1. SYDE 575: Introduction to Image Processing
Spatial-Frequency Domain:
Implementations
Textbook: Chapter 4

2. Filtering in Spatial and Spatial-Frequency Domains
Basic spatial filtering is essentially 2D discrete convolution between an image f and filter function h
Convolution in spatial domain becomes multiplication in frequency domain

3. Spatial-Frequency Implementations
We will discuss these implementations:
Low pass filters: ideal, Butterworth, Gaussian
High pass filters: ideal, Butterworth, Gaussian
Edge enhancement: high boost filtering
HVS modelling: Difference of Gaussians (DoG), Gabor, Laplacian of Gaussian
Periodic noise filtering (Section 5.4)

4. Blurring/Noise reduction
Noise characterized by sharp transitions in image intensity
Such transitions contribute significantly to high frequency components of Fourier transform
Intuitively, attenuating certain high frequency components result in blurring and reduction of image noise

5. Ideal LPF
Cuts off all high-frequency components at a distance greater than a certain distance from origin (D0: cutoff frequency)

6. Visualization
Source: Gonzalez and Woods

7. Effect of Different Cutoff Frequencies
Source: Gonzalez and Woods

8. Effect of Different Cutoff Frequencies
Source: Gonzalez and Woods

9. Effect of Different Cutoff Frequencies
As cutoff frequency decreases
Image becomes more blurred
Noise becomes more reduced
Analogous to larger spatial filter sizes
Noticeable ringing artifacts that increase as the amount of high frequency components removed is increased
Why ringing?

10. Why is there ringing?
Ideal low-pass filter function is a rectangular function
The inverse Fourier transform of a rectangular function is a sinc function
Convolution of a sinc and a step function generates ringing on both sides of the edge

11. Ringing
Source: Gonzalez and Woods

12. Butterworth LPF
Transfer function does not have sharp discontinuity establishing cutoff between passed and filtered frequencies
Cutoff frequency D0 defines point at which H(u,v)=0.5
Similar to exponential LPF

13. Butterworth LPF
Source: Gonzalez and Woods

14. Spatial Representations
Tradeoff between amount of smoothing and ringing
Source: Gonzalez and Woods

15. Butterworth LPFs of Different Orders
Source: Gonzalez and Woods

16. Gaussian LPF
This is another form of a Gaussian filter, as used by Gonzalez & Woods (textbook)
Transfer function is smooth, like Butterworth filter
Gaussian in frequency domain remains a Gaussian in spatial domain
Advantage: No ringing artifacts

17. Gaussian LPF
Source: Gonzalez and Woods

18. Gaussian LPF
Source: Gonzalez and Woods

19. Spatial-Frequency High Pass Filters (HPFs)
HPFs are effectively the opposite of LPFs
High pass filtering in the spatial-frequency domain is related to low pass filtering
HHP(u,v) = 1 – HLP(u,v)
hHP(x,y) = d(x,y) – hLP (x,y)
Note: DC gain is zero for a HPF
Handwritten notes 3-18 to 3-19

20. Impact of High Pass Filtering
Edges and fine detail characterized by sharp transitions in image intensity
Such transitions contribute significantly to high frequency components of Fourier transform
Intuitively, attenuating low frequency components and preserving high frequency components will retain image intensity edges

21. HPF Transfer Functions
Ideal HPF
Butterworth HPF
Gaussian HPF

22. HPF Transfer Functions

23. Spatial Representations of HPFs

24. Ideal HPF Filtering

25. Butterworth HPF Filtering

26. Gaussian HPF Filtering

27. Observations of HPFs
As with ideal LPF, ideal HPF shows significant ringing artifacts
Second-order Butterworth HPF shows sharp edges with minor ringing artifacts
Gaussian HPF shows good sharpness in edges with no ringing artifacts

28. Spatial-Frequency Edge Enhancement
Edge enhancement can be performed directly in the spatial-frequency domain
Example: high boost filtering (unsharp masking)
pp to 3-19 in handwritten notes

29. High frequency emphasis
Advantageous to accentuate enhancements made by high-frequency components of image in certain situations (e.g., image visualization)
Solution: multiply high-pass filter by a constant and add offset so zero frequency term not eliminated
g(x,y) = f(x,y) + k gHPF(x,y)
As discussed earlier, this is referred to as high-boost filtering

30. High Boost Filtering In spatial domain: g(x,y) = f(x,y) + k gHPF(x,y)
Impulse response:
h(x,y) = d(x,y) + k hHPF(x,y)
Transfer function in spatial-frequency domain:
H(u,v) = 1 + k HHPF(u,v)
or:
H(u,v) = 1 + kHHPF(u,v) = 1 + k(1- HLPF(u,v))
= (1+k) - kHLPF(u,v)
Recall: Set k=1 for unsharp masking
k=1 gives unsharp masking

31. Results Source: Gonzalez and Woods
Result of using 1-Laplacian for enhancement in the frequency domain.
Source: Gonzalez and Woods

32. Examples of Frequency Domain Filtering
Source: Gonzalez and Woods

33. Human Visual System Models
For a generic spatial-frequency image enhancement filter, what should the transfer function look like?
DC gain is typically reduced so 0<H(u)<1
H(u) approaches zero as u increases
H(u) > 1 for frequency range where signal dominates
Sketch:

34. Model of HVS Light entering the eye is processed by two steps
Cornea/Lens H1(u): modelled as LPF e.g., Gaussian
Retina H2(u): modelled as edge enhancement e.g., 1-Laplacian
Combined: H(u) = H1(u) H2(u)
= (1+(2pu/a)2) e-2p2u2s2
Sketch:
See notes 3-22 to calculate u of peak response

35. Difference of Gaussians
There are a number of Gaussian-based functions that mimic lateral inhibition
Difference of Gaussians takes the difference of two Gaussians with different s
H(u) = A e-2p2u2s12 - Be-2p2u2s22
With A>B and s1<< s2
Sketch in frequency and time domains
Can vary s1 and s2 to create filter bank with varying peak frequencies

36. Gabor Filter Gabor is a Gaussian band pass filter
H(u) = (A/2) e-2p2u2s12 * [ d(u-up) + d(u+up)]
In time domain, a Gaussian-modulated sinusoid (real part of Gabor filter)
h(x) = A/(s(2p)0.5) e-0.5(x/s)2cos(2pupx)
Sketch in frequency and time
Similar shape as Difference of Gaussians, but with ringing
Note: complex form of filter used for texture feature extraction

37. Laplacian of a Gaussian
Consider the Marr-Hildreth operator i.e., a Laplacian of a Gaussian
H(u) = (-j2pu)2 e-2p2u2s2 = 4p2u2 e-2p2u2s2
Sketch in time and frequency domains
What is the impact of this filter? Why?
Add detail

38. Periodic Noise Reduction
Typically occurs from electrical or electromechanical interference during image acquisition
Spatially dependent noise
Example: spatial sinusoidal noise

39. Example
Source: Gonzalez and Woods

40. cos(2pu0x) <-> 0.5[d(u + u0) + d(u – u0)]
Observations
Symmetric pairs of bright spots appear in the Fourier spectra
Why?
Fourier transform of cosine function is the sum of a pair of impulse functions
cos(2pu0x) <-> 0.5[d(u + u0) + d(u – u0)]
Intuitively, sinusoidal noise can be reduced by attenuating these bright spots

41. Bandreject Filters
Removes or attenuates a band of frequencies about the origin of the Fourier transform
Sinusoidal noise may be reduced by filtering the band of frequencies upon which the bright spots associated with period noise appear

42. Example: Ideal Bandreject Filters

43. Example
Source: Gonzalez and Woods

44. Notch Reject Filters Idea:
Sinusoidal noise appears as bright spots in Fourier spectra
Reject frequencies in predefined neighborhoods about a center frequency
In this case, center notch reject filters around frequencies coinciding with the bright spots

45. Some Notch Reject Filters
Source: Gonzalez and Woods

46. Example: Moire pattern reduction
In physics, a moiré pattern is an interference pattern created, for example, when two grids are overlaid at an angle, or when they have slightly different mesh sizes.
Source: Gonzalez and Woods

47. Homomorphic Filtering
Image can be modeled as a product of illumination (i) and reflectance (r)
Unlike additive noise, can not operate on frequency components of illumination and reflectance separately
Handwritten notes 3-29 to 3-34

48. Homomorphic Filtering
Idea: What if we take the logarithm of the image?
Now the frequency components of i and r can be operated on separately

49. Homomorphic Filtering Framework
Source: Gonzalez and Woods

50. Homomorphic Filtering: Image Enhancement
Simultaneous dynamic range compression (reduce illumination variation) and contrast enhancement (increase reflectance variation)
Illumination component characterized by slow spatial variations (low spatial frequencies)
Reflectance component characterized by abrupt spatial variations (high spatial frequencies)

51. Homomorphic Filtering: Image Enhancement
Can be accomplished using a high frequency emphasis filter in log space
DC gain of 0.5 (reduce illumination variations)
High frequency gain of 2 (increase reflectance variations)
Output of homomorphic filter

52. Example
Source: Gonzalez and Woods

53. Homomorphic Filtering: Noise Reduction
Multiplicative noise model
signal
noise
Transforming into log space turns multiplicative noise to additive noise
Low-pass filtering can now be applied to reduce noise

54. Example
Source: Jernigan, 2003

55. Example
Source: Jernigan, 2003