Biomedical Engineering Reference
In-Depth Information
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Linear approach
. The method linearizes the reflectance map in tilts or
depth. The linear model can be solved and the shape of the object can be
calculated. The basic assumption behind this idea is that the lower order
components of the reflectance maps dominate the reflectance maps.
In the rest of this section, we will discuss in detail two widely used methods:
the linear approach and the minimization approach.
5.3.1 Linear Approaches
In this approach, the basic idea is to linearize the reflectance map and solve the
depth information of the shape from the equations. Different linear functions
can be formulated in terms of surface gradient or the height of the surface. In
the following contexts we will introduce two approaches which are based on
linear equations in terms of gradients and the heights of the surface. Both start
with the use of first-order finite difference to discretize the reflectance equation.
However, they are different after the initial discretization. Pentland's algorithm
uses the Fourier transform and inverse Fourier transform to obtain the depth
map, while Tsai-Shah's algorithm uses the Newton method to derive the depth
map. We will explain these two linear approaches in the following sections.
5.3.1.1 Pentland's Linear Approach
Pentland [6,46,47] introduced a method which takes directly linearization of the
reflectance map in the surface gradient (
p
,
q
). It greatly simplifies the shape from
shading problem with scarifying part of the accuracy of the reconstruction result.
We start with the expansion of the right-hand side of the irradiance equation (5.2)
at
p
=
p
0
,
q
=
q
0
using Taylor's expansion. We have
R
(
p
,
q
)
=
R
(
p
0
,
q
0
)
+
(
p
−
p
0
)
∂
R
∂
p
(
p
0
,
q
0
)
+
(
q
−
q
0
)
∂
R
∂
q
(
p
0
,
q
0
)
.
(5.33)
For Lambertian reflectance, Eq. (5.33) at
p
0
=
0
,
q
0
=
0 can be reduced to
I
(
x
,
y
)
=
R
(0
,
0)
+
p
∂
R
∂
p
(0
,
0)
+
q
∂
R
∂
q
(0
,
0)
.
(5.34)
