Noise Filtering & Edge Detection

Jeremy Wyatt

Filtering

Last time we saw that we could detect edges by calculating the intensity change (gradient) across the image

We saw that we could implement this using the idea of filtering

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Linear filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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-1 0 1

-2 0 2

-1 0 1

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i

j

NB We count from the upper left,and in MATLAB we start at 1

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

0

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

0

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

1

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

1

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

1

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

5

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

5

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

5

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=3

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14 4

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=4

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14 4 3

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=2

j=5

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14 4 3

10

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=3

j=2

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

0 1 1 3 4 5

0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14 4 3

10 16

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=3

j=3

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14 4 3

10 16 3

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=3

j=4

Linear Filtering: the algorithm

for i=2:image_height-1

for j=2:image_width-1

end

end

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0 0 2 3 3 4

0 0 4 6 3 5

0 0 0 4 4 3

0 0 0 3 5 2

0 0 0 0 5 5

0 0 0 0 4 3

-1 0 1

-2 0 2

-1 0 1

9 14 4 3

10 16 4 -2

i+y

j+x

1 1

1 1

( , ) ( 2, 2)out iny x

i y j x y x

A (i, j) A M

y+2

x+2

i=3

j=5

Noise filtering

We can use convolution to remove noise as we mentioned, e.g. mean filter

This is a linear filter The most widely used is Gaussian filtering

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0 .01 .02 .01 0

.01 .06 .11 .06 .01

.02 .11 .16 .11 .02

.01 .06 .11 .06 .01

0 .01 .02 .01 0

Effect of mean filtering

Original 3x3 filter 5x5 filter

Horizontal Sobel operator

Abs(Gx)

Threshold=30

5x5 Mean Filter

Horizontal Sobel operator

Abs(Gx)

Threshold=30

Effect of Gaussian filtering

Original 5x5 filter Horizontal Sobel Operator

Abs(Gx)Threshold = 30

Sequenced filtersWe can replace a 2d Gaussian filter with 2, 1d Gaussian filters in

sequence

0.003 .0133 .0219 .0133 0.003

.0133 .0596 .0983 .0596 .0133

.0219 .0983 .1621 .0983 .0219

.0133 .0596 .0983 .0596 .0133

0.003 .0133 .0219 .0133 0.003

.0545 .2442 .4026 .2442 .0545

.0545

.2442

.4026

.2442

.0545

Gaussian edge detection

We can take the first derivative of the masks and then convolve with those

Then we can combine the resulting images using the formula for magnitude

However when thresholded we can see that this loses edge information

How can we keep this?

.1897 .1741 0 -.1741 -.1897

2 2( ) x yM G G G

Second order operators Thresholding the first derivative of

the smoothed signal thickens the edges and also we lose some useful edges

One solution is therefore to take the second derivative instead

A basic second order mask is the Laplacian

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0 1 0

Reading

RC Jain, Chapter 4