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 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

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