The Chinese University of Hong Kong Math 3360: Mathematical Imaging Lecture 12: Image Enhancement & Types of noises Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong

Image Enhancement What is image enhancement? How? Image enhancement is the process by which we improve an image so that it looks subjectively better. How? An image is enhanced when we: remove additive noise and interference; remove multiplicative interference; increase its contrast; decrease its blurring. Some standard methods: smoothing or low pass filtering; sharpening or high pass filtering; histogram manipulation and algorithms that remove noise while avoid blurring the image.

Image Enhancement We will consider two image enhancement problems: Image denoising Image deblurring

Image Enhancement We will consider two image enhancement problems: Image denoising Image deblurring

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: 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

Type of noises Preliminary statistical knowledge: Random variables; Random field; Probability density function; Expected value/Standard deviation; Joint Probability density function; Independence; Uncorrelated; Covariance; Autocorrelation; Cross-correlation; Cross covariance; Noise as random field etc… Please refer to Lecture Note Chapter 3 for details.

Statistical Background Please refer to Lecture Note Chapter 3 for details.

Statistical Background

Statistical Background

Statistical Background

Statistical Background

Statistical Background

Statistical Background What is noise?

Statistical Background

Statistical Background

Statistical Background

Statistical Background

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: Uncorrelated noise: White noise: The noise at each pixel (as random variables) are independent. Uncorrelated 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 iid 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:

White noise Example of white noises:

Noises as high frequency component Why noises are often considered as high frequency component? frequency frequency (a) Clean image spectrum and Noise spectrum (Noise dominates the high-frequency component); (b) Filtering of high-frequency component