1. Distance Functions and Metric Learning: Part 1 12/1/2011 version. Updated Version: http://www.cs.huji.ac.il/~ofirpele/DFML_ECCV2010_tutorial/ Michael Werman Ofir Pele The Hebrew University of Jerusalem

3. Distance ?

4. Metric ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² I d e n t i t y o f i n d i s c e r n i b l e s : D ( P ; Q ) = 0 i ® P = Q ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q )

5. Pseudo-Metric (aka Semi-Metric) ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² P r o p e r t y c h a n g e d t o : D ( P ; Q ) = 0 i f P = Q ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q )

6. Metric A n o b j e c t i s m o s t s i m i l a r t o i t s e l f ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² I d e n t i t y o f i n d i s c e r n i b l e s : D ( P ; Q ) = 0 i ® P = Q

7. Metric ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q ) U s e f u l f o r m a n y a l g o r i t h m s

8. Minkowski-Form Distances L p ( P ; Q ) = Ã X i j P i ¡ Q i j p ! 1 p

9. Minkowski-Form Distances L 2 ( P ; Q ) = s X i ( P i ¡ Q i ) 2 L 1 ( P ; Q ) = X i j P i ¡ Q i j L 1 ( P ; Q ) = m a x i j P i ¡ Q i j

11. Jensen-Shannon Divergence J S ( P ; Q ) = 1 2 K L ( P ; M ) + 1 2 K L ( Q ; M ) M = 1 2 ( P + Q ) J S ( P ; Q ) = 1 2 X i P i l o g 2 P i P i + Q i + 1 2 X i Q i l o g 2 Q i P i + Q i

15. Histogram Distance < Â 2 Â 2 ( ; ) Â 2 ( ; )

16. Histogram Distance > Â 2 L 1 ( ; ) L 1 ( ; )

22. The Quadratic-Form Histogram Distance Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) = s X i j ( P i ¡ Q i ) ( P j ¡ Q j ) A i j A = I = s X i j ( P i ¡ Q i ) 2 = L 2 ( P ; Q )

25. Pseudo-Metric (aka Semi-Metric) ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² P r o p e r t y c h a n g e d t o : D ( P ; Q ) = 0 i f P = Q ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q )

27. The Quadratic-Form Histogram Distance Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) = q ( P ¡ Q ) T W T W ( P ¡ Q ) = L 2 ( W P ; W Q )

28. The Quadratic-Form Histogram Distance We assume there is a linear transformation that makes bins independent There are cases where this is not true e.g. COLOR Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) = L 2 ( W P ; W Q )

34. Similarity-Matrix-Quantization-Invariant Property ( , ) D ) ( , = D

35. Sparseness-Invariant = D ) ( , Empty bins D ( , )

36. The Quadratic-Chi Histogram Distance Code f u n c t i o n d i s t = Q u a d r a t i c C h i ( P , Q , A , m ) Z = ( P + Q ) * A ; % 1 c a n b e a n y n u m b e r a s Z _ i = = 0 i f f D _ i = 0 Z ( Z = = 0 ) = 1 ; Z = Z . ^ m ; D = ( P - Q ) . / Z ; % m a x i s r e d u n d a n t i f A i s % p o s i t i v e - s e m i d e f i n i t e d i s t = s q r t ( m a x ( D * A * D ' , 0 ) ) ;

37. The Quadratic-Chi Histogram Distance Code f u n c t i o n d i s t = Q u a d r a t i c C h i ( P , Q , A , m ) Z = ( P + Q ) * A ; % 1 c a n b e a n y n u m b e r a s Z _ i = = 0 i f f D _ i = 0 Z ( Z = = 0 ) = 1 ; Z = Z . ^ m ; D = ( P - Q ) . / Z ; % m a x i s r e d u n d a n t i f A i s % p o s i t i v e - s e m i d e f i n i t e d i s t = s q r t ( m a x ( D * A * D ' , 0 ) ) ; T i m e C o m p l e x i t y : O ( # A i j 6 = 0 )

38. The Quadratic-Chi Histogram Distance Code f u n c t i o n d i s t = Q u a d r a t i c C h i ( P , Q , A , m ) Z = ( P + Q ) * A ; % 1 c a n b e a n y n u m b e r a s Z _ i = = 0 i f f D _ i = 0 Z ( Z = = 0 ) = 1 ; Z = Z . ^ m ; D = ( P - Q ) . / Z ; % m a x i s r e d u n d a n t i f A i s % p o s i t i v e - s e m i d e f i n i t e d i s t = s q r t ( m a x ( D * A * D ' , 0 ) ) ; W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ?

39. The Quadratic-Chi Histogram Distance Code 0 0 0 0 0 0 0 0 0 4 5 0 -3 P ¡ Q = W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ? T i m e C o m p l e x i t y : O ( S K )

40. The Quadratic-Chi Histogram Distance Code 0 0 0 0 0 0 0 0 0 4 5 0 -3 P ¡ Q = # ( P 6 = 0 ) + # ( Q 6 = 0 ) W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ? T i m e C o m p l e x i t y : O ( S K )

41. The Quadratic-Chi Histogram Distance Code A v e r a g e o f n o n - z e r o e n t r i e s 0 0 0 0 0 0 0 0 0 4 5 0 -3 P ¡ Q = i n e a c h r o w o f A W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ? T i m e C o m p l e x i t y : O ( S K )

42. The Earth Mover’s Distance

44. The Earth Mover’s Distance ≠

45. The Earth Mover’s Distance =

46. The Earth Mover’s Distance E M D D ( P ; Q ) = m i n F = f F i j g X i ; j F i j D i j s : t : X j F i j = P i X i F i j = Q j X i ; j F i j = X i P i = X j Q j = 1 F i j ¸ 0

47. The Earth Mover’s Distance E M D D ( P ; Q ) = m i n F = f F i j g P i ; j F i j D i j P i F i j s : t : X j F i j · P i X i F i j · Q j X i ; j F i j = m i n ( X i P i ; X j Q j ) f a l s e F i j ¸ 0

49. Definition: E M D D C ( P ; Q ) = m i n F = f F i j g X i ; j F i j D i j + ¯ ¯ ¯ ¯ ¯ ¯ X i P i ¡ X j Q j ¯ ¯ ¯ ¯ ¯ ¯ £ C s : t : X j F i j · P i X i F i j · Q j X i ; j F i j = m i n ( X i P i ; X j Q j ) F i j ¸ 0 f a l s e

50. S u p p l i e r E M D D e m a n d e r D i j = 0 P D e m a n d e r · P S u p p l i e r

51. D e m a n d e r S u p p l i e r P D e m a n d e r · P S u p p l i e r D i j = C E M D

57. C = 1 D i j = ( 0 i f i = j 2 o t h e r w i s e E M D D C ( P ; Q ) = m i n F = f F i j g X i ; j F i j D i j + ¯ ¯ ¯ ¯ ¯ ¯ X i P i ¡ X j Q j ¯ ¯ ¯ ¯ ¯ ¯ £ C ² E M D = L 1 i f : E M D - a N a t u r a l E x t e n s i o n t o L 1

70. The Flow Network Transformation Original Network Simplified Network

71. The Flow Network Transformation Original Network Simplified Network

72. The Flow Network Transformation Flowing the Monge sequence (if ground distance is a metric, zero-cost edges are a Monge sequence)

73. The Flow Network Transformation Removing Empty Bins and their edges

74. The Flow Network Transformation We actually finished here….

77. The Earth Mover’s Distance Approximations

82. The Earth Mover’s Distance Approximations

87. Robust Distances ( b l u e ; r e d ) = 5 6 ¢E 0 0 ( b l u e ; y e l l o w ) = 1 0 2 ¢E 0 0

90. Robust Distances - Thresholded ² L e t d ( a ; b ) b e a d i s t a n c e m e a s u r e b e t w e e n t w o f e a t u r e s - a ; b . ² T h e t h r e s h o l d e d d i s t a n c e w i t h a t h r e s h o l d o f t > 0 i s : d t ( a ; b ) = m i n ( d ( a ; b ) ; t ) .

95. A Ground Distance for SIFT d R = j j ( x i ; y i ) ¡ ( x j ; y j ) j j 2 + m i n ( j o i ¡ o j j ; M ¡ j o i ¡ o j j ) d T = m i n ( d R ; T ) T h e g r o u n d d i s t a n c e b e t w e e n t w o S I F T b i n s ( x i ; y i ; o i ) a n d ( x j ; y j ; o j ) :

96. A Ground Distance for Color Image T h e g r o u n d d i s t a n c e s b e t w e e n t w o L A B i m a g e b i n s ( x i ; y i ; L i ; a i ; b i ) a n d ( x j ; y j ; L j ; a j ; b j ) w e u s e a r e : d c T = m i n ( ( j j ( x i ; y i ) ¡ ( x j ; y j ) j j 2 ) + ¢ 0 0 ( ( L i ; a i ; b i ) ; ( L j ; a j ; b j ) ) ; T )

97. Perceptual Color Differences

106. Perceptual Color Descriptors

120. Hands-On Code Example http://www.cs.huji.ac.il/~ofirpele/FastEMD/code/ http://www.cs.huji.ac.il/~ofirpele/QC/code/

121. Tutorial: Or “Ofir Pele” http://www.cs.huji.ac.il/~ofirpele/DFML_ECCV2010_tutorial/