Scale Space         The Gaussian Approach                               Li Hui                   bugway@gmail.com         ...
1    Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   2 / 17
12    Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   2 / 17
123    Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   2 / 17
1234    Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   2 / 17
12345    Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   2 / 17
123456    Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   2 / 17
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   3 / 17
Tt                    t ≥0                                              {Tt }t∈R + ,Tt : Cb (R 2 ) → Cb (R 2 ),      ∞    ...
Tt                    t ≥0                                              {Tt }t∈R + ,Tt : Cb (R 2 ) → Cb (R 2 ),      ∞    ...
Tt                    t ≥0                                              {Tt }t∈R + ,Tt : Cb (R 2 ) → Cb (R 2 ),      ∞    ...
Marr-Hildreth-Koenderink-WitKin    1980           Marr   Hildreth[1]      Li Hui (Earth)        Scale Space   The Gaussian...
Marr-Hildreth-Koenderink-WitKin    1980           Marr   Hildreth[1]              1983     (Witkin[2],Koenderink [3])     ...
Marr-Hildreth-Koenderink-WitKin    1980           Marr    Hildreth[1]              1983     (Witkin[2],Koenderink [3])    ...
Marr-Hildreth-Koenderink-WitKin    1980           Marr    Hildreth[1]              1983     (Witkin[2],Koenderink [3])    ...
Koenderink [3]         Hummel [5]                                t                   ∂u(x, y, t)                          ...
Koenderink [3]           Hummel [5]                                t                     ∂u(x, y, t)                      ...
:Li Hui (Earth)       Scale Space   The Gaussian Approach   July 8, 2009   6 / 17
:                            +∞        +∞             u(x, y, t) =                   u(x, y, 0) · Gt (x, y)dxdy           ...
:                                  +∞        +∞             u(x, y, t) =                         u(x, y, 0) · Gt (x, y)dxd...
.Li Hui (Earth)   Scale Space       The Gaussian Approach   July 8, 2009   7 / 17
.Li Hui (Earth)   Scale Space       The Gaussian Approach   July 8, 2009   7 / 17
.Li Hui (Earth)   Scale Space       The Gaussian Approach   July 8, 2009   7 / 17
.Li Hui (Earth)   Scale Space       The Gaussian Approach   July 8, 2009   7 / 17
.Li Hui (Earth)   Scale Space       The Gaussian Approach   July 8, 2009   7 / 17
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   8 / 17
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   8 / 17
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   8 / 17
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   9 / 17
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   9 / 17
1Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   9 / 17
12Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   9 / 17
A                                   (                        )e(x,t) =               (                               )    ...
A                                    (                        )e(x,t) =                 (                                )...
A                                      (                            )e(x,t) =                    (                        ...
Qx,t =             (                                              )                       Q(x, t)A∆x  Li Hui (Earth)      ...
Qx,t =             (                                               )                       Q(x, t)A∆x                   ∂[...
Qx,t =             (                                              )                       Q(x, t)A∆x              ∂[e(x,t)...
Qx,t =             (                                              )                       Q(x, t)A∆x               ∂[e(x,t...
,u(x,t) =         (                        t                )Li Hui (Earth)       Scale Space   The Gaussian Approach     ...
, u(x,t) =             (                        t                )c=                (                                     ...
,  u(x,t) =             (                        t                )c=                 (                                   ...
,  u(x,t) =             (                        t                )c=                 (                                   ...
,  u(x,t) =             (                        t                )c=                 (                                   ...
,  u(x,t) =             (                        t                )c=                 (                                   ...
,  u(x,t) =             (                         t                )c=                 (                                  ...
,  u(x,t) =             (                         t                )c=                 (                                  ...
2∂u∂t   = k∂ u        ∂x 2 Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   13 / 17
2∂u∂t   = k∂ u        ∂x 2      K0k=    cρ Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   13 / 17
2∂u∂t   = k∂ u        ∂x 2      K0k=    cρ                                               x2                   : u(x, t) = ...
2∂u∂t   = k∂ u        ∂x 2      K0k=    cρ                                               x2                   : u(x, t) = ...
2∂u∂t   = k∂ u        ∂x 2      K0k=    cρ                                                        x2                      ...
∂u(x,y ,t)   ∂t        = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0  Li Hui (Earth)         Scale Space   The Gaussian Approach   J...
∂u(x,y ,t)   ∂t        = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0                        u(x, y, 0) = u0 (x, y), (x, y) ∈ ω  Li H...
∂u(x,y ,t)   ∂t        = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0                          u(x, y, 0) = u0 (x, y), (x, y) ∈ ω    ...
∂u(x,y ,t)   ∂t        = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0                          u(x, y, 0) = u0 (x, y), (x, y) ∈ ω    ...
Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   15 / 17
Hummel [6] Li Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   15 / 17
Hummel [6]P-M               Perona    Malik [7]                                            . Li Hui (Earth)            Sca...
Hummel [6]P-M               Perona    Malik [7]                                            .                              ...
[1]Marr D,and Hildreth E,Theory of edge detection.Proc.Roy.Soc.Lond,B207 p187-217,1980[2]A.P.Witkin.Space-scale filtering.I...
!                    Email/Gtalk: bugway@gmail.comLi Hui (Earth)   Scale Space   The Gaussian Approach   July 8, 2009   17...
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Scale Space: The Gaussion Approach

  1. 1. Scale Space The Gaussian Approach Li Hui bugway@gmail.com July 8, 2009Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 1 / 17
  2. 2. 1 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17
  3. 3. 12 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17
  4. 4. 123 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17
  5. 5. 1234 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17
  6. 6. 12345 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17
  7. 7. 123456 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17
  8. 8. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 3 / 17
  9. 9. Tt t ≥0 {Tt }t∈R + ,Tt : Cb (R 2 ) → Cb (R 2 ), ∞ Cb (R 2 ) ∞ Cb (R 2 ) u0 (x, y) (x, y, t) = (Tt u0 )(x, y), {Tt }t∈R + t Tt u0 t Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 3 / 17
  10. 10. Tt t ≥0 {Tt }t∈R + ,Tt : Cb (R 2 ) → Cb (R 2 ), ∞ Cb (R 2 ) ∞ Cb (R 2 ) u0 (x, y) (x, y, t) = (Tt u0 )(x, y), {Tt }t∈R + t Tt u0 t " " . Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 3 / 17
  11. 11. Tt t ≥0 {Tt }t∈R + ,Tt : Cb (R 2 ) → Cb (R 2 ), ∞ Cb (R 2 ) ∞ Cb (R 2 ) u0 (x, y) (x, y, t) = (Tt u0 )(x, y), {Tt }t∈R + t Tt u0 t " " . ( 10m 10cm ) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 3 / 17
  12. 12. Marr-Hildreth-Koenderink-WitKin 1980 Marr Hildreth[1] Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 4 / 17
  13. 13. Marr-Hildreth-Koenderink-WitKin 1980 Marr Hildreth[1] 1983 (Witkin[2],Koenderink [3]) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 4 / 17
  14. 14. Marr-Hildreth-Koenderink-WitKin 1980 Marr Hildreth[1] 1983 (Witkin[2],Koenderink [3]) 1986 Canny [4] Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 4 / 17
  15. 15. Marr-Hildreth-Koenderink-WitKin 1980 Marr Hildreth[1] 1983 (Witkin[2],Koenderink [3]) 1986 Canny [4] σ (0 ≤ σ < ∞) 1 −(x 2 +y 2 ) Gσ (x, y) = e 2σ2 4Πσ 2 . Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 4 / 17
  16. 16. Koenderink [3] Hummel [5] t ∂u(x, y, t) = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0 ∂t u(x, y, 0) = u0 (x, y), (x, y) ∈ ω Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 5 / 17
  17. 17. Koenderink [3] Hummel [5] t ∂u(x, y, t) = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0 ∂t u(x, y, 0) = u0 (x, y), (x, y) ∈ ω u0 (x, y) ω = (xa , xb )x(ya , yb )t ,∇2 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 5 / 17
  18. 18. :Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 6 / 17
  19. 19. : +∞ +∞ u(x, y, t) = u(x, y, 0) · Gt (x, y)dxdy −∞ −∞ −(x 2 +y 2 ) 1 Gt (x, y) Gt (x, y) = 4πt e 2tLi Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 6 / 17
  20. 20. : +∞ +∞ u(x, y, t) = u(x, y, 0) · Gt (x, y)dxdy −∞ −∞ −(x 2 +y 2 ) 1 Gt (x, y) Gt (x, y) = 4πt e 2t u(x,y,t) ( )t u0 (x, y) Gt (x, y)Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 6 / 17
  21. 21. .Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17
  22. 22. .Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17
  23. 23. .Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17
  24. 24. .Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17
  25. 25. .Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17
  26. 26. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 8 / 17
  27. 27. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 8 / 17
  28. 28. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 8 / 17
  29. 29. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17
  30. 30. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17
  31. 31. 1Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17
  32. 32. 12Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17
  33. 33. A ( )e(x,t) = ( ) = e(x,y)A∆x( ∆x ) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 10 / 17
  34. 34. A ( )e(x,t) = ( ) = e(x,y)A∆x( ∆x ) : x ∆x ∂[ex,t]A∆x x ∂t = + Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 10 / 17
  35. 35. A ( )e(x,t) = ( ) = e(x,y)A∆x( ∆x ) : x ∆x ∂[ex,t]A∆x x ∂t = +φ(x, t) = ( ) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 10 / 17
  36. 36. Qx,t = ( ) Q(x, t)A∆x Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 11 / 17
  37. 37. Qx,t = ( ) Q(x, t)A∆x ∂[e(x,t)A∆x] : ∂t ≈ Φ(x, t)A − Φ(x + ∆x, t)A + Q(x, t)A∆x Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 11 / 17
  38. 38. Qx,t = ( ) Q(x, t)A∆x ∂[e(x,t)A∆x] : ∂t ≈ Φ(x, t)A − Φ(x + ∆x, t)A + Q(x, t)A∆x∂e∂t = lim∆x→0 Φ(x,t)−Φ(x+∆x,t) + Q(x, t) ∆x Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 11 / 17
  39. 39. Qx,t = ( ) Q(x, t)A∆x ∂[e(x,t)A∆x] : ∂t ≈ Φ(x, t)A − Φ(x + ∆x, t)A + Q(x, t)A∆x∂e∂t = lim∆x→0 Φ(x,t)−Φ(x+∆x,t) + Q(x, t) ∆x∂e ∂φ∂t = − ∂x + Q Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 11 / 17
  40. 40. ,u(x,t) = ( t )Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  41. 41. , u(x,t) = ( t )c= ( ) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  42. 42. , u(x,t) = ( t )c= ( )ρ(x) = Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  43. 43. , u(x,t) = ( t )c= ( )ρ(x) = Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  44. 44. , u(x,t) = ( t )c= ( )ρ(x) = :e(x, t) = c(x)ρ(x)u(x, t) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  45. 45. , u(x,t) = ( t )c= ( )ρ(x) = :e(x, t) = c(x)ρ(x)u(x, t)c(x)ρ(x) ∂u = − ∂φ + Q ∂t ∂x Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  46. 46. , u(x,t) = ( t )c= ( )ρ(x) = :e(x, t) = c(x)ρ(x)u(x, t)c(x)ρ(x) ∂u = − ∂φ + Q ∂t ∂x φ = −K0 ∂u ∂x Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  47. 47. , u(x,t) = ( t )c= ( )ρ(x) = :e(x, t) = c(x)ρ(x)u(x, t)c(x)ρ(x) ∂u = − ∂φ + Q ∂t ∂x φ = −K0 ∂u ∂xcρ ∂u = ∂t ∂ ∂u ∂t (K0 ∂(x) ) + Q Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17
  48. 48. 2∂u∂t = k∂ u ∂x 2 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17
  49. 49. 2∂u∂t = k∂ u ∂x 2 K0k= cρ Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17
  50. 50. 2∂u∂t = k∂ u ∂x 2 K0k= cρ x2 : u(x, t) = √ 1 e − 4kt 4Πt Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17
  51. 51. 2∂u∂t = k∂ u ∂x 2 K0k= cρ x2 : u(x, t) = √ 1 e − 4kt 4Πt u(x, 0) = u0 (x) Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17
  52. 52. 2∂u∂t = k∂ u ∂x 2 K0k= cρ x2 : u(x, t) = √ 1 e − 4kt 4Πt u(x, 0) = u0 (x) x 2 +∞u(x, t) = √1 − 4kt 4Πt −∞ u0 (x)e dx Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17
  53. 53. ∂u(x,y ,t) ∂t = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 14 / 17
  54. 54. ∂u(x,y ,t) ∂t = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0 u(x, y, 0) = u0 (x, y), (x, y) ∈ ω Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 14 / 17
  55. 55. ∂u(x,y ,t) ∂t = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0 u(x, y, 0) = u0 (x, y), (x, y) ∈ ω +∞ +∞ u(x, y, t) = −∞ −∞ u(x, y, 0) · Gt (x, y)dxdy Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 14 / 17
  56. 56. ∂u(x,y ,t) ∂t = ∇2 u(x, y, t), (x, y) ∈ Ω , t > 0 u(x, y, 0) = u0 (x, y), (x, y) ∈ ω +∞ +∞ u(x, y, t) = −∞ −∞ u(x, y, 0) · Gt (x, y)dxdy −(x 2 +y 2 ) 1 Gt (x, y) Gt (x, y) = 4πt e 2t Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 14 / 17
  57. 57. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17
  58. 58. Hummel [6] Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17
  59. 59. Hummel [6]P-M Perona Malik [7] . Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17
  60. 60. Hummel [6]P-M Perona Malik [7] . Alvarez,Lions,Morel [8] Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17
  61. 61. [1]Marr D,and Hildreth E,Theory of edge detection.Proc.Roy.Soc.Lond,B207 p187-217,1980[2]A.P.Witkin.Space-scale filtering.In Proc.Of IJCAI,p1019-1021 1983[3]J.Koenderink.The structure of images.Biological Cybernation,Vol 50,p262-270 1984[4]A.Canny.A computational approach to edge detection.IEEE Trans.PAMI,vol 8,p769-698 1986[5]R.A.Hummel,Representations based on zero crossing in scale-space.CVPR p204-209 1986[6]R.A.Hummel,B.Kimia,Zucker,De-blurring Gaussian blur[J],1987[7]P.Perona,J.Malik,Scale-Space and edge detection using anisotropic diffusion. PAMI p629-639 1990 Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 16 / 17
  62. 62. ! Email/Gtalk: bugway@gmail.comLi Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 17 / 17

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