Image noise removal approach based on subpixel

Image noise removal approach based on subpixel anisotropic diffusion Yanhui Guo Heng-Da Cheng

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Journal of Electronic Imaging 21(3), 033026 (Jul–Sep 2012)

Image noise removal approach based on subpixel anisotropic diffusion Yanhui Guo University of Michigan Department of Radiology Ann Arbor, Michigan 48109 E-mail: [email protected] Heng-Da Cheng Utah State University Department of Computer Science Logan, Utah 84322

Abstract. Image noise removal is the first step in image processing, pattern recognition, and computer vision. A novel algorithm is proposed to remove noise on images based on anisotropic diffusion and subpixel approaches. Firstly, the subpixel difference of an image is defined and the Euler-Lagrange equation is solved. Then, the diffusion equation is solved numerically using an iterative approach. Finally, the noise is removed after the diffusion procedure is finished. The experiments show that the proposed algorithm yields better signal-to-noise ratio and has no blocky effect and less generated speckle noise in the results than the other methods do. In addition, it is easy to implement, takes less iterations, and has low computational complexity. © 2012 SPIE and IS&T. [DOI: 10.1117/1.JEI.21.3.033026]

1 Introduction Digital images play an important role in numerous applications such as satellite television, magnetic resonance imaging, computer tomography, etc. The images are often taken in different conditions. Noise often occurs in imaging systems, image acquisition, transmission, etc.1 It is necessary to apply an efficient denoising technique to compensate for image corruption.2 Image noise removal is often a necessary and first step before the images are analyzed. In addition, it is also one of the most widely investigated topics in image processing, pattern recognition, and image vision.3 Denoising algorithm performance mainly depends on a suitable representation to describe the original image information. Image noise removal remains a challenge since it introduces artifacts and causes blurring.1 Filter is an important method to remove noise on the image, and related literature has been published in this field. Bosco et al.4 proposed a spatio-temporal filter based on Duncan filtering to reduce the image in motion video. The spatio-temporal filter used the noise standard deviation σ to determine the wideness of the filter. Battiato et al.5 presented a spatial noise reduction technique for color filtering array (CFA) data acquired by charge coupled device (CCD)/ Paper 12121 received Apr. 3, 2012; revised manuscript received Aug. 7, 2012; accepted for publication Aug. 13, 2012; published online Sep. 17, 2012. 0091-3286/2012/$25.00 © 2012 SPIE and IS&T

Journal of Electronic Imaging

complementary metal oxide semiconductor (CMOS) image sensors. The method preserved image details using some heuristics related to the human visual system (HVS). The estimates of local texture degree and noise levels were computed to regulate the filter smoothing capability. Partial differential equations (PDEs) have been also studied as a useful tool for image noise removal. Perona and Malik (PM)6 developed an anisotropic diffusion process as a nonlinear image noise removal method, which analogized heat diffusion to adaptively remove the noise of the images. The basic idea of anisotropic diffusion is that it encourages intra-region smoothing and discourages inter-region smoothing at the edges. The decision on local smoothing is based on a diffusion coefficient, which is a function of the local image gradient. When the gradient is low, the smoothing takes place. Meanwhile, the smoothing is suppressed where the gradient is high and an edge exists.7 Many noise removal algorithms have been proposed based on anisotropic diffusion. Reference 7 used fourth-order PDEs for noise removal. A fourth-order PDE was derived by the Laplacian function. The time evolution of these PDEs seeks to minimize a cost functional which was defined as an increasing function of the absolute value of the Laplacian of the image intensity function. Reference 8 extended the traditional anisotropic diffusion filtering method by allowing isotropic smoothing within homogeneous regions and anisotropic smoothing along structure boundaries. It smoothed diffusion tensor images in which direction information needs to be restored following noise corruption and preserved around tissue boundaries. Reference 9 developed an adaptive diffusion scheme to select the threshold of the diffusion process for improving the performance. Anisotropic diffusion filtering process depends on the threshold in the diffusion process, which varies from image to image and even from region to region in an image. This adaptive diffusion scheme applied the central limit theorem to selecting the threshold. Gaussian distribution and Rayleigh distribution were employed to estimate the distributions of objects in images. Regression under such distributions separates the distribution of the major object from other objects, which was used to determine the threshold for the diffusion process. Based on a nonlinear diffusion

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

scale-space framework, Ref. 10 presented semi-implicit schemes in nonlinear diffusion filtering which are stable for all time steps. These schemes used an additive operator splitting (AOS) to guarantee the equal treatment of all coordinates. The AOS can be implemented in arbitrary dimensions, and has good rotational invariance and a computational complexity and memory requirement. Reference 11 proposed a new edge-stopping function using the relations between the anisotropic diffusion and robust statistics to preserve the sharper boundaries and improve the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the edges between the piecewise smooth regions that have been smoothed with anisotropic diffusion. Reference 12 provided the derivation of the speckle reducing anisotropic diffusion (SRAD), which is the edge-sensitive diffusion. SRAD utilized the coefficient of variation in adaptive filtering, which is a function of the local gradient magnitude and Laplacian operators. Reference 13 applied a fourth-order PDE noise suppression method to structural and functional MRI data set for noise removal. Although the above techniques have achieved a good trade-off between noise removal and edge preservation, they tend to cause the processed image to look blocky.14 They can cause the boundaries of different blocks to be falsely recognized as the edges, which actually belong to the smooth areas in the original image. Furthermore, anisotropic diffusion is often limited by the number of iterations to achieve an acceptable result. A variety of research aims to improve the computational efficiency and to develop robust diffusion algorithms. In this paper, we propose a novel noise removal algorithm based on anisotropic diffusion and subpixel approaches. The proposed method was tested on many images, and the experimental results show it can improve the computation speed and achieve better noise removal performance at the same time. The diffusion equation is an Euler—Lagrange equation, which is an increasing function of the absolute value of the intensity change. The subpixel difference of the image is defined and employed, which has a clear physical meaning and could be regarded as a generalization of the second order difference, and the Euler-Lagrange equation is solved based on it. Then, the diffusion equation is solved numerically using an iterative approach. Finally, the noise is removed after the diffusion procedure is finished. In the experiments, we apply the proposed algorithm to numerous images having noise with different levels and show that the proposed algorithm yields higher computation speed, better signal-to-noise ratio (SNR), and good visual effects. We also compare our algorithm with the common PM diffusion algorithm6 and a fourth-order anisotropic diffusion algorithm.7 Results show that the common PM method and the fourth-order method cannot smooth severely noise-contaminated images, while our algorithm can reduce noise and obtain higher SNR. In addition, the proposed algorithm is easy to implement, takes fewer iterations, and has low computational cost. The outline of the paper is as follows. In Sec. 2, at first, a simple introduction of the anisotropic diffusion and subpixel difference is given. It shows that the time evolution of the proposed model seeks to minimize a cost functional of the absolute value of the difference of the intensity function. Journal of Electronic Imaging

Then, we use subpixel difference to solve the algorithm numerically and iteratively. Several examples are presented in Sec. 3 and the conclusion is given in Sec. 4. 2 Proposed Method 2.1 Anisotropic Diffusion Adaptive smoothing is an effective image processing method that has been used for denoising, restoration, and enhancement. One of its drawbacks is that the smoothing can damage image features such as edges, lines, and textures. To avoid such problem, the smoothing has to be adaptively controlled by the amount or direction of smoothing. The anisotropic diffusion6 is a typical adaptive smoothing approach, in which the smoothing process is formulated by a PDE. The anisotropic diffusion is described as follows. Let I denote the image, t the time, and cð•Þ the diffusion coefficient function, the anisotropic diffusion is formulated as:5 ∂I ¼ div½cðj∇IjÞ∇I
: ∂t

(1)

This equation is associated with the following energy functional Z EðIÞ ¼ fðj∇IjÞdΩ; (2) Ω

where Ω is the image support, and fð•Þ ≥ 0 is an increasing function associated with the diffusion coefficient: pffiffiffi f 0 ð sÞ cðsÞ ¼ pffiffiffi ; (3) s where cð•Þ is the diffusion coefficient function. Here the magnitude of the gradient j∇Ij can be used as the variable in coefficient function. Anisotropic diffusion is regarded as an energy spreading process and aims to find the minimum of the energy functional. Since anisotropic diffusion is to diffuse smooth areas faster than less smooth ones, blocky effects will appear when the iterations are not enough. On the other hand, the anisotropic diffusion will blur the image if too many iterations are taken. In addition, speckle noise will be generated and poseprocessing is needed.7 According to the anisotropic diffusion theory, the energy function can be defined with different formulas.13 An energy function is defined over a support of Ω (Ref. 7): Z fðj∇IjÞdΩ; (4) EðIÞ ¼ Ω

where ∇ð•Þ is a second qorder derivative operator, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ∇I ¼ ð∇x I; ∇y IÞ and j∇Ij ¼ ∇x I þ ∇2y I . An Euler-Lagrange equation is constructed to seek the minimum of the energy function. For any function ξ ∈ C∞ ðΩÞ, a cost function ϕðλÞ is defined as: Z ΦðλÞ ¼ fðj∇I þ λ∇ξjÞdxdy: (5) Ω

Then

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

 Z  dΦð0Þ d ¼ fðj∇I þ λ∇ξjÞdxdy dλ dλ Ω λ¼0  Z dfðj∇I þ λ∇ξjÞ ¼  dxdy dλ Ω λ¼0 hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Z df ∇2x ðI þ λξÞ þ ∇2y ðI þ λξÞ   dxdy ¼  dλ Ω λ¼0 Z hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ∇ ξ∇ ðI þ λξÞ þ ∇ ξ∇ ðI þ λξÞ x x y y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  dxdy ∇2x ðI þ λξÞ þ ∇2y ðI þ λξÞ ¼ f0 2 2 Ω ∇x ðI þ λξÞ þ ∇y ðI þ λξÞ λ¼0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 2 0 2 2 Z f0 f ∇x I þ ∇y I ∇x I ∇x I þ ∇y I ∇y I 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ∇x ξ þ ∇y ξ5dxdy 4 2 2 2 2 Ω ∇x I þ ∇y I ∇x I þ ∇y I  Z  0 f 0 ðj∇IjÞ∇y I f ðj∇IjÞ∇x I ∇x ξ þ ¼ ∇y ξ dxdy j∇Ij j∇Ij Ω Z Z ¼ ½cðj∇Ij2 Þ∇x I
∇x ξdxdy þ ½cðj∇Ij2 Þ∇y I
∇y ξdxdy Ω Ω Z Z ∇x ½cðj∇Ij2 Þ∇x I
ξdxdy þ ∇y ½cðj∇Ij2 Þ∇y I
ξdxdy: ¼ Ω

Ω

For all ξ ∈ C∞ ðΩÞ, the Euler-Lagrange equation is ∇x ½cðj∇j2 Þ∇x I
þ ∇y ½cðj∇Ij2 Þ∇y I
¼ 0:

(7)

2.2 Subpixel Method Now, we briefly describe the subpixel approach and its application. In subpixel framework, we not only consider the pixels at the integer grids of the images, but also the pixels between the integer grids, which are called subpixels, and they are interpolated accordingly. Comparing with the methods based on pixel-level, the subpixel-level methods can solve the problems with higher precision.15 Hence, the subpixel approach was applied to many fields as an important complement of the pixel-level methods, such as image edge detection,15–17 image resolution enhancement,18 image registration,19,20 and target detection.21 According to the range of hS , the subpixel PiþhS ;j locates between the pixels Pi;j and Piþ1;j , or the pixels Piþ1;j and Piþ2;j . The relation between the subpixel PiþhS ;j and the pixels Pi;j and Piþ1;j is shown in Fig. 1(a) and 1(b). In Fig. 1(a), hS is less than 1, and in Fig. 1(b), hS is greater than 1. To obtain the subpixel intensity value, numerous interpolation approaches were studied, including linear, nonlinear interpolation, and B-spline interpolation methods.16 In this paper, a linear interpolation is employed to make the computation simple and fast, which is defined as follows: I iþhS ;j ¼ I i;j − hS ðI i;j − I iþ1;j Þ; Journal of Electronic Imaging

(9)

I i;jþhS ¼ I i;j − hS ðI i;j − I i;jþ1 Þ;

(10)

I i;j−hS ¼ I i;j − hS ðI i;j − I i;j−1 Þ;

(11)

(6)

The Euler-Lagrange equation can be solved through the following gradient descent procedure and the diffusion equation Eq. (1) is interpreted as: ∂I ¼ −∇x ½cðj∇Ij2 Þ∇x I
− ∇y ½cðj∇Ij2 Þ∇y I
: ∂t

I i−hS ;j ¼ I i;j − hs ðI i;j − I i−1;j Þ;

and

where I i;j is the intensity of a pixel Pi;j at the coordinates ði; jÞ. I iþ1;j , I i−1;j , I i;jþ1 , and I i;j−1 are the intensities of the neighbors of Pi;j , respectively. I iþhS ;j , I i−hS ;j , I i;jþhS , and I i;j−hS are the intensities of the subpixel neighbors of Pi;j , respectively. The equations are valid for both the cases in Fig. 1(a) and 1(b). The subpixel difference ∇S I, which can be seen as the generalization of the pixel difference, has a clear physical meaning and is described using a discrete Laplacian operation: ∇Sx I i;j ¼

I iþhs ;j þ I i−hs ;j − 2I i;j ; h2s

(12)

∇Sy I i;j ¼

I i;jþhs þ I i;j−hs − 2I i;j ; h2s

(13)

and

(8)

Fig. 1 The illumination of subpixels.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

∇S I i;j ¼ ∇Sx I i;j þ ∇Sy I i;j ¼

and

I iþhs ;j þ I i−hs ;j þ I i;jþhs þ I i;j−hs − 4I i;j ; h2s

where ∇S ð•Þ is the complete subpixel difference, ∇Sx ð•Þ and ∇Sy ð•Þ are the partial differences along x and y directions, respectively. hs is the step size, which can be a fractional or integer number. i ¼ 0; 1; 2; : : : ; H − 1 and j ¼ 0; 1; 2; : : : ; W − 1. H and W are the height and width of the image, respectively. The boundary conditions can be computed symmetrically. I −hS ;j ¼ I 0;j

I H−1þhS ;j ¼ I H−1;j

I i;−hS ¼ I i;0

Thirdly, we compute the subpixel differences of gSx and gSy : ∇Sx gtSxi;j ¼

I tþ1 ¼ I t þ λct ð∇I t Þ∇I t ;

I tiþhs ;j þ I ti−hs ;j − 2I ti;j h2s I ti;jþhs þ I ti;j−hs − 2I ti;j

∇Sy gtSyi;j ¼

gtSyi;jþhS þ gtSyi;j−hS − 2gtSyi;j h2S

:

(25)

h2s

;

(17)

;

h2s

gtSxi;−hS ¼ gtSxi;0

gtSxi;W−1þhS ¼ gtSxi;W−1

gtSy−hS ;j ¼ gtSy0;j

gtSyH−1þhS ;j ¼ gtSyH−1;j

gtSyi;−hS ¼ gtSyi;0

gtSyi;W−1þhS ¼ gtSyi;W−1 :

t t t I tþ1 i;j ¼ I i;j − λS ∇Sx gSxi;j − λS ∇Sy gSyi;j ;

(26)

(27)

(28)

where λS is a rate parameter and 0 < λS ≤ 1∕4. In our experiments, λS is selected as 0.05. In this paper, the anisotropic diffusion is employed to remove noise on the images. The noise removal process can be described as follows: (1) Let the input image be I, set t ¼ 1 and I t ¼ I, and initialize the step size hs . (2) Calculate subpixel difference ∇S I t at time step t. (3) Compute gtSx and gtSy and their subpixel differences ∇Sx gtSx and ∇Sy gtSy . t t (4) Update the image I tþ1 i;j ¼ I i;j − λS ∇Sx gSxi;j − t λS ∇Sy gSyi;j and t ¼ t þ 1; if t reaches the maximum iteration, stop; else go to 2.

(18)

:

gtSxH−1þhS ;j ¼ gtSxH−1;j

Finally, the numerical approximation of the differential equation [Eq. (7)] is computed as:

∇S I ti;j ¼ ∇Sx I ti;j þ ∇Sy I ti;j I tiþhs ;j þ I ti−hs ;j þ I ti;jþhs þ I ti;j−hs − 4I ti;j

gtSx−hS ;j ¼ gtSx0;j

and

and

(19)

The boundary conditions can be computed symmetrically. Secondly, we define gSx and gSy , then compute the components in Eq. (15): gSx ¼ cðj∇S Ij2 Þ∇Sx I;

(20)

gSy ¼ cðj∇S Ij2 Þ∇Sy I;

(21)

gtSxi;j ¼ cðj∇S I ti;j j2 Þ∇Sx I ti;j ;

(22)

Journal of Electronic Imaging

(24)

h2S

and

(16)

where t is the iteration number, λ is a rate parameter (0 < λ ≤ 1∕4) which is used to make the numerical scheme to be stable,6 c is the diffusion coefficient, and ∇I is the second order derivative. In traditional methods, the derivative of I is calculated using the pixels and their spatial neighborhood pixels. In the proposed method, we use the complete subpixel difference ∇S I to replace the second order derivative ∇I. At iteration t, the second order difference ∇S I t is defined as:

¼

gtSxiþhS ;j þ gtSxi−hS ;j − 2gtSxi;j

The boundary conditions of gtSx and gtSy are computed symmetrically

2.3 Numerical Solution A discrete version of the diffusion equation Eq. (7) can be solved numerically using an iterative approach, which is described as:6

∇Sy I ti;j ¼

(23)

(15)

I i;W−1þhS ¼ I i;W−1 :

∇Sx I ti;j ¼

gtSyi;j ¼ cðj∇S I ti;j j2 Þ∇Sy I ti;j :

(14)

3 Experimental Results In this section, we present the results obtained by applying our proposed novel subpixel difference model for image denoising. We test the proposed method on “Lena,” “Fishing Boat,” “Pepper,” “Plane,” “Bacteria,” and “Blood” images having Gaussian noise with different standard deviations using different step size. The denoising performance is evaluated quantitatively using SNR. A step size value is selected according to the biggest SNR value. Then, we compare the proposed method with PM and fourth-order diffusion model using the test images having Gaussian noise with different standard deviations.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

3.1 Step Size Selection The step size hS is taken as a parameter. We will show the results obtained using various values of hS on the images having different level noise. The step size is chosen so that the best denoising performance is obtained. The denoising performance is evaluated by the SNR, which is an important index to measure the images’ quality:3  SNR ¼

PH−1 PW−1

i¼0 j¼0 10log10 PH−1 PW−1 i¼0 j¼0 ðI i;j

I 2i;j − I ni;j Þ2

 ;

(29)

where I i;j and I ni;j represent the intensities of pixel ði; jÞ in the original noise-free image and the result image after applying the denoise algorithm to the noisy image, respectively. We study the denoising performance using different step sizes selected from 0.5 to 2.2. Three images, “Lena,” “Fishing Boat,” and “Pepper,” are added Gaussian noise whose mean is 0 and standard deviation is 25. The original

Fig. 2 Three images and images having Gaussian noise. (a) Original Lena image. (b) Corrupted Lena image σ ¼ 25. (c) Original Fishing Boat image. (d) Corrupted Fishing Boat image σ ¼ 25. (e) Original Pepper image. (f) Corrupted Pepper image σ ¼ 25. Journal of Electronic Imaging

images and corrupted images are shown in Fig. 2. The corrupted images are utilized to test the proposed anisotropic diffusion and subpixel model with different step sizes. The relationships between the step size and SNR of the three images are plotted in Fig. 3. From Fig. 3, we can see clearly that the proposed method achieves the biggest SNR when hS ¼ 1.95. Therefore, we select 1.95 as the step size in the following experiments. 3.2 Experimental Results and Comparison In this section, we also use “Lena,” “Fishing Boat,” “Pepper,” “Plane,” “Bacteria,” and “Blood” images to evaluate the performance of the proposed method and compare it with that of other anisotropic diffusion methods: PM algorithm6 and a fourth-order PDE approach,7 which claimed that they have achieved better performance than the traditional filtering methods on noise removal. In addition, the proposed method adopts the idea in Ref. 7 to solve the diffusion equation numerically. Six original images are added Gaussian noise with six standard deviations (σ ¼ 5; 10; 20; 30; 40; 50). The noisy images and denoised images using the above three methods are shown in Figs. 4 to 9. For the space limitation, we only show the results on the images with three standard deviations (σ ¼ 10; 30; 50). In these figures, the first columns show the corrupted images having noise with different levels. The second and third columns are the denoising results using the PM algorithm and the fourth-order PDE method, respectively. The fourth columns show the results by the proposed method. All approaches take the same number of iterations and perform on the images with the same noise level. From Figs. 4 to 9, we can see that the proposed method removes most of noise in the corrupted images and the resulting images achieve the best visual effect. In addition, there is no blocky effect and less speckle noise in the results of the proposed method. However, there are blocky effects and more generated speckle noise in the results of other approaches. When the noise standard deviation is small (σ ¼ 10), the fourth-order method performs better than the PM method, which can be seen in the second and third columns in Figs. 4(a), 4(b), 5(a), 5(b), 6(a), 6(b), 7(a), 7(b), 8(a), 8(b), 9(a), and 9(b). When the noise standard deviation increases, the performance of the PM and fourth-order methods become worse, as shown in the second and third columns

Fig. 3 SNR values using different step size on three images with Gaussian noise.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

Fig. 4 Denoise result comparisons on Lena image with different noise levels.

Fig. 5 Denoise result comparisons on Shipping Boat image with different noise levels.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

Fig. 6 Denoise result comparisons on Pepper image with different noise levels.

Fig. 7 Denoise result comparisons on Plane image with different noise levels.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

Fig. 8 Denoise result comparisons on Bacteria image with different noise levels.

Fig. 9 Denoise result comparisons on Blood image with different noise levels.

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in Figs. 4(c), 4(d), 4(e), 5(c), 5(d), 5(e), 6(c), 6(d), 6(e), 7(c), 7(d), 7(e), 8(c), 8(d), 8(e), 9(c), 9(d), and 9(e). The resulting images by the PM approach are much more blury and blocky. In addition, the results by the fourth-order algorithm are corrupted by the generated speckle noise. For example, in Fig. 4(d), the result by PM (second column) is blury and lost the most detail features in the hair area. The third column in Fig. 4(d) shows the result by the fourth-order method which has speckle noise, especially in the face area. Compared with the results by PM and fourth-order methods at all noise levels, the resulting images by the proposed method become smoother, had no blocky effects, and no generated speckle noise, which were shown in the fourth column of Figs. 4 to 9. For instance, in the fourth column in Fig. 4, the boundaries of the face and hat are still clear while the hair region does not have much speckle noise. In order to compare the noise removal performance of the three methods quantitatively, we calculate the SNR values of

the resulting images and the central processing unit (CPU) times of the three methods. All of our experiments are performed on a PC machine configured with an Intel Core 2 Duo Dual Core processor with 2.2 GHz and 2 GB RAM memory under a Matlab 7.0 platform. Figures 10 to 15 show the comparisons of three methods on the values of SNR and CPU time. In Fig. 10, the average SNR of the proposed method is 24.93, much higher than PM and fourth-order’s average SNRs, 19.52 and 19.10, respectively. The average CPU time of the propose method is 8.82 s, almost the same as that of PM method, 8.24 s, and much less than that of the fourth-order, 30.39 s. In the results on the “Fishing Boat” image (Fig. 11), the average SNR of the proposed method is 21.64, higher than PM and fourthorder’s average SNRs, 16.93 and 17.93, respectively, while the average CPU time of the proposed method is 8.89 s, almost the same as that of the PM method, 8.35 s, and much less than that of the fourth-order, 30.75 s. There are

Fig. 10 Denoise performance comparisons on Lena image. (a) SNR comparison. (b) CPU times comparison.

Fig. 11 Denoise performance comparisons on Shipping Boat image. (a) SNR comparison. (b) CPU times comparison.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

Fig. 12 Denoise performance comparisons on Pepper image. (a) SNR comparison. (b) CPU times comparison.

Fig. 13 Denoise performance comparisons on Plane image. (a) SNR comparison. (b) CPU times comparison.

Fig. 14 Denoise performance comparisons on Bacteria image. (a) SNR comparison. (b) CPU times comparison.

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Guo and Cheng: Image noise removal approach based on subpixel anisotropic diffusion

Fig. 15 Denoise performance comparisons on Blood image. (a) SNR comparison. (b) CPU times comparison.

similar results in Figs. 12 to 15. The comparisons in Figs. 10 to 15 show that the proposed method reaches the best SNR at all noise levels and takes much less CPU time than that of the fourth-order method, and similar to that of the PM method. From the comparison of the visual and quantity measures in Figs. 4 to 15, we can conclude that the proposed denoising method not only removes Gaussian noise properly and achieves the best visual result, but also has higher SNR and less CPU time. 4 Conclusion A novel denoising technique is proposed based on subpixel theory and anisotropic diffusion approach. Firstly, the subpixel difference is defined, and the Euler-Lagrange equation is solved based on the subpixel difference. Secondly, the diffusion equation is solved numerically using an iterative approach. Finally, the noise is removed when the diffusion procedure is finished. A variety of images having noise with different levels are employed to test the performance of the proposed method. The experimental results show that the proposed algorithm yields better SNR, has no blocky effect, and less generated speckle noise in the results than the other methods do. In addition, it takes fewer iteration times and has low computational cost. The proposed method will have more applications in image processing and computer vision. Acknowledgments Thanks to the editor and anonymous reviewers for their worthy suggestions on the improvement of this paper. References 1. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4(2), 490–530 (2005). 2. M. C. Motwani et al., “Survey of image denoising techniques,” in Proc. of GSPx, Santa Clara, CA, pp. 27–31 (2004). 3. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed., pp. 175–183, Prentice Hall Press (2003).

Journal of Electronic Imaging

4. A. Bosco et al., “Noise reduction filter for full-frame imaging devices,” IEEE Trans. Consum. Electron. 49(3), 676–682 (2003). 5. S. Battiato et al., “Noise reduction for CFA image sensors exploiting HVS behavior,” Sensors 9(3), 1692–1713 (2009). 6. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990). 7. Y. L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9(10), 1723–1730 (2000). 8. Z. H. Ding, J. C. Gore, and A. W. Anderson, “Reduction of noise in diffusion tensor images using anisotropic smoothing,” Magn. Reson. Med. 53(2), 485–490 (2005). 9. J. S. Jin, Y. Wang, and J. Hiller, “An adaptive nonlinear diffusion algorithm for filtering medical images,” IEEE Trans. Inf. Technol. Biomed. 4(4), 298–305 (2000). 10. J. Weickert, B. M. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7(3), 398–410 (1998). 11. M. J. Black et al., “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7(3), 421–432 (1998). 12. Y. J. Yu and S. T. Acton, “Speckle reducing anisotropic diffusion,” IEEE Trans. Image Process. 11(11), 1260–1270 (2002). 13. M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourthorder partial differential equation with application to medical magnetic resonance images in space and time,” IEEE Trans. Image Process. 12(12), 1579–1590 (2003). 14. R. T. Whitaker and S. M. Pizer, “A multi-scale approach to nonuniform diffusion,” CVGIP: Image Understanding 57(1), 99–110 (1993). 15. Y. D. Qu et al., “A fast subpixel edge detection method using SobelZernike moments operator,” Image Vision Comput. 23(1), 11–17 (2005). 16. F. Truchetet, F. Nicolier, and O. Laligant, “Subpixel edge detection for dimensional control by artificial vision,” J. Electron. Imaging 10(1), 234–239 (2001). 17. T. Hermosilla et al., “Non-linear fourth-order image interpolation for subpixel edge detection and localization,” Image Vision Comput. 26(9), 1240–1248 (2008). 18. E. P. Lyvers and O. R. Mitchell, “Pecision edge contrast and orientation estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 10(6), 927–937 (1988). 19. J. Zhang et al., “Application of an improved subpixel registration algorithm on digital speckle correlation measurement,” Opt. Laser Technol. 35(7), 533–542 (2003). 20. K. Takita et al., “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fund. Electron. Commun. Comput. Sci. E86A(8), 1925–1934 (2003). 21. D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39(7), 1392–1409 (2001). Biographies and photographs of the authors are not available.

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