoothing is inhibited. Therefore, noise/speckle in the neighborhood of an edge is remained after filtering. To overcome the shortcoming of these conventional anisotropic diffusion methods, a new anisotropic diffusion method based on directions of gradient is proposed. In this paper, Prewitt mask is performed for directions of gradient. According to the directions of gradient, diffusion is performed adaptively using weight to preserve edges and remove noises. Experiments results show that the conventional inverse proportion method[2] and RAD[3][6] cannot remain contour of edge. However, the proposed method is performed adaptively to preserve edges and remove noises. For the future, we will apply the model to special local pathological changes, where we can hope preferable results.
impulse noise
(d)
(e)
Fig. 4. Experiments Result with lena image (a) Original image (b) Noisy image (c) Proposed method (d) Inverse proportion edge stopping function (e) RAD
Table 2. Comparison of PSNR Types of diffusions
Diffusion number
(Y )
30.683 9
proposed method Inverse proportion
K ratio
PSNR (dB)
15
RAD
0.3
30.266 27.570 7
6. References
Our method has been tested by using ultrasound image with speckle noise. Speckle, a form of multiplicative, locally correlated noise, plagues imaging applications such as medical ultrasound image interpretation. For images that contain speckle, a goal of enhancement is to remove the speckle without destroying important image features. Fig. 5 (a) shows the ultrasound image of a liver with speckle noise. Experiments results show that the proposed method can eliminate noise/speckle and can remain contour of edge in ultrasound images with speckle noises.
[1] A. P. Witkin, “Scale-Space Filtering,” Proc. 8th Int. Joint Conf. Art. Intelligence, vol. 2, pp. 1019–1022, 1983. [2] P. Perona and J. Malik, “Scale-Space and Edge Detection Using Anisotropic Diffusion,” IEEE. Trans. Patt.Anal. and Machine Intell., vol. 12, no. 7, pp 629–639, 1990. [3] M. J. Black, G. Sapiro, D. H. Marimont and D. Hegger, “Robust Anisotropic Diffusion,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 421–432, Mar. 1998. [4] F. Voci, S. Eiho, N. Sugimoto and H Sekiguchi, “Estimating the Gradient Threshold in the Perona-Malik Equation,” IEEE Signal Processing Magazine, pp. 39–46, May 2004. [5] J. Canny, “A Computational Approach to Edge Detection,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 679–698, 1986.
(a) smudged edge
impulse noise
[6] Hae Yong Kim, “An Anisotropic Diffusion with Meaningful Scale Parameter”, Personal Communication. , 2006.
(b)
(c)
[7] Y. Yu and S. Acton, “Speckle reducing anisotropic diffusion,” IEEE Trans. Image Process., vol. 11, no. 11, pp. 1260-1270, Nov. 2002. [8] Hye Suk Kim, Jae Myeong Yoo, Mi Seon Park, Toan Nguyen Dinh and Guee Sang Lee, “ An anisotropic Diffusion
(d)
Fig. 5. Visual comparison of experiments result in the ultrasound image
202
Based on Diagonal Edges” The 9th International Conference on Advanced Communication Technology, pp. 384-388, 2007. [9] shujun FU, Qinqi RUAN, Shujun FU, Wenqia Wang, Yu Li, “A compound Anisotropic Diffusion for Ultrasonic Image Denosing and Edge Enhancement” ISCAS, Page(s):2779 2782 Vol. 3 Circuits and Systems, 2005. [10] Krissian, K.; Westin, C.-F.; Kikinis, R.; Vosburgh, K. G.; “Oriented Speckle Reducing Anisotropic Diffusion” Image Processing, IEEE Transactions on Volume 16, Issue 5, May 2007 Page(s):1412 – 1424 Digital Object Identifier 10.1109/TIP.2007.
203
