CHAPTER 7 IMAGE ANALYSIS Template Filters A. Dermanis

Moving templates for image filtering 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 The discrete convolution process in template filtering A. Dermanis

Typical template dimensions Non-square templates viewed as special cases of square ones A. Dermanis

      gij = hi,j;k,m fkm hi,j;k,m = hk–i,m–j gij = hk–i,m–j fkm Template filters = Localized position-invariant linear transformations of an image 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 localized gij = hi,j;k,m fkm   k=i–p m=j–p i+p j+p Using a (p+1)(p+1) template A. Dermanis

      gij = hk–i,m–j fkm gij = hk,m fi+k,j+m g00 = hk,m fk,m Template filters = Localized position-invariant linear transformations of an image 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 renamed (i = 0, j = 0, k = k, m = m) g00 = hk,m fk,m   k = –p m = –p p p A. Dermanis

Template filters = Localized position-invariant linear transformations of an image j–1 j j+1 i+1 i i–1 renamed hij g00 = hk,m fk,m   k = –p m = –p p p 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 A. Dermanis

        hk,m = 1 hk,m = 0 g00 = hk,m C = C g00 = hk,m C = 0 Low-pass filters High-pass filters hk,m = 1   k = –p m = –p p p hk,m = 0   k = –p m = –p p p homogeneous (low frequency) areas preserve their value fkm = C  g00 = hk,m C = C   k = –p m = –p p p homogeneous areas are set to zero high values emphasize high frequencies fkm = C  g00 = hk,m C = 0   k = –p m = –p p p Examples 1 25 9 Examples 1 -1 1 -2 8 4 A. Dermanis

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 33 and 55 Original Moving mean 33 Moving mean 55 A. Dermanis

An example of a high pass filter: The original image is undergoing high pass filtering with a 33 template, which enhances edges, best viewed as black lines in its negative Original high pass filtering 33 high pass filtering 33 (negative) A. Dermanis

Templates expressing linear operators Local interpolation and template formulation fkm interpolation f(x, y) hkm fkm  k, m A evaluation gij g(x, y) g(0, 0) A. Dermanis

2 2 A =  = + x2 y2 The Laplacian operator 2 2 x2 y2 A =  = + Examples of Laplacian filters with varying template sizes Original (TM band 4) Laplacian 99 Laplacian 1313 Laplacian 1717 A. Dermanis

Examples of Laplacian filters with varying template sizes Original (TM band 4) Laplacian 55 Original + Laplacian 55 A. Dermanis

The Roberts and Sobel filters for edge detection Roberts filter Sobel filter X 2+Y 2 X 2+Y 2 X Y X Y 1 -1 1 -1 -1 1 -2 2 -1 -2 1 2 Original (TM band 4) Roberts Sobel A. Dermanis