Lecture'4:' Pixels'and'Filters Lecture 3: Pixels and Filters
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1 Lecture'4:' SF323: Applications in Machine Learning Pixels'and'Filters Lecture 3: Pixels and Filters Dr.'Juan'Carlos'Niebles Stanford'AI'Lab Krisada Chaiyasarn, PhD Professor'Fei>Fei Li Stanford'Vision'Lab 1
2 What'we'will'learn'today? Images'as'functions Linear'systems'(filters) Convolution'and'correlation Some'background'reading: Forsyth'and'Ponce,'Computer'Vision,'Chapter'7 2
3 An'image'contains'discrete'number'of'pixels A'simple'example Pixel'value: Images'as'functions grayscale ' (or' intensity ):'[0,255]
4 An'image'contains'discrete'number'of'pixels A'simple'example Pixel'value: Images'as'functions grayscale ' (or' intensity ):'[0,255] color [90,'0,'53] RGB:'[R,'G,'B] [249,'215,'203] Lab:'[L,'a,'b] HSV:'[H,'S,'V] [213,'60,'67] 4
5 Images'as'functions An'Image'as'a'function f from'r2 to'rm: f( x, y ) gives'the'intensity at'position'( x, y ) Defined'over'a'rectangle,'with'a'finite'range: f: [a,b] x [c,d ]! [0,255] Domain support range 5
6 Images'as'functions An'Image'as'a'function f from'r 2 to'r M : f( x, y ) gives'the'intensity at'position'( x, y ) Defined'over'a'rectangle,'with'a'finite'range: f: [a,b] x [c,d ]! [0,255] Domain support range A'color'image:' rxy (, ) f( x, y) = g( x, y) bxy (, ) 6
7 Images'as'discrete'functions Images'are'usually'digital'(discrete): Sample the'2d'space'on'a'regular'grid Represented'as'a'matrix'of'integer'values pixel 7
8 Images'as'discrete'functions Cartesian'coordinates Notation'for' discrete' functions 8
9 What'we'will'learn'today? Images'as'functions Linear'systems'(filters) Convolution'and'correlation Some'background'reading: Forsyth'and'Ponce,'Computer'Vision,'Chapter'7 9
10 Filtering: Systems'and'Filters Form'a'new'image'whose'pixels'are'a' combination'original'pixel'values Goals:' >Extract'useful'information'from'the'images Features'(edges,'corners,'blobs ) > Modify'or'enhance'image'properties:' super>resolution;'in>painting;'de>noising 10
11 De>noising Super>resolution In>painting Bertamio et'al' 11
12 2D'discrete>space'systems'(filters) 12
13 * Filter'example'#1:'Moving'Average 2D'DS'moving'average'over'a'3' 3'window'of' neighborhood h ( f 1 * h)[ m, n] = R f [ k, l] h[ m-k, n-l] 9 k, l 13
14 Filter'example'#1:'Moving'Average ( f h)[m, n] = f [k,l] h[m k, n l] k,l 14 Courtesy'of'S.'Seitz
15 Filter'example'#1:'Moving'Average ( f h)[m, n] = f [k,l] h[m k, n l] k,l 15 15
16 Filter'example'#1:'Moving'Average ( f h)[m, n] = f [k,l] h[m k, n l] k,l 16
17 Filter'example'#1:'Moving'Average ( f h)[m, n] = f [k,l] h[m k, n l] k,l 17
18 Filter'example'#1:'Moving'Average ( f h)[m, n] = f [k,l] h[m k, n l] k,l 18
19 Filter'example'#1:'Moving'Average ( f h)[m, n] = f [k,l] h[m k, n l] k,l 19 Source:*S.*Seitz
20 Filter'example'#1:'Moving'Average In'summary: Replaces'each'pixel' with'an'average'of'its' neighborhood. Achieve'smoothing' effect'(remove'sharp' features) h[, ]
21 Filter'example'#1:'Moving'Average 21
22 Filter'example'#2:'Image'Segmentation Image'segmentation'based'on'a'simple' threshold: 255, 22
23 Classification'of'systems 'Amplitude'properties 'Linearity 'Stability 'Invertibility 'Spatial'properties 'Causality 'Separability 'Memory 'Shift'invariance 'Rotation'invariance 23
24 Shift>invariance If'''''''''''''''''''''''''''''''''''''''''''''then for'every'input'image'f[n,m]'and'shifts'n 0,m 0 24
25 Is'the'moving'average'system'is'shift'invariant?'
26 Is'the'moving'average'system'is'shift'invariant?' Yes! 26
27 Linear'Systems'(filters) Linear'filtering: Form'a'new'image'whose'pixels'are'a'weighted'sum'of' original'pixel'values Use'the'same'set'of'weights'at'each'point S is'a'linear'system'(function)'iff it's#satisfies superposition'property 27
28 What'we'will'learn'today? Images'as'functions Linear'systems'(filters) Convolution'and'correlation Some'background'reading: Forsyth'and'Ponce,'Computer'Vision,'Chapter'7 32
29 Discrete'convolution'(symbol:''''')' 'Fold'h[k,l]'about'origin'to'form'h[ k, l] 'Shift'the'folded'results'by'n,m to'form'h[n' 'k,m 'l] 'Multiply'h[n' 'k,m 'l]'by'f[k,'l] 'Sum'over'all'k,l 'Repeat'for'every'n,m h[k,l] h[>k,>l] h[n>k,m>l] n 33
30 Discrete'convolution'(symbol:''''')' 'Fold'h[k,l]'about'origin'to'form'h[ k, l] 'Shift'the'folded'results'by'n,m to'form'h[n' 'k,m 'l] 'Multiply'h[n' 'k,m 'l]'by'f[k,'l] 'Sum'over'all'k,l 'Repeat'for'every'n,m h[n>k,m>l] f[k,l]'x'h[n>k,m>l] f[k,l] Sum'(f[k,l]'x'h[n>k,m>l]) 35 n
31 Convolution'in'2D'B examples * =? Original Courtesy'of'D'Lowe 36
32 Convolution'in'2D'B examples * = Original Filtered (no change) Courtesy'of'D'Lowe 37
33 Convolution'in'2D'B examples * =? Original Courtesy'of'D'Lowe 38
34 Convolution'in'2D'B examples * = Original Shifted right By 1 pixel Courtesy'of'D'Lowe 39
35 Convolution'in'2D'B examples * =? Original Courtesy'of'D'Lowe 40
36 Convolution'in'2D'B examples * = Original Blur (with a box filter) Courtesy'of'D'Lowe 41
37 Convolution'in'2D'B examples 2 - =? Original (Note that filter sums to 1) details of the image + - Courtesy'of'D'Lowe 42
38 What'does'blurring'take'away? = original smoothed.(5x5) detail Let s'add'it'back: +*a = original detail sharpened 43
39 Convolution'in'2D' Sharpening.filter 2 - = Original Sharpening'filter: Accentuates'differences'with'local'average Courtesy'of'D'Lowe 44
40 Image'support'and'edge'effect A'computer'will'only'convolve'finite'support' signals.' That'is:'images'that'are'zero'for'n,m outside'some' rectangular'region MATLAB s'conv2'performs'2d'ds'convolution'of'finite> support'signals. * = N2' M2' (N1'+'N2' '1)' (M1'+M2' '1) N1' M1 45
41 Image'support'and'edge'effect A'computer'will'only'convolve'finite'support' signals.' What'happens'at'the'edge? f h 'zero' padding 'edge'value'replication 'mirror'extension 'more'(beyond'the'scope'of'this'class) >>'Matlab conv2'uses' zero>padding 46
42 What'we'will'learn'today? Images'as'functions Linear'systems'(filters) Convolution'and'correlation Some'background'reading: Forsyth'and'Ponce,'Computer'Vision,'Chapter'7 47
43 (Cross)'correlation'(symbol:'''''''')'' Cross'correlation'of'two'2D'signals'f[n,m]'and'g[n,m] Equivalent'to'a'convolution'without'the'flip (k,'l)'is'called'the'lag r fg [n, m] =f[n, m] g [ n, m] (g*'is'defined'as'the'complex#conjugate#of'g.'in'this'class,'g(n,m)'are'real'numbers,'hence'g*=g.)' 48
44 (Cross)'correlation' example MATLAB s' xcorr2 Courtesy'of'J.'Fessler 49
45 (Cross)'correlation' example Left Right scanline Norm.'cross'corr.'score 50
46 Convolution'vs.'(Cross)'Correlation 51
47 Convolution'vs.'(Cross)'Correlation A'convolution is'an'integral'that'expresses'the'amount' of'overlap'of'one'function'as'it'is'shifted'over'another' function.' convolution'is'a'filtering'operation Correlation compares'the'similarity of#two sets#of# data.&correlation'computes'a'measure'of'similarity'of' two'input'signals'as'they'are'shifted'by'one'another.' The'correlation'result'reaches'a'maximum'at'the'time' when'the'two'signals'match'best'. correlation'is'a'measure'of'relatedness'of'two'signals 52
48 What'we'have'learned'today? Images'as'functions Linear'systems'(filters) Convolution'and'correlation 56
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