in 3D space so that the illumination directions are non-zero when projected to the 2D horizontal space.
Figure 1. Photometric stereo SAfS at point
O
The task of SAfS is to estimate the slope at
O
along the two directions pointed by
[
�
̂
�
̂ ] and
[
�
̂
�
̂ ] . This can be expressed as
solving for two scalars
s
1
and
s
2
through SAfS and multiply them to
[
�
̂
�
̂ ] and
[
�
̂
�
̂ ] , respectively, to reach to the estimated
slope [ ] and [ ] along each direction:
[ ] = [
�
̂
�
̂ ] = [ ] ;
[ ] = [
�
̂
�
̂ ] = [ cos �
sin �] 7 The solution of Photometric stereo SAfS is a surface normal
[
� �
] such that it arrives [ ] and [ ] simultaneously when projected to the directions denoted by
[
�
̂
�
̂ ] and
[
�
̂
�
̂ ] respectively. Which means
[
� �
] is in fact the intersection of the perpendiculars of
[
�
̂
�
̂ ] and
[
�
̂
�
̂ ] originated at
[ ] and[ ]: [
� �
] = [
⊥
̂
⊥
̂ ] + [ ] = [
⊥
̂
⊥
̂ ] + [ ] 8
where [
⊥
̂
⊥
̂ ] = [ − ][
�
̂
�
̂ ] = [ ] [
⊥
̂
⊥
̂ ] = [ − ][
�
̂
�
̂ ] = [ −sin �
cos � ] In equation 8,
m
1
and
m
2
are the two scalars to be applied on [
⊥
̂
⊥
̂ ] and
[
⊥
̂
⊥
̂ ] respectively such that
[
⊥
̂
⊥
̂ ] + [ ] and
[
⊥
̂
⊥
̂ ] + [ ] intersects. In this case
[
� �
] = [ ] because the p-q coordinate frame is always aligned to
[
�
̂
�
̂ ] and
hence
�
= = . The problem can be simplified to only
solve for
m
1
that reaches magnitude
s
2
when projected to [
�
̂
�
̂ ] and this can be formulated by using vector dot product:
[ ] [
�
̂
�
̂ ]
�
̂ ,
�
̂
= 9
where
�
̂ ,
�
̂ is the length of the unit vector [
�
̂
�
̂ ] and
equals to 1. Hence, =
− c � i �
10 Therefore the solution of Photometric stereo SAfS
[
� �
] can be expressed as:
[
� �
] = [
− c � i �
] 11
3.2 Error propagation of Photometric Stereo SAfS
The combined error i.e., � of the resulted surface normal
[
� �
] from SAfS can be expressed as: � = � + �
12 where
� = squared error of
�
� = squared error of
�
From equation 11, the solution of Photometric stereo SAfS is determined by three factors:
s
1
and
s
2
which are solved through SAfS and reflectance modelling; and
� which is the azimuthal difference between the two illumination directions. Hence the
error of these components can be propagated to the solution by applying the Error Propagation Law to
[
� �
]: � =
�
�
�
� +
�
�
�
� +
�
�
��
�
�
13 � =
�
�
�
� +
�
�
�
� +
�
�
��
�
�
14 where
�
�
�
= ;
�
�
�
=
�
�
�
= −
a �
;
�
�
�
=
i �
Assuming there is no or neglectable error from � i.e.,�
�
= and combing equation 13 and 14 to 12, the final error term
is then: � = � + � =
+
a �
� +
i �
� 15
Note that equation 15 describes the effects of errors in
s
1
and
s
2
to the final solution of surface normal [
� �
]. Errors related to SAfS algorithms such as reflectance modelling error and
albedo error are represented by � and � and therefore not
explicitly presented in equation 15. To explicitly evaluate the effects of each of the SAfS error components e.g., Albedo
error, modelling error one has to perform error propagation on
s
1
and
s
2
and propagate them to equation 15. Equation 15 presents an interesting relationship between the
illumination azimuthal difference of the images i.e., � and the
final error term � : The coefficient of � behaves in a
quadratic fashion and reaches minimal when � =
where tan � = +�
and hence
a �
is practically zero. On the other hand, it approaches maximal when
� =
This contribution has been peer-reviewed. https:doi.org10.5194isprs-archives-XLII-3-W1-91-2017 | © Authors 2017. CC BY 4.0 License.
93
where tan � = and hence
a �
= +� . The coefficient
of � behaves in similar pattern where
i �
reaches minimal when
� = and maximal when
� = .
Considering both terms, � approaches minimal when � =
, which implies photometric stereo pair i.e., two images with azimuthal difference
� close to 90
o
is likely to produce better SAfS results and the performance decreases as
� deviates from 90
o
. One reasoning for such behaviour is that at 90
o
both directions are independent from each other and therefore
� will only affect
�
and � while � will only affect
�
and � ; while in other cases � will affect both [
� �
] as well as
[ �
� ] and its effect increases as the two directions get closer
to be collinear. At 0
o
or 180
o
, the two illumination directions become collinear coplanar in 3D space and therefore it is
insufficient to obtain [
� �
] accurately without external constraints.
4. EXPERIMENTAL ANALYSIS
