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?.`3.$5$=)3M3,),2$=,/V.$-H/),CH$-H.$1*5C.(
Minimize some “energy function” of that curve
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Cost(x1,x
2, . . . , xn)=(wx1→x2+wx2→x3+.. . +wxn−1→xn)+$dist(xn,x
destination)
Cost(x1,x
2, . . . , xn)=(wx1→x2+.. . +wxn−1→xn)+$·dist(xn,x
destination)
Cost(x1,x
2, . . . , xn) =(wx1→x2+wx2→x3+. . . +wxn−1→xn)
+$·dist(xn,x
destination)
v(s) : [0,1] →R2
v(s)=(x(s),y(s))
E(v)="1
0
F(s, v, v",v
"")ds
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Cost(x1,x
2, . . . , xn)=(wx1→x2+wx2→x3+.. . +wxn−1→xn)+$dist(xn,x
destination)
Cost(x1,x
2, . . . , xn)=(wx1→x2+.. . +wxn−1→xn)+$·dist(xn,x
destination)
Cost(x1,x
2, . . . , xn) =(wx1→x2+wx2→x3+. . . +wxn−1→xn)
+$·dist(xn,x
destination)
v(s) : [0,1] →R2
v(s)=(x(s),y(s))
E(v)="1
0
F(s, v, v",v
"")ds
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Cost(x1,x
2, . . . , xn)=(wx1→x2+wx2→x3+. . . +wxn−1→xn)+$dist(xn,x
destination)
Cost(x1,x
2, . . . , xn)=(wx1→x2+. . . +wxn−1→xn)+$·dist(xn,x
destination)
Cost(x1,x
2, . . . , xn) =(wx1→x2+wx2→x3+. . . +wxn−1→xn)
+$·dist(xn,x
destination)
v(s) : [0,1] →R2
v(s)=(x(s),y(s))
E(v)="1
0
F(s, v, v",v
"")ds
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
BP+1=59$#35J.$!3./C1.2
•!DC.$6/)\1*1-P
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
•!952M=1-P
•#Ma3.22
•]2./$13-./5=M)3
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
BP+1=59$#35J.$!3./C1.2
•!DC.$6/)\1*1-P
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
Eedge(v)=−!1
0
EdgeCosts(x(s),y(s))ds
EdgeCosts(x, y )=M−|∇I(x, y)|2
EdgeCosts(x, y )= 1
"1+|∇I(x,y )|2
Euser(v)=!1
0
User(x(s),y(s))ds
Eelastic(v)= 1
2!1
0
α(s)·|v!(s)|2ds
Estiffness (v)=1
2!1
0
β(s)·|v!!(s)|2ds
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsinφ−ycos φ=ρ
ρ=rsin(φ−θ)
r="x2+y2
tanθ=y /x
Cost(x1,x
2,. . . , xn)=(wx1→x2+wx2→x3+.. . +wxn−1→xn)+&dist(xn,x
destination)
Cost(x1,x
2,. . . , xn)=(wx1→x2+.. . +wxn−1→xn)+&·dist(xn,x
destination)
Cost(x1,x
2,. . . , xn) =(wx1→x2+wx2→x3+. . . +wxn−1→xn)
+&·dist(xn,x
destination)
1
Eedge(v)=−!1
0
EdgeCosts(x(s),y(s))ds
EdgeCosts(x, y )=M−|∇I(x, y)|2
EdgeCosts(x, y )= 1
"1+|∇I(x,y )|2
Euser(v)=!1
0
User(x(s),y(s))ds
Eelastic(v)= 1
2!1
0
α(s)·|v!(s)|2ds
Estiffness (v)=1
2!1
0
β(s)·|v!!(s)|2ds
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsinφ−ycos φ=ρ
ρ=rsin(φ−θ)
r="x2+y2
tanθ=y /x
Cost(x1,x
2,. . . , xn)=(wx1→x2+wx2→x3+.. . +wxn−1→xn)+&dist(xn,x
destination)
Cost(x1,x
2,. . . , xn)=(wx1→x2+.. . +wxn−1→xn)+&·dist(xn,x
destination)
Cost(x1,x
2,. . . , xn) =(wx1→x2+wx2→x3+. . . +wxn−1→xn)
+&·dist(xn,x
destination)
1
•!952M=1-P
•#Ma3.22
•]2./$13-./5=M)3
Eedge(v) = + !1
0
EdgeC osts(x(s),y(s))ds
EdgeC osts(x, y)=M−|∇I(x, y)|2
EdgeC osts(x, y)= 1
"1+|∇I(x, y)|2
Euser(v)=!1
0
User(x(s),y(s))ds
Eelastic(v)= 1
2!1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2!1
0
β(s)·|v!!(s)|2ds
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r="x2+y2
tan θ=y/x
Cost(x1,x
2, . . . , xn)=(wx1→x2+wx2→x3+. . . +wxn−1→xn)+&dist(xn,x
destination)
Cost(x1,x
2, . . . , xn)=(wx1→x2+. . . +wxn−1→xn)+&·dist(xn,x
destination)
Cost(x1,x
2, . . . , xn) =(wx1→x2+wx2→x3+. . . +wxn−1→xn)
+&·dist(xn,x
destination)
1
y=mx +b
b=mx −y
(x0,y
0)
(x1,y
1)
(m0,b
0)
(m1,b
1)
xsin φ−ycos φ=ρ
ρ=rsin(φ−θ)
r=!x2+y2
tan θ=y/x
Eelastic(v)= 1
2"1
0
α(s)·|v!(s)|2ds
Estiff ness(v)= 1
2"1
0
β(s)·|v!!(s)|2ds
Eedge(v)=−"1
0
|∇I(x(s),y(s))|2ds
Euser(v)="1
0
User(x(s),y(s))ds
1
v(s) : [0,1] →R2
v(s)=(x(s),y(s))
E(v)=!1
0
F(s, v, v!,v
!!)ds
!1
0
F(x, v, v!,v
!!)ds
Fx−
∂
∂sFx!+∂2
∂s2Fx!!
Fy−
∂
∂sFy!+∂2
∂s2Fy!!
αx!! +βx!!!! =∂
∂xEdgeC osts(x, y)
2
v(s) : [0,1] →R2
v(s)=(x(s),y(s))
E(v)=!1
0
F(s, v, v!,v
!!)ds
!1
0
F(x, v, v!,v
!!)ds
Fx−
∂
∂sFx!+∂2
∂s2Fx!! =0
Fy−
∂
∂sFy!+∂2
∂s2Fy!! =0
αx!! +βx!!!! =∂
∂xEdgeC osts(x(s),y(s))
αy!! +βy!!!! =∂
∂yEdgeC osts(x(s),y(s))
2
b,*./1=59$#)9,M)3
!,9./$<5C/53C.$!c,5M)3(
W@$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$12$*131*17.DI$-H.3
