CHAPTER 4Image Enhancement in Frequency Domain

Digital Image Processing

Spectrum of images

Spectrum of images

Spectrum of images

Spectrum of images

Spectrum of images

Basic steps for filtering in the frequency domain

Basics of filtering in the frequency domain

1. multiply the input image by (-1)x+y to center the transform to u = M/2 and v = N/2 (if M and N are even numbers, then the shifted coordinates will be integers)

2. computer F(u,v), the DFT of the image3. multiply F(u,v) by a filter function H(u,v)4. compute the inverse DFT of the result in (3)5. obtain the real part of the result in (4)6. multiply the result in (5) by (-1)x+y to cancel the

multiplication of the input image.

Notch filter

otherwise 1

N/2 (M/2, v)(u, if 0),(

) vuH

• this filter is to force the F(0,0) which is the average value of an image (dc component of the spectrum)• the output has prominent edges• in reality the average of the displayed image can’t be zero as it needs to have negative gray levels. the output image needs to scale the gray level

Low pass filter

high pass filter

Filtering examples - revisited

Transfer function 5x5 lowpass filter

Filtering Examples

Filtering Examples

Filtering Examples

Transfer function high-pass filter

Filtering Examples

Filtering in frequency domain

Add the ½ of filter height to F(0,0) of the high pass filter

Correspondence between filter in spatial and frequency domains

Lowpass filtering

Lowpass filtering1 1

2

0 0

Total Power = ( , )M N

u v

F u v

α= 92.0, 94.6, 96.4, 98, 99.5

% total power

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Lowpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Highpass filtering

Homomorphic filtering

Homomorphic filtering

Homomorphic filtering

2 20( ( . ) / )( , ) ( ) 1 c D u v D

H L Lh u v e

C controls the sharpness of the slope

Homomorphic filtering