CHAPTER 7CHAPTER 7

Template FiltersTemplate Filters

IMAGE ANALYSISIMAGE ANALYSIS

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gij = fi–1,j–1 h–1,–1 + fi–1,j h–1,0 + fi–1,j+1 h–1,1 +

+ fi,j–1 h0,–1 + fi,j h0,0 + fi,j+1 h0,1 +

+ fi+1,j–1 h1,–1 + fi+1,j h1,0 + fi+1,j+1 h1,1

gij = fi–1,j–1 h–1,–1 + fi–1,j h–1,0 + fi–1,j+1 h–1,1 +

+ fi,j–1 h0,–1 + fi,j h0,0 + fi,j+1 h0,1 +

+ fi+1,j–1 h1,–1 + fi+1,j h1,0 + fi+1,j+1 h1,1

Moving templates for image filtering Moving templates for image filtering

The discrete convolution process in template filtering

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Typical template dimensions

Non-square templates viewed as special cases of square ones

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localized gij = hi,j;k,m fkm k=i–p m=j–p

i+p j+p

Template filters = Localized position-invariant linear transformations of an image

Using a (p+1)(p+1) templateUsing a (p+1)(p+1) template

linear gij = hi,j;k,m fkm k m

position-invariant hi,j;k,m = hk–i,m–j

gij = hk–i,m–j fkm k m

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Template filters = Localized position-invariant linear transformations of an image

renamed (i = 0, j = 0, k = k, m = m)

Combination of all properties

gij = hk–i,m–j fkm k=i–p m=j–p

i+p j+p

k = k – i

m = m – j

gij = hk,m fi+k,j+m k = –p m = –p

p p

g00 = hk,m fk,m k = –p m = –p

p p

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Template filters = Localized position-invariant linear transformations of an image

renamed

j–1 j j+1

i+1i

i–1

hij

fij

g00 = h–1,–1 f–1,–1 + h –1,0 f–1,+1 + h –1,1 f–1,+1 +

+ h0,–1 f0,–1 + h0,0 f0,0 + h0,+1 f0,+1 +

+ h+1,–1 f+1,–1 + h+1,0 f+1,0 + h+1,+1 f+1,+1

g00 = h–1,–1 f–1,–1 + h –1,0 f–1,+1 + h –1,1 f–1,+1 +

+ h0,–1 f0,–1 + h0,0 f0,0 + h0,+1 f0,+1 +

+ h+1,–1 f+1,–1 + h+1,0 f+1,0 + h+1,+1 f+1,+1

g00 = hk,m fk,m k = –p m = –p

p p

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Examples

homogeneous areas are set to zerohigh values emphasize high frequencies

fkm = C

g00 = hk,m C = 0 k = –p m = –p

p p

hk,m = 0 k = –p m = –p

p p

Examples

1

25

1

9

homogeneous (low frequency) areas preserve their value

fkm = C

g00 = hk,m C = C k = –p m = –p

p p

1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1

1 1 1 1 2 1

1 8 1 2 4 2

1 1 1 1 2 1

High-pass filtersHigh-pass filters

hk,m = 1 k = –p m = –p

p p

Low-pass filtersLow-pass filters

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An example of low pass filters: The original band 3 of a TM image is undergoing low pass filtering by moving mean templates with dimensions 33 and 55

An example of low pass filters: The original band 3 of a TM image is undergoing low pass filtering by moving mean templates with dimensions 33 and 55

Original

Moving mean 33 Moving mean 55

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An example of a high pass filter:The original image is undergoing high pass filtering with a 33 template,which enhances edges, best viewed as black lines in its negative

An example of a high pass filter:The original image is undergoing high pass filtering with a 33 template,which enhances edges, best viewed as black lines in its negative

Original

high pass filtering 33 high pass filtering 33 (negative)

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evaluation

Local interpolation and template formulationLocal interpolation and template formulation

interpolation

Templates expressing linear operatorsTemplates expressing linear operators

fkm f(x, y)

A

g(x, y)g(0, 0)

gij

hkm fkmk, m

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Original (TM band 4)

Laplacian 99 Laplacian 1313 Laplacian 1717

Examples of Laplacian filters with varying template sizes Examples of Laplacian filters with varying template sizes

The Laplacian operator

2 2

x2 y2A = = +

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Original (TM band 4)

Laplacian 55 Original + Laplacian 55

Examples of Laplacian filters with varying template sizes Examples of Laplacian filters with varying template sizes

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The Roberts and Sobel filters for edge detection The Roberts and Sobel filters for edge detection

Original (TM band 4) Roberts Sobel

Roberts filter Sobel filter

0 0 0

0 1 0

0 0 -1

0 0 0

0 0 1

0 -1 0

X Y

-1 0 1

-2 0 2

-1 0 1

-1 -2 -1

0 0 0

1 2 1

X Y

X 2+Y

2 X 2+Y

2

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