1. Digital Image Processing Chapter - 4
Instructor:
P. Harikanth

2. Impulse function & Properties (Continuous)1D 2D
sifting

3. Impulse function & Properties (Discrete)

4. Fourier Transform (Continuous) 1D

5. These properties helps in interpreting the spectra of
2D Fourier Transform of images

6. Fourier Transform (Continuous) 2D

7. Magnitude spectrum

8. Convolution
Convolution theorem:

9. Sampling

10. Fourier Series:

11. Fourier Transform of Sampled function:

13. Sampling Theorem A continuous, band limited function
can be recovered completely from a
set of its samples, if the samples are
acquired at a rate exceeding twice
The highest frequency of the function

14. Ideal low-pass (or) reconstruction filter

15. Aliasing The effect of aliasing can be reduced by
smoothing the i/p function to attenuate its
higher frequencies. (Anti-Aliasing)

16. Reconstruction from sampled data

17. 2D Sampling 2D Impulse train function can be expressed as
Fourier transform of band limited function f(t,z)
Sampling rate according to Nyquist criterion

18. Aliasing in images
There are 2 principal manifestations of aliasing in images
Spatial: due to under sampling
Temporal: time intervals b/n images in a sequence of images (Ex: Wagon Wheel)

19. Effect of aliasing in images can be reduced by slightly defocussing the scene (Anti-aliasing)

20. Image interpolation and Resampling
Perfect reconstruction requires approximations which in turn leads to interpolation
Most common applications of interpolation includes
Zooming (Over sampling):pixel replication to integer no of times
Shrinking (Under sampling): row-column deletion
Super sampling is alternative option for anti aliasing

23. Moire Patterns
Result from sampling scenes with periodic or nearly periodic components
Arises routinely when scanning media print, in images with periodic components whose spacing is comparable to the spacing between samples

25. Discrete Fourier Transform (1D)
To obtain M equally spaced samples or
Inverse Discrete Fourier Transform
can be expressed as or or

26. Discrete Fourier Transform pair(2D)

27. Properties of 2D DFT
Relation between spatial and frequency intervals is
Translational property:
Rotational property:

28. Periodicity:

30. Symmetric Properties:

32. Fourier spectrum and phase angle:
Generally 2D DFT is complex, and can be expressed in polar form as

33. 2D Convolution Theorem

36. Frequency Domain Filtering Fundamentals

40. Correspondence b/n spatial and frequency domains
If we want to represent in spatial domain, and if then
Filtered output is
Where h(x,y) is a spatial filter which can be obtained by response of a frequency domain filter.
h(x,y) sometimes referred as impulse response of H(u,v). Because of allquantities in a discrete I
mplementation are finite, such filters are also referred as Finite Impulse Response filters.

41. High-pass filters can be constructed by difference of Gaussians
These equations facilitate the analysis because both are Fourier transform pair and components of Gaussian are real (NO bothering about complex no.s)
Functions behave reciprocally (σ)
As σ approaches to infinite, H(u) tends to const. function and h(x) tends to impulse function
Use response as a guide for specifying mask coefficients. More narrow the filter response (σ), more it attenuate low frequencies, results in increased blurring (In spatial, increased mask size)
High-pass filters can be constructed by difference of Gaussians

43. Image Smoothing using frequency domain filters
Main categories include
Ideal (Sharp filtering)
Gaussian (Smooth filtering)
Butterworth (Have filter order, capable of sharp and smooth)
Ideal Low-Pass Filter(ILPF):
Cutoff frequency is the point of transition between H(u,v)=1 and H(u,v)=0 (D0)
These are nonphysical filters

45. Total image power is calculated for determining cutoff frequency loci

47. The cross section of ILPF in freq
The cross section of ILPF in freq. domain is looks like box filter (Sinc func. In spatial)
Filtering in spatial means convolving h(x,y) with image (place sinc at each location)
Central lobe is responsible for blurring while outer lobes cause ringing
Spread of sinc is inversely prop. to radius of H(u,v) (larger D0) causes no blurring

48. Butterworth Low Pass Filter
BLPF with order 1 and 2 has no ringing

50. Gaussian Low Pass Filter
GLPF

55. Image Sharpening using frequency domain filters

57. Ideal High-Pass Filter(IHPF)

58. Butterworth High Pass Filter (BHPF)

59. Gaussian High Pass Filter(GHPF)

61. The Laplacian in frequency domain
Laplacian image can be obtained as

63. Un-sharp Masking, High-boost filtering and
High frequency Emphasis filtering
Where
When k=1, un-sharp masking
When k>1, high boost filtering
g(x,y) = F-1{[1+k * [1 – HLP(u,v)]] F(u,v)}
g(x,y) = F-1{[1+k * HHP(u,v)] F(u,v)}
High Frequency Emphasis filter does not reduce the avg. intensity in filtered image to zero
g(x,y) = F-1{[k1 + k2 * HHP(u,v)] F(u,v)}
Where k1>=0 controls offset and k2 >=0 controls contribution of high frequencies

65. Homomorphic Filtering
Z(u,v) = Fi(u,v) + Fr(u,v)

66. Homomorphic Filtering

68. Selective filtering
Band pass or Band Reject filtering
Notch filtering

69. Butterworth Notch Reject filtering

73. Computing IDFT using DFT algorithm
2D FFT Algorithm
Reduces to
where
nothing but