**6.3 Image Operation Based on Spatial
Neighbourhoods**

__6.3.1
Window-based image smoothing, Low - pass filters__

1. Averaging with equal weightsThe last filter can be used to remove drop-out lines in Landsat images. This is done by applying a filter

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

2. Averaging with different weights

3. Median filterThis 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,

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

1 1 1 1By taking I(i+1) - I(i) , we get:2 2 2 2 I<> edge

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)

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

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

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

Another form can be:

Sobel filter - spatial derivative

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.

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