Robust Anisotropic Diffusion Scheme for Image Noise Removal - Core

Available online at www.sciencedirect.com

ScienceDirect Procedia Computer Science 35 (2014) 522 – 530

18th International Conference on Knowledge-Based and Intelligent Information & Engineering Systems - KES2014

Robust anisotropic diffusion scheme for image noise removal Tudor Barbu* Institute of Computer Science of the Romanian Academy, Iaşi, Romania

Abstract An anisotropic diffusion-based image enhancement method is proposed in this article. The provided PDE denoising approach is derived from the well-known Perona-Malik nonlinear diffusion model, representing an improved version of it. Thus, we model a novel diffusivity function and explain the mathematical reasons behind it. The conductance parameter of our diffusion technique is automatically detected at each iteration. The proposed PDE-based smoothing technique provides satisfactory noise removal and edge enhancement results, outperforming the most diffusion-based approaches. © Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © 2014 2014 The The Authors. Authors. Published Published by by Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of KES International. Peer-review under responsibility of KES International. Keywords: PDE-based image denoising; anisotropic diffusion; Perona-Malik filter; edge preservation; edge-stopping function; diffusivity conductance parameter; noise estimation.

1. Introduction Images captured of the real world objects are often corrupted by an amount of noise during the acquisition and transmission process1,2. Feature preserving image noise reduction still represents a challenging image processing task. Since image noise removal represents a relevant issue in various image analysis and computer vision problems, it is a challenge to preserve the essential image features, such as edges, corners and other sharp structures during the smoothing process1.

* Corresponding author. E-mail address: [email protected].

1877-0509 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of KES International. doi:10.1016/j.procs.2014.08.133

Tudor Barbu / Procedia Computer Science 35 (2014) 522 – 530

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The conventional image denoising techniques, such as averaging filter, median filter or 2D Gaussian filter are efficient in reducing the amount of noise, but also have the disadvantage of blurring the image edges2,3. For this reason, numerous edge preserving techniques based on Partial Differential Equations (PDEs) have been developed in the last two decades4,5. The idea of using the PDE diffusion equations in image denoising and restoration arose from the use of the Gaussian filter in multiscale image analysis. Convolving an image with a two-dimensional Gaussian filter is equivalent to the solution of the diffusion equation in two dimensions. Various diffusion-based noise removal methods have been introduced since the early work of P. Perona and J. Malik, published in 19876. They developed an anisotropic diffusion framework for image denoising and segmentation. Their PDE scheme was able to smooth the processed image while preserving its edges, by encouraging the diffusion within image regions and prohibiting it across strong boundaries. Numerous nonlinear diffusion techniques derived from the influential Perona-Malik approach have been proposed in recent years. Many other papers perform various analysis of the algorithm provided by Perona and Malik7. We have also proposed some robust PDE-based image denoising and restoration methods in the last years, including a nonlinear diffusion based noise removal technique8-10. In this paper we provide a novel image enhancement approach using an anisotropic diffusion scheme derived from Perona-Malik filter. Related models, representing state-of-the-art of the diffusion-based image denoising algorithms, are discussed in the next section. Then, our proposed PDE smoothing approach is described in the third section. It provides a novel edge-stopping function for the nonlinear anisotropic equation, while its diffusivity conductance parameter is detected automatically using image noise estimation. A mathematical investigation of the proposed diffusion scheme is performed. It’s successfully image denoising results are presented in the fourth section. Our PDE model outperforms Perona-Malik technique and many other state-of-the-art image filtering models. Method comparisons are also performed in the same section of this paper. The article finalizes with a conclusions section and a list of references. 2. Related work The image affected by noise is smoothed by the diffusion techniques which modify it via a partial differential equation. These denoising approaches are based on the following diffusion equation: wu ° div C ( x, y, t ) u , x, y : R 2 ® wt ° u 0, x, y u 0 ¯

(1)

where u 0 is the initial noised image and C represents the diffusion tensor. If this tensor is constant C(x, y, t) = c over the entire image domain : R 2 , then we have a homogeneous diffusion. If the diffusion is spacedependent, it is called inhomogeneous. In computer vision literature, the homogeneous filtering is named isotropic, while the inhomogeneous filtering is called anisotropic7. In the homogeneous case, from wu divC ( x, y, t ) u C( x, y, t ) 'u c u one obtains div c u c 'u , therefore equation (1) wt becomes the heat equation. If the diffusion process does not depend on the evolving image u (t, x, y), it is called linear. If the diffusion is a function of that image, C ( x, y, t ) g ut , x, y , the process becomes nonlinear. Choosing the proper diffusivity, or edge-stopping, function g represents an important and difficult task. The linear diffusion models are the simplest PDE–based image denoising techniques. A 2D Gausian smoothing process is equivalent to a linear diffusion filtering10. Thus, if an degraded image is processed by

524


convolution with a Gaussian kernel, the result u(t , x, y) Gt * u 0 ( x, y) , where Gt ( x, y )

1 e 4St

x2 y2 4t

,

( x, y) R 2 , t ! 0 , represents also the solution of the heat equation given by the diffusivity coefficient c = 1. The main disadvantage of the linear PDE denoising models is their blurring effect over edges and other image features. Linear diffusion has no localization property and may dislocate the edges when moving from finer to coarser scales7,10. The nonlinear diffusion techniques avoid the blurring and localization problems of the linear filtering. A nonlinear denoising solution is the directional diffusion, which is degenerate along the gradient direction, having the effect of smoothing the image along but not across the edges11. The most popular nonlinear anisotropic diffusion technique is the influential denoising scheme developed by P. Perona and J. Malik in 19876,7. Their approach reduces diffusivity at those locations having a larger likelihood to represent image edges. The Perona-Malik filter is characterized by the following nonlinear diffusion equation: ut

wu wt

2

div g ( u ) u

(2)

with the noisy image u 0 as the initial condition. Obviously, the diffusivity coefficient that controls how much

image smoothing is performed in (x, y) is C ( x, y, t ) g u(t , x, y) 2 . The value of C (x, y, t) should be lower when (x, y) is part of an edge, and higher when it is not. The function g that controls the blurring intensity should be monotonous decreasing for this reason. Perona and Malik considered two diffusivity function variants, g : >0, f@ o >0, f@ , which are: 2

g (s )

e

s2 k2

; g (s 2 )

1 §s· 1 ¨ ¸ ©k¹

(3)

2

where k > 0 represents the diffusivity conductance, being the parameter that controls the diffusion process6. They discretized the diffusion model provided by formulas (2) and (3), as following:

>

u t O c Nt N u t c St S u t c Et E u t cWt W u t

u t 1

@

(4)

u t x 1, y u t x, y

(5)

where

Nut

u t x 1, y - u t x, y , S u t cNt

u t x 1, y u t x, y , E u t

g N u t , cSt

g S u t , cEt

u t x 1, y 1 - u t x, y , W u t

g E u t , cWt

g W u t

(6)

The Perona-Malik diffusion algorithm produces good smoothing results while preserving image edges for a long time. Thus, the boundaries remain stable for high values of time variable t. Also, edge detection based on this anisotropic diffusion method outperforms other edge detectors, including the well-known Canny detector12. There is a lot of literature based on the denoising framework proposed by Malik and Perona. Their algorithm inspired numerous researchers to analyze and further improve it. There are many papers that investigate its mathematical properties, numerical implementations and the possible applications. Thus, the stability of Perona-Malik model has been extensively studied in the last years13. Also, in the last decades there


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have been developed a lot of denoising techniques derived from Perona-Malik framework7, 9. Many of these modified versions produce better smoothing and edge enhancement results than the original algorithm. Usually, these nonlinear diffusion-based denoising schemes differ from each other through the diffusivity function. This function g may have various forms, but must satisfy several conditions, such as positivity, decreasing monotony, g (0) = 1 and convergence to 0. Let us survey some of these approaches and their edge1 , and its regularized form stopping functions. Total variation (TV) diffusivity, expressed as g ( s 2 ) s 14 1 represent popular edge-stopping functions . Charbonnier et al. proposed the Charbonnier g (s 2 ) 2 s H2 diffusivity15, given by

§ s2 · g ( s ) ¨¨1 2 ¸¸ © k ¹

1 / 2

(7)

2

J. Weickert7 proposed the following edge-stopping function for anisotropic diffusion: Cm °1 e ( s 2 / k 2 ) m , g (s ) ® °¯ 1, 2

if s ! 0

(8)

if s 0

where 1 eCm (1 2Cm m) , m ^2,3,4` , C 2 2.3366 , C3 2.9183 , C 4 3.3148 . Black et al. provided the robust anisotropic diffusion (RAD)16. They used robust estimation theory to model an edgestopping function called Tukey’s biweight:

g (s 2 )

§ s2 °¨¨1 °© 5k 2 ® ° 0, ° ¯

2

· ¸¸ , ¹

s2 d k2 5 s2 if ! k2 5 if

(9)

The anisotropic diffusion approaches may also differ through the way they choose the diffusivity conductance. Thus, parameter k should be properly chosen for an efficient denoising. When the gradient magnitude exceeds the value of k, the corresponding edge is enhanced. Some techniques, including the PeronaMalik filter, use a fixed k value that could be determined empirically. Other authors, like Li and Chen, demonstrated that using a single conductance value for the entire denoising process would be inappropriate17. Another solution is to make this parameter a function of time, k (t). One can use a high k (0) value at the beginning, then k (t) is reduced gradually, as the image is smoothed. Other approaches detect automatically the conductance diffusivity as a function of the current state of the processed image. Various noise estimation methods are used for conductance parameter detection18. Thus, the image noise can be estimated at each iteration as the difference between the average intensities of images processed by the morphological operations of opening and closing18. In this case the conductance parameter is computed as k avgu $ S - avg u x S , where S represents the structuring element of the operations. Another variant for the k parameter is obtained by estimating the noise using the p-norm of the Vu p , where m is the number of pixels and V is proportional to the image average intensity18. image: k m

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Other approaches compute the conductance diffusivity as the robust scale of the analyzed image, using statistical measures such as the median17: k V e 1.4826 median(u) u median(u) u . In the next section there is described an anisotropic diffusion scheme using novel variants for both the edge-stopping function and the diffusivity conductance parameter. 3. Nonlinear anisotropic diffusion-based image denoising model We propose a novel nonlinear anisotropic diffusion model that performs efficient image denoising while preserving successfully the image boundaries. Our diffusion-based image noise reduction scheme is given by the next parabolic equation:

wu ° ® wt ° ¯

div g K ( u ) u u 0, x, y

2

u , x, y :

(10)

u0

where u 0 is the degraded image and its domain : R 2 . The nonlinear diffusion equation given by (10) uses a novel edge-stopping function. We propose the following diffusivity function, g K (u ) : >0, f o >0, f :

g K (u ) ( s 2 )

K (u ) °D , ® E s2 J °1, ¯

if s ! 0 if s

(11)

0

where the parameters D , E >0.5, 0.8@ and J >0.5, 5 . Also, as one can see in (11), the modelled edge-stopping function is based on a conductance diffusivity depending on the current state (at the time t) of the processed image. We propose an automatic computation of this parameter, based on the image noise estimation at each time (iteration) t. Statistical measures are used by the following conductance computation: K u

median(u ) u H n(u )

F

,

(12)

where H 0, 1@, u is the Frobenius norm of image u, median(u) represents its median value and n(u) is the F number of its pixels. The proposed diffusivity function g K (u ) is properly chosen. It satisfies the main conditions related to any edge-stopping function. Obviously, we have g K (u ) (0) 1 . Also this function is always positive, because

D

K (u )

E s2 J

! 0, s R.

The

function

g K (u ) ( s 2 )

is

monotonically

decreasing,

because

K (u ) K (u ) g K (u ) ( s 2 ) 0 . dD g K (u ) ( s22 ) for s1 t s 2 , We also have lim 2 2 s of E s1 J E s2 J Besides these conditions, our diffusivity function satisfies another important one. If one considers the flux function, defined as I s s g K (u ) ( s 2 ) , the process of enhancing the image and sharpening the edges g K (u ) ( s12 )

D

527


depends on the sign of its derivative, I ' s 7,19. Therefore, if the derivative of a flux function of a diffusion model is positive ( I ' s ! 0 ), then the respective model represents a forward parabolic equation. Otherwise, for I ' s 0 , that nonlinear diffusion model is a backward parabolic equation19. In our case, the derivative of the flux function is computed as following:

I ' ( s ) g K ( u ) ( s 2 ) 2s 2 g K ( u ) ' ( s 2 )

(13)

which leads to

I ' ( s)

D K (u ) Es J 2

D K (u ) s 2 E Es 2 J Es 2 J

(14)

Therefore, we get

I ' ( s)

D K (u )

Es

2

J

3/ 2

>Es

2

J Es 2

@

DJ K (u )

Es

2

J

3/ 2

(15)

Since DJ K (u ) ! 0 , we get I ' s ! 0 for any s, which means our PDE denoising model is a forward Es 2 J 3 / 2 parabolic equation that is stabile and it is quite likely to have a solution. The existence and uniqueness of its solution represents a problem that requires a rigorous mathematical treatment. For this reason it will be further investigated in our next papers. For now, it should be said that, in general, the problem (10) is ill-posed. One can prove the existence and uniqueness of a weak solution in a certain case, related to some values of the parameters of this PDE model. Thus, we could demonstrate that our anisotropic diffusion model converges if J D 2 . If we consider the following modification of the function, g K ( u ) :

g K (u ) (s 2 )

°D ° ® ° ° ¯

K (u )

Es J D J 2

if 0 s d M if s

(16)

0

where M ! 0 is an arbitarily large but fixed value, the diffusion model based on this modified edge-stoppping function has a unique weak solution. By weak solution to the PDE equation (10) we mean a function u : 0, T u : o R such that u L2 (0, T ; H 1 (:)) L2 ((0, T ) u :, wu L2 (0, T ;( H 1 (:)) ') and for dt M H 1 (:), t >0,T @ 2 w ° u (t , x, y )M ( x, y )dxdy ³ g K ( u ) ( u (t , x, y ) )u (t , x, y )M ( x, y )dxdy ® wt : ° u ( 0 , x , y ) u 0 ( x, y ), ( x, y ) : ¯

(17)

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The effectiveness of our constructed model is also demonstrated by the efficient noise reduction and boundary enhancement results described in the next section. These results depend also on the choice of our differential model’s parameters: D , E , J and H . Some properly chosen values of these parameters are provided in that section. The continuous PDE model provided by (10) is then discretized by using a 4-nearest-neighbours discretization of the Laplacian operator, 'u . Thus, from the equation (10) one obtains wu 2 2 div g K (u ) u u u ( x, y, t 1) u ( x, y, t ) # g K (u ) u 'u , which leads to the following wt approximating scheme:

u t 1

u t O ¦ g K (u ) u p ,q (t ) qN p

u 2

p ,q

(t )

(18)

where O 0, 1 , N p is the set of pixels representing the 4-neighborhood of the pixel p, represented as a pair of coordinates (x, y), and the image gradient magnitude in a particular direction at iteration t is computed as following: u p , q (t )

u(q, t ) u( p, t )

(19)

The denoising algorithm applies the procedure given by (18) on the current image for each t ^0, 1,..., N ` , where N is the maximum number of iterations. Our noise reduction approach obtains the desired smoothed image u N from the noisy image u 0 u0 in a relatively small number of steps, so it is characterized by a quite low N value. Also, the proposed denoising technique outperforms the Perona-Malik algorithm and other anisotropic diffusion based approaches, as resulting from the method comparison performed in the following section.

Fig. 1. Method comparison: image filtered with various denoising techniques

529


4. Experiments and method comparison We have performed numerous image enhancement experiments using the described anisotropic diffusionbased method. Our denoising technique has been tested on hundreds images corrupted with various levels of Gaussian noise and produced satisfactory results. The following parameters of the denoising model provided the best results: D 0.7, E 0.65, J 0.5, H 0.3, O 0.33 and N = 15. One can see that D 2 # J , so the diffusion scheme has a unique solution. It converges fast to that solution, the number of steps N being quite low. The performance of our restoration technique has been assessed by using the norm of the error image measure, computed as

X

Y

¦¦ (u

N

( x, y ) u 0 ( x, y )) 2 , the original noisy image u 0 having a >X u Y @

x 1 y 1

dimension. Method comparisons were also performed and we found that our AD approach outperforms many other denoising techniques. It produces better noise removal results and converges faster than some PDE models, including the Perona-Malik algorithm. It also provides a much better image smoothing than the non-PDE techniques. In Fig. 1, there are displayed the following: the original >512u 512@ Lena image, the image corrupted with Gaussian noise given by P 0.21 and var = 0.02, the image filtered by our AD method, the image denoised by the 2 versions (diffusivity functions) of Perona-Malik scheme, and the filtering results of >3u 3@ 2D Gaussian, average, median and Wiener kernels. These displays and the corresponding norm of the error image values from Table 1 (see the lowest value) confirm that our approach produces better edge preserving image denoising results. Table 1. Values of the norm-of-the-error measure for various denoising algorithms Our AD

P-M 1

P-M 2

Gaussian

Average

Median

Wiener

5.1 u 10 3

6.1 u 10 3

5.9 u 10 3

7.3 u 10 3

6.4 u 10 3

6 u 10 3

5.8 u 10 3

5. Conclusions A novel image restoration system based on an anisotropic diffusion model has been provided in this article. Our nonlinear PDE-based model performs an efficient noise reduction while preserving and enhancing de image boundaries. The main contributions of this paper are the proposed edge-stopping function and conductance parameter of the diffusion equation, and also the robust mathematical treatment of the developed anisotropic diffusion model. First, we have demonstrated that our diffusivity function is properly chosen, satisfying the required conditions. Then, we have provided some mathematical discussion on the existence and uniqueness of the solution of this forward parabolic equation. The computed numerical approximation iterative algorithm represents another contribution. The performed denoising experiments and the method comparison provided very encouraging results. Our anisotropic diffusion technique executes faster and produces much better image enhancement results than other PDE-based algorithms, like influential Perona-Malik scheme, and many non-PDE denoising methods. Because of its strong edge-preserving character, it can be successfully used for edge detection or computer vision tasks like object detection.

530


Also, our future research in this domain will focus on a rigorous mathematical investigation of the convergence of this nonlinear diffusion model. Thus, we intend to provide some detailed mathematical demonstrations of the existence of a unique solution of the described PDE technique. Acknowledgements The research work described here has been supported by the project PN-II-PCE-20113-0027, financed by the Romanian Minister of Education and Technology. We acknowledge also the research support of the Institute of Computer Science of the Romanian Academy, Iaşi, ROMANIA. References 1. 2. 3. 4.

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