© 2002-2003 by Yu Hen Hu 1ECE533 Digital Image Processing

Image Enhancement in Frequency Domain

© 2002-2003 by Yu Hen Hu 2ECE533 Digital Image Processing

Image and Its Fourier Spectrum

© 2002-2003 by Yu Hen Hu 3ECE533 Digital Image Processing

Filtering in Frequency Domain: Basic Steps

Basic Steps1. Multiply pixel f(x,y) of the

input image by (-1)x+y. 2. Compute F(u,v), the DFT3. G(u,v)=F(u,v)H(u,v)4. g1(x,y)=F-1{G(u,v)}5. g(x,y) = g1(x,y)*(-1)x+y

© 2002-2003 by Yu Hen Hu 4ECE533 Digital Image Processing

Notch Filter

The frequency response F(u,v) has a notch at origin (u = v = 0).

Effect: reduce mean value.

After post-processing where gray level is scaled, the mean value of the displayed image is no longer 0.

.100

),(otherwise

vuvuF

© 2002-2003 by Yu Hen Hu 5ECE533 Digital Image Processing

Low-pass & High-pass Filtering

© 2002-2003 by Yu Hen Hu 6ECE533 Digital Image Processing

Gaussian Filters

Fourier Transform pair of Gaussian function

Depicted in figures are low-pass and high-pass Gaussian filters, and their spatial response, as well as FIR masking filter approximation.

High pass Gaussian filter can be constructed from the difference of two Gaussian low pass filters.

222

22

2

2/

2)()(

x

u

AexhAeuH

© 2002-2003 by Yu Hen Hu 7ECE533 Digital Image Processing

Gaussian Low Pass Filters

D(u,v): distance from the origin of Fourier transform

2

2

2),(exp),(

vuDvuH

© 2002-2003 by Yu Hen Hu 8ECE533 Digital Image Processing

Ideal Low Pass Filters

The cut-off frequency Do determines % power are filtered out.

Image power as a function of distance from the origin of DFT (5, 15, 30, 80, 230)

© 2002-2003 by Yu Hen Hu 9ECE533 Digital Image Processing

Effects of Ideal Low Pass Filters

Blurring can be modeled as the convolution of a high resolution (original) image with a low pass filter.

© 2002-2003 by Yu Hen Hu 10ECE533 Digital Image Processing

Ringing and Blurring

© 2002-2003 by Yu Hen Hu 11ECE533 Digital Image Processing

Butterworth Low Pass Filters

noDvuDvuH 2/),(1

1),(

© 2002-2003 by Yu Hen Hu 12ECE533 Digital Image Processing

Ideal high pass filter

Butterworth high pass filter

Gaussian high pass filter

High Pass Filters

.1

),(0),(

otherwiseDvuDif

vuH o

nvuDDvuH 2

0 ),(/11),(

2

0

2

2),(exp1),(

DvuDvuH

© 2002-2003 by Yu Hen Hu 13ECE533 Digital Image Processing

Applications of HPFs Ideal HPF

» Do = 15, 30, 80

Butterworth HPF» n = 2,» Do = 15, 30, 80

Gaussian HPF» Do = 15, 30, 80

© 2002-2003 by Yu Hen Hu 14ECE533 Digital Image Processing

Laplacian HPF 3D plots of the Laplacian

operator, its 2D images, spatial domain response

with center magnified, and Compared to the FIR mask

approximation

),(2/2/

),(22

2

vuFNvMu

yxf