IV. Image Restoration
Image restoration aims to recover an original image \( f(x,y) \) from a degraded observation \( g(x,y) \). The degradation model is typically:
\[ g(x,y) = (h * f)(x,y) + \eta(x,y) \]
where \( h(x,y) \) is the point spread function (PSF) of the system, \( * \) denotes convolution, and \( \eta(x,y) \) is additive noise. In the frequency domain:
\[ G(u,v) = H(u,v)F(u,v) + N(u,v) \]
Noise Parameter Estimation
Accurate noise statistics (variance, distribution) are critical for optimal filtering. Common estimation methods:
| Method | Principle | Key Considerations |
|---|---|---|
| Smooth Region Analysis | Compute sample mean \( \mu \) and variance \( \sigma^2 \) from visually uniform areas (e.g., sky). | Assumes noise dominates in these regions; signal variations must be minimal. |
| Autocorrelation Function | For zero-mean, uncorrelated noise, \( R_{\eta}(0,0) = \sigma_{\eta}^2 \). Estimate autocorrelation \( R_{gg}(x,y) \) and subtract signal contribution. | Requires image signal autocorrelation to decay rapidly; sensitive to edges. |
| Histogram Analysis | In smooth regions, histogram approximates noise distribution. Fit to parametric models (e.g., Gaussian, uniform). | Needs smooth regions; distribution type must be known or assumed. |
| Robust Statistics | Use median absolute deviation (MAD): \( \sigma \approx \frac{\text{MAD}}{0.6745} \) for Gaussian noise. | Robust to outliers; less sensitive to non-uniform regions. |
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Common Pitfall: Smooth region method may overestimate noise if the region contains subtle textures. Always validate with multiple methods.
Restoration Filters
Filters are designed to minimize an error criterion, typically Mean Square Error (MSE):
\[ \text{MSE} = E\left[ (f(x,y) - \hat{f}(x,y))^2 \right] \]
where \( \hat{f} \) is the restored image.
Minimum Mean Square Error (MMSE) Filtering
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Principle: Find a linear filter \( w(x,y) \) such that \( \hat{f} = w * g \) minimizes MSE.
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The optimal MMSE estimator in the linear class is the Wiener filter, derived under assumptions of stationarity and known power spectra.
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General MMSE solution: \( \hat{F}(u,v) = W(u,v) G(u,v) \), where \( W(u,v) \) is chosen to minimize \( E[|F - \hat{F}|^2] \).
Wiener Filtering
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Definition: A frequency-domain MMSE filter that optimally balances noise suppression and detail preservation.
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Filter Transfer Function:
\[ \boxed{H_w(u,v) = \frac{H^*(u,v)}{|H(u,v)|^2 + \frac{S_{nn}(u,v)}{S_{ff}(u,v)}}} \]
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\( H(u,v) \): Degradation PSF.
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\( H^*(u,v) \): Complex conjugate.
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\( S_{ff}(u,v) \): Power spectrum of the original image.
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\( S_{nn}(u,v) \): Power spectrum of the noise.
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Restored Image:
\[ \hat{F}(u,v) = H_w(u,v) G(u,v) \]
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Special Cases:
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If \( S_{nn} = 0 \) (no noise), \( H_w = \frac{1}{H} \) → inverse filter.
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If \( H = 1 \) (only noise), \( H_w = \frac{S_{ff}}{S_{ff} + S_{nn}} \) → noise smoothing filter.
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Practical Implementation: \( S_{ff} \) and \( S_{nn} \) are often estimated or approximated (e.g., constant \( K = \frac{S_{nn}}{S_{ff}} \)).
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Exam Key: The Wiener filter formula must include the conjugate \( H^* \) and the SNR term \( \frac{S_{nn}}{S_{ff}} \). Forgetting \( H^* \) yields a non-causal, unstable filter.
Complexity: Wiener filtering is \( O(MN \log(MN)) \) using FFT for an \( M \times N \) image.