The original version of this manuscript will appear in the Proceedings of the 7th International Conference on Advances in Pattern Recognition (ICAPR 2009)

1

Smoothening and Sharpening Effects of Theta in Complex Diffusion for Image Processing Jeny Rajan, K. Kannan, M.R. Kaimal Member IEEE

Abstract— In this paper we present a study on how the changing values of theta in complex diffusion affects the images. Normally it is considered that the low value of theta is suitable for image smoothening using complex diffusion, because at higher values of theta the imaginary part may feed back into the real part, creating wave-like ringing effects. Our study shows that as the value of theta increases, ringing effects starts appearing and reaches its peak at 1800 and then it starts disappearing, and the process continues in a 360 degree cycle, where the peak of the wave indicates image with maximum ringing effects (or the maximum sharpened image, property of inverse diffusion). Regarding non-linear complex diffusion we experimentally proved the smoothening is fast at higher values of theta, which can be used for image denoising purpose. Index Terms— Processing

T

Complex

Diffusion,

Denoising,

Image

I. INTRODUCTION

HE scale space approach and Partial Differential Equation (PDE) techniques have been extensively applied over the last decade in signal and image processing. Many works [1-12] are published regarding the application of 2nd, 4th and complex PDEs in processing images. One of the main interests in using the PDEs is that the theory is well established. PDEs are written in a continuous setting, referring to analogous images. The basic idea is to deform an image, a curve or a surface with a PDE and obtain the expected results as a solution to this equation. Among the PDE based methods, complex diffusion is the least studied one. The property of complex diffusion to denoise images was introduced and extensively studied by Gilboa et al. [10]. The PDE techniques have been widely used for enhancing images. The diffusion process is equivalent to a smoothing process with a Gaussian kernel. A major drawback of the linear diffusion is its uniform filtering of local signal features and noise. This problem was addressed by Perona and Malik (P-M) [1], who proposed a nonlinear diffusion process, where diffusion can take place with a variable diffusion in order to control the smoothing effects [8]. Jeny Rajan & K. Kannan is with the Medical Imaging Research Group, Healthcare Division, NeST, Technopark, Trivandrum, INDIA (corresponding author to provide phone: +91 471 4060824; fax: +91-471-2700442; e-mail: [email protected]). M.R Kaimal is with the Dept. of Computer Science, University of Kerala, Trivandrum, INDIA.

Second order PDEs especially anisotropic diffusion equations have been studied as an efficient tool for removing noise from images. This is due to its efficiency in preserving edges while smoothing images. Several papers [9], [2], [3],[4] have noted that anisotropic diffusions with diffusion coefficients given by P-M equations are ill posed in the sense that images close to each other are likely to diverge during the diffusion process. In the past few years, a number of authors have proposed analogous fourth order PDEs for edge detection and image denoising with the hope that these methods would perform better than their second order analogues [2],[3],[11]-[14]. Indeed there are good reasons to consider fourth order equations. First, fourth order linear diffusion damps oscillations at high frequencies much faster than second order diffusion. Second, there is the possibility of having schemes that include effects of curvature in the dynamics, thus creating a richer set of functional behaviors. On the other hand, the theory of fourth order nonlinear PDEs is far less developed than their second order analogues. Also such equations do not possess a maximum principle or comparison principle, and the implementation of the equations could thus introduce artificial singularities or other undesirable behavior. Another alternative is to use complex diffusion. Nonlinear complex diffusion processes (2nd order PDE's) can behave like 3rd and 4th order real PDE's enabling a variety of new options with standard 2nd order numerical schemes. Complex diffusion is explained in detail in Section II. Most of the PDE-based studies have been devoted to denoising of images, attempting to preserve the edges [8]. This is achieved by running diffusion in forward direction. Sharpening of images can be achieved by moving diffusion backward in time along the scale space. Our study shows that both forward and backward diffusion can be achieved with complex diffusion running forward with changing the values of theta from 0 to 360. The paper is organized as follows. Section II explains the linear and non linear complex diffusion introduced by Gilboa et al. Section III discuss the results and impacts of varying values of theta. Finally conclusion and remarks are added in Section IV. II.

LINEAR AND NONLINEAR COMPLEX DIFFUSION

In 1931 Schrodinger explored the possibility that one might use diffusion theory as a starting point for the derivation of the equations of quantum theory. These ideas were developed by Fuerth who indicated that the Schrodinger equation could be derived from the diffusion equation by introducing a relation

2 between the diffusion coefficient and Planck’s constant, and stipulating that the probability amplitude of quantum theory should be given by the resulting differential equation [17]. It has been the goal of a variety of subsequent approaches to derive the probabilistic equations of quantum mechanics from equations involving probabilistic or stochastic processes. The time dependent Schrodinger equation is the fundamental equation of quantum mechanics. In the simplest case for a particle without spin in an external field it has the form [18] ∂ψ =2 (1) i= =− Δψ + V ( x)ψ ∂t 2m where ψ = ψ (t , x) is the wave function of a quantum particle, m is the mass of the particle, = is Planck’s constant, V(x) is the external field potential, Δ is the Laplacian and

c(Im(I ) ) =

e iθ ⎛ Im(I ) ⎞ 1+ ⎜ ⎟ ⎝ kθ ⎠

(8) 2

where k is the threshold parameter. The effects of linear and non linear complex diffusion on images is shown in Fig 1.

(a)

(b)

(c)

i = − 1 . With an initial condition ψ |t =0 = ψ 0 ( x) , requiring that ψ (t ,⋅) ∈ L2 for each fixed t, the solution is −i

tH

ψ (t ,⋅) = e h ψ 0 , where the exponent is shorthand for the corresponding power series, and the higher order terms are defined recursively by H nψ = H H n −1ψ . The operator

(

H =−

)

=2 Δ + V ( x) 2m

I IT = C I I Rxx + C R I Ixx , I I |t =0 = 0

(4) (5)

where I RT is the image obtained at real plane and I IT is the image obtained at imaginary plane at time T and CR = cos(θ ) , CI = sin(θ ) . The relation I Rxx >> θI Ixx holds for small theta approximation [18]: I RT ≈ I Rxx ;

I It ≈ I Ixx + θI Rxx

(6)

In (6) I R is controlled by a linear forward diffusion equation, whereas I I is affected by both the real and imaginary equations. The above said method is linear complex diffusion equation. To preserve edges, non linear version of complex diffusion as written below has to be used (7) I t = ∇ ⋅ (c(Im(I )∇I ) where

(g)

(h)

(i)

(3)

From (2) and (3) we can derive the following two equations.

I RT = C R I Rxx − C I I Ixx , I R|t = 0 = I 0

(e) (f) (Linear Complex Diffusion)

(2)

called the Schrodinger operator, is interpreted as the energy operator of the particle under consideration. The first term is the kinetic energy and the second is the potential energy. The duality relations that exist between the Schrodinger equation and the diffusion theory have been studied in [19]. The standard linear diffusion equation can be written as

∂u = Div( D ⋅ ∇u ) ∂t

(d)

(j)

(k) (l) (Non linear Complex Diffusion)

Fig. 1. Image Processing with Linear and Non linear Complex diffusion. Top row shows the image processed with linear complex diffusion and bottom row shows the image processed with non linear complex diffusion. Figure shows both real and imaginary parts (at angle 2, 180 and 358 degrees)

III. EFFECTS OF THETA The changing values of theta can make a big impact on the image processed with complex diffusion. Both forward and backward diffusion can be achieved with forward complex diffusion with varying values of theta from 0 to 360 degrees. As the theta value increases, the image will sharpen more (sharpening property increases) till the theta value reaches 180 degree, and the sharpening property reduces (smoothening property increases) from 181to 360 degrees and the process continues. This is very clear in the case of linear complex diffusion. In Fig2, we can see the behavior of theta in linear complex diffusion. It can be seen that the graph value (PSNR) from 180-360 is exactly the mirror image of 0-180 degrees.

3 i.e, the sharpening (or smoothening) at 10 degree and 350 degrees are exactly same. This property is only applicable to the real plane. But in the imaginary plane the values before and after 180 will be mirrored and inverse. The values in the Images

θ 1 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

Lena Mean:0, Var:0.01 Baboon Mean:0, Var: 0.01 Pallon Mean:0, Var: 0.01 PSNR - 20.2360 PSNR - 19.9986 PSNR - 20.0238 No. of No. of No. of Time Time Time PSNR Iteratio PSNR Iteratio PSNR Iteratio in secs. in secs. in secs. n n n

23.9018 23.7007 23.0778 21.8487 20.3773 20.0115 19.7572 19.5620 19.4411 19.4028 19.4495 19.5784 19.7803 20.0394 20.4513 22.1762 23.5738 24.2331 24.3133

90 91 96 103 7 1 1 1 1 1 1 1 1 1 6 21 17 13 12

8.503 8.604 9.161 9.862 0.645 0.082 0.084 0.084 0.083 0.083 0.082 0.083 0.083 0.083 0.553 2.015 1.601 1.228 1.122

22.0064 21.8297 21.3395 20.6302 20.0207 19.7348 19.4793 19.2832 19.1618 19.1233 19.1703 19.2997 19.5025 19.7628 20.0370 20.7341 21.6009 22.1974 22.3148

20 18 15 9 2 1 1 1 1 1 1 1 1 1 2 6 8 7 7

1.895 1.692 1.413 0.839 0.178 0.083 0.083 0.083 0.083 0.082 0.082 0.083 0.083 0.083 0.181 0.554 0.742 0.651 0.661

27.0615 26.8586 26.2668 25.1178 22.4948 19.8080 19.5521 19.3557 19.2341 19.1955 19.2426 19.3722 19.5753 19.8361 21.9819 24.8755 26.2293 26.8486 26.8566

215 221 247 285 547 1 1 1 1 1 1 1 1 1 147 66 38 26 21

20.535 21.072 23.497 27.143 52.041 0.083 0.083 0.082 0.082 0.082 0.083 0.082 0.082 0.082 14.067 6.269 3.607 2.453 1.965

Table 1: Table shows the results of image denoising with different values of theta. It can be seen from the table that highest PSNR can be achieved with less number of iterations when theta is near to 360 degree.

real and imaginary planes are fine details and coarse details in the image. There is a slight difference in the behavior of theta in the case of non linear complex diffusion. Non linear complex diffusion is an edge preserving diffusion process, i.e, intra region smoothening will occur before inter region smoothening. Because of its edge preserving smoothing property it is highly preferred for noise removal. If we vary the value of theta in non linear complex diffusion, as in linear complex diffusion, the sharpening property increases from 0180 and reduces from 180-360. But the interesting thing is the smoothening is faster near 360 degrees than at 0 degrees. i.e. the image will get smoother faster near 360 degrees than at near to 0 degrees. This property of non linear complex diffusion can make an impact in image denoising. For denoising, we can reduce the diffusion time by selecting theta near to 360 degrees. The behavior of theta in non linear complex diffusion can be easily studied from the graph shown in Fig 3. The impact of theta (sharpening & smoothening) on images processed with linear and nonlinear complex diffusion is shown in Fig 1.

(a)

(b)

(c) Fig.2 Effects of theta in complex diffusion. The x axis shows the theta range and y axis the psnr value. psnr value of the original image and image after processing with complex diffusion is considered. (a) Linear Complex Diffusion theta varies from 0ө to 1440 ө, (b) & (C) Non linear Complex Diffusion theta varies from 0ө to 1440 ө and -720ө to 720 ө, respectively.

IV. CONCLUSION This paper presents a study on the impact of theta in images processed with linear and nonlinear complex diffusion. It can be seen in the case of both linear and non linear complex diffusion, the sharpening property increases up to 180 degrees and then reduces. In the case of linear complex diffusion, the effects is really symmetric from 0 to 180 and from 360 to 180. But in the case of non linear complex diffusion, smoothening is faster at higher theta values (near to 360 degrees). For application areas of complex diffusion such as image denoising, our suggestion is to use theta value near to 360 for fast convergence and for better result.

4

Image : Lena:

(a) x axis theta, y axis PSNR Image : Baboon:

(b) x axis theta, y axis iterations

(c) x axis theta, y axis time

(a) x axis theta, y axis PSNR Image :Pallon:

(b) x axis theta, y axis iterations

(c) x axis theta, y axis time

(a) x axis theta, y axis PSNR

(b) x axis theta, y axis iterations

(c) x axis theta, y axis time

Fig. 3. Graphical repersentation of Effects of theta in non linera complex diffusion(Graph of Table 1)

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