Lecture 6 Sharpening Filters

1. The concept of sharpening filter2 First and second order derivatives2. First and second order derivatives3. Laplace filter4. Unsharp maskp5. High boost filter6. Gradient mask7. Sharpening image with MatLab

Sharpening Spatial Filters

• To highlight fine detail in an image or to enhance detail that has been blurred, either in error or as a natural ff t f ti l th d f i i itieffect of a particular method of image acquisition.

• Blurring vs Sharpening• Blurring vs. Sharpening

• Blurring/smooth is done in spatial domain by pixel averaging in a i hb it i f i t tineighbors, it is a process of integration

Sh i i i t fi d th diff b th• Sharpening is an inverse process, to find the difference by the neighborhood, done by spatial differentiation.spatial differentiation.

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Derivative operator

• The strength of the response of a derivative operator is proportional to the degree ofoperator is proportional to the degree of discontinuity of the image at the point at which the operator is applied.

• Image differentiation – enhances edges and other discontinuities (noise)– deemphasizes area with slowly varying gray-level

valuesvalues.

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Sharpening edge by First and second order derivatives

• Intensity function f =f’

• First derivative f ’ =f

• Second-order derivative f ’’

f f’’

f ’’ =f-f

• f- f’’ =

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First and second order difference of 1D

• The basic definition of the first-order derivative of a one-dimensional function f(x) is the difference

)()1( xfxfxf

−+=∂∂

• The second-order derivative of a one-dimensional function f(x) is the difference

x∂

( )

)(2)1()1(2

xfxfxff++=

∂ )(2)1()1(2 xfxfxfx

−−++=∂

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First and Second-order derivative of 2D

• when we consider an image function of two variables, f(x, y), at which time we will dealing with partial d i ti l th t ti lderivatives along the two spatial axes.

yxfyxfyxf ∂∂∂ )()()(y

yxfx

yxfyxyxf

∂∂

+∂

∂=

∂∂∂

=∇),(),(),(f

(linear operator)

Gradient operator

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2

2

22 ),(),( yxfyxff

∂∂

+∂

∂=∇

(linear operator)

Laplacian operator 22 yxf

∂∂p p

(non-linear)

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Discrete form of Laplacian

)(2)1()1(2

yxfyxfyxff−−++=

∂from ),(2),1(),1(2 yxfyxfyxfx

++=∂

)(2)1()1(2

ffff∂ ),(2)1,()1,(2 yxfyxfyxfy

f−−++=

∂∂

),1(),1([2 yxfyxff −++=∇)],(4)1,()1,(

),1(),1([yxfyxfyxf

yxfyxff−−+++

++∇

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Result Laplacian mask

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Laplacian mask implemented an extension of diagonal neighbors

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Other implementation of Laplacian masks

give the same result, but we have to keep in mind that when combining (add / subtract) a Laplacian-filtered i ith th i

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image with another image.

Effect of Laplacian Operator• as it is a derivative operator,

– it highlights gray-level discontinuities in an imageit deemphasizes regions with slowly varying gray levels– it deemphasizes regions with slowly varying gray levels

• tends to produce images that have – grayish edge lines and other discontinuities, all superimposed on

a darka dark, – featureless background.

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Correct the effect of featureless background

• easily by adding the original and Laplacian image.• be careful with the Laplacian filter used• be careful with the Laplacian filter used

if th t ffi i t

⎩⎨⎧

∇∇−

=)()(),(),(

),( 2

2

ffyxfyxf

yxg

if the center coefficient of the Laplacian mask is negative

⎩⎨

∇+ ),(),()( 2 yxfyxf

yg

if the center coefficient of the Laplacian mask is of the Laplacian mask is positive

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Example

• a). image of the North pole of the moonpole of the moon

• b). Laplacian-filtered image with

1 1 1

1 -8 1

1 1 1

• c). Laplacian image scaled for display purposes

• d). image enhanced by addition with original image

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image

Mask of Laplacian + addition

• to simply the computation, we can create a mask which do both operations, Laplacian Filter and Addition the

i i l ioriginal image.

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Mask of Laplacian + addition

)](4)1()1(),1(),1([),(),( −++−=

fffyxfyxfyxfyxg

),1(),1([),(5)],(4)1,()1,(

−++−=+−+++

yxfyxfyxfyxfyxfyxf

)]1,()1,( −+++ yxfyxf

0 -1 0

-1 5 -11 5 1

0 -1 0

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Example

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Note

⎩⎨⎧

∇+∇−

=)()(),(),(

),(2

2

yxfyxfyxfyxf

yxg

0 -1 0 0 0 0

⎩ ∇+ ),(),( yxfyxf

0 -1 00 1 0-1 5 -10 -1 0

0 1 00 0 0

= + -1 4 -10 -1 0

0 1 0 0 0 0 0 -1 00 -1 0-1 9 -10 1 0

0 0 00 1 00 0 0

= +0 -1 0-1 8 -10 -1 0

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0 -1 0 0 0 0 0 1 0

Unsharp masking

)()()( yxfyxfyxf = ),(),(),( yxfyxfyxfs −=

sharpened image = original image – blurred imagesharpened image = original image – blurred image

• to subtract a blurred version of an image produces sharpening output image.

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Unsharp mask

High-boost filtering

)()()( yxfyxAfyxf −= ),(),(),( yxfyxAfyxfhb =

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

ffAyxfyxfyxfAyxfhb −−=

),(),()1( yxfyxfA s−−=

• generalized form of Unsharp masking• A ≥ 1

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High-boost filtering

),(),()1(),( yxfyxfAyxf shb −−=• if we use Laplacian filter to create sharpen image fs(x,y)

with addition of original image

),(),()(),( yfyfyf shb

with addition of original image

⎨⎧ ∇−

=),(),(

)(2 yxfyxf

yxf⎩⎨

∇+=

),(),(),( 2 yxfyxf

yxfs

⎧

⎩⎨⎧

∇+∇−

=)()(),(),(

),(2

2

yxfyxAfyxfyxAf

yxfhb

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⎩ ∇+ ),(),( yxfyxAf

High-boost Masks

A ≥ 1if A = 1 it becomes “standard” Laplacian sharpeningif A = 1, it becomes standard Laplacian sharpening

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Example

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Gradient Operator

• first derivatives are implemented using the magnitude of the gradientmagnitude of the gradient. ⎤⎡∂fmagnitude of the gradientmagnitude of the gradient.

⎥⎥⎥⎤

⎢⎢⎢⎡

∂∂∂

=⎥⎦

⎤⎢⎣

⎡=∇ f

xf

GGxf

⎥⎥⎦⎢

⎢⎣∂

⎦⎣yfGy

2122 ][)f( +=∇=∇ GGmagf yx

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⎥⎥⎤

⎢⎢⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=ff

commonly approx.

⎥⎦⎢⎣⎟⎠

⎜⎝ ∂

⎟⎠

⎜⎝ ∂ yx

yx GGf +≈∇

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the magnitude becomes nonlinearyx

Gradient Mask

• simplest approximation, 2x2z1 z2 z3

z4 z5 z6z4 z5 z6

z7 z8 z9

)( and )( 5658 zzGzzG yx −=−=

212

562

582

122 ])()[(][ zzzzGGf yx −+−=+=∇

5658 zzzzf −+−≈∇

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Gradient Mask z1 z2 z3

z z z• Roberts cross-gradient operators, 2x2

z4 z5 z6

z7 z8 z9

)( and )( 6859 zzGzzG yx −=−= )()( 6859 yx

212

682

592

122 ])()[(][ zzzzGGf −+−=+=∇ 6859 ])()[(][ zzzzGGf yx ++∇

6859 zzzzf −+−≈∇ 6859 zzzzf +≈∇

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Gradient Mask z1 z2 z3

z z z• Sobel operators, 3x3

z4 z5 z6

z7 z8 z9

)2()2( 321987 zzzzzzGx ++−++=)2()2( 741963 zzzzzzGy ++−++=

yx GGf +≈∇

the weight value 2 is to achieve smoothing by giving more important

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giving more important to the center point

Example

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Example of Combining Spatial Enhancement Methods

• want to sharpen the original image and bring out more skeletal detail.

• problems: narrow dynamic range of gray level and high noiselevel and high noise content makes the image difficult toimage difficult to enhance

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Example of Combining Spatial Enhancement Methods

• solve : 1. Laplacian to highlight fine detail2. gradient to enhance prominent edges3. gray-level transformation to increase the dynamic

range of gray levelsrange of gray levels

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