6.3 Image Operation Based on Spatial Neighbourhoods

6.3.1 Window-based image smoothing, Low - pass filters

1. Averaging with equal weights

We can also use 5x5, 7x7, etc. This filter is also called a box-car filter.

2. Averaging with different weights

The last filter can be used to remove drop-out lines in Landsat images. This is done by applying a filter only along the drop-out lines in those images.
3. Median filter
This filter is more useful in removing outliers, random noise, and speckles on RADAR imagery, than a simple average filter. It has a desirable effect of keeping edges to some extent. This filter can also be applied to drop-out line removal in some Landsat images.
If we denote an image window in the following form:
The average filter in 1 can be written as
By moving (i, j) all over an image, the original image, I, can be filtered and the new image, I', can be created.
For 2,

6.3.2 Window-based edge enhancement - High-pass filters

In order to enhance edges, differences between neighbourhood digital numbers are taken. We will start from one dimensional example:

1 1 1 1 : 2 2 2 2 I 

<> edge

By taking I(i+1) - I(i) , we get
0 0 0 1 0 0 0 I' 
We suppressed all the non-change part and left the edge out and thus an enhancement can be achieved. We can apply the differencing technique again to I', to get I''
0 0 1 -1 0 0 I" 
I" = I'(i+1) - I'(i) = I(i+2) - I(i+1) - I(i+1) + I(i)

= I(i+2) - 2I(i+1) + I(i)

1 -2 1 are the weights
The advantage of using a second order differencing is that we can locate exact position of the edge at the zero-crossing point.

We call the first differencing, taking a gradient and the second differencing, taking a. We can use the matrix

1 -2 1 as a Laplacian filter, an edge enhancement filter.
In the two-dimension form, a Laplacian filter is:

Another form can be: 

Sobel filter - spatial derivative


6.3.3 Contrast stretching through high-frequency enhancement

This is also called edge-enhancement by subtractive smoothing

Why we don't use

This contrast will not be as good as DN-KDN".

The question is, can we write DN-kDN" in a filter form? The answer is yes.

6.3.4 Linear Line Detection Templates

With 5 x 5 filters we can have more directions ex.