240-373 Image Processing

1
240-373 Image Processing
Montri Karnjanadecha montri_at_coe.psu.ac.th http//f
ivedots.coe.psu.ac.th/montri
2
Chapter 14

• The Frequency Domain

3
The Frequency Domain

• Any wave shape can be approximated by a sum of
  periodic (such as sine and cosine) functions.
• a--amplitude of waveform
• f-- frequency (number of times the wave repeats
  itself in a given length)
• p--phase (position that the wave starts)
• Usually phase is ignored in image processing

4
(No Transcript)
5
(No Transcript)
6
The Hartley Transform

• Discrete Hartley Transform (DHT)
• The M x N image is converted into a second image
  (also M x N)
• M and N should be power of 2 (e.g. .., 128, 256,
  512, etc.)
• The basic transform depends on calculating the
  following for each pixel in the new M x N array

7
The Hartley Transform

• where f(x,y) is the intensity of the pixel at
  position (x,y)
• H(u,v) is the value of element in frequency
  domain
• The results are periodic
• The cosinesine (CAS) term is call the kernel of
  the transformation (or basis function)

8
The Hartley Transform

• Fast Hartley Transform (FHT)
• M and N must be power of 2
• Much faster than DHT
• Equation

9
The Fourier Transform

• The Fourier transform
• Each element has real and imaginary values
• Formula
• f(x,y) is point (x,y) in the original image and
  F(u,v) is the point (u,v) in the frequency image

10
The Fourier Transform

• Discrete Fourier Transform (DFT)
• Imaginary part
• Real part
• The actual complex result is Fi(u,v) Fr(u,v)

11
Fourier Power Spectrum and Inverse Fourier
Transform

• Fourier power spectrum
• Inverse Fourier Transform

12
Fourier Power Spectrum and Inverse Fourier
Transform

• Fast Fourier Transform (FFT)
• Much faster than DFT
• M and N must be power of 2
• Computation is reduced from M2N2 to MN log2 M .
  log2 N (1/1000 times)

13
Fourier Power Spectrum and Inverse Fourier
Transform

• Optical transformation
• A common approach to view image in frequency
  domain
• Original image Transformed image

14
Power and Autocorrelation Functions

• Power function
• Autocorrelation function
• Inverse Fourier transform of
• or
• Hartley transform of

15
Hartley vs Fourier Transform
16
Interpretation of the power function
17
Applications of Frequency Domain Processing

• Convolution in the frequency domain

18
Applications of Frequency Domain Processing

• useful when the image is larger than 1024x1024
  and the template size is greater than 16x16
• Template and image must be the same size

19

• Use FHT or FFT instead of DHT or DFT
• Number of points should be kept small
• The transform is periodic
• zeros must be padded to the image and the
  template
• minimum image size must be (Nn-1) x (Mm-1)
• Convolution in frequency domain is real
  convolution
• Normal convolution

20
(No Transcript)
21

• Convolution in frequency domain is real
  convolution
• Normal convolution
• Real convolution

22
Convolution using the Fourier transform

• Technique 1 Convolution using the Fourier
  transform
• USE To perform a convolution
• OPERATION
• zero-padding both the image (MxN) and the
  template (m x n) to the size (Nn-1) x (Mm-1)
• Applying FFT to the modified image and template
• Multiplying element by element of the transformed
  image against the transformed template

23
Convolution using the Fourier transform

• OPERATION (contd)
• Multiplication is done as follows
• F(image) F(template)
  F(result)
• (r1,i1) (r2, i2) (r1r2
  - i1i2, r1i2r2i1)
• i.e. 4 real multiplications and 2 additions
• Performing Inverse Fourier transform

24
Hartley convolution

• Technique 2 Hartley convolution
• USE To perform a convolution
• OPERATION
• zero-padding both the image (MxN) and the
  template (m x n) to the size (Nn-1) x (Mm-1)
• image
  template

25
Hartley convolution

• Applying Hartley transform to the modified image
  and template
• image
  template

26
Hartley convolution

• Multiplying them by evaluating

27
Hartley convolution Contd

• Giving
• Performing Inverse Hartley transform, gives

28
Hartley convolution Contd
29
Deconvolution

• Convolution R I T
• Deconvolution I R -1 T
• Deconvolution of R by T convolution of R
• by some inverse of the template T (T)

30
Deconvolution

• Consider periodic convolution as a matrix
  operation. For example

31
Deconvolution

• is equivalent to
• A B
  C
• AB C
• ABB-1 CB-1
• A CB-1