10.9 Spatial Filtering

10.9 Spatial Filtering

Spatial filtering is used to obtain enhanced images or improved images by applying, filter function or filter operators in the domain of the image space (x,y) or spatial frequency (x,h). Spatial filtering in the domain of image space aims at image enhancement with so-called enhancement filters, while in the domain of spatial frequency it aims at reconstruction with so-called reconstruction filters.

a. Filtering in the Domain of Image Space
In the case of digital image data, spatial filtering in the domain of image space is usually achieved by local convolution with an n x n matrix operator as follows.

where f: input image
h: filter function
g: output image

The convolution is created by a series of shift-multiply-sum operators with an n x n matrix (n: odd number). Because the image data are large, n is usually selected as 3, although n is sometimes selected as 5, 7, 9 or 11.

Figure 10.9.1 shows typical 3 x 3 enhancement filters. Figure 10.9.2 shows the input image and several output images with various 3 x 3 operators.

b. Filtering in the domain of Spatial Frequency
Filtering in the domain of spatial filtering uses the Fourier transformation to convert from image space domain to spatial frequency domain as follows.

G(u,v) = F(u,v) H(u,v)

F: Fourier transformation of input image
H: filter function

An output image from filtering of spatial frequency, can be obtained by using an inverse Fourier transformation of the above formula.

Low pass filters, high pass filters, band pass filters etc., are typical filters with a criterion of frequency control. Low pass filters which out puts only lower frequency image data, less than a specified threshold, can be applied to remove high frequency, noise,while high pass filter are used for removing, for example, stripe noise of low frequency.