ImageProcessing-SpatialFiltering

7/30/2019 ImageProcessing-SpatialFiltering

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Digital Image Processing

Image Enhancement-Spatial Filtering

From:Digital Image Processing, Chapter 3


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Contents

Next, we will look at spatial filteringtechniques:

What is spatial filtering? Smoothing Spatial filters. Sharpening Spatial Filters.

Combining Spatial Enhancement Methods


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Neighbourhood Operations

Neighbourhood operations simply operateon a larger neighbourhood of pixels thanpoint operations

Neighbourhoods aremostly a rectanglearound a central pixel

Any size rectangleand any shape filter are possible

Origin x

y Image f (x, y)

(x, y)Neighbourhood


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Neighbourhood Operations

For each pixel in the origin image, theoutcome is written on the same location atthe target image.

Origin x

y Image f (x, y)

(x, y)Neighbourhood

Target

Origin


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The Spatial Filtering Process

j k l

m n o

p q r

Origin x

y Image f (x, y)

e processed = n*e + j*a + k *b + l *c +

m*d + o*f + p*g + q*h + r *i

Filter ( w )Simple 3*3

Neighbourhood

e

3*3 Filter

a b c

d e f

g h i

Original ImagePixels

*

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


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Spatial Filtering: Equation Form

a

a s

b

bt

t y s x f t sw y x g ),(),(),(

Filtering can be givenin equation form asshown aboveNotations are basedon the image shownto the left

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i g i t a l I m a g e

P r o c e s s

i n g

( 2 0 0 2 )


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Smoothing Spatial Filters

One of the simplest spatial filteringoperations we can perform is a smoothingoperation

Simply average all of the pixels in aneighbourhood around a central value

Especially usefulin removing noisefrom images

Also useful for highlighting gross

detail

1/9 1/9 1/9

1/9 1/9 1/9

1/9 1/9 1/9

Simpleaveragingfilter


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Smoothing Spatial Filtering

1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9

Origin x

y Image f (x, y)

e = 1 / 9 *106 +1 / 9 *104 +

1 / 9 *100 +1 / 9 *108 +

1

/ 9 *99 +1

/ 9 *98 +1 / 9 *95 +

1 / 9 *90 +1 / 9 *85

= 98.3333

Filter Simple 3*3Neighbourhood

106

104

99

95

100 108

98

90 85

1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9

3*3 Smoothing Filter

104 100 108

99 106 98

95 90 85


*

The above is repeated for every pixel in theoriginal image to generate the smoothed image


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Image Smoothing Example

The image at the top leftis an original image of size 500*500 pixels

The subsequent imagesshow the image after filtering with an averaging

filter of increasing sizes 3, 5, 9, 15 and 35Notice how detail begins

to disappear I m a g e s

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P r o c e s s

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( 2 0 0 2 )


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I m a g e s

t a k e n

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G o n z a

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d s , D


P r o c e s s

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( 2 0 0 2 )


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I m a g e s

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G o n z a

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d s , D


P r o c e s s

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( 2 0 0 2 )


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I m a g e s

t a k e n

f r o m

G o n z a

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& W o o

d s , D


P r o c e s s

i n g

( 2 0 0 2 )


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I m a g e s

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f r o m

G o n z a

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& W o o

d s , D


P r o c e s s

i n g

( 2 0 0 2 )


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I m a g e s

t a k e n

f r o m

G o n z a

l e z

& W o o

d s , D


P r o c e s s

i n g

( 2 0 0 2 )


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I m a g e s

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f r o m

G o n z a

l e z

& W o o

d s , D


P r o c e s s

i n g

( 2 0 0 2 )


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Weighted Smoothing Filters

More effective smoothing filters can begenerated by allowing different pixels in theneighbourhood different weights in the

averaging function Pixels closer to the

central pixel are moreimportant

Often referred to as aweighted averaging

1/16 2/16 1/16

2/16 4/16 2/16

1/16 2/16 1/16

Weighted

averaging filter


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Another Smoothing Example

By smoothing the original image we get ridof lots of the finer detail which leaves onlythe gross features for thresholding

I m a g e s

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Original Image Smoothed Image Thresholded Image

* Image taken from Hubble Space Telescope


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Averaging Filter Vs. Median Filter Example

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Original


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I m a g e s

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P r o c e s s

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AveragingFilter


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I m a g e s

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P r o c e s s

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( 2 0 0 2 )

MedianFilter


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Strange Things Happen At The Edges!

Origin x

y Image f (x, y)

e

e

e

e

At the edges of an image we are missingpixels to form a neighbourhood

e e

e


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Strange Things Happen At The Edges!(cont)

There are a few approaches to dealing withmissing edge pixels:

Omit missing pixels

Only works with some filters Can add extra code and slow down processing

Pad the image Typically with either all white or all black pixels

Replicate border pixels Truncate the image


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Correlation & Convolution

The filtering we have been talking about sofar is referred to as correlation with the filter itself referred to as the correlation kernel

Convolution is a similar operation, with justone subtle difference

For symmetric filters it makes no difference

e processed = v*e +

z *a + y*b + x*c +w*d + u*e +t *f + s*g + r *h

r s t

u v w x y z

Filter

a b c

d e e f g h


*


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Sharpening Spatial Filters

Previously we have looked at smoothingfilters which remove fine detailSharpening spatial filters seek to highlight

fine detail Remove blurring from images Highlight edges

Sharpening filters are based on spatial differentiation


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Spatial Differentiation

Differentiation measures the rate of change of a functionLets consider a simple 1 dimensional

example

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Spatial Differentiation

I m a g e s

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P r o c e s s

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A B


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1 st Derivative

The formula for the 1 st derivative of afunction is as follows:

Its just the difference between subsequentvalues and measures the rate of change of

the function

)()1( x f x f x

f


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1 st Derivative (cont)

01

2

3

4

5

6

7

8

-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

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

f(x)

f(x)


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2nd Derivative

The formula for the 2 nd derivative of afunction is as follows:

Simply takes into account the values bothbefore and after the current value

)(2)1()1(2

2

x f x f x f x

f


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2nd Derivative (cont)

01

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

-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

f(x)

f(x)


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1 st and 2 nd Derivative

0

1

2

3

4

5

6

78

-8

-6

-4

-2

0

2

4

6

8

-15

-10

-5

0

5

10

f(x)

f(x)

f(x)

U i S d D i i F I


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Using Second Derivatives For ImageEnhancement

The 2nd

derivative is more useful for imageenhancement than the 1 st derivative Stronger response to fine detail Simpler implementation We will come back to the 1 st order derivative

later onThe first sharpening filter we will look at is

the Laplacian Isotropic One of the simplest sharpening filters We will look at a digital implementation


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The Laplacian

The Laplacian is defined as follows:

where the partial 1 st order derivative in the x direction is defined as follows:

and in the y direction as follows:

y f

x f

f 22

2

22

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

2

y x f y x f y x f x

f

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

2

y x f y x f y x f y

f


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The Laplacian (cont)

So, the Laplacian can be given as follows:

We can easily build a filter based on this

),1(),1([2 y x f y x f f )]1,()1,( y x f y x f

),(4 y x f

0 1 0

1 -4 1

0 1 0

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The Laplacian (cont)

Applying the Laplacian to an image we get anew image that highlights edges and other discontinuities

I m a g e s

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P r o c e s s

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OriginalImage

LaplacianFiltered Image


Scaled for Display

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But That Is Not Very Enhanced!

The result of a Laplacian filteringis not an enhanced imageWe have to do more work in

order to get our final imageSubtract the Laplacian resultfrom the original image to

generate our final sharpenedenhanced image


Scaled for Display

I m

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f y x f y x g 2),(),(

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Laplacian Image Enhancement

In the final sharpened image edges and finedetail are much more obvious

I m

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- =

OriginalImage


SharpenedImage

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Laplacian Image Enhancement

I m

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Simplified Image Enhancement

The entire enhancement can be combinedinto a single filtering operation

),1(),1([),( y x f y x f y x f )1,()1,( y x f y x f

)],(4 y x f

f y x f y x g 2),(),(

),1(),1(),(5 y x f y x f y x f )1,()1,( y x f y x f

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Simplified Image Enhancement (cont)

This gives us a new filter which does thewhole job for us in one step

0 -1 0

-1 5 -1

0 -1 0

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Simplified Image Enhancement (cont)

I m

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P r o c e s s

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( 2 0 0 2 )


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Unsharp Mask & Highboost Filtering

Using sequence of linear spatial filters inorder to get Sharpening effect.

-Blur - Subtract from original image- add resulting mask to original image

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Highboost Filtering

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1 st Derivative Filtering

Implementing 1 st derivative filters is difficult inpracticeFor a function f(x, y) the gradient of f at

coordinates (x, y) is given as the columnvector:

y f x

f

G

G

y

xf

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1 st Derivative Filtering (cont)

The magnitude of this vector is given by:

For practical reasons this can be simplified as:

)f (mag f

2122 y x GG2

122

y f

x f

y x GG f

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1 st Derivative Filtering (cont)

There is some debate as to how best tocalculate these gradients but we will use:

which is based on these coordinates

321987 22 z z z z z z f

741963 22 z z z z z z

z1 z2 z3z4 z5 z6

z7 z8 z9

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Sobel Operators

Based on the previous equations we canderive the Sobel Operators

To filter an image it is filtered using bothoperators the results of which are addedtogether

-1 -2 -1

0 0 0

1 2 1

-1 0 1

-2 0 2

-1 0 1

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Sobel Example

Sobel filters are typically used for edgedetection

I m

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( 2 0 0 2 ) An image of a

contact lens whichis enhanced inorder to makedefects (at four and five oclock in

the image) moreobvious

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1 st & 2nd Derivatives

Comparing the 1 st and 2 nd derivatives wecan conclude the following:

1 st order derivatives generally produce thicker edges

2nd order derivatives have a stronger response to fine detail e.g. thin lines

1 st order derivatives have stronger responseto grey level step

2nd order derivatives produce a doubleresponse at step changes in grey level

53 Combining Spatial Enhancement


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Combining Spatial EnhancementMethods

Successful imageenhancement is typically notachieved using a single

operationRather we combine a rangeof techniques in order toachieve a final resultThis example will focus onenhancing the bone scan tothe right

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Combining Spatial EnhancementMethods (cont)

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Laplacian filter of bone scan (a)

Sharpened version of bone scan achievedby subtracting (a)and (b) Sobel filter of bone

scan (a)

(a)

(b)

(c)

(d)


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Combining Spatial EnhancementMethods (cont)

Compare the original and final images

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ImageProcessing-SpatialFiltering

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