image processing 6-spatialfiltering2.ppt

8/9/2019 Image Processing 6-SpatialFiltering2.ppt

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Course Website: http://www.comp.dit.ie/bmacnamee

Digital Image Processing

Image Enhancement(Spatial Filtering !

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#

o"

#$Spatial Filtering +e"resher

r s t

u v w

x y z

Origin x

y Image f (x, y)

e processed = v*e +

r *a + s*b + t *c +

u*d + w*f + x*g + y*h + z *i

Filter Simple 3*3

Neighbourhood

e

3*3 Filter

a b c

d e f

g h i

Original ImagePixels

*

,he abo*e is repeated "or e*er- piel in the

original image to generate the smoothed image


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o"

#$Sharpening Spatial Filters

Pre*iousl- we ha*e loo%ed at smoothing"ilters which remo*e "ine detail

Sharpening spatial filters see% to highlight

"ine detail ' +emo*e blurring "rom images

' 0ighlight edges

Sharpening "ilters are based on spatialdifferentiation


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1

o"

#$Spatial Di""erentiation

Di""erentiation measures the rate of change o"a "unction

2et3s consider a simple ) dimensional

eample

I m a

g e s t a % e n " r o m 4 o n 5 a l e 5 6 W o o d s 7

D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !


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8

o"

#$Spatial Di""erentiation

I m a g e s t a % e n " r o m 4 o n 5 a l e 5 6 W o o d s 7

D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !

9


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;

o"

#$)st Deri*ati*e

,he "ormula "or the )st deri*ati*e o" a"unction is as "ollows:

It3s


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=

o"

#$)st Deri*ati*e (cont>!

1 1 # ) $ $ $ 8 $ $ $ $ ) # ) $ $ $ $ ; ; ; ;

$ ?) ?) ?) ?) $ $ 8 ?8 $ $ $ ) ? ?) $ $ $ ; $ $ $


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@

o"

#$nd Deri*ati*e

,he "ormula "or the nd deri*ati*e o" a"unction is as "ollows:

Simpl- ta%es into account the *alues both

be"ore and a"ter the current *alue

)(2)1()1(2

2

x f x f x f x

f −−++=

∂

∂


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)$

o"

#$nd Deri*ati*e (cont>!

1 1 # ) $ $ $ 8 $ $ $ $ ) # ) $ $ $ $ ; ; ; ;

?) $ $ $ $ ) $ 8 ?) 8 $ $ ) ) ? ) ) $ $ ; ?; $ $


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))

o"

#$

Asing Second Deri*ati*es For Image

Enhancement

,he nd deri*ati*e is more use"ul "or imageenhancement than the )st deri*ati*e

' Stronger response to "ine detail

' Simpler implementation ' We will come bac% to the )st order deri*ati*elater on

,he "irst sharpening "ilter we will loo% at is

the Laplacian ' Isotropic

' Bne o" the simplest sharpening "ilters

' We will loo% at a digital implementation


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)

o"

#$,he 2aplacian

,he 2aplacian is de"ined as "ollows:

where the partial )st order deri*ati*e in the x direction is de"ined as "ollows:

and in the y direction as "ollows:

y

f

x

f f

2

2

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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)#

o"

#$,he 2aplacian (cont>!

So7 the 2aplacian can be gi*en as "ollows:

We can easil- build a "ilter 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 −

$ ) $

) ? )

$ ) $


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)

o"

#$,he 2aplacian (cont>!

9ppl-ing the 2aplacian to an image we get anew image that highlights edges and other

discontinuities


D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !

Briginal

Image

2aplacian

Filtered Image

2aplacian

Filtered Image

Scaled "or Displa-


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)1

o"

#$ut ,hat Is ot er- Enhanced

,he result o" a 2aplacian "ilteringis not an enhanced image

We ha*e to do more wor% in

order to get our "inal imageSubtract the 2aplacian result

"rom the original image to

generate our "inal sharpenedenhanced image

2aplacian

Filtered Image

Scaled "or Displa-


D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !

f y x f y x g 2),(),( ∇−=


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)8

o"

#$2aplacian Image Enhancement

In the "inal sharpened image edges and "inedetail are much more ob*ious


D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !

?

Briginal

Image

2aplacian

Filtered Image

Sharpened

Image


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);

o"

#$2aplacian Image Enhancement


D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !


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)=

o"

#$Simpli"ied Image Enhancement

,he entire enhancement can be combinedinto a single "iltering 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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)@

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#$Simpli"ied Image Enhancement (cont>!

,his gi*es us a new "ilter which does thewhole


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$

o"

#$Simpli"ied Image Enhancement (cont>!


D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !


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)

o"

#$ariants Bn ,he Simple 2aplacian

,here are lots o" slightl- di""erent *ersions o"the 2aplacian that can be used:

$ ) $

) ? )

$ ) $

) ) )

) ?= )

) ) )

?) ?) ?)

?) @ ?)

?) ?) ?)

Simple2aplacian

ariant o" 2aplacian


D i g i t a l I m a

g e P r o c e s s i n g ( $ $ !


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o"

#$Simple Con*olution ,ool In Ga*a

9 great tool "or testing out di""erent "ilters ' From the boo% HImage Processing tools in

Ga*a

' 9*ailable "rom webC, later on toda- ' ,o launch: java ConvolutionTool Moon.jpg


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#

o"

#$)st Deri*ati*e Filtering

Implementing )st deri*ati*e "ilters is di""icult inpractice

For a "unction f(x y! the gradient o" f at

coordinates (x y! is gi*en as the column*ector:

∂

∂∂

∂

=

=∇

y

f x

f

"

"

y

xf


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#$)st Deri*ati*e Filtering (cont>!

,he magnitude o" this *ector is gi*en b-:

For practical reasons this can be simpli"ied as:

)f (∇=∇ mag f

[ ] 21

22

y x "" +=

21

22

∂∂

+

∂∂

= y

f

x

f

y x "" f +≈∇


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1

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#$)st Deri*ati*e Filtering (cont>!

,here 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 ++−+++

5) 5 5#

5 51 58

5; 5= 5@


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8

o"

#$Sobel Bperators

ased on the pre*ious e&uations we canderi*e the Sobel Operators

,o "ilter an image it is "iltered using both

operators the results o" which are added

together

?) ? ?)

$ $ $

) )

?) $ )

? $

?) $ )


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;

o"

#$Sobel Eample

Sobel "ilters are t-picall- used "or edge

detection

I m

a g e s t a % e n " r o m 4 o n 5 a l e 5 6 W o o d s 7

D i g i t a l I m a g e P r o c e s s i n g ( $ $ ! An image of a

contact lens which

is enhanced in

order to make

defects (at four

and five o’clock inthe image) more

obvious


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=

o"

#$)st 6 nd Deri*ati*es

Comparing the )st and nd deri*ati*es wecan conclude the "ollowing:

' )st order deri*ati*es generall- produce thic%er

edges ' nd order deri*ati*es ha*e a stronger

response to "ine detail e.g. thin lines

' )st order deri*ati*es ha*e stronger response

to gre- le*el step

' nd order deri*ati*es produce a double

response at step changes in gre- le*el


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@

o"

#$Summar-

In this lecture we loo%ed at: ' Sharpening "ilters

)st deri*ati*e "ilters

nd

deri*ati*e "ilters ' Combining "iltering techni&ues

C bi i S ti l E h t


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#$

o"

#$

Combining Spatial Enhancement

Jethods

Success"ul imageenhancement is t-picall-

not achie*ed using a single

operation+ather we combine a range

o" techni&ues in order to

achie*e a "inal result,his eample will "ocus on

enhancing the bone scan to

the right I m a g e s t a % e n " r o m 4 o n 5 a l e 5 6 W o o d s 7

D i g i t a l I m a g e P r o c e s s i n g ( $ $ !

C bi i S ti l E h t


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#)

o"

#$


Jethods (cont>!

I m



2aplacian "ilter o"

bone scan (a!

Sharpened *ersion o"

bone scan achie*ed

b- subtracting (a!

and (b! Sobel "ilter o" bone

scan (a!

(a)

(b)

(c)

(d)

C bi i S ti l E h t


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#

o"

#$


Jethods (cont>!

I m



,he product o" (c!

and (e! which will be

used as a mas%

Sharpened image

which is sum o" (a!

and ("!

+esult o" appl-ing a

power?law trans. to(g!

(e)

(f)

(g)

(h)

Image (d! smoothed witha 1K1 a*eraging "ilter

C bi i S ti l E h t


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##

o"

#$


Jethods (cont>!

Compare the original and "inal images

m

a g e s t a % e n " r o m 4 o n 5 a

l e 5 6 W o o d s 7


image processing 6-spatialfiltering2.ppt

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