![Page 1: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/1.jpg)
Introduction
Variational Methods in Image Processing
ÚTIA AV CR
Variational Methods
![Page 2: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/2.jpg)
Introduction
Outline
1 IntroductionMotivationDerivation of Euler-Lagrange EquationVariational Problem and P.D.E.
Variational Methods
![Page 3: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/3.jpg)
Introduction Motivation E-L PDE
Outline
1 IntroductionMotivationDerivation of Euler-Lagrange EquationVariational Problem and P.D.E.
Variational Methods
![Page 4: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/4.jpg)
Introduction Motivation E-L PDE
History
The Brachistochrone Problem:
“Given two points A and B in a verticalplane, what is the curve traced out bya point acted on only by gravity, whichstarts at A and reaches B in theshortest time.”Johann Bernoulli in 1696
In one year Newton, Johann andJacob Bernoulli, Leibniz, and deL’Hôpital came with the solution.
Variational Methods
![Page 5: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/5.jpg)
Introduction Motivation E-L PDE
History
The Brachistochrone Problem:
“Given two points A and B in a verticalplane, what is the curve traced out bya point acted on only by gravity, whichstarts at A and reaches B in theshortest time.”Johann Bernoulli in 1696
In one year Newton, Johann andJacob Bernoulli, Leibniz, and deL’Hôpital came with the solution.
Variational Methods
![Page 6: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/6.jpg)
Introduction Motivation E-L PDE
History
The Brachistochrone Problem:
“Given two points A and B in a verticalplane, what is the curve traced out bya point acted on only by gravity, whichstarts at A and reaches B in theshortest time.”Johann Bernoulli in 1696
In one year Newton, Johann andJacob Bernoulli, Leibniz, and deL’Hôpital came with the solution.
Variational Methods
![Page 7: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/7.jpg)
Introduction Motivation E-L PDE
History
The problem was generalized and an analytic method wasgiven by Euler (1744) and Lagrange (1760).
Variational Methods
![Page 8: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/8.jpg)
Introduction Motivation E-L PDE
Calculus of Variations
Most of the image processing tasks can be formulated asoptimization problems, i.e., minimization of functionals
Calculus of Variations solves
minu
F (u(x)) ,
where u ∈ X ,F : X → R,X . . . Banach spacesolution by means of Euler-Lagrange (E-L) equation
Variational Methods
![Page 9: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/9.jpg)
Introduction Motivation E-L PDE
Calculus of Variations
Most of the image processing tasks can be formulated asoptimization problems, i.e., minimization of functionalsCalculus of Variations solves
minu
F (u(x)) ,
where u ∈ X ,F : X → R,X . . . Banach space
solution by means of Euler-Lagrange (E-L) equation
Variational Methods
![Page 10: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/10.jpg)
Introduction Motivation E-L PDE
Calculus of Variations
Most of the image processing tasks can be formulated asoptimization problems, i.e., minimization of functionalsCalculus of Variations solves
minu
F (u(x)) ,
where u ∈ X ,F : X → R,X . . . Banach spacesolution by means of Euler-Lagrange (E-L) equation
Variational Methods
![Page 11: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/11.jpg)
Introduction Motivation E-L PDE
Calculus of Variations
Integral functionals
F (u) =
∫Ω
f (x ,u(x),∇u(x))dx
Example
x ∈ R2 . . . space of coordinates [x1, x2]
Ω . . . image supportu(x) : R2 → R . . . grayscale image∇u(x) . . . image gradient [ux1 ,ux2 ]
Variational Methods
![Page 12: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/12.jpg)
Introduction Motivation E-L PDE
Examples
Image Registrationgiven a set of CP pairs [xi , yi ]↔ [xi , yi ]find x = f (x , y), y = g(x , y)
F (f ) =∑
i
(xi − f (xi , yi))2 + λ
∫ ∫f 2xx + 2f 2
xy + f 2yydxdy
and a similar equation for g(x , y)
Image Reconstruction
Variational Methods
![Page 13: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/13.jpg)
Introduction Motivation E-L PDE
Examples
Image Registration
Image Reconstruction
Variational Methods
![Page 14: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/14.jpg)
Introduction Motivation E-L PDE
Examples
Image Registration
Image Reconstructiongiven an image acquisition model H(·) and measurement gfind the original image u
F (u) =
∫(H(u)− g)2dx + λ
∫|∇u|2
Variational Methods
![Page 15: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/15.jpg)
Introduction Motivation E-L PDE
Examples
Image Registration
Image Reconstruction
Variational Methods
![Page 16: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/16.jpg)
Introduction Motivation E-L PDE
Examples
Image Segmentationfind a piece-wise constant representation u of an image g
F (u,K ) =
∫Ω−K
(u − g)2dx + α
∫Ω−K|∇u|2dx + β
∫K
ds
Motion Estimation
Variational Methods
![Page 17: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/17.jpg)
Introduction Motivation E-L PDE
Examples
Image Segmentation
Motion Estimation
Variational Methods
![Page 18: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/18.jpg)
Introduction Motivation E-L PDE
Examples
Image Segmentation
Motion Estimationfind velocity field v(x) ≡ [v1(x), v2(x)] in an image sequenceu(x , t)
F (v) =
∫|v · ∇u + ut |dx + α
∑j
∫|∇vj |dx + β
∫c(∇u)|v |2dx
Variational Methods
![Page 19: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/19.jpg)
Introduction Motivation E-L PDE
Examples
Image Segmentation
Motion Estimation
Variational Methods
![Page 20: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/20.jpg)
Introduction Motivation E-L PDE
Examples
Image classification
and many more
Variational Methods
![Page 21: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/21.jpg)
Introduction Motivation E-L PDE
Examples
Image classificationand many more
Variational Methods
![Page 22: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/22.jpg)
Introduction Motivation E-L PDE
Outline
1 IntroductionMotivationDerivation of Euler-Lagrange EquationVariational Problem and P.D.E.
Variational Methods
![Page 23: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/23.jpg)
Introduction Motivation E-L PDE
Extrema points
From the differential calculus follows that
if x is an extremum of g(x) : RN → R then ∀ν ∈ RN
ddε
g(x + εν)∣∣∣ε=0
= 0
= 〈∇g(x), ν〉 ⇔ ∇g(x) = 0
in 1-D (g : R → R) we get the classical condition
g′(x) = 0
Variational Methods
![Page 24: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/24.jpg)
Introduction Motivation E-L PDE
Extrema points
From the differential calculus follows thatif x is an extremum of g(x) : RN → R then ∀ν ∈ RN
ddε
g(x + εν)∣∣∣ε=0
= 0
= 〈∇g(x), ν〉 ⇔ ∇g(x) = 0
in 1-D (g : R → R) we get the classical condition
g′(x) = 0
Variational Methods
![Page 25: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/25.jpg)
Introduction Motivation E-L PDE
Extrema points
From the differential calculus follows thatif x is an extremum of g(x) : RN → R then ∀ν ∈ RN
ddε
g(x + εν)∣∣∣ε=0
= 0
= 〈∇g(x), ν〉 ⇔ ∇g(x) = 0
in 1-D (g : R → R) we get the classical condition
g′(x) = 0
Variational Methods
![Page 26: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/26.jpg)
Introduction Motivation E-L PDE
Extrema points
From the differential calculus follows thatif x is an extremum of g(x) : RN → R then ∀ν ∈ RN
ddε
g(x + εν)∣∣∣ε=0
= 0 = 〈∇g(x), ν〉
⇔ ∇g(x) = 0
in 1-D (g : R → R) we get the classical condition
g′(x) = 0
Variational Methods
![Page 27: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/27.jpg)
Introduction Motivation E-L PDE
Extrema points
From the differential calculus follows thatif x is an extremum of g(x) : RN → R then ∀ν ∈ RN
ddε
g(x + εν)∣∣∣ε=0
= 0 = 〈∇g(x), ν〉 ⇔ ∇g(x) = 0
in 1-D (g : R → R) we get the classical condition
g′(x) = 0
Variational Methods
![Page 28: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/28.jpg)
Introduction Motivation E-L PDE
Extrema points
From the differential calculus follows thatif x is an extremum of g(x) : RN → R then ∀ν ∈ RN
ddε
g(x + εν)∣∣∣ε=0
= 0 = 〈∇g(x), ν〉 ⇔ ∇g(x) = 0
in 1-D (g : R → R) we get the classical condition
g′(x) = 0
Variational Methods
![Page 29: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/29.jpg)
Introduction Motivation E-L PDE
Variation of Functional
F (u) =
∫ b
af (x ,u,u′)dx
if u is extremum of F then fromdifferential calculus follows
ddε
F (u + εv)∣∣∣ε=0
= 0 ∀v
F (u +εv) =
∫ b
af (x ,u +εv ,u′+εv ′)dx
Variational Methods
![Page 30: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/30.jpg)
Introduction Motivation E-L PDE
Variation of Functional
F (u) =
∫ b
af (x ,u,u′)dx
if u is extremum of F then fromdifferential calculus follows
ddε
F (u + εv)∣∣∣ε=0
= 0 ∀v
F (u +εv) =
∫ b
af (x ,u +εv ,u′+εv ′)dx
Variational Methods
![Page 31: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/31.jpg)
Introduction Motivation E-L PDE
Chain Rule
ddx
f (u(x), v(x)) =( ∂∂u
f (u, v))du
dx+( ∂∂v
f (u, v))dv
dx
Example
u = x , v = sin x , f = uv = x sin x
ddx
f (u, v) = v(x)1 + u(x) cos x = sin x + x cos x
Variational Methods
![Page 32: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/32.jpg)
Introduction Motivation E-L PDE
Chain Rule
ddx
f (u(x), v(x)) =( ∂∂u
f (u, v))du
dx+( ∂∂v
f (u, v))dv
dx
Example
u = x , v = sin x , f = uv = x sin x
ddx
f (u, v) = v(x)1 + u(x) cos x = sin x + x cos x
Variational Methods
![Page 33: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/33.jpg)
Introduction Motivation E-L PDE
Chain Rule
ddx
f (u(x), v(x)) =( ∂∂u
f (u, v))du
dx+( ∂∂v
f (u, v))dv
dx
Example
u = x , v = sin x , f = uv = x sin x
ddx
f (u, v) = v(x)1 + u(x) cos x = sin x + x cos x
Variational Methods
![Page 34: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/34.jpg)
Introduction Motivation E-L PDE
Chain Rule
ddx
f (u(x), v(x)) =( ∂∂u
f (u, v))du
dx+( ∂∂v
f (u, v))dv
dx
Example
u = x , v = sin x , f = uv = x sin x
ddx
f (u, v) = v(x)1 + u(x) cos x = sin x + x cos x
Variational Methods
![Page 35: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/35.jpg)
Introduction Motivation E-L PDE
Partial derivatives
Example
f (x ,u) = x sin x
dfdx
= chain rule = sin x + x cos x
but∂f∂x
= sin x
Variational Methods
![Page 36: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/36.jpg)
Introduction Motivation E-L PDE
Partial derivatives
Example
f (x ,u) = x sin x
dfdx
= chain rule = sin x + x cos x
but∂f∂x
= sin x
Variational Methods
![Page 37: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/37.jpg)
Introduction Motivation E-L PDE
per partes
∫ b
auv ′ = uv
∣∣∣ba−∫ b
au′v
Variational Methods
![Page 38: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/38.jpg)
Introduction Motivation E-L PDE
Derivation of E-L equation
ddε
F (u + εv) =ddε
∫ b
af (x ,u + εv ,u′ + εv ′)
=
∫ b
a
∂f∂u
v +∂f∂u′
v ′ chain rule
=
∫ b
a
∂f∂u
v −∫ b
a
ddx
∂f∂u′
v +∂f∂u′
v∣∣∣ba
per partes
=
∫ b
a
[∂f∂u− d
dx∂f∂u′
]v +
∂f∂u′
v∣∣∣ba
= 0
Variational Methods
![Page 39: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/39.jpg)
Introduction Motivation E-L PDE
Derivation of E-L equation
ddε
F (u + εv) =ddε
∫ b
af (x ,u + εv ,u′ + εv ′)
=
∫ b
a
∂f∂u
v +∂f∂u′
v ′ chain rule
=
∫ b
a
∂f∂u
v −∫ b
a
ddx
∂f∂u′
v +∂f∂u′
v∣∣∣ba
per partes
=
∫ b
a
[∂f∂u− d
dx∂f∂u′
]v +
∂f∂u′
v∣∣∣ba
= 0
Variational Methods
![Page 40: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/40.jpg)
Introduction Motivation E-L PDE
Derivation of E-L equation
ddε
F (u + εv) =ddε
∫ b
af (x ,u + εv ,u′ + εv ′)
=
∫ b
a
∂f∂u
v +∂f∂u′
v ′ chain rule
=
∫ b
a
∂f∂u
v −∫ b
a
ddx
∂f∂u′
v +∂f∂u′
v∣∣∣ba
per partes
=
∫ b
a
[∂f∂u− d
dx∂f∂u′
]v +
∂f∂u′
v∣∣∣ba
= 0
Variational Methods
![Page 41: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/41.jpg)
Introduction Motivation E-L PDE
Derivation of E-L equation
ddε
F (u + εv) =ddε
∫ b
af (x ,u + εv ,u′ + εv ′)
=
∫ b
a
∂f∂u
v +∂f∂u′
v ′ chain rule
=
∫ b
a
∂f∂u
v −∫ b
a
ddx
∂f∂u′
v +∂f∂u′
v∣∣∣ba
per partes
=
∫ b
a
[∂f∂u− d
dx∂f∂u′
]v +
∂f∂u′
v∣∣∣ba
= 0
Variational Methods
![Page 42: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/42.jpg)
Introduction Motivation E-L PDE
Derivation of E-L equation
ddε
F (u + εv) =ddε
∫ b
af (x ,u + εv ,u′ + εv ′)
=
∫ b
a
∂f∂u
v +∂f∂u′
v ′ chain rule
=
∫ b
a
∂f∂u
v −∫ b
a
ddx
∂f∂u′
v +∂f∂u′
v∣∣∣ba
per partes
=
∫ b
a
[∂f∂u− d
dx∂f∂u′
]v +
∂f∂u′
v∣∣∣ba
= 0
to be equal to 0 for any v ,[∂f∂u −
ddx
∂f∂u′
]= 0→ E-L equation
Variational Methods
![Page 43: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/43.jpg)
Introduction Motivation E-L PDE
Derivation of E-L equation
ddε
F (u + εv) =ddε
∫ b
af (x ,u + εv ,u′ + εv ′)
=
∫ b
a
∂f∂u
v +∂f∂u′
v ′ chain rule
=
∫ b
a
∂f∂u
v −∫ b
a
ddx
∂f∂u′
v +∂f∂u′
v∣∣∣ba
per partes
=
∫ b
a
[∂f∂u− d
dx∂f∂u′
]v +
∂f∂u′
v∣∣∣ba
= 0
to be equal to 0, we need boundary conditions,e.g., fixed u(a),u(b)→ v(a) = v(b) = 0.
Variational Methods
![Page 44: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/44.jpg)
Introduction Motivation E-L PDE
Toy caseShortest path
Find the shortest path betweenpoints A and B, assuming thatone can write y = y(x).
We want to minimize F (y(x)) =∫ b
a
√1 + y ′(x)2dx
with b.c. y(a) = α, y(b) = β.
E-L eq.: − ddx
y ′(x)√1+y ′(x)2
= 0
⇒ y ′ = C√
1 + y ′2 ⇒
y ′ = constant
y(x) is a straight line between A and B.
Variational Methods
![Page 45: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/45.jpg)
Introduction Motivation E-L PDE
Toy caseShortest path
Find the shortest path betweenpoints A and B, assuming thatone can write y = y(x).
We want to minimize F (y(x)) =∫ b
a
√1 + y ′(x)2dx
with b.c. y(a) = α, y(b) = β.
E-L eq.: − ddx
y ′(x)√1+y ′(x)2
= 0
⇒ y ′ = C√
1 + y ′2 ⇒
y ′ = constant
y(x) is a straight line between A and B.
Variational Methods
![Page 46: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/46.jpg)
Introduction Motivation E-L PDE
Toy caseShortest path
Find the shortest path betweenpoints A and B, assuming thatone can write y = y(x).
We want to minimize F (y(x)) =∫ b
a
√1 + y ′(x)2dx
with b.c. y(a) = α, y(b) = β.
E-L eq.: − ddx
y ′(x)√1+y ′(x)2
= 0
⇒ y ′ = C√
1 + y ′2 ⇒
y ′ = constanty(x) is a straight line between A and B.
Variational Methods
![Page 47: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/47.jpg)
Introduction Motivation E-L PDE
Toy caseShortest path
Find the shortest path betweenpoints A and B, assuming thatone can write y = y(x).
We want to minimize F (y(x)) =∫ b
a
√1 + y ′(x)2dx
with b.c. y(a) = α, y(b) = β.
E-L eq.: − ddx
y ′(x)√1+y ′(x)2
= 0⇒ y ′ = C√
1 + y ′2
⇒
y ′ = constanty(x) is a straight line between A and B.
Variational Methods
![Page 48: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/48.jpg)
Introduction Motivation E-L PDE
Toy caseShortest path
Find the shortest path betweenpoints A and B, assuming thatone can write y = y(x).
We want to minimize F (y(x)) =∫ b
a
√1 + y ′(x)2dx
with b.c. y(a) = α, y(b) = β.
E-L eq.: − ddx
y ′(x)√1+y ′(x)2
= 0⇒ y ′ = C√
1 + y ′2 ⇒
y ′ = constant
y(x) is a straight line between A and B.
Variational Methods
![Page 49: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/49.jpg)
Introduction Motivation E-L PDE
Toy caseShortest path
Find the shortest path betweenpoints A and B, assuming thatone can write y = y(x).
We want to minimize F (y(x)) =∫ b
a
√1 + y ′(x)2dx
with b.c. y(a) = α, y(b) = β.
E-L eq.: − ddx
y ′(x)√1+y ′(x)2
= 0⇒ y ′ = C√
1 + y ′2 ⇒
y ′ = constanty(x) is a straight line between A and B.
Variational Methods
![Page 50: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/50.jpg)
Introduction Motivation E-L PDE
E-L equation
If u(x) : RN → R is extremum of F (u) =∫
Ω f (x ,u,∇u)dx ,where ∇u ≡ [ux1 , . . . ,uxN ]then
F ′(u) =∂f∂u
(x ,u,∇u)−N∑
i=1
ddxi
(∂f∂uxi
(x ,u,∇u)
)= 0 ,
which is the E-L equation.
Variational Methods
![Page 51: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/51.jpg)
Introduction Motivation E-L PDE
E-L equation
If u(x) : RN → R is extremum of F (u) =∫
Ω f (x ,u,∇u)dx ,where ∇u ≡ [ux1 , . . . ,uxN ]then
F ′(u) =∂f∂u
(x ,u,∇u)−N∑
i=1
ddxi
(∂f∂uxi
(x ,u,∇u)
)= 0 ,
which is the E-L equation.
Variational Methods
![Page 52: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/52.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
f (x ,u,u′)∂f∂u− d
dx
( ∂f∂u′)
= 0
dfdx
=∂f∂u
u′ +∂f∂u′
u′′ +∂f∂x
∂f∂u
u′ =dfdx− ∂f∂u′
u′′ − ∂f∂x
u′∂f∂u− u′
ddx
( ∂f∂u′)
= 0
dfdx− ∂f∂u′
u′′ − ∂f∂x− u′
ddx
( ∂f∂u′)
= 0
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
Variational Methods
![Page 53: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/53.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
f (x ,u,u′)∂f∂u− d
dx
( ∂f∂u′)
= 0
dfdx
=∂f∂u
u′ +∂f∂u′
u′′ +∂f∂x
∂f∂u
u′ =dfdx− ∂f∂u′
u′′ − ∂f∂x
u′∂f∂u− u′
ddx
( ∂f∂u′)
= 0
dfdx− ∂f∂u′
u′′ − ∂f∂x− u′
ddx
( ∂f∂u′)
= 0
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
Variational Methods
![Page 54: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/54.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
f (x ,u,u′)∂f∂u− d
dx
( ∂f∂u′)
= 0
dfdx
=∂f∂u
u′ +∂f∂u′
u′′ +∂f∂x
∂f∂u
u′ =dfdx− ∂f∂u′
u′′ − ∂f∂x
u′∂f∂u− u′
ddx
( ∂f∂u′)
= 0
dfdx− ∂f∂u′
u′′ − ∂f∂x− u′
ddx
( ∂f∂u′)
= 0
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
Variational Methods
![Page 55: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/55.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
f (x ,u,u′)∂f∂u− d
dx
( ∂f∂u′)
= 0
dfdx
=∂f∂u
u′ +∂f∂u′
u′′ +∂f∂x
∂f∂u
u′ =dfdx− ∂f∂u′
u′′ − ∂f∂x
u′∂f∂u− u′
ddx
( ∂f∂u′)
= 0
dfdx− ∂f∂u′
u′′ − ∂f∂x− u′
ddx
( ∂f∂u′)
= 0
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
Variational Methods
![Page 56: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/56.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
f (x ,u,u′)∂f∂u− d
dx
( ∂f∂u′)
= 0
dfdx
=∂f∂u
u′ +∂f∂u′
u′′ +∂f∂x
∂f∂u
u′ =dfdx− ∂f∂u′
u′′ − ∂f∂x
u′∂f∂u− u′
ddx
( ∂f∂u′)
= 0
dfdx− ∂f∂u′
u′′ − ∂f∂x− u′
ddx
( ∂f∂u′)
= 0
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
Variational Methods
![Page 57: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/57.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
if ∂f∂x = 0 then
ddx
(f − u′
∂f∂u′)
= 0⇐⇒ f − u′∂f∂u′
= C
Variational Methods
![Page 58: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/58.jpg)
Introduction Motivation E-L PDE
Beltrami Identity
ddx
(f − u′
∂f∂u′)− ∂f∂x
= 0
if ∂f∂x = 0 then
ddx
(f − u′
∂f∂u′)
= 0⇐⇒ f − u′∂f∂u′
= C
Variational Methods
![Page 59: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/59.jpg)
Introduction Motivation E-L PDE
Brachistochrone
F =∫
dt , minF . . . curve of theshortest time.
F =∫ ds
v =∫ b
0
√1+(y ′(x))2
v dx12mv2 = mgy(x)⇒ v =
√2gy(x)
F =
∫ b
0
√1 + (y ′)2√
2gydx
Variational Methods
![Page 60: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/60.jpg)
Introduction Motivation E-L PDE
Brachistochrone
F =∫
dt , minF . . . curve of theshortest time.
F =∫ ds
v =∫ b
0
√1+(y ′(x))2
v dx12mv2 = mgy(x)⇒ v =
√2gy(x)
F =
∫ b
0
√1 + (y ′)2√
2gydx
Variational Methods
![Page 61: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/61.jpg)
Introduction Motivation E-L PDE
Brachistochrone
f (y , y ′) =
√1 + (y ′)2√
2gy
f − y ′∂f∂y ′
= C Beltrami identity
......
y(1 + (y ′)2) =1
2gC2 = k
The solution is a cycloid
x(θ) =12
k(θ − sin θ), y(θ) =12
k(1− cosθ)
Variational Methods
![Page 62: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/62.jpg)
Introduction Motivation E-L PDE
Brachistochrone
f (y , y ′) =
√1 + (y ′)2√
2gy
f − y ′∂f∂y ′
= C Beltrami identity
......
y(1 + (y ′)2) =1
2gC2 = k
The solution is a cycloid
x(θ) =12
k(θ − sin θ), y(θ) =12
k(1− cosθ)
Variational Methods
![Page 63: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/63.jpg)
Introduction Motivation E-L PDE
Brachistochrone
f (y , y ′) =
√1 + (y ′)2√
2gy
f − y ′∂f∂y ′
= C Beltrami identity
......
y(1 + (y ′)2) =1
2gC2 = k
The solution is a cycloid
x(θ) =12
k(θ − sin θ), y(θ) =12
k(1− cosθ)
Variational Methods
![Page 64: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/64.jpg)
Introduction Motivation E-L PDE
Brachistochrone
f (y , y ′) =
√1 + (y ′)2√
2gy
f − y ′∂f∂y ′
= C Beltrami identity
......
y(1 + (y ′)2) =1
2gC2 = k
The solution is a cycloid
x(θ) =12
k(θ − sin θ), y(θ) =12
k(1− cosθ)
Variational Methods
![Page 65: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/65.jpg)
Introduction Motivation E-L PDE
Brachistochrone
f (y , y ′) =
√1 + (y ′)2√
2gy
f − y ′∂f∂y ′
= C Beltrami identity
......
y(1 + (y ′)2) =1
2gC2 = k
The solution is a cycloid
x(θ) =12
k(θ − sin θ), y(θ) =12
k(1− cosθ)
Variational Methods
![Page 66: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/66.jpg)
Introduction Motivation E-L PDE
Cycloid
Play avi
Play avi
Variational Methods
![Page 67: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/67.jpg)
Introduction Motivation E-L PDE
Boundary conditions
using “per partes” on u(x , y), n(x , y) ≡ [n1(x , y),n2(x , y)]normal vector at the boundary ∂Ω
∂
∂εF (u + εv) =
∫(·)dxdy +
∫∂Ω
[∂f∂ux
n1 +∂f∂uy
n2
]v ds
Dirichlet b.c.u is predefined at the boundary ∂Ω→ v(∂Ω) = 0Neumann b.c.derivative in the direction of normal ∂u
∂n = 0
Example
Consider F (u) =∫
Ω12 |∇u|2 =
∫Ω
12(u2
x + u2y )
∂f∂ux
= ux , ∂f∂uy
= uy∂f∂ux
n1 + ∂f∂uy
n2 = uxn1 + uyn2 = ∂u∂n = 0
Variational Methods
![Page 68: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/68.jpg)
Introduction Motivation E-L PDE
Boundary conditions
using “per partes” on u(x , y), n(x , y) ≡ [n1(x , y),n2(x , y)]normal vector at the boundary ∂Ω
∂
∂εF (u + εv) =
∫(·)dxdy +
∫∂Ω
[∂f∂ux
n1 +∂f∂uy
n2
]v ds
Dirichlet b.c.u is predefined at the boundary ∂Ω→ v(∂Ω) = 0
Neumann b.c.derivative in the direction of normal ∂u
∂n = 0
Example
Consider F (u) =∫
Ω12 |∇u|2 =
∫Ω
12(u2
x + u2y )
∂f∂ux
= ux , ∂f∂uy
= uy∂f∂ux
n1 + ∂f∂uy
n2 = uxn1 + uyn2 = ∂u∂n = 0
Variational Methods
![Page 69: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/69.jpg)
Introduction Motivation E-L PDE
Boundary conditions
using “per partes” on u(x , y), n(x , y) ≡ [n1(x , y),n2(x , y)]normal vector at the boundary ∂Ω
∂
∂εF (u + εv) =
∫(·)dxdy +
∫∂Ω
[∂f∂ux
n1 +∂f∂uy
n2
]v ds
Dirichlet b.c.u is predefined at the boundary ∂Ω→ v(∂Ω) = 0Neumann b.c.derivative in the direction of normal ∂u
∂n = 0
Example
Consider F (u) =∫
Ω12 |∇u|2 =
∫Ω
12(u2
x + u2y )
∂f∂ux
= ux , ∂f∂uy
= uy∂f∂ux
n1 + ∂f∂uy
n2 = uxn1 + uyn2 = ∂u∂n = 0
Variational Methods
![Page 70: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/70.jpg)
Introduction Motivation E-L PDE
Boundary conditions
using “per partes” on u(x , y), n(x , y) ≡ [n1(x , y),n2(x , y)]normal vector at the boundary ∂Ω
∂
∂εF (u + εv) =
∫(·)dxdy +
∫∂Ω
[∂f∂ux
n1 +∂f∂uy
n2
]v ds
Dirichlet b.c.u is predefined at the boundary ∂Ω→ v(∂Ω) = 0Neumann b.c.derivative in the direction of normal ∂u
∂n = 0
Example
Consider F (u) =∫
Ω12 |∇u|2 =
∫Ω
12(u2
x + u2y )
∂f∂ux
= ux , ∂f∂uy
= uy∂f∂ux
n1 + ∂f∂uy
n2 = uxn1 + uyn2 = ∂u∂n = 0
Variational Methods
![Page 71: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/71.jpg)
Introduction Motivation E-L PDE
Boundary conditions
using “per partes” on u(x , y), n(x , y) ≡ [n1(x , y),n2(x , y)]normal vector at the boundary ∂Ω
∂
∂εF (u + εv) =
∫(·)dxdy +
∫∂Ω
[∂f∂ux
n1 +∂f∂uy
n2
]v ds
Dirichlet b.c.u is predefined at the boundary ∂Ω→ v(∂Ω) = 0Neumann b.c.derivative in the direction of normal ∂u
∂n = 0
Example
Consider F (u) =∫
Ω12 |∇u|2 =
∫Ω
12(u2
x + u2y )
∂f∂ux
= ux , ∂f∂uy
= uy
∂f∂ux
n1 + ∂f∂uy
n2 = uxn1 + uyn2 = ∂u∂n = 0
Variational Methods
![Page 72: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/72.jpg)
Introduction Motivation E-L PDE
Boundary conditions
using “per partes” on u(x , y), n(x , y) ≡ [n1(x , y),n2(x , y)]normal vector at the boundary ∂Ω
∂
∂εF (u + εv) =
∫(·)dxdy +
∫∂Ω
[∂f∂ux
n1 +∂f∂uy
n2
]v ds
Dirichlet b.c.u is predefined at the boundary ∂Ω→ v(∂Ω) = 0Neumann b.c.derivative in the direction of normal ∂u
∂n = 0
Example
Consider F (u) =∫
Ω12 |∇u|2 =
∫Ω
12(u2
x + u2y )
∂f∂ux
= ux , ∂f∂uy
= uy∂f∂ux
n1 + ∂f∂uy
n2 = uxn1 + uyn2 = ∂u∂n = 0
Variational Methods
![Page 73: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/73.jpg)
Introduction Motivation E-L PDE
E-L equation example
Smoothing functional:
F (u) =
∫Ω|∇u|2dx , f = u2
x + u2y
E-L equation:
F ′(u) = −∆u = −u2xx − u2
yy
Variational Methods
![Page 74: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/74.jpg)
Introduction Motivation E-L PDE
E-L equation example
Smoothing functional:
F (u) =
∫Ω|∇u|2dx , f = u2
x + u2y
E-L equation:
F ′(u) = −∆u = −u2xx − u2
yy
Variational Methods
![Page 75: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/75.jpg)
Introduction Motivation E-L PDE
More examples
Total variation of an image function u(x,y):
F (u) =
∫Ω|∇u|dx , f =
√u2
x + u2y
E-L equation:
∂f∂u− ∂
∂x∂f∂ux− ∂
∂y∂f∂uy
− ∂
∂xux√
u2x + u2
y
− ∂
∂yuy√
u2x + u2
y
= −div(∇u|∇u|
)
Variational Methods
![Page 76: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/76.jpg)
Introduction Motivation E-L PDE
More examples
Total variation of an image function u(x,y):
F (u) =
∫Ω|∇u|dx , f =
√u2
x + u2y
E-L equation:
∂f∂u− ∂
∂x∂f∂ux− ∂
∂y∂f∂uy
− ∂
∂xux√
u2x + u2
y
− ∂
∂yuy√
u2x + u2
y
= −div(∇u|∇u|
)
Variational Methods
![Page 77: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/77.jpg)
Introduction Motivation E-L PDE
More examples
Total variation of an image function u(x,y):
F (u) =
∫Ω|∇u|dx , f =
√u2
x + u2y
E-L equation:
∂f∂u− ∂
∂x∂f∂ux− ∂
∂y∂f∂uy
− ∂
∂xux√
u2x + u2
y
− ∂
∂yuy√
u2x + u2
y
= −div(∇u|∇u|
)
Variational Methods
![Page 78: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/78.jpg)
Introduction Motivation E-L PDE
More examples
Total variation of an image function u(x,y):
F (u) =
∫Ω|∇u|dx , f =
√u2
x + u2y
E-L equation:
∂f∂u− ∂
∂x∂f∂ux− ∂
∂y∂f∂uy
− ∂
∂xux√
u2x + u2
y
− ∂
∂yuy√
u2x + u2
y
= −div(∇u|∇u|
)
Variational Methods
![Page 79: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/79.jpg)
Introduction Motivation E-L PDE
More examples
Total variation of an image function u(x,y):
F (u) =
∫Ω|∇u|dx , f =
√u2
x + u2y
E-L equation:
∂f∂u− ∂
∂x∂f∂ux− ∂
∂y∂f∂uy
− ∂
∂xux√
u2x + u2
y
− ∂
∂yuy√
u2x + u2
y
= −div(∇u|∇u|
)
Variational Methods
![Page 80: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/80.jpg)
Introduction Motivation E-L PDE
Outline
1 IntroductionMotivationDerivation of Euler-Lagrange EquationVariational Problem and P.D.E.
Variational Methods
![Page 81: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/81.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 82: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/82.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0
Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 83: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/83.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.
Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 84: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/84.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 85: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/85.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 86: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/86.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 87: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/87.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 88: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/88.jpg)
Introduction Motivation E-L PDE
Steepest Descent
Classical optimization problem
g : R → R, x = minx
g(x)
Must satisfy g′(x) = 0Imagine, analytical solution is impossible.Let us walk in the direction opposite to the gradient
xk+1 = xk − αg′(xk ) ,
where α is the step length
Variational Methods
![Page 89: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/89.jpg)
Introduction Motivation E-L PDE
Steepest Descent
∀αxk+1 − xk
α= −g′(xk ) ,
Define x(t) as a function of time such that x(tk ) = xk andtk+1 = tk + α
dxdt
(tk ) = limα→0
x(tk + α)− x(tk )
α= lim
α→0
xk+1 − xk
α= −g′(xk )
Finding the solution with the steepest-descent method isequivalent to solving P.D.E.:
dxdt
= −g′(x)
Variational Methods
![Page 90: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/90.jpg)
Introduction Motivation E-L PDE
Steepest Descent
∀αxk+1 − xk
α= −g′(xk ) ,
Define x(t) as a function of time such that x(tk ) = xk andtk+1 = tk + α
dxdt
(tk ) = limα→0
x(tk + α)− x(tk )
α= lim
α→0
xk+1 − xk
α= −g′(xk )
Finding the solution with the steepest-descent method isequivalent to solving P.D.E.:
dxdt
= −g′(x)
Variational Methods
![Page 91: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/91.jpg)
Introduction Motivation E-L PDE
Steepest Descent
∀αxk+1 − xk
α= −g′(xk ) ,
Define x(t) as a function of time such that x(tk ) = xk andtk+1 = tk + α
dxdt
(tk ) = limα→0
x(tk + α)− x(tk )
α= lim
α→0
xk+1 − xk
α= −g′(xk )
Finding the solution with the steepest-descent method isequivalent to solving P.D.E.:
dxdt
= −g′(x)
Variational Methods
![Page 92: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/92.jpg)
Introduction Motivation E-L PDE
P.D.E
Variational problem
u = minu
F (u(x))
Must satisfy E-L equation
⇒ F ′(u) = 0
Find the solution with the steepest-descent method
uk+1 = uk − αF ′(uk ) ,
where α is the step length and must be determined∀α
uk+1 − uk
α= −F ′(uk ) ,
Variational Methods
![Page 93: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/93.jpg)
Introduction Motivation E-L PDE
P.D.E
Variational problem
u = minu
F (u(x))
Must satisfy E-L equation
⇒ F ′(u) = 0
Find the solution with the steepest-descent method
uk+1 = uk − αF ′(uk ) ,
where α is the step length and must be determined∀α
uk+1 − uk
α= −F ′(uk ) ,
Variational Methods
![Page 94: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/94.jpg)
Introduction Motivation E-L PDE
P.D.E
Variational problem
u = minu
F (u(x))
Must satisfy E-L equation
⇒ F ′(u) = 0
Find the solution with the steepest-descent method
uk+1 = uk − αF ′(uk ) ,
where α is the step length and must be determined
∀αuk+1 − uk
α= −F ′(uk ) ,
Variational Methods
![Page 95: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/95.jpg)
Introduction Motivation E-L PDE
P.D.E
Variational problem
u = minu
F (u(x))
Must satisfy E-L equation
⇒ F ′(u) = 0
Find the solution with the steepest-descent method
uk+1 = uk − αF ′(uk ) ,
where α is the step length and must be determined∀α
uk+1 − uk
α= −F ′(uk ) ,
Variational Methods
![Page 96: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/96.jpg)
Introduction Motivation E-L PDE
P.D.E
Make u also function of time, i.e., u(x , t)
uk (x) ≡ u(x , tk )
and tk+1 = tk + α
limα→0
uk+1 − uk
α≡ ∂u∂t
(x , tk )
Solving the variational problem with the steepest-descentmethod is equivalent to solving P.D.E.:
∂u∂t
= −F ′(u)
+boundary conditions.
Variational Methods
![Page 97: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/97.jpg)
Introduction Motivation E-L PDE
P.D.E
Make u also function of time, i.e., u(x , t)
uk (x) ≡ u(x , tk )
and tk+1 = tk + α
limα→0
uk+1 − uk
α≡ ∂u∂t
(x , tk )
Solving the variational problem with the steepest-descentmethod is equivalent to solving P.D.E.:
∂u∂t
= −F ′(u)
+boundary conditions.
Variational Methods
![Page 98: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/98.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 99: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/99.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 100: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/100.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 101: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/101.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 102: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/102.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 103: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/103.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 104: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/104.jpg)
Introduction Motivation E-L PDE
Steepest descent
Variational Methods
![Page 105: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/105.jpg)
Introduction Motivation E-L PDE
Differential Calculus x Variational Calculus
Differential Calculus Variational Calculus
Problem Spec. function function of function= functional
Necess. Cond. 1st derivative = 0 1st variation = 0Result one number (or vector) function
Variational Methods
![Page 106: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/106.jpg)
Introduction Motivation E-L PDE
Optimization Problem
Solving PDE’s is equivalent to optimization of integralfunctionals
ut − F ′(u) = 0 ⇔ min F (u)
Example
ut = ∆u ⇔ min∫
Ω|∇u|2
Does every PDE have its corresponding optimizationproblem?
Variational Methods
![Page 107: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/107.jpg)
Introduction Motivation E-L PDE
Optimization Problem
Solving PDE’s is equivalent to optimization of integralfunctionals
ut − F ′(u) = 0 ⇔ min F (u)
Example
ut = ∆u ⇔ min∫
Ω|∇u|2
Does every PDE have its corresponding optimizationproblem?
Variational Methods
![Page 108: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/108.jpg)
Introduction Motivation E-L PDE
Optimization Problem
Solving PDE’s is equivalent to optimization of integralfunctionals
ut − F ′(u) = 0 ⇔ min F (u)
Example
ut = ∆u ⇔ min∫
Ω|∇u|2
Does every PDE have its corresponding optimizationproblem?
Variational Methods
![Page 109: Variational Methods in Image Processing - CASzoi.utia.cas.cz/files/var_calculus_2.pdfVariational Methods in Image Processing ... Motivation Derivation of Euler-Lagrange Equation Variational](https://reader036.vdocuments.mx/reader036/viewer/2022070717/5edc1421ad6a402d66669814/html5/thumbnails/109.jpg)
Introduction Motivation E-L PDE
Optimization Problem
Solving PDE’s is equivalent to optimization of integralfunctionals
ut − F ′(u) = 0 ⇔ min F (u)
Example
ut = ∆u ⇔ min∫
Ω|∇u|2
Does every PDE have its corresponding optimizationproblem?
Variational Methods