Image Filtering and Noise Models - lect 6

Spatial Filtering

• Last time, we discussed spatial filtering.

• Spatial filtering is done by computers in photo editing software automatically.

Unsharp Masking

• Unsharp masking is a filtering technique that sharpens edges.

• Process:

  • Blur the original image.

  • Subtract the blurred version from the original to get sharp edges.

  • Multiply the sharp edges by a boosting factor.

  • Add the boosted edges back to the original image.

  • This strengthens or highlights the edges.

Other Linear Spatial Filters

• Averaging filters soften or blur images.

• Unsharp masking boosts edges.

• Test images:

  • Test card.

  • Dilapidated house (chosen for vertical, diagonal, and horizontal edges).

Increasing Filter Mask Size

• Increasing the averaging filter mask size increases blurring.

• Example: 17x17 pixel mask.

• Normalization factor: $1/289$ (1 divided by the number of elements in the mask) to maintain image brightness.

Ringing Artifact

• Large averaging masks can cause a ringing artifact (stripes).

• Ringing artifacts appear due to the large mask size.

• To overcome ringing, use a mask with smaller values at the perimeter and larger values near the center, following a normal distribution (bell-shaped curve).

• This achieves smoothing without ringing.

Vertical Difference Edge Detection Filter

• A vertical difference mask highlights horizontal edges.

• Important to remember this for the exam.

• The filter responds to diagonal edges as well.

• The weights in edge detection masks sum to zero, not one.

Horizontal Edge Detection

• Configuration for horizontal edge detection:

  • Consider the expression:

  $\begin{bmatrix} -1 & 2 & -1 \ -1 & 2 & -1 \ -1 & 2 & -1 \end{bmatrix}$

  • The above mask does the same thing as the previous mask.

Vertical Edge Detection

• Horizontal difference detects vertical edges.

Combining Filters

• Filters can be combined to detect all edges (vertical, diagonal, and horizontal).

• Boosting diagonal differences/edges is also possible.

• Applications:

  • Computer vision systems: edges contain most of the useful information.

  • Preprocessing images for object detection.

  • Analyzing fingerprints by boosting ridges.

Median Filter

• Non-linear spatial filters.

• The spatial filters discussed are linear spatial filters. They operate on the image.

• The median filter is good at removing specific types of image noise.

• Process:

  • Take pixel intensity values from a neighborhood.

  • Order them in descending brightness.

  • The median value replaces the center pixel value.

Non-Local Means Filter

• Developed in 2017.

• Basic concept: find regions in the image similar in structure to the region being filtered.

• Average those similar regions to estimate the denoised region.

• Applied to the whole image.

• Removes noise without causing too much distortion.

• Weighted combination estimates what the image would look like without noise.

Image Noise Models

• In forensic camera identification, noise can be a “fingerprint” for the camera.

• Image noise: a speckly pattern, unwanted.

• Analogy to audio noise (e.g., crackle on dusty records).

Correlated Noise

• Some argue it's an artifact, not noise.

• Correlated noise has structure (regular or irregular patterns).

• Examples:

  • Electrical interference causing horizontal striping.

  • Halftone distortion: colored dots from a printer (cyan, magenta, yellow).

  • Viewed from a distance, the dots merge.

  • Screwing eyes up averages the colors.

  • Also explained that printers print colors close to each other to build up the color of the photo.

Random Noise

• Photographs taken at different exposure times.

• Short exposure times lead to a speckle pattern (image noise).

• Due to shot noise or thermal noise.

  • Shot noise: particle nature of light; brightness controlled by Poisson distribution.

  • Thermal noise: random thermal motion of electrons in the camera sensor.

• High light levels cause statistical fluctuations to average out, reducing noise.

• Film cameras:

  • Silver halide grains record light intensity, but their finite size can cause a speckle pattern.

Poisson Probability

• $P(X=x) = (e^{-\lambda} * \lambda^x) / x!$

  • $P$: Probability of X photons arriving at a photo site over a time duration.

  • $\lambda$: Average number of photons expected to arrive.

  • $x$: Number of photons of interest.

  • $e$: Euler's number (~2.71828)

  • $x!$: factorial

Other Noise Sources

• Neuronal noise in the retina.

• Quantization noise in digital photographs.

Mathematical Model for Output Image

• $J = I + N$

  • $J$: Output (noisy) image.

  • $I$: Ideal (noise-free) image.

  • $N$: Noise.

• Image processing tries to recover $I$ from $J$ using filters.

• Filters approximate $I$.

Statistical Distributions of Noise

• Gaussian (Normal) Noise:

  • Generated through shot noise or thermal noise.

  • Pixel values are distributed about the original intensity value (e.g., 127).

  • Physicists call this a Gaussian distribution, statisticians call it a normal distribution.

• Uniform Noise:

  • Every pixel has equal probability.

  • May be due to quantization of intensity values.

  • Has equal probability of every shade of gray in between shades.

• Salt and Pepper Noise:

  • Pixels have either minimum (0) or maximum (255) intensity.

  • Looks like salt and pepper sprinkled on the image.

  • May be due to transmission bit errors, photodiode leakage, or dead pixels.

Color Images

• Noise can occur in red, green, and blue color planes.

• Gaussian additive noise is added to the image.

Removing Noise

• Averaging filters smooth the image, suppressing noise but also blurring edges.

• The noise follows a speckle pattern.

• Averaging filters are not very good at removing shot and thermal noise

• Non-Local Means Filter:

  • Preserves edges better than averaging filters.

  • Removes noise and preserves the edges.

  • Needs properly set parameters to avoid excessive blurring.

Removing Salt and Pepper Noise with Median Filter

• Non-linear spatial filter.

• Returns the median value of pixels in a neighborhood.

• Better at preserving edges than the mean filter.

Examples

• Averaging filter:

  • Reasonable job of suppressing noise, but white speckles may still be visible.

  • Increasing filter size blurs the image too much.

• Median filter (3x3):

  • Very good job of removing noise while preserving edges.

• Larger median filter (5x5) may not be necessary.

Gaussian additive noise typically requires more complex filters.

Spatial Filtering

• Spatial filtering is performed by computers in photo editing software.

Unsharp Masking

• Unsharp masking sharpens edges by blurring the original image, subtracting the blurred version to get sharp edges, multiplying the edges by a boosting factor, and adding the boosted edges back to the original.

Other Linear Spatial Filters

• Averaging filters soften or blur images; unsharp masking boosts edges.

• Increasing the averaging filter mask size increases blurring. Normalization factor maintains image brightness.

• Large averaging masks can cause ringing artifacts. Use masks with smaller values at the perimeter and larger values near the center to avoid ringing.

• A vertical difference mask highlights horizontal edges; horizontal difference detects vertical edges. The weights in edge detection masks sum to zero.

• Filters can be combined to detect all edges, useful in computer vision systems and image preprocessing.

Median Filter

• The median filter is a non-linear spatial filter good at removing specific types of image noise. The median value replaces the center pixel value.

Non-Local Means Filter

• Finds similar regions to the region being filtered, averaging them to estimate the denoised region, removing noise without causing distortion.

Image Noise Models

• Image noise is a speckly pattern.

Correlated Noise

• Correlated noise has structure, like electrical interference or halftone distortion.

Random Noise

• Short exposure times lead to speckle patterns due to shot noise or thermal noise. High light levels reduce noise.

Poisson Probability

• $P(X=x) = (e^{-\lambda} * \lambda^x) / x!$

Other Noise Sources

• Neuronal and quantization noise.

Mathematical Model for Output Image

• $J = I + N$

• Image processing recovers $I$ from $J$ using filters.

Statistical Distributions of Noise

• Gaussian (Normal) Noise: Pixel values distributed about the original intensity value.

• Uniform Noise: Every pixel has equal probability.

• Salt and Pepper Noise: Pixels have minimum or maximum intensity.

Color Images

• Noise can occur in red, green, and blue color planes.

Removing Noise

• Averaging filters smooth images but blur edges. Non-Local Means Filter preserves edges better.

Removing Salt and Pepper Noise with Median Filter

• Median filter preserves edges better than the mean filter.

Examples

• Median filter (3x3) effectively removes salt and pepper noise while preserving edges.