Image denoising

IMAGE DENOISING

It is the process of removing noise from an image or signal which occurs in the process of imaging due to the uncertainty of measurements or instruments.

Noise:Image is visible with the help of pixels with corresponding intensities. If any intensity value fluctuates at some pixel then noise is formed at that pixel.

In simple words, NOISE is the unwanted signals or sound produced by the sensor and digital cameras.

Mainly 2 types of noise there are:

1) Additive Noise 2) Multiplicative Noise

CONVENTIONAL IMAGE DENOISING TECHNIQUES:

• Averaging Filter• Median Filter OR 2D Gaussian Filter

These filters are efficient in reducing the amount of noise but have the disadvantages of blurring the image edges.

Due the this reason, numerous edge preserving techniques based on PDEs have beendeveloped in the last two decades.

DIFFUSION BASED IMAGE DENOISING METHODS:

The idea of using PDE diffusion equation in image denoising and restoration arose from the use of Gaussian filter in multiscale image analysis. Convolving an image with a two-dimensional Gaussian filter is equivalent to the solutionof 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 1987. They developed an anisotropic diffusion framework for image denoising and segmentation.

The image effected by noise is smoothed by the diffusion techniques whichmodify it via the following diffusion equation:

= div(C(x,y,t) .

𝑢 (0 , 𝑥 , 𝑦 )=𝑢0

where is the initial noised image and C represents the diffusion tensor.If this tensor is constant C(x,y,t) = c over the entire domain then we have a homogeneous diffusion while inhomogeneous when it is space dependent.

(1)

In COMPUTER VISION literature , the homogeneous filter is named isotropicwhile inhomogeneous filtering is called anisotropic.

In the homogeneous case ,

= div(C(x,y,t) .

One obtains, = div(c. u) = c.∆u ,∇

Therefor equation (1) becomes a heat equation which is linear.

LINEAR DIFFUSION MODELS:

The linear diffusion models are the simplest PDE-based image denoising techniques. A 2-D Gaussian smoothing process is equivalent to linear di-ffusion filtering.

The main disadvantage of the linear PDE denoising models is their blurringeffect over edges and other image features.

It has no localization property and may dislocate the edges when moving From finer to coarser scales.

NON-LINEAR DIFFUSION MODELS:If the diffusion tensor is a function of that image ,

the process becomes Non-Linear.

• Choosing the proper diffusivity or edge stopping, function g represents an Important and difficult task.

• The non-linear diffusion techniques avoid the blurring and localization problems of linear filtering.

• A non-linear denoising solutions is the directional diffusion which generates along gradient direction, having the effect of smoothing the image along but not across the edges.

The most popular non-linear anisotropic diffusion technique is theinfluential scheme developed by

P-PERONA AND J.MALIK

in 1987. Their approach reduces diffusivity at image edges and is characterized by the following diffusion equation,

with noisy image as the initial condition and the diffusivity coefficientC = which controls how much smoothing is performed in (x,y).

• Perona-Malik considered two diffusivity function variants,

𝑔 (𝑠2 )=𝑒− 𝑠

2

𝑘2 ,𝑔 (𝑠2 )= 1

1+( 𝑠𝑘 )2

(3)

where k > 0 represents the diffusivity conductance parameter that controlsthe diffusion process.

They discretized their diffusion model as;

𝑢𝑡+1=𝑢𝑡+𝜆 .[𝑐𝑁𝑡 .𝛻𝑁𝑢𝑡+𝑐𝑆𝑡 .𝛻𝑆𝑢𝑡+𝑐𝐸𝑡 .𝛻𝐸𝑢𝑡+𝑐𝑊𝑡 .𝛻𝑊 𝑢𝑡] (4)

where

𝛻𝑁𝑢𝑡=𝑢𝑥− 1 , 𝑦𝑡 −𝑢𝑥 , 𝑦

𝑡 ,

𝛻𝑆𝑢𝑡=𝑢𝑥+1 , 𝑦𝑡 −𝑢𝑥 , 𝑦

𝑡 ,

𝛻𝐸𝑢𝑡=𝑢𝑥− 1 , 𝑦+ 1𝑡 −𝑢𝑥 ,𝑦

𝑡 ,

𝛻𝑊 𝑢𝑡=𝑢𝑥+1 ,𝑦𝑡 −𝑢𝑥 ,𝑦

𝑡 ,

,

,

,

,

• There Perona-Malik diffusion algorithm produces good smoothing results while preserving image edges.

• There is a lot of literature based on the denoising framework proposed by Perona-Malik.

• Their algorithm inspired numerous researchers to analyze and further Improve it. Many of these modified versions produce better smoothing and edge enhancement than the original algorithm.

• Usually, these non-linear diffusion based denoising schemes differ from each other through the diffusivity function “g”, which may have various forms, but must satisfy several conditions.

CONDITIONS OF “g”:

• Positivity• Monotonous decreasing• g(0)=1 and • Convergence to zero

Let us survey some of these approaches and their edge stopping functions...

BUT

SORRY to say

I can’t show you their EXPERIMENTAL resultsat this time

BECAUSE, I don’t have any experience of the implementation ofcodes in MATLAB…

• Chan et al. [1]: proposed Total Variation (TV) diffusivity representing popular edge stopping function:

𝑔 (𝑠2 )= 1¿ 𝑠∨¿¿

and its regularized form

𝑔 (𝑠2 )= 1√𝑠2+𝜀2

• Charbonnier et al. [2]:

Proposed the Charbonnier diffusivity:

𝑔 (𝑠2 )=(1+ 𝑠2𝑘2 )− 12

• J.Weickert et al. [3]:

Proposed the following edge stopping function for anisotropic diffusion

𝑔 (𝑠2 )=1−𝑒−𝑐𝑚

(𝑠2 /𝑘2 )𝑚 , 𝑖𝑓 |𝑠|>01 , 𝑖𝑓 𝑠=0

where

• Black et al. [4]:

Provided the robust anisotropic diffusion (RAD).They used robust estimation theory to model an edge stopping function called Tukey’s biweight:

𝑔 (𝑠2 )=(1− 𝑠2

5𝑘2 )2

, 𝑖𝑓 𝑠2

5≤𝑘2

0 ,𝑖𝑓 𝑠2

5>𝑘2

• The anisotropic diffusion approaches may also differ through the way how 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 Perona-Malik filter, use a fixed k value that could be determined empirically.

• Some authors, like Li-chan, use a single conductance value.

• Some use this parameter a function of time, k(t).

• some use a high k(0) value at the beginning , then k(t) is reduced gradually, as the image is smoothed.

Various noise estimation methods are used for conductance parameter detection.

So,

Now I would like to present you a Recent work done by Tuder Barbu published in 2014:

“Robust anisotropic diffusion based scheme for image noise removal.”

An anisotropic diffusion scheme using novel variants for BOTH

The edge stopping function

And

The diffusivity conductance parameter.

Based on the following parabolic equation

𝜕𝑢𝜕𝑡 =𝑑𝑖𝑣 (𝑔𝑘 (𝑢 ) (|𝛻𝑢|2 ) .𝛻𝑢) ,(𝑥 , 𝑦)∈Ω

𝑢 (𝑜 ,𝑥 , 𝑦 )=𝑢0

where is the degraded image and its domain Ωϵ. Which uses a novel edge stopping function.

(5)

𝑔𝑘 (𝑢 ) (𝑠2 )= .√ 𝑘(𝑢)

𝛽 .𝑠2+𝛾,𝑖𝑓 𝑠>0

1 ,𝑖𝑓 𝑠=0

where the parameters and .

(6)

Also u saw in the previous equation, the modeled edge stopping function is based on a conductance diffusivity depending on the current state (at thetime t) of the processed image, an automatic computation of this parameteris proposed, based on the image noise estimation at each time(iteration) t.Statistical measures are used by the following conductance computation:

𝑘 (𝑢 )=𝑚𝑒𝑑𝑖𝑎𝑛(𝑢)𝜀 .𝑛(𝑢)

.∥𝑢∥𝐹

where is the Frobenius norm of image u, represents its median value and is the number of its pixels.

satisfies the main conditions related to any edge stopping function.

• in monotonous decreasing, because

Equation (5), the proposed model, is then discretized by using a 4-nearest-neighbours discretization of the laplacian operator,

Thus, from equation (5), one can obtains

𝜕𝑢𝜕𝑡 =𝑑𝑖𝑣 (𝑔𝑘 (𝑢 ) (|𝛻𝑢|

2 ) .𝛻𝑢 )⟹𝑢 (𝑥 , 𝑦 , 𝑡+1 )−𝑢 (𝑥 , 𝑦 , 𝑡 )≅ 𝑔𝑘 (𝑢 ) (|𝛻𝑢|2 ) .∆𝑢 ,

which leads to the following approximating scheme:

(7)

where is the set of pixels representing the 4-neighborhood ofThe 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:

𝛻𝑢𝑝 ,𝑞 (𝑡 )=𝑢 (𝑞 ,𝑡 )−𝑢(𝑝 , 𝑡)

The denoising algorithm applies (7) on the current image for each , where N is the maximum number of iteration.

This noise reduction approach obtains the desired smooth image from The noisy image in a relatively small number of steps,

So, it is characterized by a quite low N values, and also outperforms the Perona-Malik algorithm and other anisotropic diffusion based denoising techniques. For properly choosing the parameters, refer to:

Robust anisotropic diffusion scheme for image noise removal.

Tudor Barbu

Available online at www.sciencedirect.com

Chan TF, Wong CK. Total variation blind deconvolution. IEEE Transactions on Image Processing 1998; 7:370–375.

References:

Charbonnier P, Blanc- Feraud L, Aubert, G, Barlaud, M. Two deterministic half-quadratic regularization algorithms for computed imaging. Proc. IEEE International Conf. on Image Processing, Austin, TX, IEEE Comp. Society Press 1994; 2:168–172.

Weickert J. Anisotropic Diffusion in Image Processing. European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart, Germany; 1998.

Black M, Shapiro G, Marimont D, Heeger D. Robust anisotropic diffusion. IEEE Trans. Image Processing 1998; 7 (3): 421-432.

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THANKSTO ALL OF YOU

Abdur Rehman, MS Math, UET Peshawar.