Restoration in the Presence of Noise Only Spatial Filtering

In digital image processing, one of the most common challenges is dealing with image degradation, particularly noise. Noise can obscure important details and degrade the quality of an image, making it difficult to interpret or analyze. However, understanding how to deal with noise using spatial filtering techniques can significantly improve image clarity.

The Simplified Model of Image Noise

When noise is the primary form of degradation in an image, the model for how the image is altered can be simplified. Mathematically, we can express the noisy image in both the spatial and frequency domains:

• Spatial Domain:
  $g(x, y) = f(x, y) + n(x, y)$
  In this equation, $g(x, y)$ represents the observed image, $f(x, y)$ is the original image, and $n(x, y)$ is the noise introduced during the image capture process.

• Frequency Domain:
  $G(u, v) = F(u, v) + N(u, v)$
  Here, $G(u, v)$ is the Fourier transform of the observed image, $F(u, v)$ is the Fourier transform of the original image, and $N(u, v)$ represents the noise in the frequency domain.

The challenge with this model is that the noise components, $n(x, y)$ and $N(u, v)$, are unknown. Because of this, directly subtracting the noise from the observed image to recover the original image is not a practical solution. So, how do we deal with this?

When Periodic Noise is Present

In some special cases, such as when the noise is periodic (like a repeating pattern), it is possible to estimate the noise from the frequency spectrum of the image.

Once the noise component $N(u, v)$ is estimated, it can be subtracted from the frequency domain representation of the observed image $G(u, v)$ to obtain a cleaner version of the original image $F(u, v)$. However, this approach works only when the characteristics of the noise are known or can be easily predicted. In real-world scenarios, such detailed knowledge about the noise is rare.

Spatial Filtering: The Preferred Solution for Random Noise

In most cases, noise is random and unpredictable. For situations involving additive random noise, spatial filtering becomes the go-to method. Spatial filters work directly on the image in the spatial domain, applying different mathematical operations to each pixel based on its neighboring pixels.

Spatial filtering methods include:

1. Mean Filters – These smooth out the image by averaging pixel values in a neighborhood.
2. Median Filters – Useful for reducing “salt and pepper” noise, where random pixels are either completely black or white.
3. Gaussian Filters – Apply a weighted average where nearby pixels influence the result more than distant ones.

The arithmetic mean filter is one of the simplest types of filters used in image processing, particularly for noise reduction. Here’s a breakdown of how it works and its purpose.

Basic Concept

The arithmetic mean filter smooths an image by reducing noise through a process known as blurring. The filter operates over a specific area of the image, often referred to as a “window” or “neighborhood.” This window is rectangular and has a size of $m \times n$, where $m$ and $n$ are odd integers, ensuring the center pixel is well-defined. The window is centered at a pixel of interest, denoted by $(x, y)$, and the region within this window is represented by a set of coordinates $S_{xy}$.

The Filtering Process

To perform the filtering, the arithmetic mean filter calculates the average pixel value within the window. In mathematical terms, it sums up all the pixel values within the defined neighborhood and divides that sum by the number of pixels in the window ($m \times n$).

So, the restored value of the pixel at $(x, y)$ in the image is given by:

Where:

• $g(s, t)$ represents the pixel values of the original (or corrupted) image within the window.
• $S_{xy}$ defines the neighborhood or window around the pixel $(x, y)$.

Implementation

The arithmetic mean filter can be implemented using a spatial filter of size $m \times n$, where all the coefficients have a value of $1 / (m \times n)$. This ensures that the output pixel value is the average of all pixel values within the window.

This filter is known for smoothing local variations in the image, effectively reducing noise. However, as a result of this smoothing, some fine details or sharp edges in the image might become blurred.

The geometric mean filter is another approach used in image processing for noise reduction, but it works differently compared to the arithmetic mean filter. While both methods aim to smooth out variations and reduce noise, the geometric mean filter often preserves more image detail in the process.

Basic Concept

In the geometric mean filter, the value of each pixel in the restored image is calculated by taking the product of the pixel values within a defined neighborhood or subimage window, and then raising that product to the power of $\frac{1}{m \times n}$, where $m \times n$ is the size of the window.

If the original image is denoted by $g(s, t)$, the restored value of the pixel at location $(x, y)$ in the image is given by:

Where:

• $g(s, t)$ represents the pixel values of the original image within the neighborhood or window.
• $S_{xy}$ defines the set of coordinates in the subimage window centered around the pixel $(x, y)$.
• $m \times n$ is the size of the window.

How the Geometric Mean Filter Works

• Product of pixels: Instead of summing the pixel values like in the arithmetic mean filter, the geometric mean filter takes the product of all the pixel values in the window.
• Power of the inverse of window size: The result is then raised to the power of $\frac{1}{m \times n}$, which effectively computes the geometric mean of the pixel values within the window.

This method of averaging helps smooth the image, but in a way that tends to preserve more of the image’s fine details compared to the arithmetic mean filter.

Advantages of the Geometric Mean Filter

• Preserving details: Unlike the arithmetic mean filter, which can blur fine details in the image, the geometric mean filter tends to maintain more of the image’s structure. This makes it useful in cases where noise reduction is needed, but preserving image sharpness is important.
• Smoothing: Like the arithmetic mean filter, the geometric mean filter also smooths the image and reduces noise by averaging the pixel values in a local neighborhood. However, the way it computes the average (using the geometric mean instead of the arithmetic mean) results in less blurring.

The harmonic mean filter is another technique used in image processing to reduce noise, and it works differently from both the arithmetic and geometric mean filters. The harmonic mean filter is particularly effective in dealing with certain types of noise, such as salt noise and Gaussian noise, but it is less effective for others, like pepper noise.

Basic Concept

The harmonic mean filter operates by calculating the harmonic mean of the pixel values within a defined window or neighborhood of the image. If the original image is represented by $g(s, t)$, the restored value of the pixel at position $(x, y)$ is computed using the following formula:

Where:

• $g(s, t)$ represents the pixel values of the original (corrupted) image within the window.
• $S_{xy}$ defines the neighborhood (a rectangular window) around the pixel at $(x, y)$.
• $m \times n$ is the size of the window.

How the Harmonic Mean Filter Works

• Inverse of pixel values: Unlike the arithmetic and geometric mean filters, the harmonic mean filter works by taking the reciprocal (inverse) of the pixel values within the window.
• Sum of inverses: It then sums these reciprocals and divides the total number of pixels in the window by the sum of the inverses. This gives the harmonic mean of the pixel values.

This type of filter is particularly effective at smoothing images and reducing certain types of noise, such as salt noise and Gaussian noise, but it is not suitable for handling pepper noise.

Performance on Different Types of Noise

• Salt noise: Salt noise appears as bright white spots (high-intensity values) in the image. The harmonic mean filter performs well against salt noise because it reduces the influence of very high-intensity pixels when calculating the mean.
• Pepper noise: Pepper noise, on the other hand, appears as dark black spots (low-intensity values) in the image. The harmonic mean filter is not effective at handling pepper noise because the low-intensity (or zero) pixels cause the filter to break down, producing poor results.
• Gaussian noise: The harmonic mean filter also works well in the presence of Gaussian noise, which is a type of noise that has a bell-shaped probability distribution and affects the entire image.

Order-statistic filters are a type of spatial filter used in image processing, where the filter’s output is determined by ranking (or ordering) the pixel values within the image area covered by the filter. These filters are effective in various noise reduction tasks, especially when dealing with impulse noise. Here’s a detailed discussion on some commonly used order-statistic filters.

1. Median Filter

The median filter is the most popular order-statistic filter. It replaces each pixel’s value with the median of the intensity values in the neighborhood around that pixel. The formula for the median filter is:

Where:

• $g(s, t)$ are the pixel values in the neighborhood.
• $S_{xy}$ represents the window centered at pixel $(x, y)$.

The median filter is effective in reducing noise, especially bipolar and unipolar impulse noise, such as salt-and-pepper noise. It provides excellent noise reduction with less blurring compared to linear filters.

2. Max and Min Filters

Besides the median filter, there are other order-statistic filters based on the ranking of pixel values, such as max and min filters.

• Max Filter: The max filter selects the maximum pixel value from the neighborhood. The formula is:

The max filter is useful for detecting the brightest points in an image and is effective in reducing pepper noise (dark spots), which have very low intensity values.

• Min Filter: The min filter selects the minimum pixel value from the neighborhood. The formula is:

The min filter is helpful for identifying darkest points in an image and is effective at reducing salt noise (bright spots) since salt noise typically consists of high-intensity values.

3. Midpoint Filter

The midpoint filter computes the midpoint between the maximum and minimum pixel values within the neighborhood. The formula is:

This filter combines order statistics with averaging and is particularly effective for reducing randomly distributed noise such as Gaussian or uniform noise.

4. Alpha-Trimmed Mean Filter

The alpha-trimmed mean filter removes the d lowest and d highest intensity values from the neighborhood and averages the remaining pixels. The formula is:

Where:

• $g_r(s, t)$ are the remaining pixel values after trimming the lowest and highest $d$ values.
• $mn$ is the total number of pixels in the neighborhood, and $d$ is the number of pixels to trim.

The alpha-trimmed mean filter is useful in situations involving multiple types of noise, such as a combination of salt-and-pepper noise and Gaussian noise.

• When $d = 0$, the filter becomes the arithmetic mean filter.
• When $d = mn – 1$, the filter reduces to the median filter.

Adaptive filters differ from the Order-Statistic Filters(fixed filters) discussed earlier in that their behavior changes depending on the local characteristics of the image. These filters adapt based on statistical measures in the image, leading to better performance in noise reduction while preserving image details. Two well-known adaptive filters are the adaptive local noise reduction filter and the adaptive median filter.

1. Adaptive Local Noise Reduction Filter

The adaptive local noise reduction filter operates by adjusting its behavior according to the mean and variance of pixel values within a local neighborhood. These statistical measures are closely related to the intensity and contrast of the image, making them a good foundation for an adaptive filter. The goal is to reduce noise while preserving important details such as edges.

The filter operates on a local window $S_{xy}$ centered at pixel $(x, y)$. Four quantities play a role in determining the output:

• $g(x, y)$: the value of the noisy image at pixel $(x, y)$.
• $m_L$: the local mean of the pixel values in the neighborhood $S_{xy}$.
• $\sigma_L^2$: the local variance of the pixel values in the neighborhood $S_{xy}$.
• $\sigma_h^2$: the variance of the overall noise in the image.

The filter’s behavior is defined as:

1. If the local variance is zero (i.e., no variation within the local neighborhood), the filter returns the value of $g(x, y)$, which implies no filtering.
2. If the local variance is high (indicating an edge or significant detail), the filter should return a value close to $g(x, y)$ to preserve the edge.
3. If the local variance equals the overall noise variance, the filter returns the arithmetic mean of the pixel values in $S_{xy}$, reducing noise by averaging.

The adaptive filter formula is:

Where:

• $f_N(x, y)$ is the restored pixel value.
• $\sigma_h^2 / \sigma_L^2$ controls how much the local pixel is smoothed based on the noise level.

In practice, the filter incorporates a safeguard to avoid nonsensical results, such as negative intensity values, by adjusting the ratio if $\sigma_h^2 > \sigma_L^2$.

2. Adaptive Median Filter

The adaptive median filter extends the standard median filter by adjusting the size of the filtering window during operation. It is particularly effective in removing salt-and-pepper noise and is better suited to handle cases where the noise density is higher than the limit of the standard median filter (around 20%).

The adaptive median filter works in a rectangular window centered on the pixel being processed, and the window size changes based on the image’s local characteristics.

Algorithm Stages:

• Stage A: The filter first checks if the median value is an impulse (extreme pixel value) or not. If it is not an impulse, the algorithm proceeds to Stage B.
• Stage B: The filter checks if the center pixel is an impulse. If it is not, the pixel value remains unchanged. If the center pixel is an impulse, the filter replaces it with the median value from the current window.

If the median value is found to be an impulse in Stage A, the filter increases the window size and repeats the process. This continues until either a non-impulse median value is found or a maximum window size is reached.