The Chinese University of Hong Kong Math 3360: Mathematical Imaging Lecture 10: Types of noises Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong

Image Enhancement Linear filtering: Modifying a pixel value (in the spatial domain) by a linear combination of neighborhood values. Operations in spatial domain v.s. operations in frequency domains: Linear filtering (matrix multiplication in spatial domain) = discrete convolution In the frequency domain, it is equivalent to multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components

Spatial transform v.s. frequency transform Discrete convolution: (Matrix multiplication, which define output value as linear combination of its neighborhood) DFT of Discrete convolution: Product of fourier transform DFT(convolution of f and w) = C*DFT(f)*DFT(w) Multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components

Spatial transform v.s. frequency transform

Spatial transform v.s. frequency transform

Image components LP = Low Pass; HP = High Pass

Image components

Type of noises 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: Gaussian 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: Multiplicative noise: Homogenous 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: Uncorrected noise: White 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