Image Processing Reference

In-Depth Information

These formulations are composed of an appropriate data fitting term and the regu-

larization term to incorporate the smoothness constraint.

The formulation of the aforementioned set of problems involves a functional or

a function of functions, and their derivatives, to be integrated over a finite image

domain. The objective is to determine an appropriate unknown function that maxi-

mizes or minimizes the corresponding functional. A functional

Ψ

over a function

ω

in variable

ζ

is defined by Eq. (
6.1
).

ζ
2

ζ
1
ψ
ω(ζ),ω
(ζ ), ω
(ζ ),
···
d

Ψ(ω)
≡

ζ,

(6.1)

ω
(ζ )

ω
(ζ )

where

and

represent the first and second derivatives of the function

ω

with respect to

ζ

, respectively, and

ζ
1
and

ζ
2
define the limits of integration. Let the

function

ω

be defined within some space of functions

F

, i.e.,

ω
∈ F

, and

ζ
∈ R

, then

the functional

.

The problem of finding stationary functions for which

Ψ

defines the mapping

Ψ
: F → R

is the minimum, is of

interest in several fields of computer vision, optimization, control theory, and physics.

This task can be accomplished by solving the Euler-Lagrange equation associated

with the corresponding functional.

Consider the functional

Ψ(ω)

Ψ

defined in Eq. (
6.1
). One would like to find the unknown

function

. The Euler-Lagrange equation corresponding

to this functional is given by Eq. (
6.2
).

ω

, defined over

[
ζ
1
,ζ
2
]⊂R

∂ψ

∂ω

∂ψ

∂ω

2

∂ζ

∂ψ

∂ω
−

∂

∂ζ

∂

+

−··· =

0

.

(6.2)

2

If the functional in Eq. (
6.1
) involves only first derivative, the corresponding Euler-

Lagrange equation is given by Eq. (
6.3
).

∂ψ

∂ω

∂ψ

∂ω
−

∂

∂ζ

=

0

.

(6.3)

The regularization parameter, if any, is also incorporated into this equation, as it will

be seen in the next section.

The Euler-Lagrange equation can be extended to the functionals of multiple func-

tions, as well as multiple variables. In 2-D, the cost functional

Ψ(ω)

takes the form

Ψ(ω)
≡

ψ(ω(

x

,

y

), ω
x
(

x

,

y

), ω
y
(

x

,

y

))

d
x
d
y

,

(6.4)

where

along the
x
- and
y
-directions,

respectively. The corresponding Euler-Lagrange equation is given by Eq. (
6.5
).

ω
x
(

x

,

y

)

and

ω
y
(

x

,

y

)

are derivatives of

ω

∂ψ

∂ω
x

∂ψ

∂ω
y

∂ψ

∂ω
−

∂

∂

∂

∂

−

=

0

.

(6.5)

x

y