A PRESENTATION ON

Image Denoising

Algorithms

By:-

Mohammad Sunny

Introduction What is Image denoising?

The removing of noise from the image is called Image denoising.

The algorithms are used for Image denoising are called Image denoising algorithms.

What is Image?

A n image is generally encoded as a matrix of grayscale or color values. Each pair (i, u(i)), where u(i) is the value at i, is called a pixel.

In the case of grayscale images, i is a point on a two-dimensional (2D) grid and u(i) is a real value. In the case of classical color images, u(i) is a triplet of values for the red, green, and blue components.

What is noise? Each one of the pixel values u(i) is the result of a light

intensity measurement, usually made by a charge coupled device (CCD) matrix coupled with a light focusing system.

Each captor of the CCD is roughly a square in which the number of incoming photons is being counted for a fixed period corresponding to the obturation time.

What is Noise? When the light source is constant, the number of

photons received by each pixel fluctuates around its average in accordance with the central limit theorem.

In other words, one can expect fluctuations of order √n for n incoming photons. In addition, each captor, if not adequately cooled, receives heat photons. This is usually called “noise.”

Noise model All denoising algorithm are based on Noise

Model. Noise Model

v(i) = u(i) + n(i) ;iϵI v(i): observed value,

u(i): true value,

n(i): noise value

Method noise

𝑛( ,v) = v – (v)𝐷ℎ 𝐷ℎ

•V: noise image

•Dh: denoise method

•Dh(v) is more smooth than v (Smooth part )

•n(Dh,v): the noise guessed by the method (Non-smooth part (contains both noise and texture))

Types of Denoising Algorithms All the denoising algorithms are achieved by averaging.

The most common types are:-

Spatial domain filter

•Gaussian filtering

•Anisotropic filtering (AF)

•Neighboring filtering

•Total Variation minimization

Non-Local-Means (NL-means) algorithm

Gaussian Filtering The image isotropic linear filtering boils down to

the convolution of the image by a linear symmetric gaussian kernel.

The image method noise of the convolution with a gaussian kernel Gh is

u − Gh u = −h²Δu + o(h²),∗ for h small enough.

Gaussian Filtering Gaussian convolution is optimal in flat

parts of the image.

Drawback of Gaussian Filtering Edges and textures are blurred.

Anisotropic filtering (AF)

Attempt to avoid the blurring effect of the Gaussian. Convolve the image at only in the direction 𝑢 𝑥

orthogonal to ( ).𝐷𝑢 𝑥

u(x) − AFhu(x) = −½h²|Du|curv(u)(x) + o(h²), where the relation holds when Du(x) = 0.

Anisotropic filtering (AF)

The Straight edges are well restored.

Drawbacks of AF Flat and texture regions are degraded

Total Variation minimization In total variation minimization, the original image u

is supposed to have a simple geometric description, namely, a set of connected sets, the objects, along with their smooth contours, or edges. The image is smooth inside the objects but with jumps across the boundaries.

u(x) − TVF[λ](u)(x) = − ½λcurv(TVF[λ](u))(x). where TV (u) denotes the total variation of u and λ

is a given Lagrange multiplier.

Total Variation minimization Straight edges are maintained because of their small curvature.

Drawback of Total Variation minimization Textures can be over smoothed if λ is too small.

Neighborhood filtering The previous filters are based on a notion of

spatial neighborhood or proximity. Neighborhood filters instead take into account grayscale values to define neighboring pixels. In the simplest and more extreme case, the denoised value at pixel i is an average of values at pixels which have a grayscale value close to u(i). The grayscale neighborhood is therefore

B(i, h) = {j I | u(i) −h < u(j) < u(i) + h}∈

Neighborhood filtering This is a fully nonlocal algorithm, since pixels

belonging to the whole image are used for the estimation at pixel i.

Drawback of Neighborhood filtering Comparing only grey level values in as single pixel

is NOT so robust when these values are noisy.

1) Noisy image , 2) Gaussian convolution (h = 1.8), 3) anisotropic filter(h = 2.4), 4) total variation (λ = 0.04), 5) Neighborhood filter (ρ =7, h = 28).

NL-Means Algorithm The NL-means algorithm tries to take advantage

of the high degree of redundancy of any natural image. By this, we simply mean that every small window in a natural image has many similar windows in the same image. This fact is patent for windows close by, at one pixel distance, and in that case we go back to a local regularity assumption.

NL-Means Algorithm NL-means not only compares the grey level in a single point but also the geometrical configuration in a whole neighborhood.

More robust than neighborhood filter.

NL-Means AlgorithmP has the same grey level value of q3

But, the neighborhoods are much different.

Therefore the weight w(p, q3) is nearly 0

Thank you