1

Digital Image Processing

Lecture # 5 Spatial & Frequency Domain Image Enhancement

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

• Let’s consider a simple 1 dimensional

example

Spatial Differentiation

A B

1st Derivative

The 1st derivative of a function is given by:

Its just the difference between subsequent

values and measures the rate of change of

the function

)()1( xfxfx

f

Image Strip

0

1

2

3

4

5

6

7

8

1st Derivative

-8

-6

-4

-2

0

2

4

6

8

5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7

-1 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0

1st Derivative

2nd Derivative

The 2nd derivative of a function is given by:

Simply takes into account the values both before and after the current value

)(2)1()1(2

2

xfxfxfx

f

Image Strip

0

1

2

3

4

5

6

7

8

5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7

2nd Derivative

2nd Derivative

-15

-10

-5

0

5

10

-1 0 0 0 0 1 0 6 -12 6 0 0 1 1 -4 1 1 0 0 7 -7 0 0

2nd Derivative for Image Enhancement

The 2nd derivative is more useful for image enhancement

than the 1st derivative - Stronger response to fine detail

We will come back to the 1st order derivative later on

The first sharpening filter we will look at is the Laplacian

Laplacian Filter

2

2

2

22

y

f

x

ff

),(2),1(),1(2

2

yxfyxfyxfx

f

The Laplacian is defined as follows:

),(2)1,()1,(2

2

yxfyxfyxfy

f

Laplacian Filter

So, the Laplacian can be given as follows:

),1(),1([2 yxfyxff

)]1,()1,( yxfyxf

),(4 yxf

0 1 0

1 -4 1

0 1 0

Can we implement it using a filter/ mask?

Laplacian Filter

Laplacian Filter

Applying the Laplacian to an image we get a

new image that highlights edges and other

discontinuities

Original

Image

Laplacian

Filtered Image

Laplacian

Filtered Image

Scaled for Display

Laplacian Image Enhancement

The result of a Laplacian filtering is not an

enhanced image

Laplacian

Filtered Image

Scaled for Display 2

5

2

5

( , ) , 0( , )

( , ) , 0

f x y f wg x y

f x y f w

To generate the final enhanced image

Laplacian Image Enhancement

In the final sharpened image edges and fine

detail are much more obvious

- =

Original

Image

Laplacian

Filtered Image

Sharpened

Image

Laplacian Image Enhancement

Simplified Image Enhancement

• The entire enhancement can be combined

into a single filtering operation

),1(),1([),( yxfyxfyxf

)1,()1,( yxfyxf

)],(4 yxf

fyxfyxg 2),(),(

Simplified Image Enhancement

• The entire enhancement can be combined

into a single filtering operation

fyxfyxg 2),(),(

),1(),1(),(5 yxfyxfyxf

)1,()1,( yxfyxf0 -1 0

-1 5 -1

0 -1 0

Simplified Image Enhancement

• 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

Unsharp Masking

Use of first derivatives for image enhancement: The Gradient

• The gradient of a function f(x,y) is defined as

y

fx

f

G

G

y

xf

-1 -2 -1

0 0 0

1 2 1

-1 0 1

-2 0 2

-1 0 1

Extract horizontal edges

7 8 9 1 2 3

3 6 9 1 4 7

( 2 ) ( 2 )

( 2 ) ( 2 )

f z z z z z z

z z z z z z

Emphasize more the current point

(y direction)

Emphasize more the current point (x

direction) Pixel Arrangement

Gradient Operators

Extract vertical edges

Sobel Operator

Sobel Operator: Example

Sobel filters are typically used for edge

detection

An image of a

contact lens

which is

enhanced in

order to make

defects more

obvious

Combining Spatial Enhancement Methods

Successful image enhancement is

typically not achieved using a single

operation

Rather we combine a range of

techniques in order to achieve a final

result

This example will focus on enhancing

the bone scan

Laplacian filter of

bone scan (a)

Sharpened version of

bone scan achieved

by subtracting (a)

and (b) Sobel filter of bone

scan (a)

(a)

(b)

(c)

(d)

Combining Spatial Enhancement Methods

Combining Spatial Enhancement Methods

The product of (c)

and (e) which will be

used as a mask

Sharpened image

which is sum of (a)

and (f)

Result of applying a

power-law trans. to

(g)

(e)

(f)

(g)

(h)

Image (d) smoothed with

a 5*5 averaging filter

Combining Spatial Enhancement Methods

Compare the original and final images

Image Enhancement in Frequency Domain

31

Joseph Fourier (1768 – 1830)

– Most famous for his work “La Théorie Analitique de la Chaleur” published in 1822

– Translated into English in 1878: “The Analytic Theory of Heat”

Nobody paid much attention when the work was first published One of the most important mathematical theories in modern engineering

32

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

33

The big idea…

Approximating a square wave as the sum of sine waves

36

Frequencies in Images

37

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:

for u = 0, 1, 2…M-1 and v = 0, 1, 2…N-1.

1

0

1

0

)//(2),(),(M

x

N

y

NvyMuxjeyxfvuF

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DFT & Images

DFT

Scanning electron microscope

image of an integrated circuit

magnified ~2500 times

Fourier spectrum of the image

39

40

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The Inverse DFT

It is really important to note that the Fourier transform is completely reversible

The inverse DFT is given by:

for x = 0, 1, 2…M-1 and y = 0, 1, 2…N-1

1

0

1

0

)//(2),(1

),(M

u

N

v

NvyMuxjevuFMN

yxf

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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

43

Some Basic Frequency Domain Filters

Low Pass Filter

High Pass Filter

44

Ideal Low Pass Filter

Simply cut off all high frequency components that are a specified distance D0 from the origin

46

Ideal Low Pass Filter (cont…)

Above we show an image, it’s Fourier spectrum and a series of ideal low pass filters of radius 5, 15, 30, 80 and 230 superimposed on top of it

47

Ideal Low Pass Filter (cont…)

48

Ideal Low Pass Filter (cont…)

49

Original

image

Result of filtering

with ideal low

pass filter of

radius 5

Result of filtering

with ideal low

pass filter of

radius 30

Result of filtering

with ideal low

pass filter of

radius 230

Result of filtering

with ideal low

pass filter of

radius 80

Result of filtering

with ideal low

pass filter of

radius 15

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Ideal High Pass Filters

The ideal high pass filter is given as:

61

Results of ideal

high pass filtering

with D0 = 15

Results of ideal

high pass filtering

with D0 = 30

Results of ideal

high pass filtering

with D0 = 80

Readings from Book (3rd Edn.)

• Sharpening Filters • Frequency Analysis • Filters in Frequency Domain

69

Acknowledgements

Digital Image Processing”, Rafael C. Gonzalez & Richard E. Woods, Addison-Wesley, 2002

Peters, Richard Alan, II, Lectures on Image Processing, Vanderbilt University, Nashville, TN, April 2008

Brian Mac Namee, Digitial Image Processing, School of Computing, Dublin Institute of Technology

Computer Vision for Computer Graphics, Mark Borg

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