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Scale SpaceµµµThe Gaussian Approach
Li Hui
July 8, 2009
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 1 / 17
SN
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 2 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 2 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 2 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 2 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 2 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 2 / 17
ºÝm
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 3 / 17
ºÝm
bTt´q±t ≥ 0ëêã²wf§PTtt∈R+ ,¿÷
vTt : C∞b (R2) → Cb(R2),Ù¥C∞
b (R2)L«k.«mþäkáëY
ꢼê8ܧCb(R2)L«k.«mþ¢¼ê8
Ü"XJéuÌãu0(x , y)k(x , y , t) = (Ttu0)(x , y),K¡Ttt∈R+
ºÝm§tºÝëê§Ttu0´ã3ºÝëêteN§½þ
Aã1w§Ý"
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 3 / 17
ºÝm
bTt´q±t ≥ 0ëêã²wf§PTtt∈R+ ,¿÷
vTt : C∞b (R2) → Cb(R2),Ù¥C∞
b (R2)L«k.«mþäkáëY
ꢼê8ܧCb(R2)L«k.«mþ¢¼ê8
Ü"XJéuÌãu0(x , y)k(x , y , t) = (Ttu0)(x , y),K¡Ttt∈R+
ºÝm§tºÝëê§Ttu0´ã3ºÝëêteN§½þ
Aã1w§Ý"
Äu<*ÔNÿ"ºÝ")nA5.
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 3 / 17
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bTt´q±t ≥ 0ëêã²wf§PTtt∈R+ ,¿÷
vTt : C∞b (R2) → Cb(R2),Ù¥C∞
b (R2)L«k.«mþäkáëY
ꢼê8ܧCb(R2)L«k.«mþ¢¼ê8
Ü"XJéuÌãu0(x , y)k(x , y , t) = (Ttu0)(x , y),K¡Ttt∈R+
ºÝm§tºÝëê§Ttu0´ã3ºÝëêteN§½þ
Aã1w§Ý"
Äu<*ÔNÿ"ºÝ")nA5.
ÄuÔNºÝé'A(ºÝ10mÚºÝ10cme¡f)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 3 / 17
Marr-Hildreth-Koenderink-WitKin
1980cdMarrÚ Hildreth[1] JÑ
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 4 / 17
Marr-Hildreth-Koenderink-WitKin
1980cdMarrÚ Hildreth[1] JÑ
¤/u1983c(Witkin[2],Koenderink [3])
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 4 / 17
Marr-Hildreth-Koenderink-WitKin
1980cdMarrÚ Hildreth[1] JÑ
¤/u1983c(Witkin[2],Koenderink [3])
1986cCanny [4]U?
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 4 / 17
Marr-Hildreth-Koenderink-WitKin
1980cdMarrÚ Hildreth[1] JÑ
¤/u1983c(Witkin[2],Koenderink [3])
1986cCanny [4]U?
ÄgÑ´^σ (0 ≤ σ < ∞)ØäO\pdØ
Gσ(x , y) =1
4Πσ2 e−(x2+y2)
2σ2
©ã?1òÈãS5EºÝm.
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 4 / 17
9D.
Koenderink [3]ÚHummel [5]5ѧ3ºÝte§&Òpd
òÈdu±&Òв;D9§)µ
∂u(x , y , t)∂t
= ∇2u(x , y , t), (x , y) ∈ Ω , t > 0
u(x , y , 0) = u0(x , y), (x , y) ∈ ω
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 5 / 17
9D.
Koenderink [3]ÚHummel [5]5ѧ3ºÝte§&Òpd
òÈdu±&Òв;D9§)µ
∂u(x , y , t)∂t
= ∇2u(x , y , t), (x , y) ∈ Ω , t > 0
u(x , y , 0) = u0(x , y), (x , y) ∈ ω
ùpu0(x , y)©G§ω = (xa, xb)x(ya, yb)©G½Â
tºÝÏf,∇2.Ê.df
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 5 / 17
9D.Y
þã§):
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 6 / 17
9D.Y
þã§):
u(x , y , t) =
∫ +∞
−∞
∫ +∞
−∞u(x , y , 0) · Gt(x , y)dxdy
Ù¥Gt(x , y)pd¼êGt(x , y) = 14πt e
−(x2+y2)2t §
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 6 / 17
9D.Y
þã§):
u(x , y , t) =
∫ +∞
−∞
∫ +∞
−∞u(x , y , 0) · Gt(x , y)dxdy
Ù¥Gt(x , y)pd¼êGt(x , y) = 14πt e
−(x2+y2)2t §
dd±w§§)u(x,y,t)´±ºÝÏf(m)t1wëê§
¿dã u0(x , y)²LGt (x , y)ÈÅ(J"
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 6 / 17
XÛEºÝm
nØþE§?n¡.
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 7 / 17
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nØþE§?n¡.
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 7 / 17
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^/[/
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 7 / 17
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nØþE§?n¡.
?nlÑêâ§æ^Øf?nz:Sêâ
^/[/
©Y²ÚR?1òÈ
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 7 / 17
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nØþE§?n¡.
?nlÑêâ§æ^Øf?nz:Sêâ
^/[/
©Y²ÚR?1òÈ
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 7 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 8 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 8 / 17
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Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 8 / 17
pd
pd¼êäkéõ`û5§lêÆÝy²§æ^pdØ
EºÝm÷vþ¡Jn^
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 9 / 17
pd
pd¼êäkéõ`û5§lêÆÝy²§æ^pdØ
EºÝm÷vþ¡Jn^
«*n)ªÒ´l9DÝn)ùL§"
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 9 / 17
pd
pd¼êäkéõ`û5§lêÆÝy²§æ^pdØ
EºÝm÷vþ¡Jn^
«*n)ªÒ´l9DÝn)ùL§"
1 9D§
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 9 / 17
pd
pd¼êäkéõ`û5§lêÆÝy²§æ^pdØ
EºÝm÷vþ¡Jn^
«*n)ªÒ´l9DÝn)ùL§"
1 9D§
2 9D§
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 9 / 17
9D§
Ä3.¡ÈA\p¡9DL§(±ý§üàÃ)
e(x,t) =9UÝ(ü NÈ¥9Uþ)9U = e(x,y)A∆x(3∆x
Ýp9U)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 10 / 17
9D§
Ä3.¡ÈA\p¡9DL§(±ý§üàÃ)
e(x,t) =9UÝ(ü NÈ¥9Uþ)9U = e(x,y)A∆x(3∆x
Ýp9U)
9UÅð: 3xÚ∆xm9UmCzÑ´d6L¡üà
9UÚSÜ)9UÚå§bý¡ýéu½
x§9UCzÇÒ´∂[ex,t]A∆x∂t 9U]CzÇ =ü m6L
>.9U +ü mSØ)9U
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 10 / 17
9D§
Ä3.¡ÈA\p¡9DL§(±ý§üàÃ)
e(x,t) =9UÝ(ü NÈ¥9Uþ)9U = e(x,y)A∆x(3∆x
Ýp9U)
9UÅð: 3xÚ∆xm9UmCzÑ´d6L¡üà
9UÚSÜ)9UÚå§bý¡ýéu½
x§9UCzÇÒ´∂[ex,t]A∆x∂t 9U]CzÇ =ü m6L
>.9U +ü mSØ)9U
9Ïþµ
φ(x , t) =9Ïþ(ü mS9U6ü L¡Èm>9Uþ)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 10 / 17
9D§
Qx,t =9 (ü m3ü NÈS)9U)
éu¹§ØÄþÝ9þ¿§ü mS)
9UCqQ(x , t)A∆x
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 11 / 17
9D§
Qx,t =9 (ü m3ü NÈS)9U)
éu¹§ØÄþÝ9þ¿§ü mS)
9UCqQ(x , t)A∆x
9UÅð: ∂[e(x,t)A∆x]∂t ≈ Φ(x , t)A − Φ(x + ∆x , t)A + Q(x , t)A∆x
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 11 / 17
9D§
Qx,t =9 (ü m3ü NÈS)9U)
éu¹§ØÄþÝ9þ¿§ü mS)
9UCqQ(x , t)A∆x
9UÅð: ∂[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)∆x + Q(x , t)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 11 / 17
9D§
Qx,t =9 (ü m3ü NÈS)9U)
éu¹§ØÄþÝ9þ¿§ü mS)
9UCqQ(x , t)A∆x
9UÅð: ∂[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)∆x + Q(x , t)
∂e∂t = −∂φ
∂x + Q
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 11 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
ρ(x) =þÝ
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
ρ(x) =þÝ
\þ¡þµ
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
ρ(x) =þÝ
\þ¡þµ
·k:e(x , t) = c(x)ρ(x)u(x , t)
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
ρ(x) =þÝ
\þ¡þµ
·k:e(x , t) = c(x)ρ(x)u(x , t)
c(x)ρ(x)∂u∂t = −∂φ
∂x + Q
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
ρ(x) =þÝ
\þ¡þµ
·k:e(x , t) = c(x)ρ(x)u(x , t)
c(x)ρ(x)∂u∂t = −∂φ
∂x + Q
Fp½Æµ φ = −K0∂u∂x
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
Ï~·^§Ý5ïþ9þ,b§ÝÚ9þ¤'
u(x,t) =§Ý(L«,:3t¯K)
c ='9N(ü þÔþ,ü §Ý¤I9þ)
ρ(x) =þÝ
\þ¡þµ
·k:e(x , t) = c(x)ρ(x)u(x , t)
c(x)ρ(x)∂u∂t = −∂φ
∂x + Q
Fp½Æµ φ = −K0∂u∂x
cρ∂u∂t = ∂
∂t (K0∂u
∂(x)) + Q
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 12 / 17
9D§
∂u∂t = k ∂2u
∂x2
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 13 / 17
9D§
∂u∂t = k ∂2u
∂x2
k = K0cρ
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 13 / 17
9D§
∂u∂t = k ∂2u
∂x2
k = K0cρ
ÏL©lCþ¦: u(x , t) = 1√4Πt
e− x2
4kt
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 13 / 17
9D§
∂u∂t = k ∂2u
∂x2
k = K0cρ
ÏL©lCþ¦: u(x , t) = 1√4Πt
e− x2
4kt
bXkå^§=UþЩG u(x , 0) = u0(x)Kd)
µ
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 13 / 17
9D§
∂u∂t = k ∂2u
∂x2
k = K0cρ
ÏL©lCþ¦: u(x , t) = 1√4Πt
e− x2
4kt
bXkå^§=UþЩG u(x , 0) = u0(x)Kd)
µ
u(x , t) = 1√4Πt
∫ +∞−∞ u0(x)e− x2
4kt dx
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 13 / 17
9D§
∂u(x,y ,t)∂t = ∇2u(x , y , t), (x , y) ∈ Ω , t > 0
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 14 / 17
9D§
∂u(x,y ,t)∂t = ∇2u(x , y , t), (x , y) ∈ Ω , t > 0
Ù¥kå^µ u(x , y , 0) = u0(x , y), (x , y) ∈ ω
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 14 / 17
9D§
∂u(x,y ,t)∂t = ∇2u(x , y , t), (x , y) ∈ Ω , t > 0
Ù¥kå^µ u(x , y , 0) = u0(x , y), (x , y) ∈ ω
)µ u(x , y , t) =∫ +∞−∞
∫ +∞−∞ u(x , y , 0) · Gt(x , y)dxdy
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 14 / 17
9D§
∂u(x,y ,t)∂t = ∇2u(x , y , t), (x , y) ∈ Ω , t > 0
Ù¥kå^µ u(x , y , 0) = u0(x , y), (x , y) ∈ ω
)µ u(x , y , t) =∫ +∞−∞
∫ +∞−∞ u(x , y , 0) · Gt(x , y)dxdy
Ù¥Gt(x , y)pd¼êGt(x , y) = 14πt e
−(x2+y2)2t
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 14 / 17
":U?
pd.3z:?ÑäkÓ*Ñݧò:§>.Ó
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 15 / 17
":U?
pd.3z:?ÑäkÓ*Ñݧò:§>.Ó
Hummel [6]Ñ¿kc©J9§±EºÝm§
÷v4nuЧѱ^5L«ùm"
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 15 / 17
":U?
pd.3z:?ÑäkÓ*Ñݧò:§>.Ó
Hummel [6]Ñ¿kc©J9§±EºÝm§
÷v4nuЧѱ^5L«ùm"
P-M.dPeronaÚMalik [7]JÑ5§þ*ÑXêØÓ.
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 15 / 17
":U?
pd.3z:?ÑäkÓ*Ñݧò:§>.Ó
Hummel [6]Ñ¿kc©J9§±EºÝm§
÷v4nuЧѱ^5L«ùm"
P-M.dPeronaÚMalik [7]JÑ5§þ*ÑXêØÓ.
5ÄuÇ6Alvarez,Lions,Morel [8]
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 15 / 17
ë][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 SpaceµµµThe Gaussian Approach July 8, 2009 16 / 17
!
Email/Gtalk: [email protected]
Li Hui (Earth) Scale SpaceµµµThe Gaussian Approach July 8, 2009 17 / 17
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