Digital image processing img smoothning

Presentation over image smoothing
and sharpening

-Vinay Kumar Gupta
0700410088

What is a Digital Image?
 A digital image is a representation of a two-
dimensional image as a finite set of digital values,
called picture elements or pixels.

Why Digital Image Processing?
Digital image processing focuses on two major tasks:

 Improvement of pictorial information for human
interpretation.

 Processing of image data for storage, transmission and
representation for autonomous machine perception

Applications:
The use of digital image processing techniques has
exploded and they are now used for all kinds of tasks in
all kinds of areas:
 Image enhancement/restoration
 Artistic effects
 Medical visualisation
 Industrial inspection
 Law enforcement
 Human computer interfaces

Examples: The Hubble Telescope
Launched in 1990 the Hubble telescope can take
images of very distant objects
However, an incorrect mirror made many of Hubble’s
images useless.
Image processing techniques were used to fix this

Examples: HCI
Try to make human computer
interfaces more natural
 Face recognition
 Gesture recognition
Does anyone remember the
user interface from “Minority
Report”?

Key Stages in Digital Image Processing:
Image Morphological
Restoration Processing

Image
Enhancement Segmentation

Image Object
Acquisition Recognition

Problem Domain Representation
& Description
Colour Image Image
Processing Compression

Image Representation
Before we discuss image acquisition
recall that a digital image is
col
composed of M rows
and N columns of pixels
each storing a value

Pixel values are most
often grey levels in the
range 0-255(black-white)
f (row, col)
row

What Is Image Enhancement?
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

Spatial & Frequency Domains
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
For the moment we will concentrate on techniques that
operate in the spatial domain

Image Histograms
The histogram of an image shows us the distribution of grey
levels in the image
Frequencies

Grey Levels

Spatial filtering techniques:
 Neighbourhood operations
 What is spatial filtering?
 Smoothing operations
 What happens at the edges?

Neighbourhood Operations
Neighbourhood operations simply operate on a larger
neighbourhood of pixels than point operations
Origin x
Neighbourhoods are
mostly a rectangle
around a central pixel
Any size rectangle
and any shape filter (x, y)
Neighbourhood
are possible

y Image f (x, y)

Simple Neighbourhood Operations
Some simple neighbourhood operations include:
 Min:
Set the pixel value to the minimum in the neighbourhood
 Max:
Set the pixel value to the maximum in the neighbourhood
 Median:
The median value of a set of numbers is the midpoint value
in that set (e.g. from the set [1, 7, 15, 18, 24] 15 is the median).
Sometimes the median works better than the average

The Spatial Filtering Process
Origin x
a b c r s t
d
g
e
h
f
i
* u
x
v
y
w
z
Original Image Filter
Simple 3*3 Pixels
e 3*3 Filter
Neighbourhood
eprocessed = v*e +
r*a + s*b + t*c +
u*d + w*f +
y Image f (x, y) x*g + y*h + z*i

The above is repeated for every pixel in the
original image to generate the filtered image

Smoothing Spatial Filters
 One of the simplest spatial filtering operations we
can perform is a smoothing operation
 Simply average all of the pixels in a neighbourhood
around a central value
 Especially useful 1/ 1/ 1/
9 9 9
in removing noise
from images
 Also useful for
1/
9
1/
9
1/
9 Simple
highlighting gross averaging
detail 1/ 1/ 1/ filter
9 9 9

Smoothing Spatial Filtering
Origin x
104 100 108 1/ 1/ 1/
9 9 9
1/ 1/ 1/
99 106 98

95 90 85
* 1/
9
1/
9
1/
9

9 9 9

1/ 1/ 1/
104 100 108
Original Image Filter
9 9 9
Simple 3*3 1/
99 106 198
1/ /9
3*3 Smoothing Pixels
9 9
Neighbourhood 195
/9 1/
90 185
/9
Filter
9

e = 1/9*106 +
1/ *104 + 1/ *100 + 1/ *108 +
9 9 9
1/ *99 + 1/ *98 +
9 9
y Image f (x, y) 1/ *95 + 1/ *90 + 1/ *85
9 9 9
= 98.3333
The above is repeated for every pixel in the
original image to generate the smoothed image

Sharpening Spatial Filters
Previously we have looked at smoothing filters which
remove fine detail
Sharpening spatial filters seek to highlight fine detail
 Remove blurring from images
 Highlight edges
Sharpening filters are based on spatial differentiation

Spatial Differentiation
Differentiation measures the rate of change of a function
Let’s consider a simple 1 dimensional example

Spatial Differentiation

A B

1st Derivative
The formula for the 1st derivative of a function is as
follows:
f
f ( x 1) f ( x)
x
It’s just the difference between subsequent values
and measures the rate of change of the function.

2nd Derivative
The formula for the 2nd derivative of a function is as
follows: 2
f
2
f ( x 1) f ( x 1) 2 f ( x)
x
Simply takes into account the values both before and
after the current value

Using Second Derivatives For Image
Enhancement
The 2nd derivative is more useful for image enhancement
than the 1st derivative
 Stronger response to fine detail
 Simpler implementation
 We will come back to the 1st order derivative later on
The first sharpening filter we will look at is the Laplacian
 Isotropic
 One of the simplest sharpening filters
 We will look at a digital implementation

The Laplacian
The Laplacian is defined as follows:
2 2
2 f f
f 2 2
x y
where the partial 1st order derivative in the x direction is
defined as follows:
2
f
2
f ( x 1, y ) f ( x 1, y ) 2 f ( x, y )
x
and in the y direction as follows:
2
f
2
f ( x, y 1) f ( x, y 1) 2 f ( x, y )
y

The Laplacian (cont…)
So, the Laplacian can be given as follows:
2
f [ f ( x 1, y ) f ( x 1, y )
f ( x, y 1) f ( x, y 1)]
4 f ( x, y)
We can easily build a filter based on this

0 1 0

1 -4 1

0 1 0

The Laplacian (cont…)
Applying the Laplacian to an image we get a new image
that highlights edges and other discontinuities

Original Laplacian Laplacian
Image Filtered Image Filtered Image
Scaled for Display

But That Is Not Very Enhanced!
The result of a Laplacian filtering is not an
enhanced image
We have to do more work in order to get
our final image
Subtract the Laplacian result from the
original image to generate our final
sharpened enhanced image
Laplacian
Filtered Image
2 Scaled for Display
g ( x, y ) f ( x, y ) f

Laplacian Image Enhancement

- =

Original Laplacian Sharpened
Image Filtered Image Image

In the final sharpened image edges and fine detail are
much more obvious

Simplified Image Enhancement
The entire enhancement can be combined into a single
filtering operation
2
g ( x, y ) f ( x, y ) f
f ( x, y) [ f ( x 1, y) f ( x 1, y)
f ( x, y 1) f ( x, y 1)
4 f ( x, y)]
5 f ( x, y) f ( x 1, y) f ( x 1, y)
f ( x, y 1) f ( x, y 1)

Simplified Image Enhancement (cont…)
This gives us a new filter which does the whole job for us
in one step

0 -1 0

-1 5 -1

0 -1 0

The Big Idea

=

Any function that periodically repeats itself can
be expressed as a sum of sines and cosines of
different frequencies each multiplied by a
different coefficient – a Fourier series

The Discrete Fourier Transform (DFT)
The Discrete Fourier Transform of f(x, y), for x = 0, 1,
2…M-1 and y = 0,1,2…N-1, denoted by F(u, v), is given by
the equation:
M 1N 1
j 2 ( ux / M vy / N )
F (u , v) f ( x, y )e
x 0 y 0
for u = 0, 1, 2…M-1 and v = 0, 1, 2…N-1.

DFT & Images
The DFT of a two dimensional image can be visualised
by showing the spectrum of the images component
frequencies

DFT

The DFT and Image Processing
To filter an image in the frequency domain:
1. Compute F(u,v) the DFT of the image
2. Multiply F(u,v) by a filter function H(u,v)
3. Compute the inverse DFT of the result

Some Basic Frequency Domain Filters
Low Pass Filter

High Pass Filter

Smoothing Frequency Domain Filters
Smoothing is achieved in the frequency domain by
dropping out the high frequency components
The basic model for filtering is:
G(u,v) = H(u,v)F(u,v)
where F(u,v) is the Fourier transform of the image being
filtered and H(u,v) is the filter transform function
Low pass filters – only pass the low frequencies,
drop the high ones.

Ideal Low Pass Filter
Simply cut off all high frequency components that are a
specified distance D0 from the origin of the transform

changing the distance changes the behaviour of the filter

Ideal Low Pass Filter (cont…)
The transfer function for the ideal low pass filter can be
given as:
1 if D(u, v) D0
H (u, v)
0 if D(u, v) D0

where D(u,v) is given as:
2 2 1/ 2
D(u, v) [(u M / 2) (v N / 2) ]

Butterworth Low pass Filters
The transfer function of a Butterworth lowpass filter of
order n with cutoff frequency at distance D0 from the
origin is defined as:
1
H (u , v)
1 [ D(u , v) / D0 ]2 n

Gaussian Low pass Filters
The transfer function of a Gaussian lowpass filter is
defined as:
D2 (u ,v ) / 2 D0 2
H (u, v) e

Lowpass Filtering Examples
A low pass Gaussian filter is used to connect broken text

Sharpening in the Frequency Domain
Edges and fine detail in images are associated with high
frequency components
High pass filters – only pass the high frequencies,
drop the low ones.
High pass frequencies are precisely the reverse of low
pass filters, so:
Hhp(u, v) = 1 – Hlp(u, v)

Ideal High Pass Filters
The ideal high pass filter is given as:
0 if D(u, v) D0
H (u, v)
1 if D(u, v) D0
where D0 is the cut off distance as before

Butterworth High Pass Filters
The Butterworth high pass filter is given as:
1
H (u , v) 2n
1 [ D0 / D(u , v)]

where n is the order and D0 is the cut off distance as
before

Gaussian High Pass Filters
The Gaussian high pass filter is given as: 2
D2 (u ,v ) / 2 D0
H (u, v) 1 e
where D0 is the cut off distance as before

Frequency Domain Filtering & Spatial
Domain Filtering
Similar jobs can be done in the spatial and frequency
domains
Filtering in the spatial domain can be easier to
understand
Filtering in the frequency domain can be much faster –
especially for large images

Digital image processing img smoothning

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