1. Scale Space The Gaussian Approach
Li Hui
bugway@gmail.com
July 8, 2009
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 1 / 17

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Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 2 / 17

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8. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 3 / 17

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. 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. 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. Marr-Hildreth-Koenderink-WitKin
1980 Marr Hildreth[1]
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 4 / 17

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. 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. 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. 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. 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. :
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 6 / 17

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
2t
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 6 / 17

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. .
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17

22. .
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17

23. .
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17

24. .
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17

25. .
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 7 / 17

26. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 8 / 17

27. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 8 / 17

28. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 8 / 17

29. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17

30. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17

31. 1
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17

32. 1
2
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 9 / 17

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. 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. 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. Qx,t = ( )
Q(x, t)A∆x
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 11 / 17

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. 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. 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. ,
u(x,t) = ( t )
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17

41. ,
u(x,t) = ( t )
c= ( )
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17

42. ,
u(x,t) = ( t )
c= ( )
ρ(x) =
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17

43. ,
u(x,t) = ( t )
c= ( )
ρ(x) =
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17

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. ,
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. ,
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. ,
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
cρ ∂u =
∂t
∂ ∂u
∂t (K0 ∂(x) ) + Q
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 12 / 17

48. 2
∂u
∂t = k∂ u
∂x 2
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17

49. 2
∂u
∂t = k∂ u
∂x 2
K0
k= cρ
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17

50. 2
∂u
∂t = k∂ u
∂x 2
K0
k= cρ
x2
: u(x, t) = √ 1 e − 4kt
4Πt
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 13 / 17

51. 2
∂u
∂t = k∂ u
∂x 2
K0
k= 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. 2
∂u
∂t = k∂ u
∂x 2
K0
k= 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. ∂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. ∂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. ∂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. ∂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. Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17

58. Hummel [6]
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17

59. Hummel [6]
P-M Perona Malik [7] .
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 15 / 17

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. [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. !
Email/Gtalk: bugway@gmail.com
Li Hui (Earth) Scale Space The Gaussian Approach July 8, 2009 17 / 17