Wiener Filter for Image Restoration

The Wiener filter is a powerful technique used for image restoration, specifically designed to handle both image degradation and noise. Unlike simple inverse filtering, the Wiener filter incorporates statistical characteristics of noise and degradation into its restoration process.

Objective

The primary objective of the Wiener filter is to estimate the original, uncorrupted image f from a degraded image g while minimizing the mean square error between the estimated image and the original. The error measure is defined as:

where E{•} denotes the expected value, and \hat{f} is the estimate of the original image.

Mathematical Formulation

The Wiener filter considers both the degradation function and the noise in the image restoration process. The filter aims to find the optimal estimate of the original image by minimizing the mean square error, under the following assumptions:

• Uncorrelated Image and Noise: The image and noise are uncorrelated.
• Zero Mean: Either the image or the noise has zero mean.
• Linearity: The gray levels in the estimate are a linear function of the levels in the degraded image.

Based on these assumptions, the Wiener filter in the frequency domain is given by:

where:

• H(u, v) is the degradation function in the frequency domain.
• H^*(u, v) is the complex conjugate of H(u, v).
• |H(u, v)|^2 = H^*(u, v) H(u, v) is the magnitude squared of the degradation function.
• S_\eta(u, v) is the power spectrum of the noise N(u, v).
• S_f(u, v) is the power spectrum of the undegraded image F(u, v).

Frequency Domain to Spatial Domain

The restored image in the spatial domain is obtained by applying the inverse Fourier transform to the filtered frequency domain estimate:

Special Cases

The Wiener filter adapts to different scenarios based on the characteristics of the noise and degradation function:

• Zero Noise: If S_\eta(u, v) = 0, the Wiener filter reduces to the inverse filter. This is effective when there is no noise, but less practical in real-world scenarios with noise.
• Spectrally White Noise: When the noise is spectrally white, S_\eta(u, v) is constant, which simplifies the Wiener filter to:



where K is a specified constant representing the noise power.

Significance

The Wiener filter is significant due to its ability to handle both image degradation and noise effectively. It minimizes the mean square error, making it suitable for practical image restoration where noise and degradation are present. The filter's adaptability to different noise scenarios and its improved performance over simpler methods like inverse filtering make it a valuable tool in image processing.