Inverse Filtering in Image Restoration

Image restoration is a significant aspect of image processing, aiming to recover an original image that has been degraded or corrupted by factors like noise, blurring, or distortion. Among various techniques used for image restoration, inverse filtering is one of the most straightforward and commonly used methods, especially for removing the effects of blurring. This article explains the mathematical concepts behind inverse filtering, the steps involved in the process, and examples to illustrate its practical application.

1. Degradation Model

The degradation of an image can be mathematically represented by a degradation model. The general degradation model is described by:

Where:

• $g(x, y)$ is the degraded image.
• $f(x, y)$ is the original image (unknown).
• $h(x, y)$ is the degradation function, also called the point spread function (PSF) or blurring filter.
• $\eta(x, y)$ represents the noise added to the image.
• $*$ denotes convolution.

In the frequency domain, the above model can be rewritten using the Fourier transform:

Where:

• $G(u, v)$, $H(u, v)$, and $F(u, v)$ are the Fourier transforms of $g(x, y)$, $h(x, y)$, and $f(x, y)$ respectively.
• $N(u, v)$ is the Fourier transform of the noise $\eta(x, y)$.
• $\cdot$ denotes pointwise multiplication.

1.1. Blurring Degradation

In real-world applications, blurring is a common type of degradation and is modeled by the degradation function $h(x, y)$. For example, motion blur and Gaussian blur are frequently used degradation types. In this context, the goal of inverse filtering is to recover the original image $f(x, y)$ from the blurred image $g(x, y)$ by reversing the effect of the degradation function $h(x, y)$.

2. Inverse Filtering Concept

The basic idea of inverse filtering is to reverse the degradation process by dividing the degraded image by the degradation function in the frequency domain.

Given the degradation model in the frequency domain:

The inverse filter tries to recover $F(u, v)$ by simply dividing $G(u, v)$ by $H(u, v)$:

Where $\hat{F}(u, v)$ is the estimate of the original image’s Fourier transform.

The recovered image $\hat{f}(x, y)$ can then be obtained by applying the inverse Fourier transform to $\hat{F}(u, v)$.

2.1. Challenges in Inverse Filtering

While the concept of inverse filtering seems simple, it faces practical difficulties:

• Noise Sensitivity: Inverse filtering amplifies any noise $N(u, v)$, especially if $H(u, v)$ has very small values or zeros. This is because dividing by small values results in large values, leading to significant noise amplification.

• Ill-conditioning: If $H(u, v)$ is close to zero for some frequencies, division by such small values causes instability and unreliability in the reconstruction.

3. Example of Inverse Filtering

3.1. Problem Setup

Let’s assume a 1D image signal is blurred by a known degradation function $h(x, y)$, and the observed image is:

3.2. Frequency Domain Representation

The Fourier transforms of the functions $f(x, y)$, $h(x, y)$, and $g(x, y)$ are:

• $F(u) = \text{Fourier Transform of } f(x, y)$
• $H(u) = \text{Fourier Transform of } h(x, y)$
• $G(u) = \text{Fourier Transform of } g(x, y)$

Using inverse filtering, we can estimate the original image in the frequency domain:

Finally, the original image can be recovered by performing the inverse Fourier transform of $\hat{F}(u)$.

3.3. Example

Suppose the degradation function is a simple Gaussian blur with known parameters. After applying the inverse filter to the blurred image, we would use the Fourier transforms of both the degraded image and the blur function to estimate the original image.

Let’s say:

• The degradation function in the frequency domain $H(u)$ has small values at high frequencies.
• The Fourier transform of the observed image $G(u)$ is known.

After applying the inverse filter, the high-frequency noise in the image may be amplified, resulting in a noisy restored image. This is a known limitation of inverse filtering, leading to the development of regularized filtering or Wiener filtering to address these issues.

4. Improving Inverse Filtering with Regularization

Due to the noise sensitivity and ill-conditioning issues in inverse filtering, more advanced methods are often used to stabilize the solution. One such method is Wiener filtering, which incorporates regularization and reduces the effect of noise.

The Wiener filter modifies the inverse filtering formula to include both the degradation function and the noise characteristics, leading to a more stable solution.

Wiener Filter Formula:

Where:

• $H^*(u, v)$ is the complex conjugate of the degradation function.
• $S_N(u, v)$ and $S_F(u, v)$ are the power spectral densities of the noise and the original image, respectively.

This filter optimally balances the inverse filtering solution by minimizing the effect of noise.