Noise Models in Image processing

In image processing, images are often degraded due to various factors such as noise, motion blur, or sensor imperfections. The Image Degradation/Restoration Process models this degradation mathematically and provides techniques to recover the original image. This process is fundamental in applications like medical imaging, satellite imagery, and photography, where images need to be restored to a clearer form.

1. Image Degradation Model

The degradation process of an image can be expressed mathematically as a combination of the true (original) image, some degrading function, and additive noise. The general form of this model is:

Where:

• $g(x, y)$ is the observed degraded image.
• $f(x, y)$ is the original (ideal) image.
• $h(x, y)$ is the degradation function (also called the point spread function, or PSF), which models how the image gets blurred or distorted. It might represent, for instance, camera motion or atmospheric distortion.
• $*$ denotes convolution.
• $\eta(x, y)$ is the additive noise, representing random disturbances like sensor noise or transmission errors.

This model assumes that the degradation process can be modeled as convolution (linear operation) followed by additive noise (nonlinear operation).

2. Image Restoration Model

The goal of the image restoration process is to estimate the original image $f(x, y)$ given the degraded image $g(x, y)$, the degradation function $h(x, y)$, and the characteristics of the noise $\eta(x, y)$. Mathematically, the restoration process can be seen as solving the inverse problem of the degradation model.

In the frequency domain, the degradation model becomes simpler using Fourier transforms. Taking the 2D Fourier transform of the degradation model:

Where:

• $G(u, v)$, $H(u, v)$, $F(u, v)$, and $N(u, v)$ are the Fourier transforms of the degraded image, the degradation function, the original image, and the noise, respectively.
• $u, v$ represent the frequency variables.

In this domain, convolution becomes multiplication, which simplifies many restoration methods.

Restoration Techniques:

Several approaches can be applied to restore the degraded image. Below are the most common methods:

3. Inverse Filtering

Inverse filtering is the most straightforward approach to image restoration. In the frequency domain, the original image can be estimated as:

This means the degraded image is divided by the degradation function in the frequency domain, thereby “undoing” the effects of the degradation function.

Challenges with Inverse Filtering:

• Noise amplification: If $H(u, v)$ is close to zero for some frequencies, the result can be noisy or unstable.
• Not robust to noise: Since noise is always present in practical situations, this method might amplify noise in the restored image.

Example: Consider an image that has been blurred due to motion. If the motion blur can be modeled with a known function $h(x, y)$, we can take the Fourier transform of the blurred image, apply inverse filtering, and recover an approximation of the original image.

4. Wiener Filtering

Wiener filtering is an optimal approach to image restoration, which minimizes the mean square error between the estimated image and the true image. It is particularly effective when noise is present. The Wiener filter in the frequency domain is given by:

Where:

• $H^*(u, v)$ is the complex conjugate of the degradation function.
• $S_{\eta}(u, v)$ is the power spectral density of the noise.
• $S_f(u, v)$ is the power spectral density of the original image.
• $|H(u, v)|^2$ is the squared magnitude of $H(u, v)$.

The Wiener filter accounts for both the degradation and the presence of noise, balancing the effects of inverse filtering and noise suppression.

Advantages of Wiener Filtering:

• It handles noise optimally by considering both the degradation function and noise statistics.
• The filter does not amplify noise as much as inverse filtering.

Example: A medical image obtained through an MRI scan might suffer from noise due to low signal strength and equipment limitations. Applying the Wiener filter, where both the noise and degradation function are known or estimated, helps in restoring a cleaner, more accurate image.

5. Regularized Restoration

Another approach to handle image degradation and noise is through regularization. This method involves formulating the restoration as an optimization problem where a regularization term is added to enforce smoothness or other constraints on the restored image.

The restoration problem can be written as:

Where:

• $\| g(x, y) – h(x, y) * f(x, y) \|^2$ is the data fidelity term, ensuring that the restored image matches the observed data.
• $\lambda$ is the regularization parameter, controlling the trade-off between the data fidelity and smoothness of the solution.
• $\| D(f(x, y)) \|^2$ is the regularization term, where $D(f(x, y))$ might represent the derivative of the image, enforcing smoothness (i.e., penalizing large variations).

Advantages of Regularized Restoration:

• It helps recover stable solutions in cases where inverse filtering would fail due to noise or when the degradation function is ill-conditioned.
• Different types of regularization can enforce specific prior knowledge about the image, such as smoothness, edges, or sparsity.

Example: Consider restoring an astronomical image where details like stars or distant galaxies might be blurred. Regularized restoration can enforce smoothness to remove noise while preserving important structures in the image.

6. Constrained Least Squares Filtering

This technique is a type of regularization method where the solution is found by minimizing the squared error between the observed and restored images, subject to a smoothness constraint. It can be formulated as:

Where:

• $P(u, v)$ is the Fourier transform of a Laplacian operator that enforces smoothness.
• $\gamma$ is a regularization parameter that controls the trade-off between fitting the degraded image and enforcing smoothness.

Advantages:

• It combines noise reduction with image restoration.
• Can be adapted for different types of images and degradation functions by modifying the regularization term.

Example: An image captured by a drone flying at high altitude might suffer from atmospheric blur and noise. Constrained least squares filtering, using a proper degradation model and regularization, can help in restoring the image while reducing noise.

7. Blind Deconvolution

In many practical situations, the degradation function $h(x, y)$ is unknown. Blind deconvolution refers to restoring the image while simultaneously estimating the degradation function. This is a highly complex task but essential in cases where the point spread function (PSF) cannot be measured directly.

Blind deconvolution can be tackled using iterative algorithms that alternately estimate the image and the degradation function. An example of such an algorithm is the Richardson-Lucy deconvolution.

Advantages of Blind Deconvolution:

• It works even when the exact degradation model is unknown.
• Effective in restoring images affected by unknown blur, such as motion blur or atmospheric distortion.

Example: A photograph taken with a moving camera might be blurred, and the exact nature of the blur might not be known. Blind deconvolution can help recover both the original image and the blur function, improving the overall image quality.

Rayleigh noise is a type of statistical noise that appears in systems where signals experience random scattering, such as in radar, wireless communications, and certain image processing applications. It arises when the magnitude of a vector, composed of two orthogonal components that follow normal distributions, is observed. This type of noise is particularly useful in modeling environments where signals encounter multiple reflections or scatterings before reaching the receiver.

Mathematical Definition:

The probability density function (PDF) of Rayleigh noise is described by the following equation:

Where:

• $x$ is the noise amplitude (a non-negative random variable).
• $\sigma$ is the scale parameter that determines the spread of the distribution.
• The PDF is defined only for $x \geq 0$, meaning Rayleigh noise takes positive values.

The distribution starts at zero, increases to a peak at $x = \sigma$, and then decays exponentially for larger values of $x$.

Properties of Rayleigh Noise:

1. Asymmetric Distribution: Unlike Gaussian noise, which is symmetric around its mean, Rayleigh noise has a skewed distribution that only allows positive values.
2. Defined by One Parameter: The distribution’s shape is controlled solely by the parameter $\sigma$, which represents the scale of the noise.
3. Common in Scattering Environments: Rayleigh noise occurs in situations where a signal is scattered or reflected, such as in wireless channels or radar systems.

Example:

Suppose Rayleigh noise follows a distribution with $\sigma = 3$. The probability density function is:

To calculate the probability density at $x = 4$:

This shows that the likelihood of the noise having an amplitude of 4 is about 0.183.

Applications of Rayleigh Noise:

1. Wireless Communications: Rayleigh noise is commonly used to model fading in wireless communication channels, especially in urban environments where multiple signal paths arise due to reflection and scattering. For instance, when a mobile phone receives signals, some of the signal paths are longer due to reflections from buildings, causing random variations in the signal strength. These variations follow the Rayleigh distribution.

2. Radar Systems: Rayleigh noise appears in radar systems when reflected signals scatter off multiple objects before returning to the radar receiver. In such scenarios, the strength of the reflected signal varies unpredictably, and the Rayleigh distribution provides an accurate model for this randomness.

3. Image Processing: In medical imaging, such as ultrasound, Rayleigh noise models random variations in pixel intensity caused by scattering within tissues. The skewed nature of the distribution makes it an ideal model for noise in these images.

Noise Reduction Techniques:

1. Median Filtering: A commonly used method to reduce Rayleigh noise in images, median filtering smooths the noise while preserving edges. It works by replacing each pixel value with the median value of its neighboring pixels.

2. Adaptive Filters: In wireless communications, adaptive filters can be used to reduce Rayleigh noise by adjusting to the dynamic variations in the noise pattern over time. These filters help in minimizing signal distortion caused by fading.

3. Wavelet Denoising: This technique decomposes the signal into wavelet components and applies thresholding to remove noise. It is especially effective in cases where Rayleigh noise affects fine details, such as in radar and ultrasound images.