Advanced Concepts in Constrained Least Squares Filtering for Image Restoration

Image restoration is a critical process in digital image processing, aimed at recovering the original image from a degraded version. One of the advanced methods used for this purpose is Constrained Least Squares Filtering (CLSF). Unlike the Wiener filter, CLSF incorporates additional constraints that allow for improved restoration under certain conditions, especially when the power spectra of the noise and degradation function are not known with precision. In this article, we will explore the theory behind constrained least squares filtering, present key mathematical formulations, and demonstrate its practical application through iterative algorithms.

Constrained Least Squares Filtering: An Overview

The primary objective of CLSF is to minimize the error in estimating the undegraded image, subject to certain constraints. Mathematically, the degradation process can be represented as:

$$g = Hf + \eta$$

Where:

• $g$ is the observed (degraded) image,
• $H$ is the degradation function (blur kernel),
• $f$ is the original image to be recovered,
• $\eta$ is the additive noise.

The challenge in constrained least squares filtering is the formulation of an optimal estimate of the original image $f$, given that the degradation function $H$ is often sensitive to noise. To address this, we introduce a smoothness constraint, which penalizes high-frequency components in the estimated image. The cost function to be minimized is:

$$C = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} [\nabla^2 f(x, y)]^2$$

Where $\nabla^2$ is the Laplacian operator, a measure of image smoothness.

Optimization Under Constraint

The constrained least squares optimization can be expressed as minimizing the following functional:

$$\| g – Hf \|^2 + \gamma \|\nabla^2 f\|^2$$

Where $\| \cdot \|^2$ denotes the Euclidean norm, and $\gamma$ is the regularization parameter that controls the trade-off between fitting the observed data and maintaining smoothness in the restored image.

Frequency Domain Solution

By transforming the problem into the frequency domain, the CLSF solution can be expressed as:

$$\hat{F}(u, v) = \frac{H^*(u, v)}{|H(u, v)|^2 + \gamma |P(u, v)|^2} G(u, v)$$

Where:

• $\hat{F}(u, v)$ is the Fourier transform of the restored image,
• $H^*(u, v)$ is the complex conjugate of the Fourier transform of the degradation function,
• $P(u, v)$ is the Laplacian operator in the frequency domain,
• $G(u, v)$ is the Fourier transform of the observed image.

This solution provides a balance between minimizing the noise and enforcing smoothness in the restored image.

Iterative Adjustment of $\gamma$

One of the key challenges in applying CLSF is selecting the appropriate value of $\gamma$, which determines the regularization strength. The optimal value of $\gamma$ can be found iteratively. Let the residual vector be:

$$r = g – Hf$$

We define a function $\phi(\gamma)$ that measures the residual energy:

$$\phi(\gamma) = \frac{r^T r}{\|r\|^2}$$

The algorithm proceeds as follows:

1. Initialize $\gamma$.
2. Compute $\|r\|^2$.
3. Adjust $\gamma$ iteratively based on the residual energy until the optimal constraint is met.