UNIT 4: IMAGE RESTORATION

 

Introduction to Image Restoration

Image Restoration is the process of recovering an original image that has been degraded by known or unknown factors (such as blur, noise, or motion).
It focuses on model-based correction, not just enhancing the image visually.

• Goal: Retrieve the most accurate version of the original image.

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2. Image Formation Models

The mathematical relationship between the original image, the degradation, and the observed image.

General Model:

$$g(x, y) = h(x, y) * f(x, y) + \eta(x, y)$$

Where:

• $g(x,y)$ = degraded image

• $h(x,y)$ = degradation function (e.g., blur)

• $f(x,y)$ = original image

• $\eta(x,y)$ = noise

• $*$ = convolution operation

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3. Noise Models

Describes how random variations corrupt an image.

Common Noise Types:

• Gaussian Noise: Random values from a normal distribution.

• Salt-and-Pepper Noise: Random white and black pixels (high contrast noise).

• Poisson Noise: Related to photon counting in sensors.

• Speckle Noise: Common in radar and ultrasound images (multiplicative noise).

Noise Model Example:

$$g(x, y) = f(x, y) + \eta(x, y)$$

4. Restoration Techniques

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(a) Inverse Filtering

• Idea: Directly undo degradation by applying the inverse of the degradation function.

Formula:

$$F(u,v) = \frac{G(u,v)}{H(u,v)}$$

• Sensitive to noise → not ideal when noise is significant.

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(b) Wiener Filtering

• Idea: Reduces overall mean square error between estimated and original image.

Formula:

$$F(u,v) = \left( \frac{H^*(u,v)}{|H(u,v)|^2 + S_\eta(u,v)/S_f(u,v)} \right) G(u,v)$$

Where:

• $H^*(u,v)$ = complex conjugate

• $S_\eta$ = noise power spectrum

• $S_f$ = original signal power spectrum

• Balances both deblurring and denoising.

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(c) Least Squares Filtering

• Idea: Minimizes the mean squared difference between restored and original image.

• Often uses regularization to prevent noise amplification.

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(d) Recursive Filters

• Idea: Filter output depends on both current and past inputs/outputs.

• Efficient for large images.

• Useful for real-time restoration (low computational load).

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(e) Maximum Entropy Method (MEM)

• Idea: Chooses the solution with the maximum entropy (least amount of prior information) that fits the observed data.

• Especially useful when multiple possible restorations exist.

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(f) Blind Deconvolution

• Idea: Restores an image without prior knowledge of the degradation function.

• Requires iterative techniques to estimate both the image and the blur simultaneously.

Common Techniques:

• Iterative blind deconvolution

• Regularized blind deconvolution

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(g) Bayesian Method of Noise Removal

• Idea: Uses Bayesian probability to model prior knowledge about noise and image.

• Restoration by maximizing the posterior probability.

Bayes Rule:

$$P(f|g) = \frac{P(g|f)P(f)}{P(g)}$$

Where:

• $P(f|g)$ = probability of original image given the observed

• $P(g|f)$ = likelihood

• $P(f)$ = prior

• Especially effective when statistical models of noise and the image are available.

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5. Image Reconstruction

Process of recovering an image from incomplete, noisy, or indirect measurements.

Examples:

• CT scans (from projections)

• MRI images

• Astronomical imaging

Common Techniques:

• Radon Transform

• Iterative Reconstruction Algorithms (e.g., ART - Algebraic Reconstruction Technique)

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Summary Table

Topic	Key Point
Image Formation	Convolution + Noise
Noise Models	Gaussian, Poisson, Salt-Pepper, Speckle
Inverse Filtering	Direct deblurring (very sensitive to noise)
Wiener Filtering	Balances deblurring and noise reduction
Least Squares	Error minimization
Recursive Filters	Fast, history-dependent filtering
Maximum Entropy	Picks least-biased solution
Blind Deconvolution	No knowledge of blur needed
Bayesian Methods	Probabilistic noise removal
Image Reconstruction	Recover from incomplete data

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In Short:

Image Restoration focuses on undoing known distortions (unlike enhancement which just improves appearance).
It uses mathematical models like inverse filtering, Bayesian methods, and more, to faithfully recover the original image from the corrupted one.