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the finite differences of neighboring points on the height field, from which the
actual
z
values can be readily reconstructed. The process minimizes a kind of en-
ergy function that quantifies how the reflection of the reconstructed surface differs
from ideal diffuse reflection. The error expression is
=
i
,
j
α
(
ρ
R
(
p
ij
,
q
ij
)
−
I
(
i
,
j
))
v
ij
+
λ
u
ij
+
2
E
,
(9.2)
where
p
ij
and
q
ij
are the first-order finite differences of the height field;
z
ij
,
u
ij
,
and
v
ij
are the second-order finite differences;
R
(
,
)
p
ij
q
ij
is the computed Lam-
is the pixel value.
2
The first term in the summand
of Equation (9.2) is the difference between ideal Lambertian reflection and the
captured pixel values; the second is a smoothness constraint. The parameters
(
,
)
bertian irradiance; and
I
i
j
α
and
λ
provide the relative weights of these terms. The value
ρ
is the Lambertian
reflectance (albedo) of the surface.
The original height-from-shading algorithm assumes the surface is Lamber-
tian, has a constant albedo, and only a single image is considered. In the BTF
synthesis problem, the surface need not be Lambertian (the problem is much less
interesting if it is), the reflectance varies over the surface sample, and instead of a
single image, a collection of images is employed to reconstruct the geometry. The
authors therefore employed a modified version of the algorithmbetter suited to the
BTF images. Their approach starts with a few registered and aligned images of
the surface sample. Each point of the height-field grid therefore has several pixel
values
I
k
(
where
k
runs over the input images (only a few captured images
having similar lighting/viewing directions are used for geometry recovery). The
pixel values corresponding to a given point are not all the same: depending on
the view and light directions, the point may be in shadow or show a specular
highlight. The values needs to be de-emphasized in these cases, so the authors
employ a weighting function
i
,
j
)
determined by classifying the pixel as be-
ing in shadow or having a highlight. The first term in Equation (9.2) therefore
becomes
η
k
(
i
,
j
)
2
α
∑
k
η
k
(
i
,
j
)(
ρ
ij
R
k
(
p
ij
,
q
ij
)
−
I
k
(
i
,
j
))
,
(9.3)
ij
is the position-dependent albedo. Another smoothing term also needs
to be added to the formula for
E
to smooth the albedo values. Regarding geomet-
ric smoothing, one other modification is necessary. Because the height field may
where
ρ
2
These finite differences are the
central
differences of the height field, defined by
p
ij
=
z
i
+
1
,
j
−
z
i
−
1
,
j
/
2and
q
ij
=
z
i
,
j
+
1
−
z
i
,
j
−
1
/
2. That is,
p
and
q
are the differences of the heights
at adjacent points in the horizontal and vertical directions, respectively. The second-order finite dif-
ferences are defined analogously:
u
ij
=
z
i
+
1
,
j
−
2
z
i
,
j
+
z
i
−
1
,
j
and
v
ij
=
z
i
,
j
+
1
−
2
z
i
,
j
+
z
i
,
j
−
1
.
