Frequency Domain Filtering of Digital Images

Filtering in Frequency Domain
Upendra
Indian Institute of Information Technology, Allahabad
Image and Video Processing
February 26, 2017
Upendra (Indian Institute of Information Technology, Allahabad[4ex] Image and Video Processing)Filtering in Frequency Domain February 26, 2017 1 / 120

Background I
Time Domain and Frequency Domain Analysis
Time Domain Analysis
1 Applications: predictions, fitting regression models etc[7].
2 Different types of equipments in each field
Frequency Domain Analysis
1 Motivation: conversion of complex differentials into polynomial
equations
2 Inverse transform feasible (take care of rules though)
3 Different transforms like Fourier, Laplace, Z etc.

Periodic Signals I
1 A signal f (t) that satisﬁes
f (t) = f (t + T) ∀t ⊆ (1)
2 In general,
f (t) = f (t ± T) = f (t ± 2T) = ... = f (t ± nT) (2)
3 T ﬁxed called period
4 Smallest value of T called Principal Period
5 Principal period Vs Period ?

Background I
1 Proposed by French mathematician Jean Baptise Joseph Fourier [2]
2 Any periodic signal = sum of sines and/or cosines terms of different
frequencies.
3 Each term multiplied by a coefficient
4 Coefficients value determines the term’s contribution [3].

Dirichlet Conditions I
1 Named after Peter Gustav Lejeune Dirichlet [6].
2 Provides suﬃcient conditions for a real valued signal to be equal to its
fourier series sum
3 Conditions are
Signal must be absolutely integrable over a period
Finite number of extrema points in any given interval
Finite number of discontinuities in any given interval
4 Such a function is said to have a bounded variation over a period [6]

Deﬁnition I
Fourier Series [2][3][5]
A signal f(t) of a continuous variable ’t’ that is periodic with period ’T’,
can be expressed as
f (t) =
∞
n=−∞
cn ej 2πn
T
t
(3)
where
cn =
T
2
−T
2
f (t) e−j 2πn
T
t
for n = 0, ±1, ±2.... (4)
are the coeﬃcients.

Background I
Characteristics of Fourier Series Representation [2]
1 Holds good for all functions (complication immaterial)
2 The original function can be reconstructed completely; hence, a
lossless transformation[1][2]
3 Flexibility in terms of domain switch
4 Industries and Academic institutions alike

Problem-01 I
Find the Fourier Series Coeﬃcients of the following signal:
Figure: Calculation of Fourier Series Coeﬃcients for the above signal

Problem-02 I
Find the Fourier Series Coeﬃcients of the following signal:
Figure: Calculation of Fourier Series Coeﬃcients for the above signal

Properties of Fourier Series [1][2][3] I
Assuming
x(t) ⇐⇒ {cn} ; y(t) ⇐⇒ {dn} (5)
Linearity
Ax(t) + By(t) ⇐⇒ {Acn + Bdn} (6)
Multiplication
x(t)y(t) ⇐⇒ {
∞
k=−∞
ckdn−k} (7)

Properties of Fourier Series I
Time Shifting
x(t − t0) ⇐⇒ {e
−j2πnt0
T cn} (8)
Time Reversal
x(−t) ⇐⇒ {c−n} (9)
Conjugation
x∗
(t) ⇐⇒ {c∗
−n} (10)
Time Scaling property
x(at) ⇐⇒
∞
n=−∞
cne
j2πn(at)
T (11)
Time scaling, thus, changes the frequency components [3].

Impulse functions and Time Shift Property I
Deﬁnition
δ(t) =
1, if t = 0,
0, if t = 0.
(12)
Subjected to,
∞
−∞
δ(t)dt = 1 (13)
Physical Interpretation A spike of inﬁnite amplitude and zero duration,
having a unit area [2].

Impulse functions and Time Shift Property II
Deﬁnition
Figure: Plot of an Impulse Function

Time Shift Property-Continuous Domain I
1 The impulse function has got a time shift property (wrt integration)
given by [2][4],
∞
−∞
f (t)δ(t) = f (0) (14)
provided that the function remain continuous at t = 0
2 In general, this notion could be generalized to,
∞
−∞
f (t)δ(t − t0) = f (t0) (15)

Time Shift Property-Discrete Domain I
The unit discrete impulse function, serves the same purpose as its
continuous counterpart [2]. Mathematically,
δ(x) =
1, if x = 0,
0, if x = 0.
(16)
As such, the time shift properties become,
x=∞
x=−∞
f (x)δ(x) = f (0) (17)
x=∞
x=−∞
f (x)δ(x − x0) = f (x0) (18)

Need for Fourier Transform[8][9] I
Figure: Diﬀerent Types of Fourier Transforms. Source: Digital Image Processing
Processing(3rd Edition) by Gonzalez, R.C. and Woods, R.E, PHI
1 Inverse transform is loss-less
2 Widespread use since the advent of digital computers and Fast
Fourier Transform

Fourier Transform[8][9][10] I
Deﬁnition
The Fourier Transform of a continuous function f(t) of a continuous
variable t denoted by
F{f (t)} =
∞
−∞
f (t) e−j2πµt
dt (19)
where µ is also a continuous variable
Thus,
F{f (t)} = F(µ) (20)
F(µ) =
∞
−∞
f (t) e−j2πµt
dt (21)

Fourier Transform I
Deﬁnition
Using Euler’s Formula,
F(µ) =
∞
−∞
f (t)[cos(2πµt) − jsin(2πµt)]dt (22)
Inverse Fourier Transform
f (t) =
∞
−∞
F(µ)ej2πµt
dµ (23)
Together, F(µ) and f (t) are known as Fourier Transform pairs

Fourier Transform I
Note: The Fourier Transform is an expansion of f(t) multiplied by
sinusoidal terms whose frequencies are determined by µ.
Question
Why is the domain of Fourier Transform ’frequency’?

Fourier Spectrum I
Need and Deﬁnition
Fourier transform contains complex terms. So, we usually deal with
magnitude part
Mathematically, the Fourier Spectrum or the Frequency Spectrum is given
by,
|F(µ)| = |
∞
−∞
f (t)[cos(2πµt) − jsin(2πµt)]dt | (24)
Question
What is the physical signiﬁcance of frequency spectrum?

Questions I
Find out the Fourier Transform of the following signals
f (t) = e−a|t|
(25)
f (t) = δ(t − t0) (26)
Figure: A simple signal in time domain
Also plot the obtained Fourier Transform

Convolution I
Deﬁnition
1 Flip, multiply and then add.
2 Denoted by a operator.
3 Mathematically, the convolution of two functions f (t) and h(t) of one
continuous variable ’t’ is given by
f (t) h(t) =
∞
−∞
f (τ)h(t − τ) dτ (27)
4 Flip by - sign

Convolution I
Fourier Transform of Convolution operation[2]
F{f (t) h(t)} =
∞
−∞
∞
−∞
f (τ)h(t − τ)dτ e−2jπµt
dt (28)
In other words,
F{f (t) h(t)} =
∞
−∞
f (τ)
∞
−∞
h(t − τ)e−2jπµt
dt dτ (29)
=
∞
−∞
f (τ) H(µ)e−2πjµτ
dτ (30)
= H(µ)
∞
−∞
f (τ)e−j2πµτ
dτ (31)
= H(µ)F(µ) (32)

Convolution I
Consequence-Convolution Theorem[2]
1 First half of convolution theorem
f (t) h(t) ⇐⇒ F(µ)H(µ) (33)
2 Interchangeability of domains
spatial domain(t) ⇐⇒ frequency domain(µ) (34)
3 Another half
f (t)h(t) ⇐⇒ H(µ) F(µ) (35)

Properties of Fourier Transform[10] I
Assuming that
f (t) ⇐⇒ F(µ) (36)
We have the following properties for the Fourier Transform
Translation
f (t − t0) ⇐⇒ e−jµt0
F(µ) (37)
Modulation
ejµ0t
f (t) ⇐⇒ F(µ − µ0) (38)
Scaling
f (at) ⇐⇒
1
|a|
F
µ
a
(39)

Properties of Fourier Transform I
Duality
F(t) ⇐⇒ 2πf (−µ) (40)
Multiplication
f1(t)f2(t) ⇐⇒
1
2π
F1(µ) F2(µ)] (41)
Diﬀerentiation in Time
df (t)
dt
⇐⇒ jµ F(µ) (42)
Diﬀerentiation in Frequency
(−jt)n
f (t) ⇐⇒
dnF(µ)
dµ
(43)

Sampling and Fourier Transform of Sampled signals I
Sampling
Continuous signals into discrete signals
Sampled values then quantized
Mathematically,
f ˜
(t) = f (t)s∆T (t) =
∞
n=−∞
f (t)δ(t − n∆T) (44)
Each component of this summation is an impulse weighted by the
value of f(t) at the location of the impulse

Sampling and the Fourier Transform of Sampled signals I
Sampling
The value of each sample is given by the strength of the weigted impulse,
which we obtain by integration.
Mathematically,
fk =
∞
−∞
f (t)δ(t − k∆T) dt (45)
= f (k∆T) (46)

Fourier Transform of Sampled Function I
The Fourier Transform F˜(µ) of the sampled function f ˜(t) is
F˜
(µ) = F{f ˜
(t)} (47)
= F{f (t)s∆T (t)} (48)
= F(µ) S(µ) (49)
where,
S(µ) =
1
∆T
∞
n=−∞
δ µ −
n
∆T
(50)

The Fourier Transform of Sampled Signal I
Using
F˜
(µ) = F(µ) S(µ) (51)
we have
F˜
(µ) =
∞
−∞
F(τ)S(µ − τ) dτ (52)
=
1
∆T
∞
−∞
F(τ)
∞
n=−∞
δ µ − τ −
n
∆T
dτ (53)
=
1
∆T
∞
n=−∞
F µ −
n
∆T
(54)
Thus, Fourier Transform F˜(µ) of the sampled signal f ˜(t) is an inﬁnite,
periodic sequence of copies of F(µ), the transform of the original,
continuous signal

The Fourier Transform of the Sampled Signals I
From,
F˜
(µ) =
1
∆T
∞
n=−∞
F µ −
n
∆T
(55)
, we have
∆T as the sample duration
The separation between copies is determined by 1
∆T
This separation can determine if F(µ) is preserved in the sum
Accordingly we have oversampling, critical sampling and
under-sampling

The Fourier Transform of the Sampled Signal[2] I
Sampling under diﬀerent conditions
Figure: Transforms of the corresponding sampled function under conditions of
over-sampling, critically-sampling and undersampling. Source: Digital Image
Processing Processing(3rd Edition) by Gonzalez, R.C. and Woods, R.E, PHI

Fourier Transform in two variables I
Deﬁnition
The Fourier transform equations can be easily extended to two variables as
F(u, v) =
∞
−∞
∞
−∞
f (x, y)e−j2π(ux+vy)
dx dy (56)
Simlarly, the inverse transform is given by
f (x, y) =
∞
−∞
∞
−∞
F(u, v)ej2π(ux+vy)
du dv (57)

Discrete Fourier Transform [2] I
Deﬁnition
The Fourier Transform of a discrete function of one variable, f [x],
x=0,1,2...M − 1 is given by
F(u) =
1
M
M−1
x=0
f [x]e
−j2πux
M for u = 0, 1, 2, ..., M − 1 (58)
Simiarly, the inverse DFT is given by
f [x] =
M−1
u=0
F(u)e
j2πux
M for x = 0, 1, 2, ..., M − 1 (59)
The DFT remains a discrete quantity with same number of components as
signal.

Discrete Fourier Transform I
Key points
1 DFT remains a discrete quantity with same number of components as
the signal
2 Same applies for IDFT as well
3 DFT and IDFT always exist (unlike the continuous case)
4 Each summation term called the component of DFT
5 In general, components are complex, Why?

Representation of DFT in Euler form
Usin Euler’s formula, we express F(u) in polar coordinates,
F(u) = |F(u)| e−j φ(u)
(60)
where
|F(u)| = R2
(u) + I2
(u)
1
2
(61)
is the magnitude spectrum of the Fourier transform and
φ(u) = tan−1 I(u)
R(u)
(62)
is the phase angle or the phase spectrum.

Representation of DFT in Euler form
Power Spectrum
This is yet another important parameter given by
P(u) = |F(u)|2
(63)
= R2
(u) + I2
(u) (64)
Also referred to as spectral density
What is the physical signiﬁcance of power spectrum?

The Two-dimensional DFT and its Inverse I
Deﬁnition of Two-dimensional DFT
Since image is a 2-D signal, we now proceed to Discrete Fourier Transform
in two dimensions.
The Discrete Fourier transform of a function f (x, y) of size M x N is
given by
F(u, v) =
1
MN
M−1
x=0
N−1
y=0
f (x, y) e−j2π( ux
M
+vy
N
)
(65)
for u=0,1,2...M-1 and v=0,1,2,..N-1.
Reminder: x,y are spatial variables while u,v are frequency variables

Deﬁnition of Inverse DFT
As is the case of 1-D transform, the inverse DFT for two dimensions is
given by
f (x, y) =
M−1
u=0
N−1
v=0
F(u, v) ej2π( ux
M
+vy
N
)
(66)
for x=0,1,2...M-1 and y=0,1,2,....N-1

Representation of 2-D DFT in Euler form
Fourier Spectrum
|F(u, v)| = R2
(u, v) + I2
(u, v)
1
2
(67)
Phase Spectrum
φ(u, v) = tan−1 I(u, v)
R(u, v)
(68)
Power Spectrum
P(u, v) = |F(u, v)|2
(69)
= R2
(u, v) + I2
(u, v) (70)

Properties of Two-dimensional DFT I
Translation Property
f (x, y)e
j2π
u0x
M
+
v0y
N
↔ F(u − u0, v − v0) (71)
Similarly,
f (x − x0, y − y0) ↔ F(u, v) e
−j2π
ux0
M
+
vy0
N
(72)

Translation to the center of the frequency rectangle
f (x, y)(−1)x+y
↔ F(u −
M
2
, v −
N
2
) (73)
And,
f (x −
M
2
, y −
N
2
) ↔ F(u, v)(−1)u+v
(74)

Convolution Property
f (x, y) h(x, y) ↔ F(u, v)H(u, v) (75)
And,
f (x, y)h(x, y) ↔ F(u, v) H(u, v) (76)

Translation to the center of the frequency rectangle
f (x, y)(−1)x+y
↔ F(u −
M
2
, v −
N
2
) (77)
Input image function usually multiplied by (−1)x+y prior to Fourier
Transform[1]. Why?
Origin of frequency rectangle shifts to the center of the frequency
rectangle.

The value of the transform at (u,v)=(0,0) is given by
F(0, 0) =
1
MN
M−1
x=0
N−1
y=0
f (x, y) (78)
which is the average value of f(x,y)(also called dc component of the
spectrum).
Corollary If the image is f(x,y), the value of Fourier Transform at the
origin is equal to the average gray level of the image.

Aliasing[11][2] I
Deﬁnition
Occurs when high frequency components ”masquarade” as low frequency
components(called aliased freqencies)
1 A consequence of under-sampling
2 Corrupts the sampled image
3 Additional frequency components are introduced into the sampled
image
4 Moire’s pattern introduced in the images (spatially sampled signal)

Aliasing I
Moire’s Patterns
Figure: A sine waveform being sampled at frequency less than twice the maximum
frequency. Source:
http: // users. wfu. edu/ matthews/ misc/ DigPhotog/ alias/

Aliasing I
(a) Original Image (b) Resized Image (c) Moire’s pattern in
image due to aliasing
Figure: Moire’s pattern in images due to aliasing. Source:
http: // users. wfu. edu/ matthews/ misc/ DigPhotog/ alias/

Aliasing I
Moire’s Patterns[2]
Figure: Some more examples of Moire’s pattern. Source: Digital Image

Anti-Aliasing[11] I
1 Attenuate higher frequencies (relative to what?)
2 Needs to be done before sampling since it cannot be undone after the
fact[1].Hence, effective software antialias filters do not exist.
3 Various strategies like notch filters, intentional blurring[6] in front of
CCD etc.
Question: Any alternative to Antialias filter?

Basics of Filtering in the frequency domain I
Basic Steps
Filtering takes place in the following steps [1]
1 Multiply the image function by (−1)x+y . Why?
2 Multiply F(u,v) of the image by ﬁlter transfer function H(u,v)
3 Compute the inverse DFT of the above product
4 Obtain the real part
5 Multiply the above result by (−1)x+y

Basics of Filtering in the frequency domain I
Basic Steps
Figure: Basic steps for ﬁltering in the frequency domain. Source: Digital Image

Image Smoothing using lowpass filter I
Key points
Edges and noises(sharp transitions) in an image contributes
significantly to the high frequency content of its Fourier transform
Smoothing(blurring) achieved by high frequency attenuation
Types of low pass filters to be discussed
1 Ideal Low pass filters
2 ButterWorth filters
3 Gaussian Filters

Ideal Lowpass Filter[2]
A 2-D lowpass filter that passes without attenuation all frequencies within
a radius of D0 and at the same time cuts off all other frequencies
completely. Mathematically, it is defined as
H(u, v) =
1, if D(u, v) ≤ D0,
0, if D(u, v) > D0.
(79)
Here, D(u, v) is the distance between a point (u,v) in the frequency
domain and the center of the frequency rectangle
D(u, v) = u −
P
2
2
+ v −
Q
2
2 1
2
(80)

Ideal Lowpass Filter
Figure: (a) Perspective plot of an ideal low pass ﬁlter (b) Filter displayed as an
image (c) Filter radial cross section. Source: Digital Image Processing

Key Point
1 ILPF is radially symmteric about the origin
2 The point of transition from H(u, v) = 1 to H(u, v) = 0 called cutoff
frequency
3 Ideal behavior cannot be realized by electronics; mathematically
feasible
In order to establish a set of cutoff frequency loci, we compute circles that
enclose specified amounts of total image power PT .
Mathematically
PT =
P−1
u=0
Q−1
v=0
P(u, v) (81)

Figure: (a) and (b) show a test pattern image and its spectrum. The circles
superimposed on the spectrum hve radii of 10, 30, 60, 160 and 460 pixels
respectively. These circles enclose α percent of image power, for
α = 87.0, 93.1, 95.7, 97.8 and 99.2 respectively. The spectrum falls oﬀ rapidly,
with 87 % of the total power being enclosed by a relatively small circle of radius
10.

Figure: (b)-(f) Results of filtering using ILPFs with cutoff frequencies set at radii
values 10, 30, 60, 160 and 460. The power removed by these filters was 13, 6.9,
4.3, 2.2 and 0.8 % of the total respectively. Source: Digital Image Processing

Severe blurring in (b) means majority of the sharp detail information
in the picture is contained in the 13 percent power removed by the
filter.
With increasing radius, lesser power is removed; hence, less blurring
Ringing gets finer in texture as the amount of high frequency
component removed decreases.
Ringing, a characteristic of less popular ideal filters
Little edge information lost meant less blurring with increasing α

Figure: (a)Representation in the spatial domain of an ILPF of radius 5 and size
1000 x 1000. (b) Intensity proﬁle of a horizontal line passing through the center
of the image. Source: Digital Image Processing Processing(3rd Edition) by
Gonzalez, R.C. and Woods, R.E, PHI

Blurring and ringing properties can be explained though convolution
theorem
Cross section of ILPF in spatial domain bound to appear as a sinc
function(why?)
Filtering in the spatial domain by convolving h(x,y) with the image
Each pixel as a discrete impulse with strength proportional to its
intensity
Convolving a sinc function with an impulse simply copies the sinc at
the location of the impulse
Center lobe of the sinc is the principal cause for blurring

Colvolving a sinc function with every pixel in the image: a nice model
to guess the response of ILPF
Spread of sinc inversely proportional to radius of H(u,v); means for
larger D0, sinc approaches an impulse function
In the extreme case, when sinc becomes an impulse function, no
blurring upon convolution

Butterworth Lowpass Filter[2]
A butterworth low pass filter of order n and with cutoff frequency D0 from
the origin is defined as
H(u, v) =
1
1 + D(u,v)
D0
2n
(82)
Here, the terms D(u, v) and D0 have the usual meaning.

Butterworth Lowpass Filter
Figure: (a) Perspective plot of a Butterworth lowpass ﬁlter transfer function. (b)
Filter displayed as an image. (c) Filter radial cross sectionsof orders 1 through 4.
Source: Digital Image Processing Processing(3rd Edition) by Gonzalez, R.C. and
Woods, R.E, PHI

10.

Figure: Results of ﬁltering using BLPFs of order (n)=2, with cutoﬀ frequencies at
the radii shown above. Source: Digital Image Processing Processing(3rd Edition)
by Gonzalez, R.C. and Woods, R.E, PHI

Figure: Spatial representations of BLPFs of order 1,2,5 and 20, and the
corresponding intensity profiles through the center of the filters (the size in all
cases in 1000 x 1000 and the cutoff frequency is 5). Ringing increases with filter
order

Ringing imperceptible in lower orders, signiﬁcant for higher orders.
For lower orders, the ringing remains less compared to ILPF
Ringing becomes prominent and comparable for orders above 20
Order 2 most popular since it strikes a balance between ﬁltering and
ringing.

Gaussian Lowpass Filters[1]
The Gaussian lowpass ﬁlter in two dimensions is given by
H(u, v) = e
−D2(u,v)
2D2
0 (83)
Here, the terms D(u, v) and D0 have the usual meaning.
When D(u, v) = D0, the GLPF is down to 0.607 of its maximum value.

Gaussian Lowpass Filters[2]
Figure: (a) Perspective plot of a GLPF transfer function. (b) Filter displayed as
an image. (c) Filter radial cross sections for various values of D0. Source: Digital
Image Processing Processing(3rd Edition) by Gonzalez, R.C. and Woods, R.E,
PHI

Gaussian Lowpass Filter
10.

Gaussian Lowpass Filters

Image Smoothing using lowpass filter II
Figure: (a) Original Image. (b)-(f) Results of filtering using GLPFs with cutoff
frequencies at the radii show above. Source: Digital Image Processing

The inverse Fourier transform of GLPF is Gaussian[1]
A spatial Gaussian filter obtained by computing the IDFT of H(u,v)
will have no ringing[1]
A smooth transition in blurring as a function of increasing cutoff
frequency obtained
GLPF achieved slightly less smoothing than the BLPF of order 2 for
same cutoff frequency
Assures no ringing[2][3]; However, if a tight control of frequency
transition required, then a BLPF is preferred.

Sharpening Frequency Domain Filters[2] I
Highpass Filters
1 Edges and other abrupt changes associated with high frequency
components.
2 Sharpening means accentuating these high frequency features
3 Assumptions
Only zero phase shift ﬁlters
Filters are radially symmetric
All ﬁlter functions assumed to be of the size PxQ

Sharpening Frequency Domain Filters I
Ideal Highpass Filters
An ideal HPF is given by
Hhp(u, v) = 1 − Hlp(u, v) (84)
Idea? Fairly Intuitive
When the low pass ﬁlter attenuates a particular frequency, highpass ﬁlter
simply allows it.

Ideal highpass Filter
Figure: Perspective plot, image representation and cross section of a typical ideal
highpass ﬁlter. Source: Digital Image Processing (3rd Edition) by Gonzalez, R.C.
and Woods, R.R.,PHI

Ideal Highpass Filters
A 2-D highpass filter (IHPF) is defined as
H(u, v) =
0, if D(u, v) ≤ D0,
1, if D(u, v) > D0.
(85)
where D0 is the cutoff distance measured from the origin of the frequency
rectangle.
Question: Why are ideal filters not physically realizable

Characteristics of Ideal Highpass Filters
Figure: Spatial representation of a typical ideal highpass ﬁlter an corresponding
gray level proﬁles. Source: Digital Image Processing (3rd Edition) by Gonzalez,
R.C. and Woods, R.R.,PHI

Characteristics of Ideal Highpass ﬁlters
1 Same ringing characteristics[12][13]
2 Smaller lines and objects appear almost solid white
3 With increasing D0, edges become much cleaner and less distorted
and smaller objects get ﬁltered properly.

Characteristics of Ideal Highpass Filters
Figure: Results of ideal highpass ﬁltering the image with D0=15,30 and 80
respectively. Ringing[12][2] quite evident in (a) and (b). Source: Digital Image
Processing (3rd Edition) by Gonzalez, R.C. and Woods, R.R.,PHI

Butterworth Highpass Filters
The transfer function of the Butterworth highpass ﬁlter (BHPF) of order n
and with cutoﬀ frequency locus at a distance D0 from the origin is given by
H(u, v) =
1
1 + D0
D(u,v)
2n
(86)

Characteristics of Butterworth Highpass Filters
Figure: Perspective plot, image representation and cross section of a typical
Butterworth highpass ﬁlter. Source: Digital Image Processing (3rd Edition) by
Gonzalez, R.C. and Woods, R.R.,PHI

Figure: Spatial representation of a typical Butterworth highpass ﬁlter an
corresponding gray level proﬁles. Source: Digital Image Processing (3rd Edition)
by Gonzalez, R.C. and Woods, R.R.,PHI

1 Smoother than IHPFs
2 For smaller objects, performance of IHPF and low order BHPF is
almost same
3 Transition into higher cutoﬀ frequencies is much smoother with the
BHPF.

Figure: Results of highpass ﬁltering the image using a BHPF or order 2 with
D0=15, 30 and 80 respectively. The results are much smoother than those
obtained with an ILPF. Source: Digital Image Processing (3rd Edition) by

Gaussian Highpass Filters
The transfer function of the Gaussian Highpass filter (GHPF) with cutoff
frequency locus at a distance D0 from the origin is given by
H(u, v) = 1 − e
−D2(u,v)
2D2
0 (87)
Results are thus much smoother compared to Butterworth filter.

Characteristics of Gaussian Highpass Filters
Figure: Perspective plot, image representation and cross section of a typical
Gaussian highpass ﬁlter. Source: Digital Image Processing (3rd Edition) by

Characteristics of Gaussian Highpass Filters
Figure: Spatial representation of a typical Gaussian highpass ﬁlter an
corresponding gray level proﬁles. Source: Digital Image Processing (3rd Edition)
by Gonzalez, R.C. and Woods, R.R.,PHI

Characteristics of Highpass Filters
Figure: Results of highpass ﬁltering the image using a GHPF of order 2 with
D0=15,30 and 80 respectively.Source: Digital Image Processing (3rd Edition) by

Unsharp Masking and Highboost Filtering in Spatial
Domain I
Introduction[2]
1 For sharpening the images
2 Idea is to substract an unsharped version of the image from the
original image
3 Process called unsharp masking
4 In printing and publishing industry

Unsharp masking and Highboost Filtering in Spatial
Domain I
Basic steps
The process of unsharp masking involves
Blur the image (using a lowpass ﬁlter). Denote it by f −(x, y)
Subtract the blurred image from the original (this diﬀerence called the
mask)
gmask(x, y) = f (x, y) − f (x, y) (88)
Add the weighted portion of mask to the original
g(x, y) = f (x, y) + k ∗ gmask(x, y) (89)

Domain I
Summary
From,
g(x, y) = f (x, y) + k ∗ gmask(x, y) (90)
The parameter k is used for generality.
1 When k = 1, we have unsharp masking
2 When k > 1 we have highboost ﬁltering

Unsharp Masking and HIghboost Filtering in Spatial
Domain I
Illustration

Unsharp Masking and HIghboost Filtering in Spatial
Domain II
Illustration
Figure: 1-D illustration of the mechanics of unsharp masking. (a) Original Signal.
(b) Blurred signal with original shown dashed for reference. (c) Unsharp Mask.
(d)Sharpened signal obtained by by adding (c) to (a). Source: Digital Image

Unsharp Masking and Highboost Filtering in Frequency
Domain I
Introduction
From the discussion wrt to spatial domain, we have
gmask(x, y) = f (x, y) − fLP(x, y) (91)
where
fLP(x, y) = f −1
HLP(u, v)F(u, v) (92)
Thus, the modiﬁed image could be written as
g(x, y) = f (x, y) + k ∗ gmask(x, y) (93)

Domain I
Derivation
Continuing from the above discussion, we have
g(x, y) = F−1
[1 + k ∗ [1 − HLP(u, v)]]F(u, v) (94)
Expressing the same results in terms of a highpass ﬁlter, we have
g(x, y) = F−1
[1 + k ∗ HHP(u, v)]F(u, v) (95)
The term in the square brackets better known as high frequency emphasis
ﬁlter. The HPFs set the dc term to 0 but not in this case.

Homomorphic Filtering[2] I
Introduction
1 Uses illumination-reﬂectance model to improve the appearance of the
image
2 Common procedures include simultaneous intensity rane compression
and contrast enhancement

Homomorphic Filtering I
Background
From the illumination-reﬂectance model, an image f (x, y) can be
expressed as the product of illumination and reﬂectance terms.
Mathematically,
f (x, y) = i(x, y)r(x, y) (96)
However, the same cannot be subsituted with the frequency counterparts.
Why?
Solution: Go for the log

Background
We deﬁne,
z(x, y) = lnf (x, y) (97)
= lni(x, y) + lnr(x, y) (98)
Then,
F(z(x, y)) = F(lni(x, y)) + F(lnr(x, y)) (99)
Equivalently,
Z(u, v) = Fi (u, v) + Fr (u, v) (100)

Filtering
With the above transformation, we can filter Z(u, v) using a filter H(u, v)
so that the output is
S(u, v) = H(u, v)Z(u, v) (101)
= H(u, v)Fi (u, v) + H(u, v)Fr (u, v) (102)
The filtered image in the spatial domain will then be
s(x, y) = F−1
S(u, v) (103)
= F−1
H(u, v)Fi (u, v) + F−1
H(u, v)Fr (u, v) (104)

Filtering
By deﬁning,
i (x, y) = F−1
H(u, v)Fi (u, v) (105)
and
r (x, y) = F−1
H(u, v)Fr (x, y) (106)
we have
s(x, y) = i (x, y) + r (x, y) (107)
Also, by reversing the logarithm, the ﬁltered image obtained could be
g(x, y) = es(x,y)
(108)
= ei (x,y)
er (x,y)
(109)
i0(x, y)r0(x, y) (110)

Filtering
The basic steps of homomorphic ﬁltering could be represented as
Figure: Summary of steps in homomorphic ﬁltering. Source: Digital Image

Filtering
1 Applicable for homomorphic systems
2 The illumination and refectance components could be separated
3 The ﬁlter then operates on individual components
Note: Illumination components associated with slow spatial variations
while reﬂectance components are usually associated with abrupt spatial
variations.

Filtering
The above constraints are taken care by homomorphic filters. In other
words, a homomorphic filter controls the illumination and reflectance
components.
The net result is simultaneous dynamic range compression and contrast
enhancement

Periodic Noise Reduction by Frequency Domain Filtering I
Introduction
1 Freuqency domain analysis suited to noise analysis
2 Periodic noise: Burst of noise in FT
3 Selective filters to isolate noise
4 Common filters used are bandreject, bandpass and notch filters

Periodic Noise Reduction by Frequency Domain Filtering[2]
I
Bandreject Filters
1 for noise removal when the location of noise components known
2 Example: an image corrupted by additive periodic noise that can be
approximated as two-dimensional sinusoids
3 Because FT of sine consists of two imaginary impulses mirrored about
origin. Imaginary, hence, complex conjugates to one another

Bandreject Filters
Figure: From left to right, perspective plots of ideal, Butterworth (of order 1),
and Gaussian bandreject ﬁlters. Source: Digital Image Processing Processing(3rd
Edition) by Gonzalez, R.C. and Woods, R.E, PHI

Bandreject Filters
Figure: (a) Image corrupted by the sinusoid noise. (b) Spectrum of (a). (c)
Butterworth bandreject ﬁlter (white represents 1). (d) Results of ﬁltering.
(Original image courtsey of NASA)

Bandreject Filters
1 image corrupted by sinusoids
2 Noise components can be seen as symmetric dots in the FT (in this
case, on a circle)
3 Butterworth bandreject ﬁlter of order 4
4 Radius appropriate to enclose completely the noise impluses
5 Small details and textures restored successfully

Bandpass Filters
1 Opposite to bandreject ﬁlter
HBP(u, v) = 1 − HBR(u, v) (111)
2 Can sometimes remove too much image details.
3 Useful in isolating the eﬀects on an image by frequency bands.

Bandpass Filters
Figure: Noise pattern of the image obtained by bandpass ﬁltering. Source: Digital
Image Processing Processing(3rd Edition) by Gonzalez, R.C. and Woods, R.E,
PHI

Bandpass Filters
1 Most image details lost
2 Noise patterns recovered accurately
3 Thus, bandpass ﬁltering helps isolate the noise patterns.

Notch Filter
1 Rejects (or passes) frequencies in predefined neighbourhoods about a
certain frequency
2 Notch filters appear in symmetric pairs about the origin
3 Usually, they are used to pass the frequencies in the notch area
4 Mathematically, notchpass and notchreject filters are related as
HNP(u, v) = 1 − HNR(u, v) (112)

Notch ﬁlters
Figure: Perspective plots of (a) ideal, (b)Butterworth (order 2), (c) Gaussian
notch ﬁlters. Source: Digital Image Processing Processing(3rd Edition) by
Gonzalez, R.C. and Woods, R.E, PHI

Inverse Filtering[2] I
Introduction
1 First step towards image restoration
2 We assume the degrading function to be H
3 Here, we an estimate of the transform simply by dividing the
transform of the degraded image G(u, v), by the degradation function
ˆF(u, v) =
G(u, v)
H(u, v)
(113)

Inverse Filtering I
Introduction
The previous equation can also be written as
ˆF(u, v) = F(u, v) +
N(u, v)
H(u, v)
(114)
In the above equation, N(u, v) is unknown.
Consequence: Even if we know the degraation function, we cannot recover
the undegraded image.
To add to this, if H(u, v) is small, then it cannot virtually dominate the
value of ˆF(u, v)

Inverse Filtering I
Introduction
Possible Solution
Limit the ﬁlter frequencies near the origin since H(0, 0) is highest near the
origin.

References I
1 http://nptel.ac.in/courses/111103021/15
2 Digital Image Processing (3rd Edition) by Gonzalez, R.C. and Woods,
R.R.,PHI
3 http://web.stanford.edu/class/ee104/lecture4.pdf
4 Digital Image Processing (3rd Edition) by Willian k. Pratt, John
Wiley and Sons
5 MIT OpenCourseWare
http://math.mit.edu/~gs/cse/websections/cse41.pdf
6 https://en.wikipedia.org/wiki/Dirichlet_conditions
7 Web Tutorialshttps://6002x.mitx.mit.edu/
8 Stanford University Text
web.stanford.edu/class/ee102/lectures/fourtran

References II
9 Nptel Tutorials(IIT Madras)
http://nptel.ac.in/courses/IIT-MADRAS/Principles_Of_
Communication/pdf/Lecture05_FTProperties.pdf
10 Princeton University Courseware https://www.princeton.edu/
~cuff/ele201/kulkarni_text/frequency.pdf
11 Web Tutorials
http://users.wfu.edu/matthews/misc/DigPhotog/alias/
12 Web Resources imaging.cs.msu.ru/en/research/ringing
13 M. Khambete and M. Joshi, ”Blur and Ringing Artifact Measurement
in Image Compression using Wavelet Transform ”, World Academy of
Science, Engineering and Technology , 2007.

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