Math 3360: Mathematical Imaging

Prof. Ronald Lok Ming LuiDepartment of Mathematics,

The Chinese University of Hong Kong

Lecture 11:Types of noises

Class schedules

Lecture 1: Introduction to Image Processing

Lecture 2: Basic idea of image transformation

Lecture 3: Image decomposition & Stacking operator

Lecture 4: Singular Value Decomposition for Image decomposition & Error analysis

Lecture 5: Haar & Walsh Transform

Lecture 6: Examples of Haar & Walsh Transform; R-Walsh transform

Lecture 7: Discrete Fourier transform

Lecture 8: Even Discrete Cosine Transform (JPEG)

Lecture 9: EDCT + ODCT+ EDST + ODST; Introduction to Image enhancement

Lecture 10: Introduction to Linear filtering & Statistical images

Lecture 15 to Lecture 17: Image deblurring

Lecture 18 to Lecture 21: Image segmentation

Lecture 22 to Lecture 24: Image registration

Lecture 11: Image denoising: Linear filtering model in the spatial domain;

Image denoising: Nonlinear filtering model in the spatial domain;

Relationship with the convolution

Lecture 12: Image denoising: Linear filtering in the frequency domain

Image denoising: Anisotropic diffusion

Lecture 13: Image denoising: Total variation (TV) or ROF model

Lecture 14: Image denoising: ROF model part 2

Type of noises

Recap: Preliminary statistical knowledge: Random variables; Random field; Probability density function; Expected value/Standard deviation; Joint Probability density function; Linear independence; Uncorrelated; Covariance; Autocorrelation; Cross-correlation; Cross covariance; Noise as random field etc…

Please refer to Supplemental note 6 for details.

Type of noises

Impulse noise: Change value of an image pixel at random; The randomness follows the Poisson distribution = Probability

of having pixels affected by the noise in a window of certain size

Poisson distribution:

Gaussian noise: Noise at each pixel follows the Gaussian probability density

function:

Type of noises

Additive noise: Noisy image = original (clean) image + noise

Multiplicative noise: Noisy image = original (clean) image * noise

Homogenous noise: Noise parameter for the probability density function at each

pixel are the same (same mean and same standard derivation)

Zero-mean noise: Mean at each pixel = 0

Biased noise: Mean at some pixels are not zero

Type of noises

Independent noise: The noise at each pixel (as random variables) are linearly

independent

Uncorrected noise: Let Xi = noise at pixel i (as random variable); E(Xi Xj) = E(Xi) E(Xj) for all i and j.

White noise: Zero mean + Uncorrelated + additive

idd noise: Independent + identically distributed; Noise component at every pixel follows the SAME probability

density function (identically distributed) For Gaussian distribution,

Gaussian noise

Example of Gaussian noises:

White noise

Example of white noises:

Image components

Noises as high frequency component

Why noises are often considered as high frequency component?

(a) Clean image spectrum and Noise spectrum (Noise dominates the high-frequency component);(b) Filtering of high-frequency component

Linear filter = Convolution

Linear filtering of a (2M+1)x(2N+1) image I (defined on

[-M,M]x[-N,N]) = CONVOLUTION OF I and H H is called the filter. Different filter can be used:

Mean filter Gaussian filter Laplcian filter

Variation of these filters (Non-linear) Median filter Edge preserving mean filter

Linear filter

Type of filter

In Photoshop

Mean filter

Mean filter

Impulse noise

After mean filter

Mean filter

Gaussian noise

After mean filter

Mean filter

Real image After mean filter

Gaussian filter

Define a function using Gaussian function

Definition of H

Gaussian filter

Real image After mean filter

Gaussian filter

Real image After mean filter

Gaussian filter

Real image After Gaussian filter

Gaussian filter

Real image After mean filter

Gaussian filter

Real image After Gaussian filter

Laplace filter

)

Laplace filter (High pass filter)

Laplace filter

Laplace filterOriginal

Laplace filter

Laplace filterOriginal

Laplace filter

Laplace filterOriginal

Median filter

Median Nonlinear filter Take median within a local window

Median filter

Real image

After mean filter

Median filter

Salt & Pepper

Mean filter

Median filter

Median filter

Noisy image Median filter

Median filter

Median filter

Noisy image Median filter

Median filter

Median filter

Noisy imageCan you guess what it is?

Median filter