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GEODETSKI VESTNIK | letn. / Vol. 63 | št. / No. 2 j
KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA
3
ANGULAR ADJUSTMENT OF SURVEYED BUILDING POLYGON USING ROBUST M-ESTIMATION METHODS
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Edward Osada, Malgorzata Mendela-Anzlik
UDK: 004.6:711 Klasifikacija prispevka po COBISS.SI: 1.01 Prispelo: 13. 10. 2018 Sprejeto: 10. 5. 2019
DOI: 10.15292/geodetski-vestnik.2019.02.250-259 SCIENTIFIC ARTICLE Received: 13.10. 2018 Accepted: 10. 5. 2019
IZVLEČEK ABSTRACT
Položaji točk poligona, ki določajo geometrično obliko stavbe, so pogojeni z naključnimi merskimi napakami kotov geometrije objekta. Če tako primerjamo dejanske in načrtovane kote poligona stavbe, ugotovimo, da niso enaki. Izmerjene vrednosti kotov poligona stavbe je mogoče popraviti v okviru standardnega odklona položaja poligonskih točk in s tem uskladiti vrednosti merjenih kotov z načrtovanimi. Za izravnavo kotov poligona je predvidena izravnava po metodi najmanjših kvadratov robustno glede na kote poligona, ki najbolj odstopajo od načrtovanih vrednosti. Kot rezultat izravnave dobimo kote, ki ustrezajo načrtovanim vrednostim, hkrati pa prepoznamo kote, ki značilno odstopajo od načrtovanih vrednosti, in sicer skupaj z ocenjenim odstopanjem. Pri tem postopku se nekoliko spremeni vrednost koordinat poligonskih točk, vendar v mejah njihovega standardnega odklona.
Positions of polygon points determining a geometric shape of a building object are affected by random measurement errors of the object's angles. Polygon angles are thus not equal when comparing actual and design corners of a building object. However it is possible to make an attempt to correct polygon angles within the standard deviation of the position of polygon points and lead to their congruency with the known design angles of the field object. In this paper, we propose the least squares method for angular adjustment of a polygon which is robust to angles much outlying from their design values. As a result the adjusted polygon has angles equal to the design angles and also detected angles outlying from the design ones, with determined values of deviations. Meanwhile, the coordinates of polygon points slightly changed within their standard deviations.
KLJUČNE BESEDE KEY WORDS
prostorska podatkovna baza, stavba, geometrija objekta, izravnava poligona objekta, robustna ocena
Edward Osada, Matgorzata Mendela-Anzlik | KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVI
| 250 | POLYGON USING ROBUST M-ESTIMATION METHODS | 250-259 |
geospatial database, building object geometry, adjustment of the polygon object, robust estimation
M-OCENJEVANJA j ANGULAR ADJUSTMENT OF SURVEYED BÜILDIN
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1 INTRODUCTION
The polygon geometry of a building object is described by consecutive points which determine the geometry of an object (Burrough, 1998; Longley et al., 2006; O' Sullivan, 2006). The coordinates of these points are obtained from direct field measurements as well as indirect measurements, such as airborne, satellite and drone photogrammetry and also through digitization of paper maps (Ghilani and Wolf, 2012; Wei, 2014; Mendela-Anzlik and Borkowski, 2017). These coordinates are affected by random measurement errors. The angles that are computed on the basis of the coordinates of a polygon object are also affected by random position errors of the points. The angles of a polygon object are thus not equal when comparing actual and design angles of a field object.
These angles can be however corrected within the standard deviations of the position of points and
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lead to their congruency with the known design angles of a field object. In this paper, we propose the least squares method for angular adjustment of a polygon which is robust to angles much outlying from their design values. As a result the adjusted polygon has angles equal to the design angles and also detected angles outlying from the design ones. Meanwhile, the positions of polygon points slightly move within their standard deviations. A similar solution is applied in a certain new method of a direct field measurement of building objects based on GNSS linear network, localized on the measured object (Osada, Karsznia K. and Karsznia I., 2018). The robust estimation methods have a wide range g of applications in geodesy and surveying, e.g. (M^kolski and Osada, 2005; Wu, Qiu and Wang, 2005; Banas, 2012; Muszynski, Zienkiewicz and Baryla, 2015; Adamczyk, 2017; Akram, Liu and Qian, 2018), while building polygon adjustment in a geographic information system (GIS) is not widely discussed in the literature. However, it is worth to mention about an interesting approach, presented in the paper (Jin, Tong and Zhang, 2018). There is proposed the partial total-least-squares adjustment method for condition equations (PTLSC), which allows for adjustment of the interior angles of the digitized buildings to right angles as well as maintenance of the correlations among the elements in the observation vector and the coefficient matrix.
Concerning geospatial databases, the identified building shape often need to be generalized and regularized with the use of appropriate GIS tools (Burdeos, Makinano-Santillan and Amora, 2015; Mendela-Anzlik, 2015) or other approaches dedicated for buildings of both simple and more complicated geometry (Jones and Ware, 2005; Zhang and Stoter, 2013; Lokhat and Touya, 2016). However, none of them assumes the standard deviation of the position of the points as one of the parameters in a computational process. Though it is possible to use Python modules, such as SciPy or NumPy, integrated with GIS software (e.g. ArcGIS), for implementation of the appropriate algorithms that can take into account this positional accuracy parameter. Additionally, some solutions concerning the geometry of rectangles (Preparata and Shamos, 1985) could be also useful for the above mentioned task. As far as geospatial databases are concerned, there is still a need to come up with new ideas and also develop the existing solutions for angular adjustment of building polygons. Thus, the proposed method can be used for obtaining the geometrical shape of corrected buildings polygons as objects of geospatial databases and also as a final stage of situ-ational measurement of a building object, before giving a database to the Office for Geodetic and Cartographic Documentation.
Edward Osada, Matgorzata Mendela-Anzlik | KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDING
POLYGON USING ROBUST M-ESTIMATION METHODS | 250-259 |	| 251 |
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^ 2 THE PROPOSED METHOD
ES
The polygonal geometry of a building is specified by measured coordinates of the building corners 2	y ,)> i ~ 2,n (Figure 1). The angles of the polygon at the points P.can be computed from their
§ coordinates according to Ogundare (2015) and Ghilani (2018), (Figure 1):
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a, = arctg( 7,-1 y' )-arctg( J'+1 J' )	(1)
X — X;
)2
Pi+i(xi+i,yi+i
Pi(*i,yd
Pi-i(xi-hyi-i)
Figure 1: The polygonal geometry of a building.
The accuracy of the position of the points P. is determined by their standard deviations (Ogundare, 2015):
=4< +<	(2)
where c , c are standard deviations of the measured coordinates x., y..
xi yi	• Jt
If c , c are unknown and c is known then the standard deviations of the coordinates are usually
xi yi P i computed assuming
cx = c , (2):
ii
^ lj2	(3)
The standard deviations ca. of the angles a. are computed from (1) based on the random error propagation law (Ogundare, 2015; Ghilani, 2018). In the simplest case assuming equal errors cp. of the points
P. and equal errors of their coordinates c = c (3) it is obtained:
.	xi yi
ap, -J^i+i + dti-1 + di+i,i-i ' \2 d^+idi i-i
where d, __1, d, .+1, d-+1 __1 are distances between points (P., PH), (P., Pi+1), (P+1, Pi_1) of the polygon (Figure 1).
The computed angles of the polygon a. ± ca (1) (4) are not equal to the actual as well as to the theoretical design angles f of the building, taken from the existing building geometry by default.
The angles a. are functions of the coordinates of three points P , P., P (1), (Figure 1). Their small differential changes da are connected with the small differential changes (v , v ), (v, v ), (v , v ) of the coordinates of the points (x_1, y_1), (x, y), (x;+1, yi+1), according to the differentials:
Edward Osada, Matgorzata Mendela-Anzlik | KOTNAIZRAVNAVAIZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDING
| 252 | POLYGON USING ROBUST M-ESTIMATION METHODS | 250-259 |
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da, =	+ (	2 2 _
1 /2	j2 xt	j2 j2 yt
di,i-1	di,i+1	di,i-1 di,i+1
+ yt-i- yi	x-i- x	yt+i- yt , x,+i
+--^-v„----v„----v.
(5)
j2 xt-i j2 yt-i j2 xt+i j2 di,t-1 di,i-1 di,t+1 dti+1
So, the theoretical design angles ß can be connected with the computed polygon angles a (1) according to the formula:
ß, + vß, = a + da.	(6)
where the differentials da. are given by (5) and Vß. denote assumed small random corrections of the angles ß. with zero value expectation and standard deviation Oß.
Fitting the theoretical design polygon angles (ß1, ß , ... ßj to the points (P , P , ... PJ can be implemented by the weighted least squares method as the result of minimization of the standardized residual corrections angles and coordinates:
B
'v ^
2 rv v
<y„
+
V Pt /

f
+
V xt /
vy.
K°y. y
(7)
v , v )Tis vector of all unknown corrections, while the coefficients matrix B and vector w = (f, - a , f,
Xn yn	1	1 2
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The set of observational equations (6) composed for all polygon angles (J, J, ... J) has a well-known
form as the condition adjustment model (Leick et al., 2015): Bv + w = 0, where v = (v , v , v ... v„,
'	¡i xi yi J
- a2, ..., ¡5 - aJTare known. It is solved under condition (7) or in the matrix notation v7^ 1v = min, where 2 is diagonal covariance matrix of the angles and coordinates diag(2) = (oJ , o , o ... oJ , o , o ). The solution is given by (Leick et al., 2015): v = P-1Br(BP'1B7)-1w, where P = 2-1.
The adjustment of the polygon starts from the computed standard deviations of coordinates o , o
xj yj
(3), assuming that the design angles J are error-free = 0. The ratio r = | V/o, (rp = | vj/o^ , rX = | v,y o , r, = lvl/o ) of the random corrections of observations v to their standard deviations o is tested:
VX Y I Y vy	v
r < c (slightly). The parameter c defines the range of acceptable random errors | v\ < cov, usually c = 3. The so-called three sigma error | v\ < 3ov is often used as a deterministic criterion for rejecting individual observations from sets of data (Ghilani and Wolf, 2012, Osada et al. 2017).
As a result, the location of the adjusted polygon is affected by random errors of the coordinates of the polygon points whereas the design polygon angles (J, J, ... J) are preserved. The results of the adjustment will not be correct (r > c), Figure 2, if the real shape of the object deviates from the design shape. In such a case, in order to detect the outlying angles from their theoretical design values (J, J, ... J) it is proposed to use the adjustment of the polygon that is robust to outlying angles (Figure2). In this robust adjustment, the angles (J, J, ... J) are considered as measured with very small errors in the first step of the iteration. In this case, the design shape of the polygon is kept at the points for which there is no physical deformation of the object's shape, while angular deformations are detected at the remaining points. The detected congruent angles usually obtain very small random corrections v. Hence, in the final adjustment of the polygon with conditions on the detected congruent angles (Figure 2), these angles are treated as error-free. Finally the adjusted polygon (P1, P , ... PJ contains new coordinates x. + vx, y . + vand new angles J + v^. The angles for which vJ = 0 are detected as equal to the theoretical
Edward Osada, Matgorzata Mendela-Anzlik | KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDING
POLYGON USING ROBUST M-ESTIMATION METHODS | 250-259 |	| 253 |
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^ design angles f, whereas the angles for which vf ^ 0 are detected as outlying from the theoretical design d angles f , with the deviations equal to v^.
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— Figure 2: Flowchart of the proposed method.
3 TEST OF THE METHOD
Figure 3: The polygon points 1, 2, ...16of the test object (51° 08' 33.2", 17° 07' 40.8")
The test of the proposed method was carried out on the example of a building located in Wroclaw, Poland
(Figure 3). The coordinates x,y of the corners 1, 2, 3, ..., 16 (Figure 3) are given in Table 1. The known
theoretical design angles f and their corresponding polygon angles a (1) as well as their differences f - a
are given in Table 2. The standard deviation of the position of the points is equal to CTp = 0.010 m, whence
ct = ct = 0.0071 m (3). The computed standard deviations of the angles ct (4) are bigger than the
xj yj	ai
angle differences f - a at the points 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15 (Table 2). It means that
probably at this thirteen points the polygon angles a. can be replaced by the theoretical design angles ff.
Edward Osada, Matgorzata Mendela-Anzlik | KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVI
| 254 | POLYGON USING ROBUST M-ESTIMATION METHODS | 250-259 |
M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDIN
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At points 2, 11 and 16 the standard deviations of the angles a are smaller than the differences f. - a.. ^ It means that probably at these three points the polygon angles a cannot be replaced by the theoretical Cj design angles f , the polygon angles can be much outlying from the design angles.
S
4 RESULTS	==
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4.1 Adjustment of the polygon with conditions on polygon angles
The adjustment of the polygon considering error-free conditional angles (a = 0 grad), (Figure 2) leads ^ to a significant deformation of the polygon max^ rY) = 37.313 and the a posteriori variance of unit ri
eight
rows(B)
= 9.788 that significantly exceeds the expected value ct0 = 1 (Table 2). This result sug-
gests the existence of a physical non-perpendicularity of some of the building sides. In the following chapter (4.2), it is proposed the adjustment robust to outlying angles which makes it possible to detect the outlying angles from their theoretical projected values.
4.2 Adjustment of the polygon robust to outlying angles
Further adjustment of the polygon with conditions on the object angles (Figure 2) is carried out iteratively. At every step the weightsp^ of the angles fare modifiedpf pf( vf) using a weight function f(vj, for example:
1) Huber weight function (Huber, 1981): 1	I v I < ra„
f M =
v I > ror.
(8)
where r = 1.5 (Erenoglu and Hekimoglu, 2009). 2) Huber modified weight function (Osada et al., 2017):
f (Va) =
where r = 1.5.
I v < ro
a	a
\v„ > ro„
1 +
(9)
3) Hampel weight function (Hampel et al., 1986):
f (Va) =
a0a
V
I vl < ao„
{c - b ) jva 0
a0a < val< b0a
b0a <\va\< COa
(10)
I va\> COa
where a = 1.5, b = 3 and c = 6 (Erenoglu and Hekimoglu, 2009).
Edward Osada, Matgorzata Mendela-Anzlik | KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDIN
N USING ROBUST M-ESTIMATION METHODS | 250-259 |
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v
v
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4) Krarup weight function (Erenoglu and Hekimoglu, 2009):
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f k) =
I V < TO I a |	a
f IvJ^
exp
(11)
where r = 3. 5) Kraus weight function (Kraus, 2000):
f (Va) =

I Va\< TOa
1
\v„ > TO„
1 +
(12)
where a, c are empirically selected parameters. 6) Yang weight function (Yang et al., 1999),
f (Va) =
aoa
b —
oa
b - a
I Va\< aOa
aOa < Va\< bOa
I Va\> bOa
(13)
where a and b are chosen as 1.0 — 1.5 and 3.0 — 6.0, respectively.
In the case of the weight function (9) starting from initial small values of the angular standard deviation CJp = 0.0005, 0.0010, 0.0015... grad after a few iterations, the adjustment process has stabilized at the acceptable level of ct0 = 0.995 for cr^ = 0.0020 grad (Table 2). Finally only 2 outlying angles at the points 11 and 16 are detected, their corrections v ^ are equal to 1.2252 grad and -1.2245, respectively (Table 2). All other 18 angles obtain very small random corrections max(Vg) = 0.0002 grad (Table 2). Hence, in the final adjustment (Figure 2) these congruent angles are treated as error-free (a^ = 0 grad).
The results of the detection of the outlying angles using all of the weight functions (8)—(13) are practically the same (Table 2). The modified Huber (9), Hampel (10), Krarup (11) and Kraus (12) functions give also very small, almost completely identical corrections at all other detected congruent angles (Table 2).
0
4.3 Adjustment of the polygon with conditions on the detected congruent polygon angles
Finally, introducing zero-mean-error values for detected congruent angles to design angles c = 0 grad, and c = 10 grad for two detected outlying angles at the points 11 and 16, the adjusted polygon is not deformed: , (Table 2). The adjusted polygon contains the theoretical design angles fi. at the fourteen points 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, for which the corrections v are equal to zero,
Pi
v f = 0 (Table 2). Only at the two points, 11 and 16, the angles are detected as outlying from the theoretical
Edward Osada, Matgorzata Mendela-Anzlik | KOTNAIZRAVNAVAIZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDING
| 256 | POLYGON USING ROBUST M-ESTIMATION METHODS | 250-259 |
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design angles fi , fi (Table 2). The deviations of the angles with respect to the theoretical design angles are Vp = 1.2257 grad and v = -1.2257 grad (Table 2). The coordinate differences Ax, Ay. of the adjusted positions of the polygon points x. + Vx, y + v. and the known database positions of these points x ,, y . do not
exceed two times of the starting standard deviation values of the coordinates a = a = 0.0071 m (Table 1).
xj yj
Table 1: The measured and adjusted positions of the polygon points [m]
Point	Measured	positions	Adjusted positions		Differences	
	x		x + v	y+v.	Ax	Ay
1	7866.422	9011.471	7866.422	9011.469	0.000	-0.002
2	7857.797	9009.556	7857.783	9009.555	-0.014	-0.001
3	7860.151	8998.804	7860.163	8998.814	0.012	0.010
4	7855.610	8997.812	7855.616	8997.806	0.006	-0.006
5	7859.228	8981.528	7859.223	8981.528	-0.005	0.000
6	7852.404	8980.020	7852.402	8980.017	-0.002	-0.003
7	7853.298	8975.950	7853.304	8975.948	0.006	-0.002
8	7854.147	8976.131	7854.149	8976.136	0.002	0.005
9	7854.818	8973.129	7854.815	8973.129	-0.003	0.000
10	7853.970	8972.942	7853.970	8972.941	0.000	-0.001
11	7860.371	8944.064	7860.368	8944.060	-0.003	-0.004
12	7872.500	8946.500	7872.496	8946.503	-0.004	0.003
13	7872.305	8947.441	7872.308	8947.439	0.003	-0.002
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14	7876.298	8948.241	7876.300	8948.243	0.002	0.002
15	7876.491	8947.300	7876.488	8947.307	-0.003	0.007
16	7880.458	8948.116	7880.460	8948.107	0.002	-0.009
Table 2: The polygon angles corrections
The angles									Values of the angles [grad]						
				ä -ri	s	Kö	-c c it o	els	Adj	ustment of the polygon robust to			outlying an	gles	Adjusted with detected congruent angles conditions
Vertex	E ar e —	E ar -a a it	etr or e	Computet from coordinates	Difference ¡3-a	Standard deviation G	1 J us di < U	ge n a (J o	Huber modified	Hampel	Krarup	Krauss	Yang	Huber	
1 2	2 3	16 1	100 100	99.9706 100.1878 100.1878	0.0294 -0.1878	0.1454 0.1848	100 100		100 100	100 100	100 100	100 100	100.0002 100.0003 100.0003	99.9909 100.1524 100.1524	100 100
3 4	4	2	300	300.0293 OQ nnxn	-0.0293	0.2974 0.2844	300		299.9998 00 QQQQ	299.9997 00 QQQ7	299.9998 00 QQQQ	299.9998 00 QQQQ	299.9992 00 QQQQ	299.9803 QQ Q/1/1Q	300
5 /T	5 6	3 4	100 300	99.7737 300.0726 1 AA AQAV	0.2263 -0.0726 A AQAV	0.1975	100 300		99.9998 299.9999 QQ QQQQ	99.9997 299.9998 QQ QQQQ	99.9998 299.9999 QQ QQQQ	99.9998 299.9999 QQ QQQQ	99.9989 299.9996 QQ QQQ<	99.8449 300.0578 1 nn m2.1	100 300
8776	7 8 11 9	5 6 6 7	100 100 200 300	100.0807 100.3931 199.8685 299.3726 3AA 1Q1Q	-0.0807 -0.3931 0.1315 0.6274	0.3557 1.4969 0.3080 1.5228	100 100 200 300		99.9999 100 199.9999 300	99.9998 100 199.9998 300	99.9999 100 199.9999 300	99.9999 100 199.9999 300	99.9995 99.9999 199.9995 299.9999 TQQ QQQQ	100.0232 99.9756 199.9393 299.9398 JAA AI QV	100 100 200 300
9 10	10 11	8 9	300 100	300.1819 99.9309 1 QQ Q7<a	-0.1819 0.0691	1.5229 1.4662	300 100		300 100	300 100	300 100	300 100	299.9999 99.9999	300.0187 100.0128 AAA nnao	300 100
10 11	11 12	6 10	200 100	199.9753 101.2685 QQ 07	0.0247 -1.2685 A aon?	0.1808 0.1116	200 100		200 101.2252	200 101.2266	200 101.2267	200 101.2252	200 101.2195	200.0032 101.2301 QQ QQfl/i	200 101.2257
12 12	13 16 1 A 14	11 11	100 200	99.6097 199.8639 afin /ilCi'X	0.3903 0.1361	1.3295 0.1875	100 200		100 200	100 200	100 200	100 200	100 199.9999	99.9806 199.9333 JAA AI 1 /.	100 200
13 14	15	12 13 1 A	300 300	300.4203 299.7095 no O/C2/C	-0.4203 0.2905	1.3619 1.3620	300 300		300 300	300 300	300 300	300 300	300 299.9999 00 0000	300.0114 299.892 on QAi 0	300 300
15 15 16	16 16 1	14 11 15 S	100 200 100	99.9636 199.6972 99.0351	0.0364 0.3028 0.9649	1.3624 0.3237 0.3150	100 200 100	100 100 100 200 200 200 98.7755 98.7745 98.7740 The general quality indicators V6 A AAOA A AAOC A AA1A				100 200 98.7755 A AAA A	99.9999 199.9998 98.7834	99.9019 199.7661 98.9879 A AAAA	100 200 98.7743 1 (1-6 2)
			tartin	g value G CT0	[grad]		10 9.78	8	0.0020 0.955	0.0025 0.913	0.0020 0.900	0.0020 0.954	0.0050 1.184	0.2000 1.064	10 0.903
maxi/x, y 37.313 2.776 2.//6 2.//8 2.//6 2.195 2.353 2.//5															
max(rj	140.27 0.910 0.914 0.916 0.910 1.285 6.359	0.963
1 the angles are computed from adjusted coordinates
2) f = 10-6 grad for detected congruent 18 angles and f 10 grad for detected 2 outlying angles_
Edward Osada, Matgorzata Mendela-Anzlik | KOTNA IZRAVNAVA IZMERJENEGA POLIGONA STAVBE Z ROBUSTNO METODO M-OCENJEVANJA | ANGULAR ADJUSTMENT OF SURVEYED BUILDING
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^	5 CONCLUSIONS
|=	Due to the random errors of georeferenced measurements, the polygon angles of the building object
2	are not congruent with both actual and design angles of the building object. However, it is possible to
§	make an attempt to bring about congruency with these angles. One of the possibilities, proposed in this
oc	paper, is the adjustment of the polygon using the least squares method robust to polygon angles, much
cc
uj outlying from their design values. The conducted experiment on real data has confirmed the efficacy of H this method. All of the six used methods of robust estimation (Hampel, Krarup, Kraus, Yang, Huber and Huber modified) have detected the same outlying angles. The adjusted polygon of the building object H has angles at fourteen points equal to the design angles and detected angles at two points that are much g outlying from the design ones, with determined values of deviations. In accordance with the expectation, qc the positions of all sixteen polygon points slightly moved within two times of values of their standard deviations. The reason for the much outlying detected angles at two points (11 and 16) can result from the measurement error of their coordinates or errors of the object construction. The determination of a reason would require to perform the control measurement of coordinates of these two points. The proposed method can be used to correct the shape of directly measured building objects as well as to adjust the shape of buildings that are a part of geospatial databases, for which the standard deviations of the polygon points are known.
References:
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Osada E., Mendela-Anzlik M. (2019). Angular adjustment of surveyed building polygon using robust M-estimation methods.
Geodetski vestnik, 63 (2), 250-259. DOI: https://doi.org/10.15292/geodetski-vestnik.2019.02.250-259
Prof. Edward Osada, Ph.D.
University of Lower Silesia, Faculty of Social and Technical Sciences
Department of Geodesy Wagonowa 9,53-609 Wroclaw, Poland email: edward.osada@dsw.edu.pl
Mafgorzata Mendela-Anzlik, Ph.D.
University of Lower Silesia, Department of Geodesy,
Faculty of Social and Technical Sciences,
Wagonowa 9, 53-609 Wroclaw, Poland
email: malgorzata.mendelaanzlik@gmail.com; malgorzata.
mendela@dsw.edu.pl
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