Image Enhancement using Frequency Domain Filters

By -
M.Phil (CSE)
M.S University.

 Image Enhancement
 Spatial Domain & Frequency Domain
 Basic Steps in frequency domain
 Smoothing
 Sharpening

Image enhancement is the process of making images more
useful
The reasons for doing this include:
– Highlighting interesting detail in images
– Removing noise from images
– Making images more visually appealing

There are two broad categories of image enhancement
techniques
– Spatial domain techniques
– Direct manipulation of image pixels
– Frequency domain techniques
– Manipulation of Fourier transform or wavelet transform of an image

1. Multiply the input image by (-1)x+yto center the
transform to u = M/2 and v = N/2 (if M and N are even
numbers, then the shifted coordinates will be integers)
2. Computer F(u,v), the DFT of the image from (1)
3. Multiply F(u,v) by a filter function H(u,v)
4. Compute the inverse DFT of the result in (3)
5. Obtain the real part of the result in (4)
6. Multiply the result in (5) by (-1)x+yto cancel the
multiplication of the input image.

The relationship between blurring mask and derivative mask
with a high pass filter and low pass filter can be defined
simply as.
– Blurring masks are also called as low pass filter
– Derivative masks are also called as high pass filter

A blurring mask has the following properties.
– All the values in blurring masks are positive
– The sum of all the values is equal to 1
– The edge content is reduced by using a blurring mask
– As the size of the mask grow, more smoothing effect will take
place

A derivative mask has the following properties.
– A derivative mask have positive and as well as negative values
– The sum of all the values in a derivative mask is equal to zero
– The edge content is increased by a derivative mask
– As the size of the mask grows , more edge content is increased

S.No Smoothing Sharpening
1 Ideal Lowpass Ideal Highpass
2 Butterworth Lowpass Butterworth Highpass
3 Gaussian Lowpass Gaussian Highpass

 Smoothing(blurring) is achieved in the frequency domain by high-
frequency attenuation; that is, by lowpass filtering.
 Here, we consider 3 types of lowpass filters:
 Ideal lowpassfilters
 Butterworth lowpassfilters
 Gaussian lowpassfilters
 These three categories cover the range from very sharp(ideal), to very
smooth(Gaussian) filtering.

 The Butterworth filter has a parameter called the
filter order.
 For high order values, the Butterworth filter
approaches the ideal filter. For low order values,
Butterworth filter is more like a Gaussian filter.
 Thus, the Butterworth filter may be viewed as
providing a transition between two “extremes”.

 The most basic of filtering operations is called “lowpass”.
 A lowpass filter is also called a “blurring” or smoothing filter.
 The simplest lowpass filter just calculates the average of a pixel
and all of its eight immediate neighbours.
 Lowpass is also called as blurring mask

 A 2-D low pass filter that passes without attenuation all
frequencies within a circle of radius D0 from the origin and
“cuts off” all frequencies outside this circle is called an ideal
lowpass filter(ILPF); it is specified by the function.
 D0 is a positive constant and D(u,v) is the distance between
a point (u,v) in the frequency domain and the center of the
frequency rectangle; that is,
D(u, v)  [(u  P / 2)2
 (v  Q / 2)2
]1/ 2

 The ideal lowpass filter is radially symmetric about the
origin, which means that the filter is completely defined by
a radial cross section.
 Rotating the cross section by 360° yields the filter in 2-D.
 For an ILPF cross section, the point of transition between
H(u,v)=1 and H(u,v)=0 is called the cutoff frequency D0.
 Simply cut off all high frequency components that are at a
specified distance D0 from the origin of the transform,
changing the distance changes the behaviour of the filter.

 The Butterworth lowpass filter is a type of signal processing
filterdesigned to have as flat a frequency response as possible in
the passband.
 The transfer function of a Butterworth lowpass filter of order n
with cutoff frequency at distance D0 from the origin is defined
as:

A)Perspective plot of an Butterworth lowpass filter transfer function
B)Filter displayed as an image
C)Filter radius cross section of orders 1 through 4

The transfer function of a Gaussian lowpass filter is defined as:
(u ,v) / 2
H (u, v)  e D
A)Perspective plot of a GLPF transfer function B)Filter displayed as an image C)Filter radius cross section for various values of D0

 Main advantage of a Gaussian LPF over a Butterworth
LPF is that we are assured that there will be no ringing
effects no matter what filter order we choose to work with.

Different Low pass Gaussian filters used to remove blemishes in
the image

Edges and fine detail characterized by sharp transitions in
image intensity
Such transitions contribute significantly to high frequency
components of Fourier transform
Intuitively, attenuating certain low frequency components and
preserving high frequency components result in sharpening

– Intended goal is to do the reverse operation of low-pass
filters
– When low-pass filer attenuates frequencies, high-pass filter
passes them
Hhp (u,v) 1 Hlp (u,v)
– When high-pass filter attenuates frequencies, low-pass filter
passes them

frequency
and Low
frequency
– High pass
components
pass
components
– the low pass
components
frequency
denotes
smooth regions.

This is a common example
of high pass filter.
When 0 is placed
inside, we get edges, which
gives us a sketched image.
An ideal low pass filter in
frequency domain is given
below.

The Butterworth high pass filter is given as:
where n is the order and D0is the cut off distance as before
0
1
1[D / D(u,v)]2n
H (u, v) 

Results of
Butterworth
high pass
filtering of
order 2 with
0D = 15
Results of
Butterworth
high pass
filtering of
order 2 with
filtering of order 2 with D0 = 30
0D = 80
Results of Butterworth high pass
10/25/16

– Gaussian high pass filter has the same concept as ideal high
pass filter, but again the transition is more smooth as
compared to the ideal one.

The Laplacian is defined as:
(dot product)
Approximate
derivatives:

Laplacian Mask
detect zero-crossings

 The aim of image enhancement is to improve the information in
images for human viewers, or to provide `better' input forother
automated image processing techniques.
 There is no general theory for determining what is `good' image
enhancement when it comes to human perception. If it looks good, it
is good!

Image Enhancement using Frequency Domain Filters

Image Enhancement using Frequency Domain Filters

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