空間フィルタリング (spatial ... - clg.niigata-u.ac.jp

1 (spatial filtering) (spatial filter) ( linear filter) (nonlinear filter) (1) (1) wwnwwmnmhnjmifjig,,,f (i, j) g (i, j) h (m, n) (2w+1) (2w+1) dyyxfyfxf21 5 5 1 5 5 1 5 5 1 5 5 1 1 1 1 1 5 5 5 5 1 1 4 4 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 4 4 4 1 1 1 4 4 4 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 (0, 0) i j (3, 0) f(i, j) f(i, j) f(7, 2) = 4 pixel by pixel (i, j) f(4, 3) g(4,3) = 1 a 1 b 1 c 1 d 1 e 1 f 4 g 4 h 4 i b c a e f d h i g h(m, n) 3 3 w = 1 -1 -1 -1 -1 9 -1 -1 -1 -1 50 50 50 50 100 100 120 120 120 240 (i, j) i, j 50 (-1)+50 (-1)+50 (-1) +50 (-1)+100 9+100 (-1) +120 (-1)+120 (-1) +120 (-1) = 240 (smoothing) (edge extraction) (sharpening) 0100200010020001002000100200010020001002 0001002000100200 (averaging filter)

2 3 3 5 5 1111111111251251251251251251251251251251 2512512512512512512512512512512512512512 51253 3 5 5 7 7 91= 1 1 4 1 1 4 1 1 4 1 1 4 4 4 4 4 1 1 1 1 1 1 4 4 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 (0, 0) i j 1 2 3 1 2 3 4 4 1 1 1 2 3 4 1 4 4 4 4 4 4 4 3 2 1 0 4 3 2 1 0 1 1 1 1 1 1 1 1 1 19g(2,1) = (1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 4) 9 18 9 2 g(3,5)= (1 1 1 4 1 4 1 1 1 4 1 4 1 1 1 4 1 4) 9 27 9 3 f(i,j) g(i,j) 2 3 1 1 1 1 1 1 1 1 1 91 1 1 1 2 1 1 1 1 1 1011 1 1 3 1 1 1 1 1 111 3 3 1 1 1 2 2 1 2 3 1 3511141141141144444111111441111111444444 444444444444(0, 0)ij1 1 2 1 2 1 2 2 1 2 1 1 1 1 1 1 5 5 (weighted averaging filter) f(i,j) 1 (Gaussian filter) 2 w = 10, = w/2 = 5 0 2 Gaussian distribution (2w + 1)(2w + 1) = w/2 22222exp21, yxyxhg3 3 5 5 1256425662564256125642561625616256425662 5662564256162561625642561256425662564256 2561 5 5 1 9 9 1 17 17 1 9 9 3 5 5 3 17 17 2 9 9 5 17 17 3 5 5 15 (statistics filter) (order-statistics filter) (mean filter) (variance filter) (max filter) (min filter) (median filter)

3 Median filter 2 2 1 2 3 1 3 9 2 3 2 2 1 2 1 2 2 1 1 2 1 2 1 2 3 2, 2, 2, 2, 2, 3, 3, 3, 9 1, 1, 2, 2, 2, 2, 3, 3, 9 2 2 2 1, 2, 2, 2, 2, 2, 3, 3, 9 4 3 2 1 0 8 7 6 5 9 2 2 2 1 100 100 140 100 100 0 100 245 140 10 100 140 140 250 140 140 100 20 100 100 5 100 140 0 100 100 250 100 100 100 235 140 140 20 140 140 140 140 250 140 140 140 140 140 20 140 140 140 140 i j 100 100 140 100 100 140 100 100 140 100 100 140 140 140 140 140 100 100 100 100 100 100 140 140 100 100 100 100 100 100 100 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 0 6 0 6 3 3 f(1,1) = 100 : { 10, 20, 100, 100, 100, 100, 100, 100, 100} g(1,1) = 100 f(3,2) = 0 { 0, 100, 100, 140, 140, 140, 140, 245, 250} g(3,2) = 140 f(2,5) = 250 { 5, 100, 100, 100, 100, 140, 140, 140, 250} g(2,5) = 100 f(3,4) = 140 { 0, 100, 140, 140, 140, 140, 245, 250, 250} g(3,4) = 140 median f(i,j ) g(i,j ) hxfhxfdxdfdxxdfxfh 0limfx()x xf x f (x) h + 0 h 0 h 0 h + 0 fx()x x f y f fx()xi+1 i 1 i f (i+1) f (i) f (i) f (i 1) f(i, j)

4 I j i, j i, j+1 i, j-1 i-1, j i+1, j i+1, j-1 i+1, j+1 i-1, j-1 i-1, j+1 i-2, j i-2, j+1 i, j+2 i+2, j jifjiffxR,,1 jifjiffxL,1, 21fffxLxRx 12ffxx jifjiffyU,,1 jifjiffyD,1, 21fffyDyUy 12ffyy f xf yf Gmag 22),(),(,jifjifjiGyxmag Dxf(i,j)Dyf(i,j) ),(),(tan,1jifjifjiGxydir Gmag(i,j)Gdir(i,j) Prewitt filter Prewitt filter Prewitt filter Sobel filter Sobel filter Sobel filter vs. 2 2 2 2 22222yx f (x, y) yxfyyxfxyxf,,,22222 2 X 2 Y 2 22dxxfddxxdfdxdxf fx()x xf x xf x2 1 f(x) yxfdyyxfdyyxfydxxfydxxfdrdydyyxfyxfdyyxf dxydxxfyxfydxxfdyyxfdyyxfydxyxfydxxfxyxf yyxfxyxf,4,,,,4,,2,,,2,,,,,,,,22222222 0,2 yxf 0,2 yxf 0,2 yxf i j i, j i, j+1 i, j-1 i-1, j i+1, j i-2, j i, j+2 i+2, j i, j-2 jifjifjifjifjifjifjifjifjiffxx,1,2,1,1,, ,1,,12 1,,21,1,,,1,,1,2 jifjifjifjifjifjifjifjifjiffyy jifjifjifjifjifjifjifjifjifjifjif,41,1,, 1,11,,21,,1,2,1 fffyx222 2 2 (Laplacian filter) 2 1 2 2 + 2 (zero crossing)

5 X 0 x 0 x 0 1 2 x 0 f x 0 g x 0 2 = jifjifjig,,,2 2 1,1,,1,1,5,41,1,,1,1, jifjifjifjifjifjifjifjifjifjifjif (sharpening filter) 4 8 2014 101101101101102101101101101 -1-1-1-19-1-1-1-11111-811110000100001210 00-1-2-12012 2. 3. 4. 5. 2010 111121111 0 0 0 10 10 0 10 10 0 0 0 10 0 10 0 10 10 0 10 0 0 0 0 0 0 1. 1 2. 2 3. 4 4. 5 5. 20 2009 1. 2. 3. 4. 5. 2008 1161621161621641621161621169191919191919 191910101-410101251251251251251251251251 2512512512512512512512512512512512512512 5125125125351351351351351351352352352351 3513523533523513513523523523513513513513 51351 2007 1161621161621641621161621169191919191919 1919101 01 51 01 0 1 1 1 000111-101-202-101