Biomedical Engineering Reference
In-Depth Information
Denoting by
τ
the tilt of the illuminant and by
γ
the slant of the illuminant,
the above equation can be rewritten as
I
(
x
,
y
)
=
cos
γ
+
p
cos
τ
sin
γ
+
q
sin
τ
cos
γ.
(5.35)
Using forward difference formula (5.23), we have
p
=
∂
∂
x
Z
(
x
,
y
)
=
Z
(
x
+
1
,
y
)
−
Z
(
x
,
y
)
,
(5.36)
q
=
∂
∂
y
Z
(
x
,
y
)
=
Z
(
x
,
y
+
1)
−
Z
(
x
,
y
)
.
By taking Fourier transform on the two sides of Eq. (5.36), we can get the
following results:
p
=
∂
F
−
F
Z
(
w
1
,w
2
)(
−
i
w
1
)
,
∂
x
Z
(
x
,
y
)
q
=
∂
F
−
F
Z
(
w
1
,w
2
)(
−
i
w
2
)
.
∂
x
Z
(
x
,
y
)
(5.37)
Substituting Eq. (5.37) into Eq. (5.35) and taking Fourier transform on both
sides, we obtain
F
I
(
w
1
,w
2
)
=
F
Z
(
w
1
,w
2
)(
−
i
w
1
) cos
τ
sin
σ
+
F
Z
(
w
1
,w
2
)(
−
i
w
2
) sin
τ
sin
σ,
(5.38)
where
F
I
(
w
1
,w
2
) and
F
Z
(
w
1
,w
2
) are the Fourier transform of the input image
I
(
x
,
y
) and depth map
Z
(
x
,
y
), respectively.
After rearranging Eq. (5.38), we obtain
F
I
(
w
1
,w
2
)
F
Z
(
w
1
,w
2
)
=
2
π
(
2
) sin
γ
(
w
1
cos
τ
+
w
2
sin
τ
)
.
(5.39)
2
2
w
1
+
w
By taking the inverse Fourier transform, we can obtain the depth map
Z
(
x
,
y
)
=
F
−
1
(
F
Z
(
w
1
,w
2
))
.
(5.40)
It is obvious that this approach does not need iterative computation and
can provide an approximate solution quickly. However, like all the other linear
approaches, this method makes an assumption that the reflectance map is locally
linear. Comparing Eq. (5.35) with the normal reflectance equation:
cos
γ
+
p
cos
τ
sin
γ
+
q
sin
τ
cos
γ
R
(
p
,
q
)
=
1
+
p
2
.
(5.41)
+
q
2
