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Modeling the thickness of vaults in the church of santa maria de magdalena (Valencia, Spain) with laser scanning techniques
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Original article
Modeling the thickness of vaults in the church of santa maria de magdalena (Valencia, Spain) with laser scanning techniques José Herráez a,∗ , Pablo Navarro b , José Luis Denia a , María Teresa Martín c , Jaime Rodríguez c a
Department of Cartographic Engineering, Geodesy and Photogrammetry, High School of Civil Engineering (UPV), 46022 Valencia, Spain Department of Architectural Projects, High School of Architecture (UPV), 46022 Valencia, Spain c Area of Cartographic Engineering, Geodesy and Photogrammetry, Polytechnic High School of Lugo (USC), 27002 Lugo, Spain b
a r t i c l e
i n f o
Article history: Received 27 May 2013 Accepted 26 November 2013 Available online xxx Keywords: Cultural heritage Structural restoration Church reinforcement Scanner laser
a b s t r a c t Restoring cultural heritage is an extremely important job due to its immeasurable value. However, it also requires even greater attention in the case of a building. The actions taken on these constructions not only guarantee their preservation from the point of view of their historical value, but they also guarantee their stability as structures. The need to control historic buildings, analyzing their defects and their possible consequences, is decisive in preventing significant damage. This work demonstrates calculating the thickness of vaults in a church. It is easy to determine the interior and exterior 3D geometry of the church using scanner laser techniques. By combining both geometries, it is easy to determine the difference between the interior vaults and the roof of the church. However, the interior of both geometries is completely unknown, and it is not possible to act on their condition for structural consolidation purposes. This work shows the methods used to determine the interior sections of the vault thicknesses by referencing the internal geometry of both models with the interior of the church, using plumb line system scanning. The results obtained show accuracy better than 6 millimeters. © 2013 Elsevier Masson SAS. All rights reserved.
1. Research aims Research aims consist on developing a methodology to get compete 3D model of a church, including not visible parts under the roof, to measure the thickness of the vaults for structural restoring. To get a complete 3D model, it is necessary to reference scanner laser point clouds to the same coordinate system. As there are not common points to join the point clouds, the development of a plumb line system with spheres will be carried out. Final 3D model obtained gets sections of the vaults with 6 mm of accuracy. 2. Introduction It is currently common to use photogrammetric and scanner laser techniques to generate 3D models in the area of civil engineering and architecture [1,2]. The scanner laser is undoubtedly the quickest and most powerful tool, although there are other possible techniques based primarily on photogrammetry [3], and it is
∗ Corresponding author. E-mail addresses: [email protected] (J. Herráez), [email protected] (P. Navarro), [email protected] (J.L. Denia), [email protected] (M.T. Martín), [email protected] (J. Rodríguez).
especially increasingly common to find works that combine both techniques [4,5]. 3D modeling for an entire building is achieved by combining the interior and exterior scans of it. This makes it simple to obtain a 3D model of everything that is visible. Works have been carried out on a variety of structures, such as buildings [6], towers [7], or bridges [8], for example. Specifically, focusing on the scope of this work, there are references to actions in churches [9]. In all of the cases mentioned, 3D modeling is highly precise from a structural point of view [10] and enables subsequent actions for restoration purposes [11]. However, the interior parts that are not visible cannot be modeled, so they cannot be included in the final complete model. That means that the problem arises when the interest focuses on a part of the building that cannot be modeled as a part of the overall whole. This problem exists whenever there is no connection between the hidden zone and the visible zones, given that the respective point clouds cannot be linked. This work shows 3D modeling of the inside of the vaults of a church for the purpose of designing the structural consolidation project. To do this, 3D models of the interior and the exterior of the church are generated, in addition to the 3D model of the interior of the vaults. The interior and exterior models of the church are easy to merge. Meanwhile, referencing the interior model with both
1296-2074/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.culher.2013.11.015
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Fig. 1. a: interior of the church (vaults); b: exterior of the church (roof); c: interior of the vaults (under the roof).
geometries is impossible without common elements. To solve the problem, a system of physical plumb lines was designed, scanned from inside the vaults and inside the church, making it possible to join the models and obtain the desired sections with accuracy better than 6 millimeters.
3. Generating 3D models 3.1. The church of Santa María de Magdalena. Statement of the problem The church of Santa María de Magdalena dates to the 16th century and is located in the city of L’Ollería (Valencia, Spain). Its exact date of construction is unknown, although it must have been built between the year 1522 (when the old church was destroyed by fire during the Revolt of the Brotherhoods) and 1566 (the date it is first mentioned in church documents). Its construction took place in at least two phases. The church’s plan consists of a single nave with four bays and a polygonal apse, with chapels between the buttresses and a tower located to one side of the entrance. Adjoining sacristy, communion chapel, and trasagrario (a special chapel for guarding the Eucharist) structures were added during later periods. The church’s walls are made of irregular stone fragments held together with lime mortar, with dressed limestone blocks used in the corners and buttresses. The nave is topped by crossribbed vaults with tiercerons and framing, finished with sculpted
keystones. Both the main facade and the tower are crowned with a decorative battlement. Despite its small size and irregular carving, the portal is very interesting because of its design in the form of a Roman triumphal arch with double doors flanked by semiinset columns. Between the columns are scalloped niches with a rich collection of decorative sculptures based on faces, cupids, and harpies. The structural problems that led to restoration of the church were derived from a partial collapse during a 1748 earthquake, which caused cracking in the church’s vaults and bell tower. Although the church was reconstructed, it has now become necessary to carry out a detailed study of the structure’s stability. This is taking place under the “Master Plan” and consists of an in-depth study of the church’s constructed elements in order to develop a proposal for restoration, conservation, and monitoring of changes in the building. This in-depth study includes 3D mapping and modeling of the church to guide planning of future actions. In relation to the vaults, a structural study is being performed in order to analyze their stability. This has required a determination of the thicknesses in each vault throughout the entire section.
3.2. 3D modeling. Design of the solution We have three 3D models generated with point clouds captured with a scanner laser: the model of the entire interior of the church (Fig. 1a, the model of the entire exterior of the church (Fig. 1b)
Fig. 2. a: tripod with plumb line and sphere inside the vault; b: physical plumb line from inside the church; c: plumb line with spheres from inside the church.
Please cite this article in press as: J. Herráez, et al., Modeling the thickness of vaults in the church of santa maria de magdalena (Valencia, Spain) with laser scanning techniques, Journal of Cultural Heritage (2013), http://dx.doi.org/10.1016/j.culher.2013.11.015
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and the model of the interior of the vaults (Fig. 1c). Each of them was obtained by means of multiple scans, followed by filtering and recording. To generate a single 3D model that contains the three previous models, those models need to be referenced to a single system of absolute coordinates. The 3D geometry referencing consists of determining a single three-dimensional system of coordinates that all the models in a single element refer to. The interior and exterior referencing of the church is simple, requiring nothing more than identifying common recognizable points in both 3D models from both positions (doors, windows, etc.). However, the interior model of the vaults can be in any position between the two previous models, and it is not possible to determine its absolute position and, as a result, the thicknesses of both the vaults and the roof. To determine the absolute position, 3 options were evaluated: • by contact of common surfaces in the point cloud. This was discarded, as it was impossible to achieve a common surface of little more than 1 sq.m, in very precarious working conditions; • measuring some points by topographic methods and referencing them to the system of absolute coordinates. This was also discarded because it was impossible to achieve good visuals both inside the church and toward any building in the surrounding area; • there are some holes with a diameter of 7–10 cm beside the vault keystones (which were likely used to hold some sort of lamp) through which it is impossible to measure but that could hold visible elements from each part that make it possible to identify common elements.
3
Z
Centre Sphere 1s Inside of the vault Thickness of the vaults
Thickness of the vaults
Inside of the Church
D Y
X
Centre Sphere 1i
Plumb line
Fig. 3. Layout of one of the plumb lines, with a sphere on the inside of the vault (sphere 1s) and another on the inside of the church (sphere 1i), going through the thickness of the vault through the hole.
where, Rj : radius of the sphere,
4. Referencing 3D models Based on the last option, tripods were placed inside the vaults from which plumb lines were hung through the holes (Fig. 2a). These plumb lines were made of 3 mm thick cord, with a 4 kg weight hung from the end to guarantee tension and a lack of movement (Fig. 2b). Once the plumb lines were installed, the vertical was perfectly defined in the scans from the inside of the church and from the inside of the vaults. However, it was necessary to make some sort of mark on the plumb lines to enable the identification of a common element. To do so, two visible spheres were used in each scan, respectively (Fig. 2a and c). Defining a system of coordinates focused on a sphere and parallel to the system of absolute coordinates, the coordinates of the centers of each pair of spheres will have the same X and Y coordinates, with only the Z coordinate varying (Fig. 3). This coordinate will be determined by the exact distance between the two spheres (D) over the vertical defined by the plumb lines. Once the coordinates of both spheres have been calculated with respect to the same system, it will be simple to locate the corresponding point clouds from the scans to reference the models based on them. However, given that the spheres are not visible simultaneously, it is necessary to carry out an initial auxiliary scan, used to calculate the distance D between them. To do so, it is necessary to remove the sphere that is on the inside through the vault hole, leaving both inside the church. With both spheres inside the church, the first scan is run, which will give the distance between them as a result. To do that, it is necessary to determine their centers. To calculate the centre of a spherical element within a point cloud, we adjust all the points scanned on its surface based on the equation of a sphere (1): Rj =
XSi − XCj
2
+ YSi − YCj
2
+ ZSi − ZCj
2
(1)
XCj , YCj , ZCj : coordinates of the centre of a sphere j, (XSi , YSi , ZSi ): coordinates of a point i on the surface of the sphere j. Based on an approximate value for the centre of each sphere, the value calculated will be given by (2):
XCj , YCj , ZCj = XAj , YAj , ZAj + ıx , ıy , ız
(2)
where, XAj , YAj , ZAj : approximate coordinates of the centre of a sphere j, ıx , ıy , ız : calculation differentials. Substituting (2) in (1) we obtain (3): Rj =
XMi − XSi − ıx
2
+ YMi − YSi − ıy
2
+ ZMi − ZSi − ız
+ ıRj
2 (3)
where, (XMi , YMi , ZMi ): coordinates measured by the laser of each point i on the surface of a sphere j, ␦Ri : error of the measured point with respect to the surface of a perfect sphere. Developing the expression (3), each point belonging to a sphere provides an equation that has the differentials ıx , ıy , ız as unknowns, making it possible to generate a matrix system that is resolved by means of a bundle adjustment. Once the coordinates have been calculated for the centers of each pair of spheres on each plumb line, the distance D will be given according to (4): j+1
Dj
=
XCj − XCj+1
2
+ YCj − YCj+1
2
+ ZCj − ZCj+1
2
(4)
Once the distance between both spheres has been calculated, the calculation of the coordinates of the sphere that is inside the vault
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based on the coordinates of the sphere that is inside the church or vice versa, gives (5):
XCj+1 , YCj+1 , ZCj+1 =
j+1
XCj , YCj , ZCj ± Dj
(5)
In the expression (5), we will use the “+” sign for the calculation of the upper spheres, while the “−” sign will be used for the lower spheres. Referencing the coordinates of the points scanned from the inside of the vault requires simply applying the coordinates of the sphere (5) and the rotation between both point clouds that will be defined by the vertical of the plumb line, based on (6):
XCj , YCj , ZCj
t
= (1 − Si )−1 ∗ (1 + Si ) ∗ XCj , YCj , ZCj
t
+ Tx , Ty , Tz
where : t XCj , YCj , ZCj : coordinates of the sphere j in the model of the interior of the vault, , Y , Z XCj Cj Cj
t
: coordinates of the sphere j in the model of the
interior of the outer vault, t Tx , Ty , Tz : translation vector between the two models,
(Si ) =
0 −ai bi
ai 0 −ci
−bi ci 0
Sphere 1i 1s 2i 2s 3i 3s
Scanned points 30 24 49 27 136 84
Calculated radius
Xc (m)
Yc (m)
Zc (m)
D (m)
0.0215 0.0215 0.0205 0.0205 0.0215 0.0205
6.347 6.345 4.673 4.669 2.882 2.883
16.548 16.551 11.189 11.190 5.790 5.786
8.707 12.054 8.044 11.387 8.560 11.886
3.347 3.341 3.326
t
(6)
Table 1 Calculation of the centers of the spheres (with the plumb line hanging inside the church) and the distance between each pair of them.
is the Cayley matrix.
5. Results The method shown makes it possible to reference the 3D models of the interior of the church, the exterior and the interior of the vaults with plumb lines and spheres. Taking advantage of the existence of 3 holes in the vaults, 3 plumb lines were installed. This makes the calculation redundant and offers greater accuracy. 5.1. Generating the 3D models The work was done with a Leica Scanstation II laser. Given the distance at which they were performed, the precision of the scans is 6 mm, both for measurement of points and for the spheres, which fits the requirements for the work perfectly. The precision for the centering of the spheres on the plumb line is estimated as 2–3 mm. The structural work being carried out requires a strong vault with a uniform geometry that will approximate the existing average thicknesses as much as possible, with a precision of 1–2 cm being sufficient. This level of precision can undoubtedly be achieved with the methodology and instrumentation employed. The models generated to determine the thicknesses are: 3D model of the interior of the church (vaults), 3D model of the exterior of the church (roof), 3D model of the interior of the vaults (under the roof) and 3D models of the plumb lines through the holes near the vault keystones (from the inside of the vaults and from the inside of the church). In order to obtain the interior geometry of the church, 8 scans with full 360◦ rotation were performed in order to provide good coverage for later joining of the clouds (Fig. 4). The ascent from the interior of the church to the top of the bell tower also required 17 more scans, in order to record the stairway and the faces of the tower so that its interior could be joined with the exterior model of the structure as a whole. The exterior geometry required 16 scans, also with full 360◦ rotation (Fig. 5). The reason why so much work was necessary was because of the complexity of the church’s surroundings. The urban location in an area with a high density of other buildings nearby created this complexity, in contrast to a more open location. The joining of the exterior and interior models took place by means of the church’s doorways, with 3 full rotation scans required
for this purpose. These are shown as 3 x’s in Fig. 6a, two in the main doorway and one in the side entrance. In total, the 3D model from inside the church was made with 25 scans, obtaining a cloud made up of 43 million points once filtered and after merging the clouds. The 3D model of the exterior roof of the church was made with 16 scans, obtaining a cloud made up of 52 million points once filtered and after merging the clouds (plus 3 scans in the doorways for the joining). Also, the 3D model of the interior corridor of the vaults was made with 6 scans, obtaining a cloud made up of 6 000 000 points once filtered and after merging the clouds (Fig. 6). The total number of scans used for all of the 3D models was 44. For their recording in pairs we obtained 168 cloud constraints (this involves a redundancy of 125 since the minimum number necessary is 43). The adjustments have been high precision, since the residual errors obtained have been very low: the residuals show errors of 0.000 m in 32% of the cases (in other words, the residual error has been less than 0.0005 m); less than 0.001 m in 74% of the cases; and less than 0.002 m in 100% of the cases. Once the 3D models were generated for the three main elements, the referencing was made by scanning the 3 suspended plumb lines, passed through 3 holes. Each plumb line has 2 spheres, one visible from the interior of the church and the other from inside the vaults, in each hole. 5.2. Referencing the 3D models It is first necessary to determine the distance between the spheres of each plumb line. The centers calculated by (3) for each pair of spheres on each plumb line with the cord hanging inside the church and the calculated distances are shown in Table 1. The names of the spheres correspond to the number of the plumb line accompanied by a subscript, s or i, which indicates whether the sphere is the upper one (located inside the vault) or the lower one (located inside the church). Once the distances have been determined, the plumb lines are raised so that one sphere is inside the vault and another is inside the church. Referencing the scanned clouds based on the centers of the spheres is determined by the calculated centers of the upper and lower spheres in both coordinate systems (Table 2). The table shows the coordinates of the same spheres in the model in the
Table 2 a: coordinates of the spheres with respect to the interior of the vaults; b: coordinates of the spheres in the model of the interior of the church. (a)
(b)
Sphere
Xc (m)
Yc (m)
Zc (m)
Sphere
Xc (m)
Yc (m)
Zc (m)
1s 1i 2s 2i 3s 3i
4.525 4.525 9.113 9.113 13.733 13.733
−3.548 −3.548 −6.758 −6.758 −10.111 −10.111
−0.811 −4.158 −0.821 −4.162 −0.785 −4.111
1s 1i 2s 2i 3s 3i
−0.694 −0.694 −5.748 −5.748 −10.852 −10.852
0.794 0.794 3.218 3.218 5.773 5.773
12.608 9.261 12.596 9.255 12.632 9.306
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Fig. 4. a: positions of the scanning stations in the church interior (plan view); b: 3D model of the exterior of the church with vaults scanned from the interior.
Fig. 5. a: positions of the exterior and linking scanning stations in plan view: 16 exterior stations (dots) and 3 stations in the doorways (x’s); b: 3D model of the church with the exterior roof. Table 3 Cayley’s matrix parameters. Parameter
Value
a b c
−5.77702544 0.01085382 −0.01827810
interior of the vaults and in the model in the interior of the church, calculating the record through the Cayley matrix (Table 3). The components of the translation vector are implicit in the Cayley matrix, having used the coordinates of the spheres reduced to mass center of points. 5.3. Measuring thicknesses and obtaining sections By applying the calculated parameters, we merge the models and obtain the sections of the thicknesses sought (Fig. 7) with millimetric accuracy (Table 4):
The arches have a different thickness in each vault, maintaining an approximate average value in each one of them. The average thickness is 0.149 m for the first vault, 0.146 m for the second, 0.165 m for the third and 0.102 m for the fourth. Based on the results, the first two vaults have a very similar average thickness (with small variations of around 1cm between arches in each of them), while the third vault has a significantly greater thickness (+11%) and the fourth has a thickness that is much less (−32%). The final result for each of the vaults is a full section, which will later be analyzed during the restoration work (Fig. 8).
6. Analysis and discussion It is worth considering the extent to which the methodology employed in this study can be generalized and extrapolated to other cases. To do this, it is appropriate to analyze the reliability of the reconstruction based on the number of spheres required as well as their distribution.
Fig. 6. 3D model of the interior corridor of the vaults (under the roof).
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Fig. 7. a: complete longitudinal section (with the 3 plumb lines); b: vault 1 (with plumb line); c: vault 2 (with plumb line); d: vault 3 (with plumb line; e: vault 4.
After processing the clouds we have obtained two 3D models: one corresponding to the interior-exterior of the church and the other corresponding to the interior of the vault corridor below the roof. Both models are established in an independent manner, which means that the problem to be resolved is that of using common points (the spheres) to position the model of the vault corridor within the model of the church. There are 2 spheres available in both models, which combined with the conditions of verticality (which aligns the spheres along the Z axis of the absolute coordinate system for both models) is sufficient to resolve the problem. A third sphere provides us with verification of the adjustment performed (3 spheres on each of 2 plumb lines, or else 3 plumb lines with 2 spheres on each). In this way, the greater the number of spheres the more redundancy the calculation system will have. Based on the above, the positioning of one model within another can have two distinct sources of error. These can be revealed by
analyzing the minimal case, with two plumb lines very close to each other, with two spheres on each: • an error derived from calculation of the sphere coordinates, which involves a vertical displacement of one model with respect
Table 4 Thicknesses of the four vaults in each of their seven arches, respectively (taking the central point of each arch in each case). Vault
Arch
Thickness (m)
Vault
Arch
Thickness (m)
1
1 2 3 4 5 6 7
0.157 0.166 0.148 0.143 0.138 0.151 0.137
2
1 2 3 4 5 6 7
0.141 0.146 0.156 0.151 0.148 0.139 0.142
3
1 2 3 4 5 6 7
0.193 0.175 0.182 0.160 0.157 0.138 0.152
4
1 2 3 4 5 6 7
0.097 0.093 0.102 0.114 0.108 0.091 0.106
Fig. 8. Section of one of the vaults (the hole through which the plumb line with two spheres is hanging can be seen).
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Fig. 9. Error in vertical positioning.
Fig. 10. Error in orientation.
to the other (Fig. 9), and which results in an erroneous thickness calculation; • an error derived from calculation of the planimetric orientation, which involves an error in plan-view orientation of one model with respect to the other (Fig. 10). This also results in an erroneous thickness calculation. As can be seen in Figs. 9 and 10, if we lack redundancy in the calculation, a positioning error may be serious although of constant value, while a rotational error will be propagated and increase with distance. If a third sphere is available on each plumb line, the first type of error is ruled out since a serious error in calculation for one of the spheres would be detected (with only 2 spheres this would not be possible). The same would occur with the second type of error, since availability of a third sphere allows two orientation conditions to be established, which ensures planimetric overlapping. The analysis of possible error by vertical rotation of one of the models is directly eliminated by working with the scanner in leveled mode (with verification of this afterward). With regard to the geometric distribution of the points, increasing the number of plumb lines will increase the method’s reliability, while increasing the number of spheres aligned on the same plumb line will improve the planimetric orientation calculation for the model. However, it is much more optimal to install more plumb lines (the orientation of a plane will always be better when different vectors of non-aligned points are used, rather than many vectors of aligned points). The choice of one system or the other will depend upon the scale of the element as well as the number of spheres that can be used and the possible locations for them. The worst possible case will be one where the plumb lines are located very close to each other, which favors the propagation of any error. If we add the scale of the element into this consideration, this propagation will be even more unfavorable the larger the dimensions are. This means that: • it is better to install multiple plumb lines than to add multiple spheres on the same plumb line: putting more spheres on the same plumb line does not provide more precision for location within the coordinates, since any error that exists will affect the absolute position (given that the relative positions among them will not reveal errors). Installing more plumb lines, on the other hand, allows for verification;
• the position of the spheres is very relevant: if only two plumb lines are installed then it is best to locate them as far apart as possible, which will allow the definition of vectors to be optimal. If a third plumb line is installed then verification can be performed in the case of an error. Based upon all of the above, in our case we opted for installation of three plumb lines with two spheres on each of them, covering all dimensions of the element. Using three plumb lines with 6 spheres ensures the absence of errors in the coordinates, and it also eliminates any doubt regarding propagation of an orientation error when the two models are joined. Application of this methodology can be extrapolated, under certain circumstances, to situations where 3D models must be joined but where no common elements exist. The circumstances required for the methodology presented to be reproduced are those where the two upper or lower spheres from both models are always visible in a single scan. Reconstruction of the full model will entail independent scans (of each complete model separately), which will be joined later by the partial scan that contains the spheres.
7. Conclusions The procedure used is quick and easy. With just the help of some plumb lines and spheres it is possible to reference several threedimensional point clouds in space in a way that is highly automated, with an accuracy of 6 millimeters. Generating 3D models is a problem resolved both by scanner laser techniques and by many others from the fields of topography, photogrammetry, etc. However, referencing between the models generated is a problem that requires a specific solution in each case. In situations in which there are no common elements, it is necessary to create conditions that enable us to establish a system of absolute coordinates. Topography, in this area, is very limited, given that it is very difficult to locate a sufficient number of fixed points. Photogrammetry, meanwhile, solves the problem of the number of points, but it can not be applied in all types of conditions (in our study, lighting was a key problem). The scanner laser, on the other hand, overcomes all of these inconveniences, making it possible to work under any lighting conditions and generating a sufficient number of points in any circumstance.
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The use of plumb lines is effective due both to the simplicity of their use and to the provided accuracy. Likewise, using spheres on the plumb lines is undoubtedly the best choice, given that the geometry of a sphere is more easily recognized by automatic techniques than any other 3D geometry. Application of this methodology can be extrapolated to any study where 3D models without common points must be joined (where an element of connection can be introduced between them), or else where the scale and/or geometry of the models involved does not allow full scanning and where a common point shared by two scan clouds may be necessary in order to reconstruct a single space. The final accuracy obtained was of 6 millimeters, so the thicknesses obtained in the sections results appropriate for subsequent structural consolidation studies. References [1] A. Nunez, F. Buill, J. Regot, A. Mesa, Generation of virtual models of cultural heritage, Journal of Cultural Heritage 13 (1) (2012) 103–106. [2] N. Yastikli, Documentation of cultural heritage using digital photogrammetry and laser scanning, Journal of Cultural Heritage 8 (4) (2007) 423–427.
[3] P. Salonia, S. Scolastico, A. Pozzi, A. Marcolongo, T. Messina, Multi-scale cultural heritage survey: quick digital photogrammetric systems, Journal of Cultural Heritage 10 (1) (2009) E59–E64. [4] K. Al-Manasir, S. Fraser, Registration of terrestrial laser scanner data using imagery, Photogrammetric Record 21 (115) (2006) 255–268. [5] D. Gonzalez-Aguilera, P. Rodriguez-Gonzalvez, J.J. Gomez-Lahoz, An automatic procedure for co-registration of terrestrial laser scanners and digital cameras, ISPRS Journal of Photogrammetry and Remote Sensing 64 (3) (2009) 308– 316. [6] A. Pesci, E. Bonali, C. Galli, E. Boschi, Laser scanning and digital imaging for the investigation of an ancient building: Palazzo d’Accursio study case (Bologna, Italy), Journal of Cultural Heritage 13 (2) (2012) 215–220. [7] A. Pesci, G. Casula, E. Boschi, Laser scanning the Garisenda and Asinelli towers in Bologna (Italy): detailed deformation patterns of two ancient leaning buildings, Journal of Cultural Heritage 12 (2) (2012) 117–127. [8] R. Jiang, D. Jáuregui, K. White, Close-range photogrammetry applications in bridge measurement: literature review, Measurement 41 (2008) 823– 834. [9] G. Faella, G. Frunzio, M. Guadagnuolo, A. Donadio, L. Ferri, The church of the nativity in Bethlehem: non-destructive tests for the structural knowledge, Journal of Cultural Heritage 13 (4) (2012) 27–41. [10] J. Gordon, D. Lichti, Modeling terrestrial laser scanner data for precise structural deformation measurement, Journal of Surveying Engineering – ASCE 133 (2) (2007) 72–80. [11] H.S. Park, H.M. Lee, H. Adeli, I. Lee, A new approach for health monitoring of structures: terrestrial laser scanning, Computer-Aided Civil and Infrastructure Engineering 22 (1) (2007) 19–30.
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Available online at
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Original article
Modeling the thickness of vaults in the church of santa maria de magdalena (Valencia, Spain) with laser scanning techniques José Herráez a,∗ , Pablo Navarro b , José Luis Denia a , María Teresa Martín c , Jaime Rodríguez c a
Department of Cartographic Engineering, Geodesy and Photogrammetry, High School of Civil Engineering (UPV), 46022 Valencia, Spain Department of Architectural Projects, High School of Architecture (UPV), 46022 Valencia, Spain c Area of Cartographic Engineering, Geodesy and Photogrammetry, Polytechnic High School of Lugo (USC), 27002 Lugo, Spain b
a r t i c l e
i n f o
Article history: Received 27 May 2013 Accepted 26 November 2013 Available online xxx Keywords: Cultural heritage Structural restoration Church reinforcement Scanner laser
a b s t r a c t Restoring cultural heritage is an extremely important job due to its immeasurable value. However, it also requires even greater attention in the case of a building. The actions taken on these constructions not only guarantee their preservation from the point of view of their historical value, but they also guarantee their stability as structures. The need to control historic buildings, analyzing their defects and their possible consequences, is decisive in preventing significant damage. This work demonstrates calculating the thickness of vaults in a church. It is easy to determine the interior and exterior 3D geometry of the church using scanner laser techniques. By combining both geometries, it is easy to determine the difference between the interior vaults and the roof of the church. However, the interior of both geometries is completely unknown, and it is not possible to act on their condition for structural consolidation purposes. This work shows the methods used to determine the interior sections of the vault thicknesses by referencing the internal geometry of both models with the interior of the church, using plumb line system scanning. The results obtained show accuracy better than 6 millimeters. © 2013 Elsevier Masson SAS. All rights reserved.
1. Research aims Research aims consist on developing a methodology to get compete 3D model of a church, including not visible parts under the roof, to measure the thickness of the vaults for structural restoring. To get a complete 3D model, it is necessary to reference scanner laser point clouds to the same coordinate system. As there are not common points to join the point clouds, the development of a plumb line system with spheres will be carried out. Final 3D model obtained gets sections of the vaults with 6 mm of accuracy. 2. Introduction It is currently common to use photogrammetric and scanner laser techniques to generate 3D models in the area of civil engineering and architecture [1,2]. The scanner laser is undoubtedly the quickest and most powerful tool, although there are other possible techniques based primarily on photogrammetry [3], and it is
∗ Corresponding author. E-mail addresses: [email protected] (J. Herráez), [email protected] (P. Navarro), [email protected] (J.L. Denia), [email protected] (M.T. Martín), [email protected] (J. Rodríguez).
especially increasingly common to find works that combine both techniques [4,5]. 3D modeling for an entire building is achieved by combining the interior and exterior scans of it. This makes it simple to obtain a 3D model of everything that is visible. Works have been carried out on a variety of structures, such as buildings [6], towers [7], or bridges [8], for example. Specifically, focusing on the scope of this work, there are references to actions in churches [9]. In all of the cases mentioned, 3D modeling is highly precise from a structural point of view [10] and enables subsequent actions for restoration purposes [11]. However, the interior parts that are not visible cannot be modeled, so they cannot be included in the final complete model. That means that the problem arises when the interest focuses on a part of the building that cannot be modeled as a part of the overall whole. This problem exists whenever there is no connection between the hidden zone and the visible zones, given that the respective point clouds cannot be linked. This work shows 3D modeling of the inside of the vaults of a church for the purpose of designing the structural consolidation project. To do this, 3D models of the interior and the exterior of the church are generated, in addition to the 3D model of the interior of the vaults. The interior and exterior models of the church are easy to merge. Meanwhile, referencing the interior model with both
1296-2074/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.culher.2013.11.015
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Fig. 1. a: interior of the church (vaults); b: exterior of the church (roof); c: interior of the vaults (under the roof).
geometries is impossible without common elements. To solve the problem, a system of physical plumb lines was designed, scanned from inside the vaults and inside the church, making it possible to join the models and obtain the desired sections with accuracy better than 6 millimeters.
3. Generating 3D models 3.1. The church of Santa María de Magdalena. Statement of the problem The church of Santa María de Magdalena dates to the 16th century and is located in the city of L’Ollería (Valencia, Spain). Its exact date of construction is unknown, although it must have been built between the year 1522 (when the old church was destroyed by fire during the Revolt of the Brotherhoods) and 1566 (the date it is first mentioned in church documents). Its construction took place in at least two phases. The church’s plan consists of a single nave with four bays and a polygonal apse, with chapels between the buttresses and a tower located to one side of the entrance. Adjoining sacristy, communion chapel, and trasagrario (a special chapel for guarding the Eucharist) structures were added during later periods. The church’s walls are made of irregular stone fragments held together with lime mortar, with dressed limestone blocks used in the corners and buttresses. The nave is topped by crossribbed vaults with tiercerons and framing, finished with sculpted
keystones. Both the main facade and the tower are crowned with a decorative battlement. Despite its small size and irregular carving, the portal is very interesting because of its design in the form of a Roman triumphal arch with double doors flanked by semiinset columns. Between the columns are scalloped niches with a rich collection of decorative sculptures based on faces, cupids, and harpies. The structural problems that led to restoration of the church were derived from a partial collapse during a 1748 earthquake, which caused cracking in the church’s vaults and bell tower. Although the church was reconstructed, it has now become necessary to carry out a detailed study of the structure’s stability. This is taking place under the “Master Plan” and consists of an in-depth study of the church’s constructed elements in order to develop a proposal for restoration, conservation, and monitoring of changes in the building. This in-depth study includes 3D mapping and modeling of the church to guide planning of future actions. In relation to the vaults, a structural study is being performed in order to analyze their stability. This has required a determination of the thicknesses in each vault throughout the entire section.
3.2. 3D modeling. Design of the solution We have three 3D models generated with point clouds captured with a scanner laser: the model of the entire interior of the church (Fig. 1a, the model of the entire exterior of the church (Fig. 1b)
Fig. 2. a: tripod with plumb line and sphere inside the vault; b: physical plumb line from inside the church; c: plumb line with spheres from inside the church.
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and the model of the interior of the vaults (Fig. 1c). Each of them was obtained by means of multiple scans, followed by filtering and recording. To generate a single 3D model that contains the three previous models, those models need to be referenced to a single system of absolute coordinates. The 3D geometry referencing consists of determining a single three-dimensional system of coordinates that all the models in a single element refer to. The interior and exterior referencing of the church is simple, requiring nothing more than identifying common recognizable points in both 3D models from both positions (doors, windows, etc.). However, the interior model of the vaults can be in any position between the two previous models, and it is not possible to determine its absolute position and, as a result, the thicknesses of both the vaults and the roof. To determine the absolute position, 3 options were evaluated: • by contact of common surfaces in the point cloud. This was discarded, as it was impossible to achieve a common surface of little more than 1 sq.m, in very precarious working conditions; • measuring some points by topographic methods and referencing them to the system of absolute coordinates. This was also discarded because it was impossible to achieve good visuals both inside the church and toward any building in the surrounding area; • there are some holes with a diameter of 7–10 cm beside the vault keystones (which were likely used to hold some sort of lamp) through which it is impossible to measure but that could hold visible elements from each part that make it possible to identify common elements.
3
Z
Centre Sphere 1s Inside of the vault Thickness of the vaults
Thickness of the vaults
Inside of the Church
D Y
X
Centre Sphere 1i
Plumb line
Fig. 3. Layout of one of the plumb lines, with a sphere on the inside of the vault (sphere 1s) and another on the inside of the church (sphere 1i), going through the thickness of the vault through the hole.
where, Rj : radius of the sphere,
4. Referencing 3D models Based on the last option, tripods were placed inside the vaults from which plumb lines were hung through the holes (Fig. 2a). These plumb lines were made of 3 mm thick cord, with a 4 kg weight hung from the end to guarantee tension and a lack of movement (Fig. 2b). Once the plumb lines were installed, the vertical was perfectly defined in the scans from the inside of the church and from the inside of the vaults. However, it was necessary to make some sort of mark on the plumb lines to enable the identification of a common element. To do so, two visible spheres were used in each scan, respectively (Fig. 2a and c). Defining a system of coordinates focused on a sphere and parallel to the system of absolute coordinates, the coordinates of the centers of each pair of spheres will have the same X and Y coordinates, with only the Z coordinate varying (Fig. 3). This coordinate will be determined by the exact distance between the two spheres (D) over the vertical defined by the plumb lines. Once the coordinates of both spheres have been calculated with respect to the same system, it will be simple to locate the corresponding point clouds from the scans to reference the models based on them. However, given that the spheres are not visible simultaneously, it is necessary to carry out an initial auxiliary scan, used to calculate the distance D between them. To do so, it is necessary to remove the sphere that is on the inside through the vault hole, leaving both inside the church. With both spheres inside the church, the first scan is run, which will give the distance between them as a result. To do that, it is necessary to determine their centers. To calculate the centre of a spherical element within a point cloud, we adjust all the points scanned on its surface based on the equation of a sphere (1): Rj =
XSi − XCj
2
+ YSi − YCj
2
+ ZSi − ZCj
2
(1)
XCj , YCj , ZCj : coordinates of the centre of a sphere j, (XSi , YSi , ZSi ): coordinates of a point i on the surface of the sphere j. Based on an approximate value for the centre of each sphere, the value calculated will be given by (2):
XCj , YCj , ZCj = XAj , YAj , ZAj + ıx , ıy , ız
(2)
where, XAj , YAj , ZAj : approximate coordinates of the centre of a sphere j, ıx , ıy , ız : calculation differentials. Substituting (2) in (1) we obtain (3): Rj =
XMi − XSi − ıx
2
+ YMi − YSi − ıy
2
+ ZMi − ZSi − ız
+ ıRj
2 (3)
where, (XMi , YMi , ZMi ): coordinates measured by the laser of each point i on the surface of a sphere j, ␦Ri : error of the measured point with respect to the surface of a perfect sphere. Developing the expression (3), each point belonging to a sphere provides an equation that has the differentials ıx , ıy , ız as unknowns, making it possible to generate a matrix system that is resolved by means of a bundle adjustment. Once the coordinates have been calculated for the centers of each pair of spheres on each plumb line, the distance D will be given according to (4): j+1
Dj
=
XCj − XCj+1
2
+ YCj − YCj+1
2
+ ZCj − ZCj+1
2
(4)
Once the distance between both spheres has been calculated, the calculation of the coordinates of the sphere that is inside the vault
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based on the coordinates of the sphere that is inside the church or vice versa, gives (5):
XCj+1 , YCj+1 , ZCj+1 =
j+1
XCj , YCj , ZCj ± Dj
(5)
In the expression (5), we will use the “+” sign for the calculation of the upper spheres, while the “−” sign will be used for the lower spheres. Referencing the coordinates of the points scanned from the inside of the vault requires simply applying the coordinates of the sphere (5) and the rotation between both point clouds that will be defined by the vertical of the plumb line, based on (6):
XCj , YCj , ZCj
t
= (1 − Si )−1 ∗ (1 + Si ) ∗ XCj , YCj , ZCj
t
+ Tx , Ty , Tz
where : t XCj , YCj , ZCj : coordinates of the sphere j in the model of the interior of the vault, , Y , Z XCj Cj Cj
t
: coordinates of the sphere j in the model of the
interior of the outer vault, t Tx , Ty , Tz : translation vector between the two models,
(Si ) =
0 −ai bi
ai 0 −ci
−bi ci 0
Sphere 1i 1s 2i 2s 3i 3s
Scanned points 30 24 49 27 136 84
Calculated radius
Xc (m)
Yc (m)
Zc (m)
D (m)
0.0215 0.0215 0.0205 0.0205 0.0215 0.0205
6.347 6.345 4.673 4.669 2.882 2.883
16.548 16.551 11.189 11.190 5.790 5.786
8.707 12.054 8.044 11.387 8.560 11.886
3.347 3.341 3.326
t
(6)
Table 1 Calculation of the centers of the spheres (with the plumb line hanging inside the church) and the distance between each pair of them.
is the Cayley matrix.
5. Results The method shown makes it possible to reference the 3D models of the interior of the church, the exterior and the interior of the vaults with plumb lines and spheres. Taking advantage of the existence of 3 holes in the vaults, 3 plumb lines were installed. This makes the calculation redundant and offers greater accuracy. 5.1. Generating the 3D models The work was done with a Leica Scanstation II laser. Given the distance at which they were performed, the precision of the scans is 6 mm, both for measurement of points and for the spheres, which fits the requirements for the work perfectly. The precision for the centering of the spheres on the plumb line is estimated as 2–3 mm. The structural work being carried out requires a strong vault with a uniform geometry that will approximate the existing average thicknesses as much as possible, with a precision of 1–2 cm being sufficient. This level of precision can undoubtedly be achieved with the methodology and instrumentation employed. The models generated to determine the thicknesses are: 3D model of the interior of the church (vaults), 3D model of the exterior of the church (roof), 3D model of the interior of the vaults (under the roof) and 3D models of the plumb lines through the holes near the vault keystones (from the inside of the vaults and from the inside of the church). In order to obtain the interior geometry of the church, 8 scans with full 360◦ rotation were performed in order to provide good coverage for later joining of the clouds (Fig. 4). The ascent from the interior of the church to the top of the bell tower also required 17 more scans, in order to record the stairway and the faces of the tower so that its interior could be joined with the exterior model of the structure as a whole. The exterior geometry required 16 scans, also with full 360◦ rotation (Fig. 5). The reason why so much work was necessary was because of the complexity of the church’s surroundings. The urban location in an area with a high density of other buildings nearby created this complexity, in contrast to a more open location. The joining of the exterior and interior models took place by means of the church’s doorways, with 3 full rotation scans required
for this purpose. These are shown as 3 x’s in Fig. 6a, two in the main doorway and one in the side entrance. In total, the 3D model from inside the church was made with 25 scans, obtaining a cloud made up of 43 million points once filtered and after merging the clouds. The 3D model of the exterior roof of the church was made with 16 scans, obtaining a cloud made up of 52 million points once filtered and after merging the clouds (plus 3 scans in the doorways for the joining). Also, the 3D model of the interior corridor of the vaults was made with 6 scans, obtaining a cloud made up of 6 000 000 points once filtered and after merging the clouds (Fig. 6). The total number of scans used for all of the 3D models was 44. For their recording in pairs we obtained 168 cloud constraints (this involves a redundancy of 125 since the minimum number necessary is 43). The adjustments have been high precision, since the residual errors obtained have been very low: the residuals show errors of 0.000 m in 32% of the cases (in other words, the residual error has been less than 0.0005 m); less than 0.001 m in 74% of the cases; and less than 0.002 m in 100% of the cases. Once the 3D models were generated for the three main elements, the referencing was made by scanning the 3 suspended plumb lines, passed through 3 holes. Each plumb line has 2 spheres, one visible from the interior of the church and the other from inside the vaults, in each hole. 5.2. Referencing the 3D models It is first necessary to determine the distance between the spheres of each plumb line. The centers calculated by (3) for each pair of spheres on each plumb line with the cord hanging inside the church and the calculated distances are shown in Table 1. The names of the spheres correspond to the number of the plumb line accompanied by a subscript, s or i, which indicates whether the sphere is the upper one (located inside the vault) or the lower one (located inside the church). Once the distances have been determined, the plumb lines are raised so that one sphere is inside the vault and another is inside the church. Referencing the scanned clouds based on the centers of the spheres is determined by the calculated centers of the upper and lower spheres in both coordinate systems (Table 2). The table shows the coordinates of the same spheres in the model in the
Table 2 a: coordinates of the spheres with respect to the interior of the vaults; b: coordinates of the spheres in the model of the interior of the church. (a)
(b)
Sphere
Xc (m)
Yc (m)
Zc (m)
Sphere
Xc (m)
Yc (m)
Zc (m)
1s 1i 2s 2i 3s 3i
4.525 4.525 9.113 9.113 13.733 13.733
−3.548 −3.548 −6.758 −6.758 −10.111 −10.111
−0.811 −4.158 −0.821 −4.162 −0.785 −4.111
1s 1i 2s 2i 3s 3i
−0.694 −0.694 −5.748 −5.748 −10.852 −10.852
0.794 0.794 3.218 3.218 5.773 5.773
12.608 9.261 12.596 9.255 12.632 9.306
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Fig. 4. a: positions of the scanning stations in the church interior (plan view); b: 3D model of the exterior of the church with vaults scanned from the interior.
Fig. 5. a: positions of the exterior and linking scanning stations in plan view: 16 exterior stations (dots) and 3 stations in the doorways (x’s); b: 3D model of the church with the exterior roof. Table 3 Cayley’s matrix parameters. Parameter
Value
a b c
−5.77702544 0.01085382 −0.01827810
interior of the vaults and in the model in the interior of the church, calculating the record through the Cayley matrix (Table 3). The components of the translation vector are implicit in the Cayley matrix, having used the coordinates of the spheres reduced to mass center of points. 5.3. Measuring thicknesses and obtaining sections By applying the calculated parameters, we merge the models and obtain the sections of the thicknesses sought (Fig. 7) with millimetric accuracy (Table 4):
The arches have a different thickness in each vault, maintaining an approximate average value in each one of them. The average thickness is 0.149 m for the first vault, 0.146 m for the second, 0.165 m for the third and 0.102 m for the fourth. Based on the results, the first two vaults have a very similar average thickness (with small variations of around 1cm between arches in each of them), while the third vault has a significantly greater thickness (+11%) and the fourth has a thickness that is much less (−32%). The final result for each of the vaults is a full section, which will later be analyzed during the restoration work (Fig. 8).
6. Analysis and discussion It is worth considering the extent to which the methodology employed in this study can be generalized and extrapolated to other cases. To do this, it is appropriate to analyze the reliability of the reconstruction based on the number of spheres required as well as their distribution.
Fig. 6. 3D model of the interior corridor of the vaults (under the roof).
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Fig. 7. a: complete longitudinal section (with the 3 plumb lines); b: vault 1 (with plumb line); c: vault 2 (with plumb line); d: vault 3 (with plumb line; e: vault 4.
After processing the clouds we have obtained two 3D models: one corresponding to the interior-exterior of the church and the other corresponding to the interior of the vault corridor below the roof. Both models are established in an independent manner, which means that the problem to be resolved is that of using common points (the spheres) to position the model of the vault corridor within the model of the church. There are 2 spheres available in both models, which combined with the conditions of verticality (which aligns the spheres along the Z axis of the absolute coordinate system for both models) is sufficient to resolve the problem. A third sphere provides us with verification of the adjustment performed (3 spheres on each of 2 plumb lines, or else 3 plumb lines with 2 spheres on each). In this way, the greater the number of spheres the more redundancy the calculation system will have. Based on the above, the positioning of one model within another can have two distinct sources of error. These can be revealed by
analyzing the minimal case, with two plumb lines very close to each other, with two spheres on each: • an error derived from calculation of the sphere coordinates, which involves a vertical displacement of one model with respect
Table 4 Thicknesses of the four vaults in each of their seven arches, respectively (taking the central point of each arch in each case). Vault
Arch
Thickness (m)
Vault
Arch
Thickness (m)
1
1 2 3 4 5 6 7
0.157 0.166 0.148 0.143 0.138 0.151 0.137
2
1 2 3 4 5 6 7
0.141 0.146 0.156 0.151 0.148 0.139 0.142
3
1 2 3 4 5 6 7
0.193 0.175 0.182 0.160 0.157 0.138 0.152
4
1 2 3 4 5 6 7
0.097 0.093 0.102 0.114 0.108 0.091 0.106
Fig. 8. Section of one of the vaults (the hole through which the plumb line with two spheres is hanging can be seen).
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Fig. 9. Error in vertical positioning.
Fig. 10. Error in orientation.
to the other (Fig. 9), and which results in an erroneous thickness calculation; • an error derived from calculation of the planimetric orientation, which involves an error in plan-view orientation of one model with respect to the other (Fig. 10). This also results in an erroneous thickness calculation. As can be seen in Figs. 9 and 10, if we lack redundancy in the calculation, a positioning error may be serious although of constant value, while a rotational error will be propagated and increase with distance. If a third sphere is available on each plumb line, the first type of error is ruled out since a serious error in calculation for one of the spheres would be detected (with only 2 spheres this would not be possible). The same would occur with the second type of error, since availability of a third sphere allows two orientation conditions to be established, which ensures planimetric overlapping. The analysis of possible error by vertical rotation of one of the models is directly eliminated by working with the scanner in leveled mode (with verification of this afterward). With regard to the geometric distribution of the points, increasing the number of plumb lines will increase the method’s reliability, while increasing the number of spheres aligned on the same plumb line will improve the planimetric orientation calculation for the model. However, it is much more optimal to install more plumb lines (the orientation of a plane will always be better when different vectors of non-aligned points are used, rather than many vectors of aligned points). The choice of one system or the other will depend upon the scale of the element as well as the number of spheres that can be used and the possible locations for them. The worst possible case will be one where the plumb lines are located very close to each other, which favors the propagation of any error. If we add the scale of the element into this consideration, this propagation will be even more unfavorable the larger the dimensions are. This means that: • it is better to install multiple plumb lines than to add multiple spheres on the same plumb line: putting more spheres on the same plumb line does not provide more precision for location within the coordinates, since any error that exists will affect the absolute position (given that the relative positions among them will not reveal errors). Installing more plumb lines, on the other hand, allows for verification;
• the position of the spheres is very relevant: if only two plumb lines are installed then it is best to locate them as far apart as possible, which will allow the definition of vectors to be optimal. If a third plumb line is installed then verification can be performed in the case of an error. Based upon all of the above, in our case we opted for installation of three plumb lines with two spheres on each of them, covering all dimensions of the element. Using three plumb lines with 6 spheres ensures the absence of errors in the coordinates, and it also eliminates any doubt regarding propagation of an orientation error when the two models are joined. Application of this methodology can be extrapolated, under certain circumstances, to situations where 3D models must be joined but where no common elements exist. The circumstances required for the methodology presented to be reproduced are those where the two upper or lower spheres from both models are always visible in a single scan. Reconstruction of the full model will entail independent scans (of each complete model separately), which will be joined later by the partial scan that contains the spheres.
7. Conclusions The procedure used is quick and easy. With just the help of some plumb lines and spheres it is possible to reference several threedimensional point clouds in space in a way that is highly automated, with an accuracy of 6 millimeters. Generating 3D models is a problem resolved both by scanner laser techniques and by many others from the fields of topography, photogrammetry, etc. However, referencing between the models generated is a problem that requires a specific solution in each case. In situations in which there are no common elements, it is necessary to create conditions that enable us to establish a system of absolute coordinates. Topography, in this area, is very limited, given that it is very difficult to locate a sufficient number of fixed points. Photogrammetry, meanwhile, solves the problem of the number of points, but it can not be applied in all types of conditions (in our study, lighting was a key problem). The scanner laser, on the other hand, overcomes all of these inconveniences, making it possible to work under any lighting conditions and generating a sufficient number of points in any circumstance.
Please cite this article in press as: J. Herráez, et al., Modeling the thickness of vaults in the church of santa maria de magdalena (Valencia, Spain) with laser scanning techniques, Journal of Cultural Heritage (2013), http://dx.doi.org/10.1016/j.culher.2013.11.015
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The use of plumb lines is effective due both to the simplicity of their use and to the provided accuracy. Likewise, using spheres on the plumb lines is undoubtedly the best choice, given that the geometry of a sphere is more easily recognized by automatic techniques than any other 3D geometry. Application of this methodology can be extrapolated to any study where 3D models without common points must be joined (where an element of connection can be introduced between them), or else where the scale and/or geometry of the models involved does not allow full scanning and where a common point shared by two scan clouds may be necessary in order to reconstruct a single space. The final accuracy obtained was of 6 millimeters, so the thicknesses obtained in the sections results appropriate for subsequent structural consolidation studies. References [1] A. Nunez, F. Buill, J. Regot, A. Mesa, Generation of virtual models of cultural heritage, Journal of Cultural Heritage 13 (1) (2012) 103–106. [2] N. Yastikli, Documentation of cultural heritage using digital photogrammetry and laser scanning, Journal of Cultural Heritage 8 (4) (2007) 423–427.
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Please cite this article in press as: J. Herráez, et al., Modeling the thickness of vaults in the church of santa maria de magdalena (Valencia, Spain) with laser scanning techniques, Journal of Cultural Heritage (2013), http://dx.doi.org/10.1016/j.culher.2013.11.015