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Arzu Soycan and Metin Soycan

DIGITAL ELEVATION MODEL PRODUCTION FROM SCANNED TOPOGRAPHIC CONTOUR MAPS VIA THIN PLATE SPLINE INTERPOLATION Arzu Soycan Metin Soycan* Yıldız Technical University Civil Engineering Faculty Geodesy and Photogrametry Engineering Division Istanbul-Turkey

:‫اﻟﺨﻼﺻـﺔ‬ ‫ آﻤ ﺎ ﺗ ـُﻌﺘﺒﺮ‬.‫( واﻟﺒﺮﻣﺠﻴﺎت اﻟﺤﺎﺳﻮﺑﻴﺔ اﻟﻤﺼﺎﺣﺒﺔ ﻟﻬﺎ ﻣﻦ اﻻﺑﺘﻜﺎرات اﻟﻤﻬﻤﺔ ﻓﻲ اﻟﺘﻄﺒﻴﻘﺎت اﻟﺘﺨﻄﻴﻄﻴ ﺔ واﻟﺨ ﺮاﺋﻂ‬GIS) ‫ﺗﻌﺘﺒﺮ ﻧﻈﻢ اﻟﻤﻌﻠﻮﻣﺎت اﻟﺠﻐﺮاﻓﻴﺔ‬ (‫ وﻓﻲ هﺬا اﻟﺴﻴﺎق ﻳﻤﻜﻦ اﺳ ﺘﺨﺪام اﻟﺨ ﺮاﺋﻂ اﻟﻮﺻ ﻔﻴﺔ )اﻟﻄﺒﻮﻏﺮاﻓﻴ ﺔ‬.‫( اداة ﻓﻌﺎﻟﺔ ﻓﻲ رﺑﻂ اﻟﻤﻌﻠﻮﻣﺎت اﻟﺠﻐﺮاﻓﻴﺔ وﻏﻴﺮ اﻟﺠﻐﺮاﻓﻴﺔ اﺛﻨﺎء ﺗﺤﻠﻴﻞ اﻟﻤﻌﻠﻮﻣﺎت‬GIS) ‫ وذﻟ ﻚ ﻋ ﻦ ﻃﺮﻳ ﻖ اﻟﺘﺤﻮﻳ ﻞ اﻟﺮﻗﻤ ﻲ‬-GIS ‫( – واﻟ ﺬي ﻳ ﺸﻜﻞ ﻋﻨ ﺼﺮًا ﻣﻬﻤ ًﺎ ﻓ ﻲ‬DEM) ‫ آﻤﺼﺪر رﺋ ﻴﺲ ﻟﻠﻤﻌﻠﻮﻣ ﺎت ﻻﻧﺘ ﺎج أﻧﻤ ﻮذج اﻻرﺗﻔ ﺎع اﻟﺮﻗﻤ ﻲ‬TM ‫( واﻋﺘﻤﺎده ﺎ ﻋﻠ ﻰ ﻧﻘ ﺎط اﻟﻌﻴﻨ ﺔ‬TM) ‫ واﻟﺘﻲ ﺣﺼﻠﻨﺎ ﻋﻠﻴﻬ ﺎ ﺑﺎﺳ ﺘﺨﺪام‬DEM ‫ ﺗﻬﺪف هﺬﻩ اﻟﺪراﺳﺔ ﻟﺘﺤﺪﻳﺪ دﻗﺔ‬.‫اﻟﻴﺪوي أو اﻟﺘﺤﻮﻳﻞ اﻟﻤﺘﺠﻬﺔ ﻟﻠﺨﻄﻮط اﻟﻜﻮﻧﺘﻮرﻳﺔ‬ ‫ ﻟﻠﻤﻨ ﺎﻃﻖ‬DEM ‫ ﺣ ﺼﻠﻨﺎ ﻋﻠ ﻰ ﻋ ﺪد ﻣ ﻦ‬.‫ ﻓﻲ اﻟﺨﺮاﺋﻂ اﻟﻄﺒﻮﻏﺮاﻓﻴﺔ‬1/1000 ‫ اﺳﺘﺨﺪﻣﻨﺎ ﻟﻬﺬا اﻟﻐﺮض آﻨﺘﻮرات ذات ﻋﺪة ﻧﺴﺐ ﻣﻦ‬.‫وﺣﺠﻢ اﻟﺸﺒﻜﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ‬ ‫ اﻟﻨﺎﺗﺠ ﺔ ﻋ ﻦ اﻟﺨﻄ ﻮط اﻟﻜﻨﺘﻮرﻳ ﺔ وﺷ ﺒﻜﺎت ذات اﻗﺘ ﺮان‬DEM‫ أوﻟﻴﻨﺎ اهﺘﻤﺎﻣًﺎ ﺧﺎﺻ ًﺎ ﻟ ـ‬.‫اﻟﻤﻘﺎﺑﻠﺔ ﺑﺎﺳﺘﺨﺪام ﻣﺠﻤﻮﻋﺎت اﻟﻤﻌﻠﻮﻣﺎت ذات اﻋﺪاد ﻣﺨﺘﻠﻔﺔ ﻣﻦ اﻟﻨﻘﺎط‬ .‫( ﻟﻠﺘﻮﺻﻴﻒ اﻟﻤﻤﺘﺪ – أي اﻟﻤﻮاﺋﻤﺔ اﻟﺴﻄﺢ – ﻧﻮرد ﻓﻲ هﺬا اﻟﺒﺤﺚ ﺧﻮارزﻣﻴﺔ اﻟﺤﻠﻮل اﻟﻤﻘﺘﺮﺣﺔ وﻧﺘﺎﺋﺞ ﺗﻄﺒﻴﻘﺎﺗﻬﺎ وﻣﺪى دﻗﺔ هﺬا اﻻﺳﻠﻮب‬RBF) ‫داﺋﺮي‬ ABSTRACT GIS (Geographical Information System) is one of the most striking innovation for mapping applications supplied by the developing computer and software technology to users. GIS is a very effective tool which can show visually combination of the geographical and non-geographical data by recording these to allow interpretations and analysis. DEM (Digital Elevation Model) is an inalienable component of the GIS. The existing TM (Topographic Map) can be used as the main data source for generating DEM by amanual digitizing or vectorization process for the contours polylines. The aim of this study is to examine the DEM accuracies, which were obtained by TMs, as depending on the number of sampling points and grid size. For these purposes, the contours of the several 1/1000 scaled scanned topographical maps were vectorized. The different DEMs of relevant area have been created by using several datasets with different numbers of sampling points. We focused on the DEM creation from contour lines using gridding with RBF (Radial Basis Function) interpolation techniques, namely TPS as the surface fitting model. The solution algorithm and a short review of the mathematical model of TPS (Thin Plate Spline) interpolation techniques are given. In the test study, results of the application and the obtained accuracies are drawn and discussed. The initial object of this research is to discuss the requirement of DEM in GIS, urban planning, surveying engineering and the other applications with high accuracy (a few decimeters). Key words: radial basis functions, thin plate splines, topographic map, digital elevation model, scattered data interpolation

* Corresponding Author E-mail: [email protected] Paper Received: 21 November 2007; Accepted 22 March 2008

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DIGITAL ELEVATION MODEL PRODUCTION FROM SCANNED TOPOGRAPHIC CONTOUR MAPS VIA THIN PLATE SPLINE INTERPOLATION 1.

INTRODUCTION

TM derived elevation information is an important data source especially for GIS applications. The use of contours in TM is one of the most common and conventional method for representation of the elevations. TM could be integrated in GIS by vectorization process as digital information or directly in raster format. Mostly, users need digital information about the terrain such as planimetric coordinates or elevations. Therefore, raster data cannot satisfy the user expectations in real time for the practical engineering and geosciences applications [1-6]. Thus, transformation from the TM to the DEM in spatial coordinate system is an inevitable study. DEM is the 3D representation of a part of the earth surface in global or local scale as digital format with the help of SRP (Sampling Reference Points) and the modeling algorithm. Today, DEM have been widely used in many applications, such as urban planning, civil engineering, landscape building, mining engineering, military planning, aircraft simulation, radio communications planning, visibility analysis, hydrological modeling, and routine mapping applications. The selection of the data collection method depends on accuracy, time of production and cost components. Different methods have been used to collect DEM data in the applications: o

Terrestrial topographical surveying;

o

GPS supported observation;

o

Photogrametric measurement techniques;

o

Digitizing the existing topographical map.

In recent years, many techniques have become popular for elevation data collection such as: o

Orthophoto maps;

o

Remote sensing images;

o

Lidar “light detection and ranging” technology;

o

Interferometric SAR (syntactic aperture radar) data;

o

Digital elevation databases.

Modeling algorithm based on SRP with known planimetric positions and elevations yields an interpolation problem. Interpolation is the prediction of unobserved point elevations by using observation values from SRP distributed around the unobserved point in the concept of DEM. The various methods used for interpolation of elevation are based on the elevations of for the determination of surface models of DEM. Selected interpolation models must be realistic and well adjusted with the topographic characteristics of surface [7–11]. The development of computer and software technology with improved mathematical functions offers to the user several interpolation methods for scattered data. In the technical literature, RBF interpolation models have a priority, with their widespread applications in many areas [12–17]. One of the advantages of the model is its flexibility within the single RBF. The RBF methods are modern ways to approximate multivariate functions, especially in the absence of grid data. RBF method has also been used to interpolate irregularly spaced data, which computes the signeddistance function prior to generating the RBF interpolant. Different type functions can be used as the depending on the surface and data characteristic. The most useful RBF, which provides good accurate approximations, is thin-plate splines. The TPS is an effective tool for scattered data interpolation problems. Many scientific researches show that the TPS basis function can be applied confidently in most cases [12, 18-21]. In this paper, the accuracy of the DEM constructed from scattered sample data that derived from contour polylines of the TM via TPS interpolation is examined. The achieved DEMs are tested about their accuracies by means of cross validation analysis, elevation comparison for independent check points and topographic surveys derived elevation comparison for homogenously selected test points.

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2.

TPS INTERPOLATION

Considering f: IRd→IR a real valued function of d variables that is to be approximated by s: IRd→IR, given the values, {f(xj): j = 1,2,………..,n}, where {xj: j = 1,2,………..,n} is a set of distinct points in IRd called the nodes of interpolation. The well-known equation for the RBF can be defined as,

((

n

s( x) = ∑ a j E x − x j j =1

) ) , x ∈ IR d , a j ∈ IR ,

(1)

where x∈ IRd are finitely or infinitely many regularly or arbitrarily distributed “centers”, aj are suitable real coefficients usually fixed by interpolation conditions at the centers, the norm is Euclidean and E is the radial basis functions (Table 1). If the data are given through f: IRd→IR at the centers, the interpolation conditions become s (xj) =f(xj) for all xj [12–14, 22–24]. Table 1. The Frequently-Used Functions for RBF Interpolation RBF

Function Descriptions

Multiquadratic

E (r ) = r 2 + c 2

Inverse Multiquadratic

E (r ) = 1 / r 2 + c 2

Rational Quadratic Spline

E ( r ) = r 2 /(1 + r 2 )

Natural Cubic Spline

E (r ) = r 3

TPS

E ( r ) = r 2 log r 2

Multilog

E ( r ) = log r 2

Marcov Spline

E (r ) = −r

Exponential Spline

E (r ) = e −r

( )

TPS is a physically based 2D interpolation scheme for arbitrarily spaced tabulated data (xi, yi, f(xi, yi)). The TPS is the two-dimensional analog of the cubic spline in one dimension. The spline surface represents a thin metal sheet that is constrained not to move at the grid points. The idea is to build a function f(x, y) whose graph passes through the tabulated data and minimizes the bending energy function [25]. Bending energy is defined here as the integral over IR 2 of the squares of the second derivatives,

(

)

2 2 2 ∫∫IR 2 f xx + 2 f xy + f yy dxdy ,

for tabulated points

{(xi , yi , zi )}in=1

(2)

the minimizing function is of the form, n

((

f ( x, y ) = ∑ a j E x − x j , y − y j j =1

) )+ b0 + b1x + b2 y ,

(3)

( )

where E ( r ) = r 2 log r 2 and . indicates length of a vector. The coefficients aj and bj are determined by requiring exact interpolation. This leads to n

zi = ∑ Eij a j + b0 + b1 xi + b2 yi , j =1

((

for 1≤i≤n where Eij = E xi − x j , yi − y j

(4)

) ) . In matrix form this is r r r z = Aa + Bb ,

(5)

where A = [Eij ] is an n × n matrix and where B is an n × 3 matrix whose rows are [1 xi yi]. An additional condition is that, n n r r n B t a = 0 , ∑ ai = ∑ ai .xi = ∑ ai . yi = 0 , i =1

i =1

i =1

(6)

these two equations can be solved to obtain separately, The Arabian Journal for Science and Engineering, Volume 34, Number 1A

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(

)

r r r a = A −1 z − Bb , r −1 r b = B t A−1B B t A−1 z

(

)

(7)

It can be generalized to the composite solution algorithm as below:

⎡0⎤ ⎡ bo ⎤ ⎡0 0 ⎢ ⎥ ⎢ ⎥ ⎢ b 0 ⎢ ⎥ ⎢ 1⎥ ⎢0 0 ⎢0⎥ ⎢ b2 ⎥ ⎢0 0 ⎢ ⎥ ⎢ ⎥ ⎢ L = ⎢ z1 ⎥ ; X = ⎢ a1 ⎥ ; W = ⎢ 1 x1 ⎢z ⎥ ⎢a ⎥ ⎢1 x 2 ⎢ 2⎥ ⎢ 2⎥ ⎢ ⎢ ... ⎥ ⎢ ... ⎥ ⎢... ... ⎢z ⎥ ⎢a ⎥ ⎢1 x n ⎣ n⎦ ⎣ n⎦ ⎣

0 0

1 x1

1 x2

... ...

0

y1

y2

...

y1

0

E12

...

y2 ...

E21 ...

0 ...

... ...

yn

En1

En 2 ...

1 ⎤ ⎥ xn ⎥ yn ⎥ ⎥ E1n ⎥ ; L = X .W E2 n ⎥ ⎥ ... ⎥ 0 ⎥⎦

(8)

where W is a design matrix which is symmetric with diagonal elements zero, X is an unknown coefficients matrix with the components ai and bi, an L is a matrix based on the known elevations of the control points in spatial system. To determine the a0, a1, a2, mi and ni coefficients, the matrix equations can be solved as:

(

X = WTW

)

−1

WT L ,

(9)

a new point elevation can be interpolated as depending on the x, y coordinates, Eij values and the computed coefficients b0, b1, b2, ai, by using Equation (4). It is also possible to provide a smoothing term. In this case, the interpolation is not exact. The modification is to use the equation, Pz = (A + λI) Pa + B Pb,

(10)

where λ> 0 is a smoothing parameter and I is the n × n identity matrix [25]. The most important advantage of this method is the fitting of the surface extremely optimal form by using homogenously distributed control points with accurately coordinates information both in horizontal and vertical dimension. The surface passes from the used control points and these points do not contain any residuals. TPS is described as a function of distances between control points used in interpolation, so the effects of a distant control points on the elevations of a new interpolation points decrease. Especially very successful solution can be achieved in the interpolation area by using TPS. It is possible to do a partly extrapolation near the border of the interpolation area. The easiness of the computation connected with the number of the SRP as depending on the inverse problem when the determination of the unknown coefficients [12–17, 26]. 3.

THE TEST STUDY

3.1. Data Sets The mathematical algorithm in the previous section is carried out in order to derive a continuous surface by using vectorized contour polylines of TMs. Five different 1/1000 scaled topographic paper map sheets (Figure 1) were used for test study, the sizes of the maps are 70×90cm and contour intervals are 1m. The original map sheets were produced from 1/4000 scaled airphotos in the photogrammetric mapping project of Istanbul Metropolitan Areas by Istanbul Municipality. Different type topographic maps have been considered in test study. The maps of 1 and 2 have closely spaced and frequent contours in high density. The terrains consist of highly sloping and smooth areas in together. However, geometry of contours is changing regularly and they represent smooth terrain type. On the other hand, the maps 3 and 4 show similarity to 1 and 2 but they have more simple contours with wide interval. Thus, these represent more smooth topography and fewer slopes. Finally, the terrain of map 5 is highly rough where there are frequent changes in elevation. The geometry of the contour lines is very frequent and complicated due to local changes in terrain characteristics.

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Figure 1. Scanned TM sheets


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DEM Computations and Productions First of all, map sheets were scanned by a scanner with an optical resolution of 600×600 dpi so it was achieved in raster image format. Contour widths of the original maps in Figure 1 were plotted with 0.13mm to 0.25mm, so the raster contour line thickness varies from 3 to 6 pixels for all of it.

SCANNING PROCESS Acquisition of Contour Map in Raster Image Format

RASTER TO VECTOR CONVERSION Thresholding Grayscaling Filtering Thinning Edge Detection

SELECTION AND DIGITIZING EQUIVELENT CONTROL POINTS WITH PLANIMETRIC COORDINATES FOR TRANSFORMATION

COMPUTATION OF THE TRANSFORMATION PARAMETERS BETWEEN RASTER AND GROUND COORDINATE SYSTEM Affine Transformation Methods

DIGITIZING OF CONTOUR LINES AS POLYLINE IN RASTER COORDINATE SYSTEM

TRANSFORMATION OF THE POINTS OF THE POLYLINES TO GROUND COORDINATE SYSTEM AND UPDATING OF THEIR HEIGHT INFORMATION

FORM OF THE 3D DEM INPUT DATA FILE FOR SURFACE FITTING Different input data set with the number of sampling points 100%, 50%,25%...... respectively

QUALITY CONTROL OF INTERPOLATION AND ACCURACY ASSESSMENT Cross validation analysis Independent check points Surveyed points

PERFORMING OF INTERPOLATION TECHNIQUE TO DETERMINATION OF THE LAND SURFACE Thin Plate Spline Different grid size with search ellipse and smoothing parameters

DEMONSTRATION OF DEM

Figure 2. Flow-chart of the applied algorithm for DEM production

In the next stage, the contour lines on the raster were digitized as polylines by the help of a semi-automatic vectorization process. A number of control points were selected on the raster images and map sheet for transformation between raster and spatial coordinate systems. In principle, the control points were selected on the corners and the center of the TM sheets. Affine coordinate transformation was used for transforming the raster coordinate system to a spatial coordinate system via multiple equivelent control points with known coordinates in

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both spatial and raster image coordinate systems. As the result of the affine transformations, F0, F1, F2, G0, G1, G2 transformation coefficients were estimated (Table 2). Transformation accuracies were computed by Root Mean Square (RMS) error of residuals to a few cm. Table 2. Transformation results between raster and spatial coordinate systems #CONTROL POINTS MAP1

5

MAP2

5

MAP3

5

MAP4

5

MAP5

5

x = F0+F1.u+F2.v F0=4554051.535 F1=-3.26377×10-5 F2=-0.124350155 F0=4554033.414 F1=-3.68099×10-5 F2=-0.12130668 F0=4554737.318 F1=-1.80407×10-5 F2=-0.122579186 F0=4554732.520 F1=-1.10185×10-5 F2=-0.122948911 F0=4560049.289 F1=-9.50715×10-6 F2=-0.12702789

y = G0+G1.u+G2.v G0=387620.063 G1=0.124344747 G2=-1.57271×10-5 G0=388155.160 G1=0.121293061 G2=-2.21654×10-5 G0=387640.658 G1=0.122566952 G2=-8.73534×10-6 G0=388158.912 G1=0.122974229 G2=4.05902×10-6 G0=406067.244 G1=0.126982555 G2=-1.12589×10-6

RMSx

RMSy

RMStot

0.040

0.030

0.049

0.028

0.059

0.065

0.024

0.040

0.046

0.001

0.042

0.042

0.029

0.011

0.031

Later, the points on the vectorized contour polylines were transformed to a spatial coordinate system by using the achieved transformation coefficients and elevation data were added to them. Thus, 3D input data were constituted for each map sheets. Table 3. Data sets information #TOTAL POINTS #OUTLIERS #CHECK POINTS #SURVEYED POINTS DATA SET 1 DATA SET 2 DATA SET 3 DATA SET 4 DATA SET 5 DATA SET 6

MAP1 63782 47 2155 315 61580 30790 15395 7698 3849 1924

MAP2 56471 21 1909 368 54541 27271 13635 6818 3409 1704

MAP3 50280 33 1699 213 48548 24274 12137 6069 3034 1517

MAP4 45556 29 1540 296 43987 21994 10997 5498 2749 1375

MAP5 51932 35 1741 401 50156 25078 12539 6270 3135 1567

The total numbers of points varies between 45556 and 63782 for polylines of each of the map sheets. Some points were identified as outliers from CV analysis and removed from the initial input data. Moreover, random check points were extracted from the achieved points in order to make an accuracy analysis. These points were not included in interpolation process. Thus, the remaining points were used as the full input data set (Table 3 “data set 1”). Several sampling points in different densities were considered from the full input data set. As the first data set 100% of points were used completely for all. Then 50%, 25%, 12.5%, 6.25%, 3.124% random points were selected respectively from the full input data in order to evaluate DEM accuracy in the variation of the SRP density and the variation of grid size. In general, interpolation methods fall under two main categories, a global approach and a local approach. In the global approach, each interpolated value is influenced by all the control points and therefore limited by the size of the data set. In addition to being computationally expensive, a correction in any of the elevations of a data point will modify the rest of the interpolated values. As the result of these advantages, we used the local approach. In principle, we represented of DEM in grid data format, so DEM of the test study were produced by converting the irregularly spaced input data points to interpolated grid node values. Five different grid sizes from 1m to 20m (1, 2.5, 5, 10, 20) were applied for each data set handled in previous paragraphs. Thus, the interpolation of 6 different sampling of 5 topographic map sheets for 5 different grid spacing, namely total of 150 different data sets was carried out.


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50%

25%

12.5%

6.25%

3.124%

MAP5

MAP4

MAP3

MAP2

MAP1

100%

Figure 3. Distribution of SRP used in interpolation

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In the test study, Golden Software Surfer 8.0 was used for gridding process. We considered the local approach by the help of isotrop search ellipse with a 100m sized radius; the search ellipse defines the local neighborhood of points to consider during the interpolation. Data points outside the search ellipse were not considered during gridding. Moreover the smoothing parameter was considered at the gridding stage of the input data. According to previous researches, several empirical equations may be used to compute the smoothing parameter. However, there is no universally accepted method for computing an optimal value for this factor. RBF gridding algorithm of Surfer 8.0 use the default value for smoothing parameter as (length of diagonal of the data extent)2 / (25 × number of data points). 3.2. Accuracy Assestments of Achieved DEM’s To evaluate the quality of interpolated elevations, RMS statistics were used for DEM’s producted by using each individual data set. RMS error is the most widely used statistics as a measure [11, 27–30], it measures the dispersion of the frequency distribution of deviations between the original value and the estimated value. RMS error for the deviations between original and interpolated elevations can be given as: RMS =

1 n 2 ∑ (zi − zi′ ) n i =1

(11)

where n is the number of points zi is the original or known elevation, zi’ is interpolated from the check points. The interpolation accuracy can be measured by the differences between interpolated and the original input data points elevations. The RMS error characterizes the interpolation accuracy of the the relevant points. However, this approach does not provide information about the accuracy in TPS interpolation. Because the interpolation surface passes through the given input points, these points have no difference in elevation. In order to achieve the objective of this paper, interpolation was evaluated using variation of the SRP density and variation of grid size. We used three different analyses to measure the accuracy. First of all we generate an accuracy statistic by CV (Cross Validation) that is widely used in interpolation applications. Generally, a cross validation statistic can be considered a measure of the surface error for assessing the quality of the interpolations. The accuracy analysis by cross-validation is based on removing one input data points in the beginning step, realizing the interpolation for the removed point using the remaining data and calculating the difference between the original value of the removed data point and its interpolated value [31]. The procedure is repeated for every sample. The overall input data for each sampling set were used in our CV analysis to identify problems in sampling data points. The other accuracy measure is the use of independent check points which are not used in the interpolation. This known as the jack-knifing technique based on removing of random data from sampling data points, and using the remaining data to perform the interpolation. The elevations of the removed random points are then interpolated for each check point, the difference between the original and the interpolated values are calculated and the overall accuracy is tested with the help of the RMS errors [11, 27–30]. For this purpose, normally distributed and random selected check points were used for each map sheet (Table 3 “check points”). Although cross validation analysis and jack-knifing technique are very important indicators to evaluate the appropriateness of the applied interpolation for DEM, they reveal only the inner accuracy relating to the model. Separately, the outer accuracy of DEM must be tested by using different sampling data which derive from a different source. In the last accuracy analysis, DEM of the test area were compared with elevation information from the test surveying data in the same area that were produced from precise topographic surveys. Topographic surveys were performed at different normally distributed points, which represent the test area characteristics for each map sheet (Table 3 “surveyed points”).


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Table 4. RMS Errors of Different Grid Size and Sampling Points for Each Data Set DATA

CATEGORY 1

MAP1

MAP2

CATEGORY 2

MAP3

CATEGORY 3

MAP4

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MAP5

Samplings%

CV

JACK-KNIFING 1m

2.5m

5m

10m

SURVEYED POINTS 20m

1m

2.5m

5m

10m

20m

100

0.026 0.040 0.055 0.097 0.195 0.371 0.133 0.140 0.169 0.261 0.444

50

0.045 0.057 0.068 0.101 0.200 0.374 0.141 0.148 0.172 0.266 0.448

25

0.080 0.094 0.101 0.124 0.204 0.376 0.163 0.169 0.188 0.268 0.451

12.5

0.135 0.150 0.153 0.166 0.220 0.384 0.215 0.218 0.230 0.285 0.459

6.25

0.216 0.219 0.220 0.227 0.261 0.385 0.285 0.286 0.293 0.328 0.461

3.125

0.313 0.329 0.329 0.335 0.350 0.442 0.413 0.414 0.419 0.435 0.530

100

0.030 0.043 0.054 0.084 0.157 0.326 0.112 0.121 0.153 0.246 0.480

50

0.045 0.054 0.062 0.087 0.160 0.328 0.118 0.126 0.155 0.251 0.483

25

0.071 0.086 0.090 0.106 0.163 0.331 0.152 0.157 0.178 0.253 0.487

12.5

0.114 0.138 0.140 0.148 0.187 0.331 0.220 0.223 0.234 0.286 0.488

6.25

0.175 0.187 0.188 0.194 0.222 0.358 0.288 0.290 0.298 0.336 0.524

3.125

0.274 0.260 0.262 0.266 0.282 0.397 0.391 0.394 0.398 0.421 0.580

100

0.020 0.019 0.026 0.051 0.094 0.175 0.103 0.106 0.124 0.176 0.317

50

0.031 0.029 0.035 0.053 0.096 0.180 0.102 0.107 0.123 0.182 0.323

25

0.052 0.053 0.056 0.066 0.097 0.181 0.134 0.137 0.147 0.187 0.330

12.5

0.082 0.082 0.083 0.089 0.113 0.185 0.179 0.181 0.186 0.217 0.333

6.25

0.123 0.121 0.120 0.123 0.138 0.195 0.235 0.234 0.239 0.260 0.354

3.125

0.180 0.185 0.185 0.181 0.196 0.232 0.342 0.342 0.333 0.358 0.416

100

0.022 0.024 0.033 0.058 0.108 0.206 0.088 0.094 0.120 0.188 0.342

50

0.034 0.034 0.041 0.060 0.109 0.208 0.093 0.100 0.121 0.190 0.346

25

0.058 0.059 0.062 0.075 0.111 0.209 0.117 0.122 0.138 0.191 0.346

12.5

0.090 0.087 0.089 0.096 0.121 0.210 0.156 0.159 0.169 0.206 0.349

6.25

0.125 0.115 0.117 0.121 0.138 0.216 0.197 0.199 0.206 0.233 0.357

3.125

0.179 0.169 0.169 0.172 0.183 0.247 0.284 0.285 0.290 0.306 0.409

100

0.045 0.071 0.089 0.143 0.248 0.431 0.268 0.277 0.317 0.414 0.572

50

0.087 0.098 0.111 0.154 0.248 0.426 0.271 0.280 0.319 0.412 0.567

25

0.144 0.149 0.154 0.182 0.258 0.423 0.299 0.307 0.340 0.427 0.568

12.5

0.210 0.217 0.220 0.234 0.281 0.423 0.372 0.377 0.396 0.457 0.581

6.25

0.304 0.300 0.302 0.308 0.335 0.445 0.447 0.449 0.461 0.507 0.608

3.125

0.422 0.397 0.398 0.401 0.417 0.499 0.544 0.545 0.553 0.579 0.659



CATEGORY 1 CROSS VALIDATION

0.4

CV RMS Error(m)

CV RMS Error(m)

0.2

0.5

CV RMS Error(m)

0.4 0.3

0.2

0.1 0.2

0.1

0.1

0.0

0.0

#samplings 25

50

75

100

J.K. RMS Error(m)

0.5

JACK-KNIFING

CATEGORY 3

0.3

0

0.0

#samplings 0

25

50

75

J.K. RMS Error(m)

0.3

#samplings 0

100

25

50

75

100

75

100

75

100

J.K. RMS Error(m)

0.6

0.4

0.5 0.2

0.3 0.2

0.4 0.3

0.1

0.1

0.2

0.0 0 0.6

SURVEYED POINTS

CATEGORY 2

25

#samplings 50

0.0 75

0

100 0.5

S.P. RMS Error(m)

25

#samplings 50

0.1 75

100

0 0.7

S.P. RMS Error(m)

0.5

0.4

0.6

0.4

0.3

0.5

0.3

0.2

0.4

0.2

0.1

0.3

0.1

0.0

#samplings 0

25

50

75

100

25

50

75

100

#samplings 50

S.P. RMS Error(m)

0.2

#samplings 0

25

#samplings 0

25

50

Figure 4. Analysis results of interpolation as depending on number of SRP

All the accuracy examination results are reported in Table 4. As the result of this examination we recognized that map 1 yields very similar results to map 2, and map 3 is similar to map 4. So we categorized maps as in three categories based on their similarity and achieved accuracy. We considered the accuracy of category 1 as the mean of the RMS values of map 1 and 2, for category 2 that of the RMS values of map 3 and 4. Finally, map 5 alone represents category 3. Figure 4 shows the RMS error of the differences between the original and interpolated elevation value for each sampling data set and grid size, using respectively CV, jack-knifing technique, and topographic survey comparisons for each category.

•

According to the cross validation analysis, RMS errors of the elevation differences are 0.026m to 0.313m, 0.030m to 0.274m, 0.020m to 0.180m, 0.022m to 0.179m, 0.045m to 0.422m respectively for map 1,2,3,4, and 5, based on different numbers of sampling points.

•

According to the jack-knifing analysis for independent check points, RMS errors of the elevation differences are 0.040m to 0.422m, 0.043m to 0.397m, 0.019m to 0.232m, 0.024m to 0.247m, 0.071m to 0.499m, respectively for map 1,2,3,4, and 5, based on different numbers of sampling points and grid size.

•

Finally, the RMS error varies, 0.133m to 0.530m, 0.112m to 0.580m, 0.103m to 0.416m, 0.088m to 0.409m, 0.268m to 0.659m, as the result of the surveyed point statistics for each data set of DEM’s.


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Various International Map Accuracy Standards define the maximum RMS error as (1/3) × (Contour interval) for accuracy criterion. So, the maximum acceptable RMS error is 0.333m. According to test results,

•

1 and 2.5m gridding with 50% and 100% samplings are close to each other and these are more accurate than the others. Furthermore, these results demonstrate that there is no significant difference between the RMS errors and correlation coefficients for 1m and 2.5m gridding size with 100% and 50%. This is also clear from Figure 4 and Table 4.

•

The magnitude of RMS values (a few decimeters) shows that use of the infrequent sampling points such as 3.125% and large grid size 20m are not sufficient for the production of accurately DEM for each available data set.

•

For a general conclusion, 100%, 50%, and 25% sampling points with 1m, 2.5m, and 5m grid size can be used confidently to produce a good DEM in allowable accuracy limits.

For each data set, TPS can produce the best DEM but it is quite a bit slower. Using TPS with large data sets does not result in significantly different gridding times. For example, if data file contains 3000 or 30000 data points, the gridding time is not significantly different. Either data set might take a considerable amount of time to grid, but they take approximately the same amount of time. 4.

RESULTS AND CONCLUSION

Practical applications, such as civil engineering, large-scale mapping, detailed hydrological modeling, precision agriculture, and typical surveying applications need a high resolution (1–5m) DEM with micro scale. Accurate DEM information can be acquired from large-scale topographical maps easily by the invention and development of automatic vectorization processes. The accuracy of interpolated elevation varies depending on the scale of the topographical map, scanner resolution and settings, errors during the vectorization, raster to spatial transformation errors, number of sampling points, grid size, geometry and thickness of contour lines, and topographic characteristics of the terrain. Outliers, systematic, and random errors affect the final accuracy of the creating DEM. Original digitizing by digitizer tool or manually on a raster dataset takes up lots of time. In addition to this, detecting the outliers or systematic errors is not possible in this process. For example, when the original map is deformed, the effects mentioned above affect directly the DEM’s accuracy. On the other hand, vectorization provides highly accurate data access in a short time. All kind of outliers can be eliminated by filtering, CV, and the other error analysis methods. The most important property of the TPS is the fitting of the surface from control points by trying several variations. In short, a very smooth surface can be fitted and the residuals on the control points can be completely eliminated. Besides, TPS interpolation is suitable for inhomogeneous data and it can be used for extrapolation, depending on the data limit. However, the mathematical models are complex and their computation is difficult compared to other interpolators. The most important disadvantage of the TPS interpolation is that it forms errors on the corner regions of the interpolation area. As the result of all these examinations, we can defend the proposition that TPS interpolations can be effectively used for generation of DEM from each type of vectorized TM at the expected accuracy level. Users should consider these suggestions to achieve a good DEM that is within good accuracy standards:

132

•

The raster image should be achieved by at least 300 dpi optical resolution from TMs. If the contours are close together, namely distances of the elevation contours are close or terrain is rough, the optical resolution should be increased.

•

The parameters such as resolution, bit depth, threshold value, and gamma correction of the scanner should be selected as optimal. So it will influence the quality of the raster output image.

•

Raster image should be improved by using image-processing parameters, such as thresholding, grayscaling, filtering, thinning, and edge detection before the vectorization process.

•

The vectorization process has to be performed with the maximum points possible. By omitting the mean elevation contours for decreasing the obtained high accurate point group, a significant loss of accuracy arises. It is more suitable to select the points in the same elevation contours to decrease the point group.

•

Any details except the elevation contours should not be kept on the TMs. The other objects or details should be identified and removed from the data set.

•

An appropriate transformation method should be used for the transformation of raster coordinates to the spatial coordinate system. The accuracy of the transformation should not exceed 15–20 cm.

January 2009



•

The most important case for determining the number of points, which will be used for creating the DEM, is the scale of the TM, the ground type, and the geometric shape of the contours relating to the topographic structures (smooth, wavy, rough). The contours, which are especially located on the characteristic regions like hill, stream, valley, slope, cliff or mound, should be depicted by closer points. For orderly alternating smoothing contours and the level grounds, fewer points can be sufficient. At the processing stage of the study, according to experience, a general evaluation for point density can be done like this: o

For the highly smooth areas; one point per 100m2,

o

For the wavy areas; one point per 50m2,

o

For the rough areas; one point per 20m2,

o

For the very rough areas and special characteristic regions; one point per 10m2.

•

The other critical issue in gridding process is to select an adequate grid size. Grid size depends on the scale of the map and the distances between neighbor contours. For highly sloping areas it may be a great problem if the geometry of the contours are complicated. This should be examined before interpolation of DEM. After producing the surface by the gridding method, the output products depend on grid data format. Therefore, it is anticipated that the grid data denoted the real ground surface.

•

Achieved accuracies of the obtained DEM should be analyzed. First, the employed interpolation and the compatibility of the interpolation parameters should be tested with the CV Analysis. By excluding a definite partial (e.g. percentage 5 of the homogenously distributed) of the dataset used for interpolation, the differences between the original elevations and found elevations from the interpolations should be made explicit. The two methods of analyses declared above are effective for determining the outliers and the accuracy of the interpolation and dataset. They do not give any idea about the DEM’s outer accuracy. Therefore, the creating DEM must be tested with the same or higher accuracy elevation information, which is found via different sources. In this test study, elevations of the homogenously distributed test points can be used. At same time, profile controls of the DEM’s can be done for different regions. The most important points for all of the analyses mentioned before are the amount of the RMS errors.

•

The RMS error should not be more than the 1/3 of the contour intervals.

REFERENCES [1]

P. Arrighi and P. Soille, “From Scanned Topographic Maps to Digital Elevation Models”, Proceedings of Geovision’99, International Symposium on Imaging Applications in Geology, University of Liege, Belgium, ed. D. Jongmans, E. Pirard, and P. Trefois, 1999.

[2]

O. Bonin and F. Rousseaux, “Digital Terrain Model Computation from Contour Lines: How to Derive Quality Information from Artifact Analysis”, GeoInformatica, 9(3)(2005), pp. 253–268.

[3]

W. Chao-Hsiung, “Automated Identification of Scanned Contour Information For Digital Elevation Models”, PhD Thesis. The Graduate School University of Maine. Edited by M. Caetano and M. Painho, 1993.

[4]

M.B. Gousie, “Contours to Digital Elevation Models: Grid-Based Surface Reconstruction Methods”, PhD Thesis, Rensselaer Polytechnic Institute, 1998, 126 pp.

[5]

A. Sindayihebura, M.V. Meirvenne, and S. Nsabimana, “Comparison of Methods for Deriving a Digital Elevation Model from Contours and Modelling of the Associated Uncertainty”, 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences, 2006.

[6]

D. Xin, X. Zhou, and H. Zheng, “Contour Line Extraction from Paper-based Topographic Maps”, Journal of Information and Computing Science, 1(5)(2006), pp. 275–283.

[7]

F. Aguilar, M.A Aguilar, F. Aguera, and J. Sanchez, “The Accuracy of Grid Digital Elevation Models Linearly Constructed from Scattered Sample Data”, International Journal of Geographical Information Science, 20(2)(2006), pp. 169–192.

[8]

D.A. Algarni and M.E.I. Hassan, “Comparison of Thin Plate Spline, Polynomial, CI-Function and Shepard’s Interpolation Techniques with GPS-derived DEM”, JAG, 3(2)(2001), pp. 155–161.

[9]

A. Almansa, F. Cao, Y. Gousseau, and B. Rougé, “Interpolation Of Digital Elevation Models Using AMLE And Related Methods”, IEEE Transactions On Geoscience And Remote Sensing, 40(2)( 2002), pp. 314–325.

[10]

W.Z. Shi and Y. Tian, “A Hybrid Interpolation Method for the Refinement of a Regular Grid Digital Elevation Model”, International Journal Of Geographical Information Science, 20(1)(2006), pp. 53–67.


January 2009

133


134

[11]

M. Soycan and A. Soycan, “Surface Modeling For GPS-Levelling Geoid Determination”, Bulletin of International Geoid Service–Newton’s Bulletin, 1(2003), pp. 41–52.

[12]

M.D. Buhmann, “Radial Basis Functions”, Acta Numerica (Cambridge University Press), 2000, pp. 1–38.

[13]

J.C. Carr, W.R. Fright, and R.K. Beatson, “Surface Interpolation with Radial Basis Functions for Medical Imaging”, IEEE Transactions On Medical Imaging, 16(1)(1997), pp. 96–107.

[14]

D. Lazzaro, and L.B. Montefusco, “Radial Basis Functions for the Multivariate Interpolation of Large Scattered Data Sets”, Journal of Computational and Applied Mathematics, 140(2002), pp. 521–536.

[15]

Y. Ohtakea, A. Belyaevb, and H.P. Seidelb, “3D Scattered Data Interpolation and Approximation with Multilevel Compactly Supported RBFs”, Graphical Models 67, 2005, pp. 150–165.

[16]

J. Pouderoux and J.C. Gonzato, “Adaptive Hierarchical RBF Interpolation for Creating Smooth Digital Elevation Models”, in GIS’04, Washington, DC, USA, November 12–13, 2004.

[17]

J. G. Wang and G. R. Liu, “A Point Interpolation Meshless Method Based on Radial Basis Functions”, International Journal For Numerical Methods In Engineering, 54(2002), pp. 1623–1648.

[18]

D.N. Fogel and L.R. Tinney, “Image Registration using Multiquadric Functions, the Finite Element Method, Bivariate Mapping Polynomials and Thin Plate Spline”, National Center for Geographic Information and Analysis Technical Report 96-1, 1996.

[19]

A.D. Hartkamp, K. De Beurs, A. Stein, and J.W. White, “Interpolation Techniques for Climate Variables”, Natural Resources Group Geographic Information Systems Series 99-01, 1999.

[20]

M.F. Hutchinson, “Interpolation of Rainfall Data with Thin Plate Smoothing Splines - Part II: Analysis of Topographic Dependence”, Journal of Geographic Information and Decision Analysis, 2(2)(1998), pp. 152 –167.

[21]

S. Taghvakish and J. Amini, “Optimum Weight in Thin Plate Spline for Digital Surface Model Generation”, FIG Working Week 2004. TS26 Positioning and Measurement Technologies and Practices II, Athens, Greece, May 2227, 2004.

[22]

B.J.C. Baxter, “The Interpolation Theory of Radial Basis Functions”, PhD Thesis, Cambridge University, 1992, 142 pp.

[23]

M.D. Buhmann, Radial Basis Functions. Cambridge: Cambridge University Press, 2003.

[24]

A. Iske, “Radial Basis Functions: Basics, Advanced Topics and Meshfree Methods For Transport Problems.” Rend. Sem. Mat. Univ. Pol. Torino. Splines and Radial Functions, 61(3)(2003).

[25]

D. Eberly, Thin Plate Splines. Geometric Tools, Inc. http://www.geometrictools.com, (www.cs.ualberta.ca/~nray1/CMPUT466_551/Resources/ThinPlateSplines.pdf) (2002).

[26]

S.A. Sarra, “Integrated Multiquadric Radial Basis Function Approximation Methods”, Computers and Mathematics with Applications, 51(8)(2006), pp. 1283–1296.

[27]

G. Gonçalves, “Analysis of Interpolation Errors in Urban Digital Surface Models Created from Lidar Data”, 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences, ed. M. Caetano and M. Painho, 2006, pp. 160–168.

[28]

A.S. Meneses, F. Ramírez, B. García, J. Cabrejas, and M. González-Audícana, “Quality Control In Digital Terrain Models”, Journal Of Surveying Engineering, ASCE, 2005, pp. 118–124.

[29]

J. Oksanen, “Digital Elevation Model Error in Terrain Analysis. Academic Dissertation in Geography”, Faculty of Science University of Helsinki. Publications Of The Finnish Geodetic Institute, No:134, 2006, 51 pp.

[30]

Q. Weng, “Quantifying Uncertainty of Digital Elevation Models Derived from Topographic Maps”, Advances in Spatial Data Handling, edited by D. Richardson and P. van Oosterom, New York: Springer-Verlag, 2002, pp. 403– 418.

[31]

J. Hofierka, T. Cebecauer, and M. Šúri, “Optimisation of Interpolation Parameters Using a Cross-validation “Digital Terrain Modelling”, Lecture Notes in Geoinformation and Cartography, 2007, pp. 67–82.

January 2009
