Primary Convolutions In Image Processing

Article 2

As discussed in the article 1, (If you missed it you can quickly go and see it here https://medium.com/@madarapremawardana/primary-convolutions-in-image-processing-b64917c9da80 ) here we’ll go with the linear, mean and mode filters.

Photo by Ice Tea on Unsplash

In image processing filters are mainly used to suppress

1. High frequencies in the image — Smoothing images removing roughness or edgy-ness.
2. Low frequencies — Enhancing or detecting edges in the image

There are basically two types of filters as Low pass filters (smoothing) and moving window operations. Low pass filters remove high spatial frequency noise from a digital image ad Moving window operations affects one pixel of the image at a time, changing its value by some function of a “local” region of pixels. (“covered” by the window).

1. Low pass filters — Reconstruction filtering, Enhancement filtering
2. Moving window operations — Neighborhood-averaging filters, median filters, Mode filters

Linear Filtering

Neighborhood-averaging filters

Replace the value of each pixel, a[i,j] say, by a weighted average of the pixels in some neighborhood around it. (i.e.: a weighted sum of a[i+p,j+q], with p = -k to k, q = -k to k for some positive k; the weights are non-negative with the highest weight on the p = q = 0 term. If all the weights are equal then this is a mean filter is “linear”)

Median Filters

Replaces each pixel value by the median of its neighbors. (i.e. the value such that 50% of the values in the neighborhood are above, and 50% are below. )

Mode filters

Each pixel value is replaced by its most common neighbor. This is a particularly useful filter for classification procedures where each pixel corresponds to an object which must be placed into a class.

Mean Filtering (AKA Neighborhood Averaging)

The idea of mean filtering is simply to replace each pixel value in an image with the mean (‘average’) value of its neighbors, including itself. This has effect of eliminating pixel values which are unrepresentative of their surroundings.

Let f(x,y) is a noisy image then the smoothed image

g(x,y) can be obtained by

Where S is a neighborhood of (x,y) and n is the number of pixels in S.

Results of Mean Filtering with different

sized Filters

Results of Mean Filtering with a mask of size nxn n=3,5,7,15,25

Some problems with Mean Filtering

Mean Filtering tends to blur to the image and hence the sharpness of edges may be reduced.

Example:

Since 255 was taken into calculation, a good pixel (45) has been changed to 63.

Median Filtering

In Median Filtering, pixel values are replaced by the median value of their neighborhood.

Example:

The shaded pixel will be replaced by the median of its 3x3

Neighborhood :37, 37, 39, 40, 41, 41, 42, 42, 43 Which is 41.

Notes

Median filtering has the following properties

1. Reduces the variance of the intensities in the image.

2. Intensity oscillations with a period less than the window width are smoothed.

3. Preserves the sharpness and location of edges.

a). Original Image b). noisy image c). Mean filtered image of b d). Median filtered image of b

Mode Filtering

Mode filtering involves assigning to the central pixel the most common value inside the local window around the pixel.

That is the mode of the histogram of local values.

Example:

The shaded pixel will be replaced by the mode of its 3x3

Neighborhood: 42, 37, 37, 39, 40, 41, 41, 42, 42

Which is 42.