The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 38, Part II
ESTIMATION OF MODEL ERROR OF LINE OBJECTS
Wenzhong Shia ; Sio-Kei Cheong b; Eryong Liua, c
a
Department of Land Surveying and Geo-Informatics,The Hong Kong Polytechnic University, Hong Kong
b
Department of Land Management, Renmin University of China, China
c
School of Environment and Spatial Informatics, China University of Mining and Technology, Xuzhou, China
Commission II
KEY WORDS: model error, truncation error, lines, positional error, geographic information science
ABSTRACT:
This paper presents a method for estimating the model error, specifically, a truncation error, of lines. The generic method for
describing truncation error for
n−
dimensional spatial features is developed first based on the numerical analysis theory. The
methods for describing truncation error of the lines in three-dimensional and two-dimensional space, which are the two
frequently used cases, are then derived as special cases of the generic method. This research is a further development of
earlier work on line error modelling, which were mainly focusing on the propagated error rather than model error.
the representative feature for straight line feature error. A
1. INTRODUCTION
third-order curve is also taken as an examplefor truncation error
modeling.
Line feature error is one of the fundamental issues in the areas
of modeling uncertainties in spatial data and spatial data quality
control. Placing unrealistically high trust in the accuracy of data
in GIS may mislead and seriously inconvenience GIS users.
2 METHOD FOR DESCRIBING TRUNCATION ERROR
OF STRAIGHT LINE FEATURES IN AN
Normally error of lines can be classified as the following
n−
DIMENSIONAL SPACE
categories: (a) model error which is related to the interpolation
models for a line, and (b) propagated error which is related to
Straight line features in geographic information science can
the error of the original nodes that composite the line. The first
include line segments, polylines, and polygons. The polyline is
type error can, in many cases, become a dominant part of the
taken as a generic form of a straight line feature in the following
overall line feature error. The second type of error has been
discussion.
widely researched in the past, while study on model error of
2.1 Definition of a polyline in an
lines in GIS are relatively less reported. This research intend to
dimensional space
n−
investigate the model error of the lines, as a further supplement
A polyline in
to the realy studies on propagated error of lines.
n−
Qn1 =[ x11 , x12 ,L, x1n ]
T
dimensional space is composed of vectors:
Qn 2 =[ x21 , x22 ,L, x2 n ]
T
,
Qni =[ xi1 , xi 2 ,L, xin ]
T
,L,
,L,
Straight line features in GIS include line segments, polylines,
Qnm =[ xm 1 , xm 2 ,L , xmn ]
polygons, with the latter being regarded as a special
composition points (as shown in Figure 1). The polyline is
representation of a polyline, in which the start point is identical
represented
to the end point. Line segments are also a special representation
Qn1Qn 2 ,Qn 2Qn 3 ,L, Qni Qn ,i +1 ,L,Qn ,m−1Qnm .
T
of a polyline – the number of the component line segments
being equal to one. Hence, in this study, a polyline is taken as
179
,
where
by
i =1,2,L, m
the
and
m
line
is the number of
segments
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 38, Part II
Qni
Qn1
Qnm
Qn2
Qn,m−1
Figure 1. An example of a polyline in
Any point on the line segment
Qni Qn ,i +1 ( i =1,2,L, m −1)
is then
n
( )
model
for
straight
line
features
in
-dimensional space. These two truncation error model cases
systems.
⎡ p (r ) ⎤
⎢ i1 ⎥
⎢ pi 2 ( r ) ⎥
⎢
⎥
⎢ M ⎥
⎢ p (r)⎥
⎣ in ⎦
3 THE TRUNCATION ERROR MODEL FOR STRAIGHT
(1)
LINE FEATURES IN THREE-DIMENSIONAL SPACE
(i =1,2,L,m −1， r∈[0,1])
The true polyline of
error
may be frequently used in real world geographic information
⎡x ⎤
⎢ i1r ⎥
⎢x ⎥
= ⎢ i 2 r ⎥ = 1− r Qni + rQn ,i +1
⎢ M ⎥
⎢⎣ xinr ⎥⎦
⎡ (1− r ) x + rx
⎤
i1
i +1,1 ⎥
⎢
⎢(1− r ) xi 2 + rxi +1,2 ⎥
= ⎢
⎥
M
⎢
⎥
⎢ (1− r ) x + rx
⎥
in
i +1,n ⎦
⎣
dimensional space
truncation
represented by
Qnir
n−
3.1 Definition of a polyline in three-dimensional space
Qn1Qn 2 LQni LQnm
can be represented by
⎡ μi 1 r ⎤ ⎡ f i 1 ( r ) ⎤
⎥
⎢μ ⎥ ⎢
f (r )
φnir = ⎢ i 2 r ⎥ = ⎢⎢ i 2 ⎥⎥ ( i =1,2,L, m −1， r∈[ 0,1])
⎢ M ⎥
M
⎥
⎢
⎥ ⎢
⎣⎢ μinr ⎦⎥ ⎣⎢ fin ( r ) ⎦⎥
A polyline in three-dimensional space is composed of vectors:
Q31 =[ x11 , x12 , x13 ]
T
(2)
Q32 =[ x21 , x22 , x23 ]
T
,
Q3 m =[ xm 1 , xm 2 , xm 3 ]
T
,
where
i =1,2,L, m
Q3i =[ xi1 , xi 2 , xi 3 ]
T
,L,
and
m
,L,
is the number of
composition points.
2.2 The truncation polyline error in an
n−
dimensional space
Any point on the line segment
Some assumptions about the functions
j =1,2,L, n ; r∈[ 0,1])
⎡x ⎤
⎢ i 1r ⎥
Q3 ir = ⎢ xi 2 r ⎥ = 1− r Q3 i + rQ3,i +1
⎢
⎥
⎣ xi 3 r ⎦
( )
interpolation error – a model error.
f ij ( r )∈C 2 [0,1] ,
Rij ( r ) = f ij ( r ) − pij ( r ) ≤
≤
1
M
8 2 ij
Where
⎡ (1− r ) xi 1 + rxi +1,1 ⎤
⎢
⎥
= ⎢(1− r ) xi 2 + rxi +1,2 ⎥
⎢ (1− r ) x + rx
⎥
i3
i +1,3 ⎦
⎣
⎡ pi 1 ⎤
⎢ ⎥
⎢ pi 2 ⎥
⎣⎢ pi 3 ⎦⎥
(4)
(i =1,2,L,m −1， r∈[0,1])
the following is then obtained
( )
1
M r 1− r
2 2 ij
The true polyline of
(3)
Tij
M 2 ij = max f ij'' ( r )
r∈[ 0,1]
is then
represented by
( ) (i =1,2,L, m;
fij r
need to be made in order to estimate
It is assumed that
Q3 i Q3,i +1 ( i =1,2,L, m −1)
Q31Q32 LQ3 i LQ3 m
can be represented by
⎡ μi 1 r ⎤ ⎡ f i 1 ( r ) ⎤
⎢
⎥
φ3 ir = ⎢⎢ μi 2 r ⎥⎥ = ⎢ fi 2 ( r ) ⎥ ( i =1,2,L, m −1， r∈[0,1])
⎢⎣ μi 3 r ⎥⎦ ⎢⎣ f i 3 ( r ) ⎥⎦
.
(5)
Hence, the polyline truncation error can be represented by
3.2 The truncation error of a polyline in three-dimensional
⎡ Ti 1 ⎤
⎢T ⎥
⎢ i 2 ⎥ ( i =1,2,L, m −1)
⎢ M ⎥
⎢ ⎥
⎣⎢ Tin ⎦⎥
space
.
Some assumptions about the functions
j =1,2,3; r∈[ 0,1])
Next, truncation error models for straight line features in three-,
need to be made, in order to enable the
estimation of the interpolation error.
two- dimensional spaces are derived based on the generic
180
f ij ( r )( i =1,2,L, m ;
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 38, Part II
f ij ( r )∈C 2 [0,1] ,
It is assumed that
giving
4.2 The truncation error of a polyline in two-dimensional
1
Rij ( r ) = fij ( r ) − pij ( r ) ≤ M 2 ij r (1− r )
2
1
≤ M 2 ij Tij
8
where
M 2 ij = max fij'' ( r )
r∈[0,1]
(6)
space
f ij ( r )( i =1,2,L, m ;
Some assumptions about the functions
.
j = 1, 2; r ∈ ⎡⎣0,1⎤⎦ )
need to be made in order to be able to estimate
Hence, the polyline truncation error can be represented by
the interpolation error.
⎡ Ti 1 ⎤
⎢ ⎥
⎢Ti 2 ⎥ ( i =1,2,L, m −1)
⎢⎣Ti 3 ⎥⎦
It is assumed
.
f ij ( r )∈C 2 [0,1] ,
giving:
1
Rij ( r ) = fij ( r ) − pij ( r ) ≤ M 2 ij r (1− r )
2
1
≤ M 2 ij Tij
8
4 THE TRUNCATION ERROR MODEL FOR STRAIGHT
Where
LINE FEATURES IN TWO-DIMENSIONAL SPACE
A polyline in two-dimensional space is composed of vectors:
T
i =1,2,L, m
,
Q22 =[ x21 , x22 ]
T
and
m
,L,
Q2i =[ xi1 ,xi 2 ]
T
,L,
Q2 m =[ xm1 , xm 2 ]
T
,
.
Hence, the polyline truncation error can be represented by
4.1 Definition of a polyline in two-dimensional space
Q21 =[ x11 , x12 ]
M 2 ij = max fij'' ( r )
r∈[0,1]
(9)
⎡ Ti 1 ⎤
⎢T ⎥ ( i =1,2,L, m −1)
⎣ i2⎦
where
.
is the number of composition points.
Any point on the line segment
Q2 i Q2,i +1 ( i =1,2,L, m −1)
is then
5 THE TRUNCATION ERROR MODEL FOR A CURVE
LINE IN AN
represented by
⎡x ⎤
Q2 ir = ⎢ i 1 r ⎥ = 1− r Q2 i + rQ2,i +1
⎢⎣ xi 2 r ⎥⎦
( )
DIMENSIONAL SPACE
5.1 Define a curve line in an
⎡ (1− r ) xi 1 + rxi +1,1 ⎤ ⎡ pi 1 ( r ) ⎤
=⎢
⎥ ⎢
⎥
⎣⎢(1− r ) xi 2 + rxi +1,2 ⎦⎥ ⎣ pi 2 ( r ) ⎦
The curve line is formed by the
Qn1 =[ x11 , x12 ,L, x1n ]
T
dimensional space
can be represented by
⎡ μ ⎤ ⎡ f (r ) ⎤
φ2 ir = ⎢ i1r ⎥ = ⎢ i1 ⎥ ( i =1,2,L, m −1， r∈[ 0,1])
⎢⎣ μi 2 r ⎥⎦ ⎢⎣ fi 2 ( r ) ⎥⎦
,
where
i =1,2,L, m
Qn2
Qnm
Qni
Qn,m−1
Figure 2. An example of a curve line in
Qni Qn ,i +1 ( i =1,2,L, m −1)
is then
represented by
181
n
-dimensional space
Qni =[ xi1 , xi 2 ,L, xin ]
T
,L,
and
composition points (as shown in Figure 2).
(8)
Qn1
-dimensional vectors:
T
Qnm =[ xm1 , xm 2 ,L, xmn ]
Q21Q22 LQ2 i LQ2 m
n
Qn 2 =[ x21 , x22 ,L, x2 n ]
,
T
Any point on the curve line
n−
(7)
(i =1,2,L,m−1， r∈[0,1])
The true polyline of
n−
m
,L,
is the number of
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 38, Part II
Qnir
⎡ x ⎤ ⎡ ai 1r 3 + bi 1r 2 + ci 1r + di 1 ⎤
⎥
⎢ i1r ⎥ ⎢
⎢ x ⎥ ⎢ a r 3 + bi 2 r 2 + ci 2 r + di 2 ⎥
⎥
= ⎢ i 2r ⎥ = ⎢ i2
⎥
M
⎢ M ⎥ ⎢
⎢
⎥
⎢⎣ xinr ⎥⎦
3
2
⎣⎢ ain r + bin r + cin r + din ⎦⎥
Any point on the curve line
⎡ p (r)⎤
⎢ i1 ⎥
⎢ pi 2 ( r ) ⎥
⎢
⎥
⎢ M ⎥
⎢ p (r )⎥
⎣ in ⎦
φ nir
Qn1Qn 2
⎡ x ⎤ ⎡ ai 1r 3 + bi 1r 2 + ci 1r + di 1 ⎤
⎥
⎢ i1r ⎥ ⎢
Q3 ir = ⎢ xi 2 r ⎥ = ⎢ ai 2 r 3 + bi 2 r 2 + ci 2 r + di 2 ⎥
⎥
⎢
⎥ ⎢
⎣ xi 3 r ⎦ ⎢⎣ ai 3 r 3 + bi 3 r 2 + ci 3 r + di 3 ⎥⎦
(10)
⎡ μ ⎤ ⎡ f (r ) ⎤
⎢ i1r ⎥ ⎢ i1 ⎥
⎢ μ ⎥ ⎢ f ( r )⎥
= ⎢ i2r ⎥ = ⎢ i 2 ⎥
⎢ M ⎥ ⎢ M ⎥
⎢⎣ μinr ⎥⎦ ⎢ f in ( r ) ⎥
⎣
⎦
⎡ p (r)⎤
⎢ i1 ⎥
⎢ pi 2 ( r ) ⎥
⎢
⎥
⎣⎢ pi 3 ( r ) ⎥⎦
(13)
(i =1,2,L,m−1， r∈[0,1])
can be represented by
LQni LQnm
is then
represented by
(i =1,2,L,m −1， r∈[0,1])
The true curve line of
Q3 i Q3,i +1 ( i =1,2,L, m −1)
The true curve line of
(11)
φ
(i =1,2,L,m−1， r∈[0,1])
3 ir
Q31Q32
can be represented by
LQ3 i LQ3 m
⎡ μ ⎤ ⎡ f (r)⎤
⎢ i1r ⎥ ⎢ i1 ⎥
= ⎢ μi 2 r ⎥ = ⎢ f i 2 ( r ) ⎥ i =1,2,L, m −1， r∈[ 0,1]
⎥
⎢
⎥ ⎢
⎣ μi 3 r ⎦ ⎢⎣ fi 3 ( r ) ⎦⎥
(
)
(14)
6.2 The truncation error of a curve line in three-dimensional
5.2 The truncation error of a curve line in
n−
dimensional
space
space
For simplicity, it is assumed that the cubic interpolation is
piecewise cubic Hermite interpolation.
For simplicity, it can be assumed that the cubic interpolation is
piecewise cubic Hermite interpolation.
Some assumptions about the functions
Some assumptions about the functions
j =1,2,L, n ; r∈[ 0,1])
j =1,2,3; r∈[ 0,1])
f ij ( r )( i =1,2,L, m ;
need to be made in order to be able to estimate
f ij ( r )( i =1,2,L, m ;
need to be made in order to be able to estimate
the error of interpolation.
the interpolation error.
It is assumed that
f ij ( r )∈C 4 [0,1] ,
It is assumed
giving
Rij ( r ) = f ij ( r ) − pij ( r ) ≤
Rij ( r ) = fij ( r ) − pij ( r )
≤
( )
1
2
M r 1− r
4! 4 ij
Where
2
≤
M 4 ij = max
r∈[ 0,1]
1
M
384 4 ij
f ij( 4)
≤
(12)
1
M
384 4 ij
Tij
Where
(r) .
giving
( )
1
2
M r 1− r
4! 4 ij
2
(15)
Tij
M 4 ij = max f ij( 4) ( r )
r∈[0,1]
.
Hence, the curve line truncation error can be represented by
Hence, the truncation error of a curve line can be represented by
⎡ Ti 1 ⎤
⎢T ⎥
⎢ i 2 ⎥ ( i =1,2,L, m −1)
⎢ M ⎥
⎢ ⎥
⎣⎢ Tin ⎦⎥
f ij ( r )∈C 4 [0,1] ,
⎡ Ti 1 ⎤
⎢ ⎥
⎢Ti 2 ⎥ ( i =1,2,L, m −1)
⎢⎣Ti 3 ⎥⎦
.
.
7 THE TRUNCATION ERROR MODEL FOR A CURVE
LINE IN A TWO-DIMENSIONAL SPACE
6 THE TRUNCATION ERROR FOR A CURVE LINE IN A
THREE-DIMENSIONAL SPACE
7.1 Definition of a curve line in two-dimensional space
6.1 Definition of a curve line in three-dimensional space
The curve line is formed by two-dimensional vectors:
Q21 =[ x11 , x12 ]
T
The curve line is formed by the three-dimensional vectors:
Q31 =[ x11 , x12 , x13 ]
T
Q32 =[ x21 , x22 , x23 ]
T
,
Q3 m =[ xm 1 , xm 2 , xm 3 ]
T
,
where
i =1,2,L, m
Q3i =[ xi1 , xi 2 , xi 3 ]
T
,L,
and
m
i =1,2,L, m
,L,
,
Q22 =[ x21 , x22 ]
and
T
m
,L,
Q2i =[ xi1 , xi 2 ]
represented by
composition points.
182
,L,
Q2 m =[ xm1 , xm 2 ]
T
,
where
is the number of composition points.
Any point on the curve line
is the number of
T
Q2 i Q2,i +1 ( i =1,2,L, m −1)
is then
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 38, Part II
Q2 ir
⎡ x ⎤ ⎡ a r 3 +b r 2 + c r + d ⎤
i1
i1
i1 ⎥
= ⎢ i1r ⎥ = ⎢ i1
⎣⎢ xi 2 r ⎦⎥ ⎢⎣ ai 2 r 3 + bi 2 r 2 + ci 2 r + di 2 ⎥⎦
⎡p r ⎤
⎢ i1 ( ) ⎥
⎣⎢ pi 2 ( r ) ⎥⎦
out that of the model error of a linear interpolated line is larger
than that from a nonlinear interpolation method, such as a cubic
(16)
interpolation method. Such as analysis result can be used as a
(i =1,2,L,m−1， r∈[0,1])
reference for the selection interpolation methods for lines..
The true curve line of
Q21Q22 LQ2 i LQ2 m
can be represented by
⎡ μi 1 r ⎤ ⎡ f i 1 ( r ) ⎤
⎥ ( i =1,2,L, m −1， r∈[0,1])
⎥ =⎢
⎣ μi 2 r ⎦ ⎣ f i 2 ( r ) ⎦
φ2 ir = ⎢
By integration of the analysis result of model error in this study,
and the propagated error of the lines in the early study, we can
(17)
have better estimation on the overall error of lines.
7.2 The truncation error of a curve line in two-dimensional
A further extension of this study will be to investigate
space
truncation error of other interpolation methods, such as hybrid
interpolation method.
For simplicity, it is assumed that the cubic interpolation is
piecewise cubic Hermite interpolation.
ACKNOWLEDGMENT
Some assumptions about the functions
j =1,2; r∈[0,1])
f ij ( r )( i =1,2,L, m ;
need to be made in order to be able to estimate the
The work presented in this paper is supported by, Hong Kong
Research Grants Council (Project No.: PolyU 5276/08E), The
error of interpolation.
Hong Kong Polytechnic University (Project No.: G-YX0P
It is assumed that
f ij ( r )∈C [0,1] ,
Rij ( r ) = f ij ( r ) − pij ( r ) ≤
≤
1
M
384 4 ij
Where
4
giving
( )
1
2
M r 1− r
4! 4 ij
G-YG66 G-YF24 1-ZV4F G-U 753), and the National Natural
Science Foundation, China (Project No. 40629001).
2
(18)
Tij
M 4 ij = max f ij( 4) ( r )
r∈[ 0,1]
.
Hence, the curve line truncation error can be represented by
⎡ Ti 1 ⎤
⎢T ⎥ ( i =1,2,L, m −1)
⎣ i2⎦
.
8 CONCLUSIONS AND DISCUSSION
To provide a full picture about the overall error of a line, we
need to quantify (a) the error of the model that is used to
interpolate the line, and (b) the propagated error from the error
of the component nodes of the line. This paper presented a
research on estimation the error of the interpolation model of
the lines, which is a step further to the early studies on
propagated error of lines.
A method for estimate the model error, truncation error, of the
lines have been developed in this study based on the
approximation theory. A line feature in GIS can be either
interpolated by linear or nonlinear functions, our research find
183
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 38, Part II
184