### Section 4.3

```SYDE 575: Introduction to
Image Processing
Spatial-Frequency Domain:
Implementations
Textbook: Chapter 4
Filtering in Spatial and SpatialFrequency 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)
ì1, if D(u,v) £ D0
H (u , v ) = í
î0, if D(u,v) > D0
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
H (u , v ) =



1
1 + [ D(u , v ) / D0 ]
2n
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
Butterworth LPF
Source: Gonzalez and Woods
Spatial Representations

Tradeoff between amount of smoothing and
ringing
Source: Gonzalez and Woods
Butterworth LPFs of Different
Orders
Source: Gonzalez and Woods
Gaussian LPF
H (u , v ) = e




-D
2
2
( u, v)/ 2 D0
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
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 HPF
ì0
H (u , v ) = í
î1

Butterworth HPF
H (u , v ) =

if D(u , v ) £ D0
if D(u , v ) > D0
1
1 + [ D0 / D(u , v )]
Gaussian HPF
H (u , v ) = 1 - e
- D2 ( u, v)/ 2 D02
2n
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
High frequency emphasis



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)<1
2) H(u) approaches zero as u increases
3) 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)
2u2s2
2
-2p
= (1+(2pu/a) ) e
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
2 u2 s 2
2 u2 s 2
-2p
-2p
1
2
H(u) = A e
- Be
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)
2u 2s 2
-2p
1
e
* [ d(u-up) + d(u+up)]
• In time domain, a Gaussian-modulated sinusoid
(real part of Gabor filter)
2
0.5
-0.5(x/s)
h(x) = A/(s(2p) ) e
cos(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
2u2s2
2u2s2
2
-2p
2
2
-2p
H(u) = (-j2pu) e
= 4p u e
• 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
W
ì
if D(u , v ) < D0 ï1
2
ï
W
W
ï
H (u , v ) = í0 if D0 - £ D (u , v ) £ D 0 +
2
2
ï
W
ï
if D (u , v ) > D0 +
ï1
2
î
Example
Source: Gonzalez and Woods
Notch Reject Filters

Idea:
 Sinusoidal noise appears as bright spots
in Fourier spectra
 Reject frequencies in predefined
 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
Á[ f ( x, y )] ¹ Á[ i( x, y )] Á[ r ( x, y )]
Homomorphic Filtering

Idea: What if we take the logarithm of the
image?
ln f ( x, y ) = ln i ( x , y ) + ln r ( x , y )

Now the frequency components of i and r
can be operated on separately
Á[ ln f ( x, y )] = Á[ ln i ( x , y )] + Á[ ln r ( x , y )]
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
g ( x , y ) » i ( x , y ) ( r ( x , y ))
2
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
ln f ( x, y ) = ln s ( x , y ) + ln n (x , y )

Low-pass filtering can now be applied to
reduce noise
Example
Original
Multiplicative Noise
Source: Jernigan, 2003
Example
Homomorphic
LPF
Source: Jernigan, 2003
```