syde 575: introduction to image processing
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Spatial-Frequency Domain: Implementations Textbook: Chapter 4. SYDE 575: Introduction to Image Processing. Filtering in Spatial and Spatial-Frequency Domains. Basic spatial filtering is essentially 2D discrete convolution between an image f and filter function h. - PowerPoint PPT PresentationTRANSCRIPT
SYDE 575: Introduction to Image Processing
Spatial-Frequency Domain: Implementations
Textbook: Chapter 4
Filtering in Spatial and Spatial-Frequency Domains
Basic spatial filtering is essentially 2D discrete convolution between an image f and filter function h
( , ) ( , ) ( , )g x y f x y h x y= *
Convolution in spatial domain becomes multiplication in frequency domain
( , ) ( , ) ( , )G u v F u v H u v=
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)
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
Ideal LPF
Cuts off all high-frequency components at a distance greater than a certain distance from origin (D0: cutoff frequency)
0
0
1, if D (u,v) D( , )
0, if D (u,v) DH u v
£ì=í >î
Visualization
Source: Gonzalez and Woods
Effect of Different Cutoff Frequencies
Source: Gonzalez and Woods
Effect of Different Cutoff Frequencies
Source: Gonzalez and Woods
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?
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
Ringing
Source: Gonzalez and Woods
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
[ ]2
0
1( , )
1 ( , ) /n
H u vD u v D
=+
Butterworth LPF
Source: Gonzalez and Woods
Spatial Representations
Source: Gonzalez and Woods
Tradeoff between amount of smoothing and ringing
Butterworth LPFs of Different Orders
Source: Gonzalez and Woods
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
2 20( , ) / 2( , ) D u v DH u v e -=
Gaussian LPF
Source: Gonzalez and Woods
Gaussian LPF
Source: Gonzalez and Woods
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
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
HPF Transfer Functions
Ideal HPF0
0
0 if ( , )( , )
1 if ( , )
D u v DH u v
D u v D
£ì=í >î
Butterworth HPF
[ ]2
0
1( , )
1 / ( , )n
H u vD D u v
=+
Gaussian HPF2 2
0( , ) / 2( , ) 1 D u v DH u v e -= -
HPF Transfer Functions
Spatial Representations of HPFs
Ideal HPF Filtering
Butterworth HPF Filtering
Gaussian HPF Filtering
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
Spatial-Frequency Edge Enhancement
• Edge enhancement can be performed directly in the spatial-frequency domain
• Example: high boost filtering (unsharp masking)
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
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
Results
Source: Gonzalez and Woods
Examples of Frequency Domain Filtering
Source: Gonzalez and Woods
Human Visual System Models
• For a generic spatial-frequency image enhancement filter, what should the transfer function look like?
1) DC gain is typically reduced so 0<H(u)<12) H(u) approaches zero as u increases3) H(u) > 1 for frequency range where signal
dominates
• Sketch:
Model of HVS
• Light entering the eye is processed by two steps
1) Cornea/Lens H1(u): modelled as LPF e.g., Gaussian
2) 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:
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-2p2u2s2
2
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
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
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?
Periodic Noise Reduction
Typically occurs from electrical or electromechanical interference during image acquisition
Spatially dependent noise Example: spatial sinusoidal noise
Example
Source: Gonzalez and Woods
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
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
Example: Ideal Bandreject Filters
Ideal bandreject filter
0
0 0
0
1 if ( , )2
( , ) 0 if ( , )2 2
1 if ( , )2
WD u v D
W WH u v D D u v D
WD u v D
ì < -ïïï= - £ £ +íïï > +ïî
Example
Source: Gonzalez and Woods
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
Some Notch Reject Filters
Source: Gonzalez and Woods
Example: Moire pattern reduction
Source: Gonzalez and Woods
Homomorphic Filtering
Image can be modeled as a product of illumination (i) and reflectance (r)
( , ) ( , ) ( , )f x y i x y r x y=
Unlike additive noise, can not operate on frequency components of illumination and reflectance separately
[ ] [ ] [ ]( , ) ( , ) ( , )f x y i x y r x yÁ ¹Á Á
Homomorphic Filtering
Idea: What if we take the logarithm of the image?
[ ] [ ] [ ]ln ( , ) ln ( , ) ln ( , )f x y i x y r x yÁ =Á +Á
ln ( , ) ln ( , ) ln ( , )f x y i x y r x y= +
Now the frequency components of i and r can be operated on separately
Homomorphic Filtering Framework
Source: Gonzalez and Woods
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)
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
( )2( , ) ( , ) ( , )g x y i x y r x y»
Example
Source: Gonzalez and Woods
Homomorphic Filtering: Noise Reduction
Multiplicative noise model
( , ) ( , ) ( , )f x y s x y n x y=
signal noise
Transforming into log space turns multiplicative noise to additive noise
ln ( , ) ln ( , ) ln ( , )f x y s x y n x y= + Low-pass filtering can now be applied to
reduce noise
Example
Source: Jernigan, 2003
O riginal M ultiplicative Noise
Example
Source: Jernigan, 2003
Hom om orphic LPF