International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 5 - November 2015
Impact of the Integrability constraint on Radar Shape-fromShading
Mobarak, Babikir A.
Civil Engineering Dept., Faculty of Engineering, Al-Baha University
Al baha, Saudi Arabia
Abstract — In Synthetic Aperture Radar (SAR) remote
sensing, DEMs are used to resample the SAR images
to well known coordinate systems. Also, they are used
to correct for distortions originated from imaging
sensor. Radar Shape from Shading (RSFS) is one of
different techniques that used to extract heights from
SAR data sets. The most important key to this
technique is enforcing of the integrability constraint in
frequency-domain. Fourier transformation is used to
transforms the data from time-domain to frequencydomain. The performance of this constraint was
evaluated on RADARSAT-1 image using both
graphical and statistical analysis. To show the effect
of integrability constraint, the RSFS technique was
evaluated without and with enforcing the integrability.
The RMSE and R2 were found to be 31.98m and 0.705
and 17.47m and 0.972, respectively, showing the effect
of the constraint.
Keywords — SFS, DEM,
Integrability, SFS Constraints.
SAR
Imageries,
I. INTRODUCTION
A. SHAPE FROM SHADING
Point density or accuracy of DEMs generated from
direct techniques such as ordinary surveying,
photogrammetry, and remote sensing for some parts of
the Earth, such as a tropical area, is still insufficient
for some applications. This is due to some difficulties
resulted from the nature of these areas or the limitation
of the technique or system used. Shape-from-shading
(SFS) technique deals with the recovery of shape from
a gradual variation of shading in the image [2]. The
classic assumption underlining SFS is that the surface
under investigation has a Lambertian reflectivity.
The first systematic study of SFS was reported by [5]
and his colleagues [8]. SFS deals with the process of
finding the object’s 3D shape from a single image of
that object. The use of a single image cannot always
ensure the uniqueness of the shape of an object.
Therefore, there will be relatively little effect devoted
to exploiting the exact 3D shape reconstruction from
the shading information of one image. This problem is
resolved by introducing ancillary information to the
SFS process. The basic assumption underlying SFS is a
uniform surface reflectivity (Lambert). Several studies
investigating Lambertian reflectance model have been
carried out on SFS ([7], [13], and [10]).
ISSN: 2231-5381
From a computational viewpoint, SFS involves
solving the image irradiance equation to recover a set
of surface normals or surface slopes [14]. Reference [5]
was the first researcher, who had formulated SFS
problem and found the solution as a nonlinear firstorder partial differential equation (PDF). This equation
is known as the image irradiance equation and is the
basic equation for any SFS technique. It relates the
image irradiance to the scene radiance as shown in
Equation 1 below:
E ( x,y)
ˆ ( x,y))
R( n
(1)
Where E(x,y) is the image irradiance at a point (x,y),
R is the reflectivity, and n̂ represents the three
components of unit surface normal.
The recovered surface can be expressed in four
types [3]; surface height (elevation) z(x,y), surface
normal (nx, ny, nz), surface slope (p,q), and surface slant
Φ and tilt ϴ . The depth can be considered either as the
relative distance from the camera or antenna to the
surface points, or the relative surface height above the
xy plane. This implies that equation 1 can also be
written as follows:
E ( x,y)
R( p, q)
(2)
Where (p,q) = (dz/dx , dz/dy)
B. RADAR SHAPE FROM SHADING
SAR sensor can obtain remote sensing data in all
weather conditions due to its capability of penetrating
clouds and working day or night. The singular nature
of active radar instruments allow microwaves to
interact with surface features, means that information
obtained can be dependent on moisture content,
salinity and physical characteristics such as shape, size,
and orientation [6]. There are many difficulties in
acquiring images by conventional optical sensors,
especially in cloud-covered regions and/or season.
Currently, there are new techniques developed for
height extraction from SAR data. These include radar
stereogrammetry,
microwave
altimetry,
SAR
interferometry, and Light Detection and RangingLIDAR [11]. Point density or accuracy of DEMs
generated from direct techniques such as ordinary
surveying, photogrammetry, and remote sensing for
some parts of the Earth, such as a tropical area, is still
insufficient for some applications. This is due to some
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 5 - November 2015
difficulties resulted from the nature of these areas or
the limitation of the technique or system used.
The major objection to RSFS is the ambiguity from
uncertain backscatter properties. Involving some
constraints like brightness, smoothness, and
integrability can remove or reduce this ambiguity.
C. INTEGRABILITY
This constraint is important if the surface is to be
reconstructed from the recovered field of “surface
normal” [12]. This is due to the fact that the estimated
height will depend on the integration path if the
constraint is not satisfied. The integrability constraint
ensures valid surfaces, that is p y q x . It ensures that
heights can be integrated along any path, because
these values are independent of path of integration.
Enforcing this constraint practically smoothens the
estimated surface. This condition can be described by
(
p y qx ) 2 dxdy
(3)
Reference [4] proposed a method for enforcing
integrability in [1] algorithm by projecting non
integrable surface slope estimates onto nearest
integrable surface slopes. One way to obtain such
projection is by Fourier transformation. Their results
showed that the accuracy and efficiency had improved
over the Brooks and Horn’s algorithm.
II. MATERIALS AND METHODOLOGY
Figure 1: Subset from RADARSAT-1 S7 Mode Image
c.
Methodology
Digital image pre-processing was carried out to
prepare the RADARSAT-1 SAR image first. Then the
geometric correction was done to RADARSAT-1
image using some GCPs. After that the Radar
brightness (β0) and backscatter coefficients (σ◦) were
calculated using equations 4 and 5, respectively.
βor,a =10*log10((DNr,a2 + A3) / A2r)
σor,a
o
=β
r,a +10*log10(sin
θ r)
(4)
(5)
Where:
DNr,a is a digital number at range (r) and azimuth (a),
The study area is the KASSALA state in the east of A3 is a small constant (often 0),
SUDAN. It lies between longitudes 35.59693ºE and A2r is a range dependent look-up table that contains a
36.44708ºE and latitudes 15.11390ºN and 16.17869ºN. terrain type model.
The most interesting features included within the area θr is the incident angle at range (r).
of study are ALGASH River and the TAKAH
Then, speckle filtering was applied to the
Mountain.
backscatter coefficients to remove (or reduce) the
speckle “inherent with radar data” from the image.
b.
Data
Enforcing of the Integrability
The data sets for this study consist of one
Integrability constraint was enforced after
RADARSAT-1 image, covering the study area, and
transformation of the data from time-domain to
ground control points (GCPs). A subset from image of
frequency-domain using Fourier transformation. The
RADARSAT-1 was extracted to examine the
integrability was enforced within the radar SFS model
performance of the integrability. The area of the
proposed by [9]. Integrability requires that the
subset was approximately 10 km2. Figure 1 below
recovered surface height at any particular point is
represents a subset obtained from RADARSAT-1 S7
independent of the direction of integration and is
mode image.
defined as expressed in Equation 3.
a.
Area of Study
p
y
or
ISSN: 2231-5381
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q
x
py
(6)
qx
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 5 - November 2015
Integrability requires that the recovered surface
height at any particular point is independent of the
direction of integration and is defined as expressed in
Equation 3.4.
~
( )Z ( )
~
( )Z ( )
y
~
p( )
~( )
q
a
a
(7)
x
where,
can be computed using the transformation
equation below
~
Z( )
a
cnj
x
cnj
ˆ( )
( )p
a
a
2
ax ( )
y
( )qˆ ( )
2
( )
y
where,
) is
the Fourier transform of the integrable height
Z(x,y),
p̂ ( ) and q̂( ) are the Fourier transform of the
surface slope estimates in x and y-direction,
a x ( ) and a y ( ) are the Fourier coefficients of a
discreet differentiation operator in x and y,
cnj
a
x
( ) and
cnj
a
x
( ) are the conjugates of a x ( ) and
a y ( ) , and
ω = (ωx, ωy) represents the two-dimension frequency
coordinates.
Equation 8 minimizes the following distance
simultaneously as shown in Equation 9, while
satisfying the conditions expressed by Equation 6.
d ( p̂, q̂) , ( ~
p , q~ )
2
~
p p̂
2
q~ q̂ dxdy
(9)
Fourier transform of ( pˆ ,qˆ ) was performed to get
(p̂( ), q̂( )) and then substitution of them into
Equation 8 was done to obtain the Fourier transforms
~
of integrable surface height Z ( ) .
Enforcing the integrability constraint was carried
out by calculating the Fourier transforms of the
nearest integrable surface slopes (~
p( ), ~
q( )) using
Equation 10 below
~
ax ( ) Z ( )
~
ay ( )Z ( )
~
p( )
~( )
q
(10)
Where,
ax
j sin(
(
In the last step inverse Fourier transforms of slope
estimates (~
p( ), ~
q( )) was performed to obtain
the new integrable surface slopes ( p
ˆ ,qˆ ) .
III. RESULTS AND DISCUSSION
~
Z( )
~
Z(
are the input image row and column numbers,
respectively.
x
,
y
x
)
) , ay
j sin(
y
), j
1
,
( 2 m / M ,2 n / N ) , m = -
M/2,…,0,…,M/2 , n = -N/2,…,0,…,N/2, and M and N
ISSN: 2231-5381
The objective of this section of study was to assess
the importance of the integrability constraint on
surface heights recovery using RSFS technique. The
integrability
constraint was enforced as a projection
(8)
constraint. An integrable set of surface slope estimates
was constructed from non integrable ones. In the case
of the developed SFS algorithm operated without the
integrability constraint, any number of surface height
reconstructions could be produced depending upon the
path of integration. Consequently, the surface
reconstructed could be noisy if the surface slope
estimates were noisy.
The performance of the integrability constraint was
tested on RADARSAT-1 image of Standard mode
(S7). The evaluation of the accuracy was based on the
comparison of absolute surface heights derived by the
enforcing of the integrability and the real height
obtained from GCPs. Evaluation was carried out
qualitatively and quantitatively. Qualitative evaluation
was conducted by investigating the shape of surface
topography reconstructions and their actual values in
the study area. Quantitative evaluation was done
statistically through calculations of RMSE and R2.
Some 123 GCPs were used for this purpose. It was
considered that quantitative measures would
supplement and extend the qualitative analysis.
Figures 2 and 3 represent the resultant image of
plotting absolute heights, reconstructed from RSFS
algorithm without and with enforcing the integrability
constraint, respectively. The most interesting
observation is that there are significant differences
between the unconstrained (Figure 2) and constrained
model results (Figure 3). Few obvious distortions are
observed in Figure 2. The overall surface height
values range between 425m and 800m. In addition, the
location of the peak of Taka mountain is shifted to the
lower-left corner. The value of this point, as estimated
from the figure, is approximately 800m.
Referring to Figure 4, it is observed that the range
between minimum and maximum surface heights of
non-integrable surface height is smaller than that of
the integrable one (400m-1009m). Also, it is clear that
the reconstructed surface height produced by
enforcing the integrability constraint is smoother than
that obtained without enforcing the integrability
constraint.
Table Error! No text of specified style in document.1
shows the RMSE and R2 results of surface heights
recovered for both the constrained and unconstrained
RSFS algorithms. It is clear that the integrability
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 5 - November 2015
constraint reduces the RMSE and R2 respectively from
31.98m to 17.47m and from 0.705 to 0.972, giving the
differences of 14.51 and 0.267.
Table Error! No text of specified style in document.1: RMSE and
R2 of Reconstructed Surface Height from Constrained and
Unconstrained SFS algorithm
Model
Unconstrained
Constrained
RMSE
31.98
17.47
0.705
0.972
R
2
The finding of integrability constraint is consistent
with that of [8], who found that most of surface
smoothing came from integrability constraint.
IV. CONCLUSION
Figure 2: Plot of Absolute Height from SFS without
the Integrability Constraint
It is interesting to note that enforcing the
integrability constraint has a significant effect on the
final absolute height accuracy after comparison with
the results obtained without enforcing this constraint.
The validation of the later results gave RMSE and R2
of 31.98m and 0.705, respectively. This finding
indicated that most of surface smoothing was
attributed to the integrability constraint.
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Figure 3: Plot of Absolute Height from SFS with the
Integrability Constraint
[5]
[6]
[7]
[8]
[9]
[10]
[11]
Figure Error! No text of specified style in
document.: Height Differences between Constrained
and Unconstrained Model
[12]
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ISSN: 2231-5381
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Dr. Babikir A. Mobarak received his B. Tech. and
M.Sc. degrees in Surveying Engineering and
Geodetic Survey from Sudan University of Science
and Technology (SUST), Khartoum, SUDAN in 1996
and 2003, respectively, and the Ph.D. degree in GIS
and Geomatic Engineering from Universiti Putra Malaysia (UPM),
Serdang, Malaysia, in 2011.
From 1997 to 2003, he was TA, in Surveying Eng. Dept. College of
Eng. (SUST). From 2003 to 2005, he was also in Surveying Eng.
Dept. College of Eng. (SUST) as a Lecturer. From 2011 to 2014 he
was Assistant Professor, School of Surveying, College of Eng.
(SUST).
Now he is an Assistant Professor in Civil Engineering Department,
Faculty of Engineering, Al Baha University, Al Baha, Saudi Arabia.
ISSN: 2231-5381
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