HW 6, EECS 203A

Problem 5.13, Digital Image Processing, 2nd edition by Gonzalez, Woods.

Nasser Abbasi, UCI. Fall 2004

Question

Obtain equations for the bandpass filters corresponding to the bandreject filters in eqs 5.4-1 through 5.4-3

Solution

A bandpass filter can be obtained from a bandreject filter by the following relation

eqs 5.4-1 through 5.4-3 give the bandpass reject equations, hence to obtain the bandpass equations we need to substitue in the above.

ideal bandpass

Here, every where the bandreject has a value of 1, we make it zero, and every where it is 0 we make it 1. Ideal bandreject is given by

Hence an ideal bandpass is given by

This is illustrated in the following diagram on next page

bandpass Butterworth filter

The Butterworth reject filter of order is given by

Where is the distance from center of 2D spectrum to any point. is the width of the band, is the distance from center of spectrum to the center of the band. (note: and should be written as , , but for clarity of expression, I did not add these).

Hence, the butterworth bandpass filter is

bandpass Gaussian filter

The Gaussian band reject filter is given by

Hence Gaussian band pass filter is