Two-Dimensional Filters What is this chapter about? Manipulation of images often entails omitting or enhancing details of certain spatial frequencies. This is equivalent to multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components. When we do that, we say that wefilter the image, and the function we use is called a filter. This chapter explores some of the basic properties of 2D filters and presents some methods by which the operation we wish to apply to the Fourier transform of the image can be converted into a simple convolution operation applied to the. | Image Processing The Fundamentals. Maria Petrou and Panagiota Bosdogianni Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-99883-4 Electronic ISBN 0-470-84190-7 Chapter 5 Two-Dimensional Filters What is this chapter about Manipulation of images often entails omitting or enhancing details of certain spatial frequencies. This is equivalent to multiplying the Fourier transform of the image with a certain function that kills or modifies certain frequency components. When we do that we say that we filter the image and the function we use is called a filter. This chapter explores some of the basic properties of 2D filters and presents some methods by which the operation we wish to apply to the Fourier transform of the image can be converted into a simple convolution operation applied to the image directly allowing us to avoid using the Fourier transform itself. How do we define a 2D filter A 2D filter is defined in terms of its Fourier transform called the system function. By taking the inverse Fourier transform of H jj v we can calculate the filter in the real domain. This is called the unit sample response of the filter and is denoted by h k l . How are the system function and the unit sample response of the filter related v is defined as a continuous function of fit v . The unit sample response h k I is defined as the inverse Fourier transform of H jj v but since it has to be used for the convolution of a digital image it is defined at discrete points only. Then the equations relating these two functions are h k l H e dpdis JJ jy 1 oo oo H p v EE h n m e-j m l m ä7T n oo m oo 156 Image Processing The Fundamentals X Filter Inverse S 1 5-------- K band we want to keep bands we want to eliminate Filter needed Filtered Signal . of the filtered signal 0 Figure Top row a signal and its Fourier transform. Middle row the unit sample response function of a filter on the left and the filter s system function on the right. Bottom row On the left the filtered .
