Spatial Filtering Image Processing

8/9/2019 Spatial Filtering Image Processing

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Spatial Filtering

Sankalp Kallakuri

[email protected]

Lecture 3


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Spatial Filtering

Involves moving a kernel over the image to enhance theimage

Kernel size is


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Spatial Filtering

f(x-1,y-1) f(x,y-1) f(x+1,y-1)

f(x-1,y) f(x,y) f(x+1,y)

f(x-1,y+1) f(x,y+1) f(x+1,y+1)

m(-1,-1) m(0,-1) m(1,-1)

m(-1,0) m(0,0) m(1,0)

m(-1,1) m(0,1) m(1,1)

Mask coefficients

Image pixels under mask

R = m(-1,-1) f(x-1,y-1) + m(0,-1) f(x,y-1) + m(1,-1) f(x+1,y-1) + ..

a

as

b

bt

tysxftsmyxg ),(),(),(


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Spatial Filtering

The operation is similar to Convolution. hence the masks are alsocalled convolution masks.

Non linear operations such as finding median may also be done ona neighborhood.

Near the edges parts of the masks may lie beyond the imageboundary.

To avoid this either a smaller filtered image is accepted.

Or zeros are padded along the image boundary.


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Smoothing Spatial Filters

Averaging or Low pass filters.

Used to remove unwanted detail.

Smooth out unwanted sharp noise transitions.

May smooth out edges which should be sharp.

1 2 12 4 2

1 2 1

Examples

1 1 11 1 1

1 1 1

1/9 x 1/16 x

Box filter Weighted Average


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Smoothing Spatial Filters

The box filter does a simple average.

The weighted average filter weighs the contribution by the

distance from the central image pixel.

a

as

b

bt

a

as

b

bt

tsm

tysxftsm

yxg

),(

),(),(

),(

Mathematical expressionOriginal

3 6 9 12


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Order Statistic Filters

Median, Max and Min Filters

Median Filters are particularly effective with salt peppernoise or impulse noise.

The image pixels under the mask need to be sorted.

Min and max filters are used to find the brightest anddarkest points.

original Noise added Median filtered image


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Sharpening Spatial Filters

Use of first and second order derivatives.

We need to know how they behave at discontinuities,edges, ramps, noise points.

First Derivative

1) zero in flat segments2) must be non zero at onset of edge or ramp.3)non zero along ramps

Second Derivative1) zero in flat segments2) must be non zero at onset of edge or ramp.3) must be zero along ramps of constant slope.


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Sharpening Spatial Filters

)()1( xfxfxf

First order derivative approximation

second order derivative approximation

)(2)1()1(2

2

xfxfxf

x

f

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

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

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


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Summary of First and Second OrderDerivatives

First Order Derivatives Generally Produce ThickerEdges.

Second Order Derivatives have a stronger response to

finer detail.

First Order response is generally stronger for a greylevel step.

Second Order derivatives produce a double response atstep changes in grey level .


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2nd Derivative - Laplacian

isotropic 2nd order operator2

2

2

22

y

f

x

ff

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

2

yxfyxfyxfx

f

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

2

yxfyxfyxf

y

f

),(4)]1,()1,(),1(),1([2 yxfyxfyxfyxfyxff

Partial DoubleDerivative in Y direction

Partial DoubleDerivative in X direction

Digital implementation of the laplacian operator


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Laplacian

0 1 0

1 -4 1

0 1 0

1 1 1

1 -8 1

1 1 1

Isotropic for 900 Isotropic for 450

Effect of diagonal elements is not considered in the first case

The image of the laplacian filter is usually added to the original toimprove the Features of the original.

),( yxg),(),(

2 yxfyxf

),(),(2 yxfyxf {=

-ve center coeff

+ve center coeff


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Example : Laplacian

Laplacian Matlab Laplacian Hand Coded

Original Difference


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Combined Implementation

),1()1([),(),( yxfxfyxfyxg

),(4)]1,()1,( yxfyxfyxf

)]1,()1,(

),1()1([),(5

yxfyxf

yxfxfyxf),( yxg

0 -1 0

-1 5 -10 -1 0

-1 -1 -1

-1 9 -1-1 -1 -1

Isotropic for 900 Isotropic for 450


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Example: combined implementation

Original

laplacian

Combined


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Unsharp Masking and High Boost Filtering

),(),(),( yxfyxfyxfs

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

),(),(),( yxfyxAfyxfhb

),(),()1(),( yxfyxfAyxf shb

Unsharp masking is to substract a smoothed out image from the original

High boost is to boost up the original by factor A

We can express the high boost in terms of the original and the sharpenedImage.

0 -1 0

-1 A+4 -1

0 -1 0

High boost filtering for A 1

{),(),( 2 yxfyxf

),(),( 2 yxfyxf =hbf

A

A

-ve center coeff

+ve center coeff


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Use of First Derivatives - Gradient

f

Gy

Gx

y

f

x

f

=

The gradient as a vector

f2/122

][ yx GG

2/122

y

f

x

f

=

Magnitude of gradient

|||| yx GGf Gradient approximatedBy absolute values


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Gradient Masks

0 -1

1 0

1 0

0 -1

-1 -2 -1

0 0 0

1 2 1

-1 0 1

-2 0 2

-1 0 1

Robert crossoperators

SobelOperators


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HW 2-B

Repeat HW 2 with a triangle cropping.

Equilateral with the coordinates ofcentroid same as coordinates of 3L/4,3L/4.

One edge of the triangle is at 125 degreesto the bottom edge of the image.

Edge length of the triangle is 30. DUE at same time as HW -2.

Spatial Filtering Image Processing

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