Graphics Reference
In-Depth Information
In the continuous formulation, satisfying the integrability constraint (
3.25
) corre-
sponds to the variational problem of minimising the functional
∂z(x,y)
∂x
p(x,y)
2
∂z(x,y)
∂y
q(x,y)
2
dx dy
−
+
−
(3.27)
with respect to the surface gradients
p(x,y)
and
q(x,y)
. The Euler equation of this
problem is given by
∂p
∂x
+
∂q
∂y
,
2
z
∇
=
(3.28)
2
z
denotes the Laplace operator applied to
z
, where (
3.28
) must hold at
each position in the image (Horn,
1986
). For discrete pixels, the Laplace operator
can be approximated by the expression
∇
where
∇
2
z
uv
≈
κ
ε
2
(
¯
z
uv
−
z
uv
),
(3.29)
where
ε
is the lateral extent of the pixels, which is conveniently set to
ε
=
1ifall
pixels are quadratic and of the same lateral extent, and
κ
=
4 when the local average
z
uv
is computed based on the four nearest neighbours of the pixel (Horn,
1986
).
According to Horn (
1989
), setting the derivative of the error term
f
with respect
to
p
uv
and
q
uv
to zero and combining (
3.28
) and (
3.29
) then yields the following
iteration scheme:
¯
∂z
∂x
(n)
uv
+
1
γ
(I
R
I
)
∂R
I
∂p
p
(n
+
1
)
uv
=
−
∂z
∂y
(n)
uv
+
γ
(I
−
R
I
)
∂R
I
1
q
(n
+
1
)
uv
=
(3.30)
∂q
∂p
∂x
(n
+
1
)
∂q
∂y
(n
+
1
)
.
ε
2
κ
z
(n
+
1
)
uv
z
(n)
uv
=¯
−
+
uv
uv
After each update of the surface gradients
p
uv
and
q
uv
, the corresponding height
map
z
uv
is computed by means of a discrete approximation to the solution of
the Euler-Lagrange differential equations of the corresponding variational problem
(Horn,
1989
). For a single light source and oblique illumination, this algorithm gives
a good estimate even of the surface gradients perpendicular to the direction of in-
cident light, as it adjusts them to the integrability constraint without affecting the
intensity error
e
i
given by (
3.19
).
Horn (
1989
) remarks that even if the iterative update rule (
3.24
) is initialised
with a solution that perfectly fits with the observed pixel grey values, i.e.
e
i
=
0, the
algorithm will nevertheless yield a different, smoother surface, since the constraint
of small partial derivatives of the surface is a strong restriction and not always phys-
ically reasonable. In contrast, the integrability constraint is a much weaker restric-
tion and always physically correct. If the corresponding iterative update rule (
3.30
)
is initialised with a surface for which the associated intensity error
e
i
is zero, the
algorithm will retain this 'perfect' solution. However, the algorithm based on the
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