noise models

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Digital Image Processing

Image Restoration

Noise models and additive noise removal

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Image Restoration

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Image Restoration

What is noise (in the context of image processing) and how can it be modeled?

What are the main types of noise that may affect an image? What are the possible solutions?

Subjective Vs Objective (Enhancement Vs Restoration)

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Degradation Model for a Digital Image

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Noise Models

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Noise and Noise Models

Gaussian (normal) Impulse (salt-and-pepper) Uniform Rayleigh Gamma (Erlang) Exponential

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Effect of Noise on Images & Histograms

Gaussian

Exponential

Impulse (salt-and-pepper)

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Effect of Noise on Images & Histograms

Rayleigh

Gamma (Erlang)

Uniform

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Noise Models: Gaussian Noise

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Noise Models: Rayleigh Noise

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Noise Models: Erlang (Gamma) Noise

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Noise Models: Exponential Noise

Where

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Noise Models: Uniform Noise

The mean and variance are given by

otherwise 0

if ,1)(

bzaabzp

12)(,

2

22 abba

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Noise Models: Impulse (Salt and Pepper) Noise

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Effect of Noise on Images & Histograms

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Effect of Noise on Images & Histograms

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Effect of Noise on Images & Histograms

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Periodic Noise (Example)

Spatially Dependent Case

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Applicability of various noise models

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Estimation of noise parameters

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Estimation of noise parameters (example)

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Estimation of noise parameters (example)

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Estimation of noise parameters

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Restoration of noise-only degradation

Filters to be considered

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Mean Filters: Arithmetic mean filter

Causes a certain amount of blurring (proportional to the window size) to the image, thereby reducing the effects of noise. Can be used to reduce noise of different types, but works best for Gaussian, uniform, or Erlang noise.

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Mean Filters: Geometric mean filter

– A variation of the arithmetic mean filter– Primarily used on images with Gaussian noise– Retains image detail better than the arithmetic mean

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Mean Filters: Harmonic mean filter

– Another variation of the arithmetic mean filter– Useful for images with Gaussian or salt noise– Black pixels (pepper noise) are not filtered

Harmonic mean filter

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Arithmetic and geometric mean filters (example)

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Mean Filters: Harmonic mean filter

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Mean Filters: Harmonic mean filter

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Mean Filters: Contra-harmonic mean filter

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Classification of contra-harmonic filter applications

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Contra-harmonic mean filter (example)

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Contra-harmonic mean filter (example)

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Rank / Order / Order Statistics Filters

– Known as Rank filters, Order filters OR Order Statistics filters

– Operate on a neighborhood around a reference pixel by ordering (ranking) the pixel values and then performing an operation on those ordered values to obtain the new value for the reference pixel

– They perform very well in the presence of salt and pepper noisebut are more computationally expensive as compared to mean filters

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Rank / Order Statistics Filters: Median filter

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Rank / Order Statistics Filters: Median filter

– Most popular and useful of the rank filters.

– It works by selecting the middle pixel value from the ordered setof values within the m × n neighborhood (W) around the reference pixel.

• If mn is an even number, the arithmetic average of the two values closest to the middle of the ordered set is used instead.

– Many variants, extensions, and optimized implementations in the literature.

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Median filter (Example)

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Rank / Order Statistics Filters: Max and Min filter

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Rank / Order Statistics Filters: Max and Min filter

– Max filter also known as 100th percentile filter– Min filter also known as zeroth percentile filter– Max filter helps in removing pepper noise– Min filter helps in removing salt noise

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Max and Min filter (Example)

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Rank / Order Statistics Filters: Midpoint filter

– Calculates the average of the highest and lowest pixel valueswithin a window

– What would it do with salt and pepper noise ?

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Midpoint filter (Example)