Let
defined by (
p
,
q
,-1) be the normal vector of a point (
X
,
Y
,
Z
) existing on
continuous and compact surface,
defined by
-1) the position vector of a light source, then, Lambertian reflectance
is described as
where
denotes the angle intersected by
and
. The purpose of SFS approach is to recover the surface
Z
(
X
,
Y
), or exactly the tangent surface of
Z
(
X
,
Y
), using
and existing constraint(s).
As has been discussed in previous researches[
1
,
20
,
14
], there exist a parameter-family of planes
having equal height in the direction of light source due to the constant
angle
on each plane as is shown in Fig.
1
. The edge of continuous surface cut by a equal height plane becomes a
continuously connected spatial curve, i.e. the level curve. Each level
curve has a corresponding iso-intensity contour on projected image plane
(Fig.
1
). At each point (
X
,
Y
,
Z
), a tangent vector
of level curve, another vector
existing in the direction of steepest descent/ascent variation of depth,
and the normal vector
constitute locally orthogonal coordinate system; and vectors
and
define corresponding tangent patch. Classical CSEM is a way which
recovers the tangent surface in the direction of
[
17
]. Whereas LCPM is another way which expands level curves in the
direction of
by curvature dependent speed of time for the recovery of tangent
surface[
20
,
14
].
Regardless of the ways of propagation, we should rely on a constraint:
the angle
. Then the solution of
determined using
and
becomes a ambiguous cone at each point (
X
,
Y
,
Z
)[
14
]. Consequently, the solution of
becomes another cone as is shown in Fig.
2
. This cone, generated by
which contacts the tangent surface, is called the
Monge
cone (see pp.24-33 of [
7
] for detail). A cone is composed of stacked circles, and a family of
circles having their center on a continuous curve is contacted by two
envelop curves as is shown in Fig.
3
. A family of circles, when each circle is a sliced edge of Monge cone,
is contacted by a family of envelop curves at a proportionally given
distance from the center of each cone. Since each cone has non-zero
length determined by the length of normal vector, the stack of envelop
curves becomes a paired family of envelop surfaces having angle
of separation by the definition of the Monge cone. The envelop surfaces
are the solution of this SFS problem as is shown in Fig.
4
because each envelop surface, at all point of a level curve, satisfies
the condition of all Monge cones, i.e. the solution of eq. (
2
), simultaneously.
Typically two tangent surfaces can be obtained between two adjacent
planes of level curves. The integration of one set of tangent surfaces
generally gives unique solution of continuous surface
Z
(
X
,
Y
) in LCPM when there is no topological change because the freedom of
selecting the orientation of three vectors
doesn't change the shape of integrated surface although it changes the
propagating direction, downward or upward, of level curves. However, it
is distinct that we always have two possible solutions of resultant
surface as already discussed in [
1
,
17
] because the distinguishability of correct solution between two is not
guaranteed. Generally, the existence of the solution at singular points
and in occluding boundary is not guaranteed by the global approach[
21
,
20
].
The existence of global solution comes from the compact existence of
Monge cones on a level curve. This means that the SFS problem is bound
by two constraints, i.e. the intersection angle
and the continuity, not by one constraint.
A local tangent patch, defined by the envelop surface of at least two
Monge cones, exists at any point on a level curve due to the continuity
of this curve. As is shown in Fig.
5
, this patch
has angle
of intersection w.r.t. the plane
which is bound by this curve. This is a local interpretation of the
global concept of solution. Therefore, each patch
is constrained by angle
which can be determined from eq. (
2
), by
X
,
Y
position, and by a iso-intensity curve which corresponds to the
projected curve of each level curve. The direction of
, which can be obtained from this iso-intensity curve, first fixes the
axis of rotation of a tangent patch; then this patch is rotated w.r.t.
the plane
by this axis to the amount of angle
. This rotation provides unique existence of local solution having two
degrees of freedom.
Although the patch
is constrained by the plane
, it doesn't mean that the corresponding normal vector
is variant w.r.t. the variation of
. The variants are both the angle
and the direction of
, i.e. the level curve, because the geometrical relation between
and
determines the level curves in the process of shading. The vector
is invariant, and consequently the integrated surface is invariant as
well; and which satisfies the conservation law of Hamilton-Jacobi
equations[
2
,
20
].
Seong Ik CHO