Presented By :Haitham Abdel-atty Abdullah

Supervised By : Prof .Dr . Mostafa Gadal-Haqq

The process with which we reconstruct a signal from a noisy one.

Removing unwanted noise in order to restore the original image.

Method of estimating the unknown signal from available noisy data”.

Remove noise Preserve useful information

Image de-noising is an important pre-processing step for image analysis.

Denoising

𝑥𝑥

𝑛

𝑦 = 𝑥 + 𝑛

𝑛 = 𝑦 − 𝑥Noise

Additive Noise Model Multiplicative Noise Model

Noise signal gets added to the original signal to produce a corrupted noisy signal.

Noise signal gets multiplied to the original signal to produce a corrupted noisy signal.

The Additive noise model follows the following rule:

The Multiplicative noise model follows the following rule:

Gaussian Noise Is evenly distributed over the signal This means that each pixel in the

noisy image is the sum of the true pixel value and a random Gaussian distributed noise value.

Impulse Noise The impulse noise is of two types, Fixed value and random value.

Fixed value (salt and pepper noise)

It is generally caused due to errors in transmission.

It has only two possible values For an 8-bit image, the typical value for pepper noise is 0 and for salt noise 255.

The corrupted pixels are set alternatively to minimum or to maximum values giving the image Salt and Pepper

Random value

The random valued impulse noise can have any value between 0 and 255 hence its removal is very difficult.

Uniform Noise The uniform noise cause by quantizing the pixels of image to a number

of distinct levels is known as quantization noise.

Uniform noise can be analytically described by :

Periodic Noise

Source: electrical or electromechanical interference during image acquisition

Easy to observe in frequency domain.

Image with periodic noise frequency domain

De-noising Methods

SPATIAL DOMAIN

LINEAR FILTERNON- LINEAR FILTER

FREQUENCY DOMAIN

Mean Filter

Weighted Mean Filter

Median Filter

LPF on Fourier Transform

In spatial domain , we deal with images as it is (simply we directly deal with the image matrix).

Linear filters are used to remove certain type of noise.

The linear filters work best with salt and pepper noise, and Gaussian noise.

Mean filters.

Simple to design .

These filters also tend to blur the sharp edges . destroy the lines and other fine details of image .

One of the simplest spatial filtering operations.

Replaces the centre value of the window with the average values of its all nearest pixels values together with itself.

Advantage: Easy to implement Used to remove the impulse noise.

Disadvantage: It does not preserve details of image. Some

details are removes of image with using the mean filter.

1/91/9

1/9

1/91/9

1/9

1/91/9

1/9

Origin x

y Image f (x, y)

New value = 1/9*106 + 1/9*104 +1/9*100 + 1/9*108 + 1/9*99 + 1/9*98 +

1/9*95 + 1/9*90 + 1/9*85

= 98.3333

FilterSimple 3*3

Neighbourhood106

104

99

95

100 108

98

90 85

104 100 108

99 106 98

95 90 85

Original Image Pixels

*

The above is repeated for every pixel in the original image to generate the smoothed image

Extending border values outside with values at boundary.

(a) Original Image (b) Image corrupted by Gaussian noise .

(c)De noising by mean filter

Simple averaging (Mean filter) of neighboring pixels lead to over-smoothing

Instead of weighting all neighboring pixels equally, assign higher weights to pixels that are closer to the pixel being convolved.

Weighted Mean FilterMean Filter

Can preserve edges .

Very effective at removing impulsive noise .

They are more powerful than linear filters because they are able to reduce noise levels without blurring edges.

Difficult to design than linear filters.

Median Filter.

Median filter is one of the most important filters to remove random valued impulse noise.

In this filter the value of corrupted pixel in noisy image is replaced by median value of corresponding window.

The median is calculated by first sorting all the pixel values into ascending order and then replace the pixel being calculated with the middle pixel value

Salt and pepper noise.

99

Origin x

y Image f (x, y)

Sort the pixels values106

104

99

95

100 108

98

90 85

104 100 108

99 106 98

95 90 85

Original Image Pixels

The above is repeated for every pixel in the original image.

9985 104100 108989590 106

21

Original

Salt & pepper%20

Denoising by Median filter

Original

Salt & pepper%60

Denoising by Median filter

Frequency domain de-noising method is a kind of Transform Domain, filtering where low pass filters (LPF.) is used by using Fast Fourier Transform (FFT).

The reason for doing the filtering in the frequency domain is generally because it is computationally faster to perform two 2D Fourier transforms and a filter multiply than to perform a convolution in the image (spatial) domain.

A low-pass filter attenuates high frequencies and retains low frequencies unchanged.

Here de-noising is done by designing a cut-off frequency .

Convolution in the Spatial domain is equivalent to multiplication in the frequency domain.

Original Image Fourier Transform Apply LPF on FT Inverse Fourier Transform

Original Image with noise Fourier Transform Inverse Fourier Transform

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