Photogrammetric software as an alternative to 3D laser ...826106/FULLTEXT01.pdf · DEGREE PROJECT,...
Transcript of Photogrammetric software as an alternative to 3D laser ...826106/FULLTEXT01.pdf · DEGREE PROJECT,...
DEGREE PROJECT, IN , SECOND LEVELMEDIA TECHNOLOGY
STOCKHOLM, SWEDEN 2015
Photogrammetric software as analternative to 3D laser scanning in anamateur environment
MARKUS WARNE
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF COMPUTER SCIENCE AND COMMUNICATION (CSC)
EXAMENSARBETE VID CSC, KTH
Fotogrammetrisk programvara som alternativ
till laser 3D-‐skanning i amatörmiljö
Photogrammetric software as an alternative to
3D laser scanning in an amateur environment
Markus Warne
Examensarbete i medieteknik
Handledare på CSC var Vasiliki Tsaknaki
Handledare på CUT var Krzysztof Skabek
Examinator: Haibo Li
Uppdragsgivare: Politechnika Krakowska (Cracow University of Technology)
Photogrammetric software
as an alternative to 3D laser
scanning in an amateur
environment
Abstract Photogrammetric software today is at a level where it is accessible to the mainstream
public and without larger effort is able to reconstruct digital 3D models from
photographic input. This thesis investigates the performance of photogrammetricly
reconstructed models and evaluates them by comparing the results to their
corresponding reconstructed models from a 3D laser scanner with a focus on smaller
objects in an amateur environment. The evaluation is performed on four different
objects, which are all individually compared to their scanned counterpart. They are
compared both with a subjective judgment of quality and by numerically measuring the
point-‐to-‐point distance on the models. From the results conclusions are drawn that the
methods can produce similar results albeit there are many performance factors
discovered for a good reconstructions with photogrammetry. The properties of the
physical object and the quality of the visual input data stand out as the most important
factors.
Fotogrammetrisk
programvara som alternativ
till laser 3D-‐skanning i
amatörmiljö
Abstrakt Den fotogrammetriska programvaran som existerar idag är tillgänglig för allmänheten
men framförallt kapabel att återskapa digitala 3D-‐modeller utan större ansträngning.
Denna rapport utforskar och utvärderar möjligheterna att återskapa dessa objekt för att
sedan jämföra hur dessa står sig gentemot motsvarande återskapade objekt med en 3D
laser skanner. Fokus ligger på att se hur mindre objekt kan återskapas i en amatörmiljö.
Testerna genomförs på fyra olika objekt genom att först återskapa dessa digitalt m.h.a.
fotogrammetri för att sedan jämföra dessa inviduellt med motsvarande modeller
återskapade m.h.a. 3D-‐skanning. Utvärderingen sker subjektivt med en bedömning av
kvalité men även genom att mäta avstånden från punk till punkt på modellerna. Från
resultaten kan slutsatserna dras att det går att nå likvärdiga resultat med fotogrammetri
som 3D-‐skanning men dessa beror på ett antal kritiska faktorer. Objektets fysiska
egenskaper samt kvalitén av den visuella data som används framstår som nyckelfaktorer
för att lyckas med en bra digitalt återskapad modell.
1 Introduction ................................................................................................................................................... 1
1.1 Goal of the thesis ................................................................................................................................. 2
1.2 Research questions ............................................................................................................................ 2
1.3 Limitations ............................................................................................................................................. 3
2 Background .................................................................................................................................................... 4
2.1 Related research ................................................................................................................................. 4
2.2 Triangulation ........................................................................................................................................ 5
2.3 3D laser scanning ................................................................................................................................ 5
2.3.1 How laser triangulation sensors work .............................................................................. 5
2.4 Photogrammetry ................................................................................................................................. 5
2.4.1 The basics of Photogrammetry ............................................................................................ 6
3 Method ............................................................................................................................................................. 8
3.1 Literature study ................................................................................................................................... 8
3.2 Quantitative evaluation .................................................................................................................... 8
3.3 Qualitative observations .................................................................................................................. 8
4 Evaluation setup .......................................................................................................................................... 9
4.1 Data generation -‐ Photogrammetric approach ...................................................................... 9
4.1.1 Photo environment setup and camera parameters .................................................... 9
4.1.2 Photogrammetric processing: Agisoft’s Photoscan .................................................. 10
4.1.2.1 Camera alignment .......................................................................................................... 10
4.1.2.2 Dense point cloud ........................................................................................................... 10
4.1.2.3 Mesh construction .......................................................................................................... 11
4.1.3 Photogrammetric processing: Autodesk’s 123D Catch .......................................... 11
4.2 Data generation -‐ laser scanning approach .......................................................................... 12
4.2.1 Environment and setup ........................................................................................................ 12
4.2.2 Konica Minolta Vivid 9i ......................................................................................................... 12
4.2.3 Model reconstruction from point cloud ........................................................................ 12
4.3 Data comparison .............................................................................................................................. 13
4.3.1 Alignment ................................................................................................................................... 13
4.3.2 Measurement ............................................................................................................................ 13
5 Results ........................................................................................................................................................... 15
5.1 Case 1 – Quadric object ................................................................................................................. 15
5.1.1 3D print reconstructed with PhotoScan ........................................................................ 16
5.1.2 3D print reconstructed with 123D Catch ...................................................................... 17
5.2 Case 2 – Angel figure ...................................................................................................................... 18
5.2.1 3D model reconstructed with PhotoScan ..................................................................... 19
5.2.2 3D model reconstructed with 123D Catch ................................................................... 20
5.3 Case 3 – Monkey figure ................................................................................................................. 21
5.3.1 3D model reconstructed with PhotoScan ..................................................................... 22
5.3.2 3D model reconstructed with 123D Catch ................................................................... 23
5.4 Case 4 – Wooden cat ....................................................................................................................... 24
5.4.1 3D model reconstructed with PhotoScan ..................................................................... 24
5.4.2 3D model reconstructed with 123D Catch ................................................................... 26
6 Analysis and discussion ......................................................................................................................... 27
6.1 Comparing the results of the photogrammetric reconstructions and the laser
scanned reconstructions .......................................................................................................................... 27
6.2 Strengths and weaknesses of the photogrammetric reconstruction software
applications ................................................................................................................................................... 28
6.3 The photogrammetric reconstruction process as a whole ............................................ 29
6.4 Can photogrammetry yield similar results to 3D laser scanning when used in an
amateur home setting for smaller objects? ..................................................................................... 30
7 Conclusion ................................................................................................................................................... 31
7.1 Future research ................................................................................................................................ 32
8 References ................................................................................................................................................... 33
9 Appendix ...................................................................................................................................................... 35
9.1 Case 1 – deviation results ............................................................................................................. 35
9.1.1 Photoscan model compared to 3D scanned model .................................................. 35
9.1.2 123D Catch model compared to 3D scanned model ................................................ 35
9.2 Case 2 – deviation results ............................................................................................................. 36
9.2.1 Photoscan model compared to 3D scanned model .................................................. 36
9.2.2 123D Catch model compared to 3D scanned model ................................................ 36
9.3 Case 3 – deviation results ............................................................................................................. 37
9.3.1 Photoscan model compared to 3D scanned model .................................................. 37
9.3.2 123D Catch model compared to 3D scanned model ................................................ 37
9.4 Case 4 – deviation results ............................................................................................................. 38
9.4.1 Photoscan model compared to 3D scanned model .................................................. 38
9.4.2 123D Catch model compared to 3D scanned model ................................................ 38
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1 Introduction Recently, there has been a growing interest in 3D printing technologies, with a number
of hardware and applications with a focus on the context of everyday use (Crum 2014).
This development indicates a future where 3D printing could be a tool for personal use
for everyone, not just experts. The 3D printing techniques are quickly advancing to
become more accessible to the general public with new models created specifically for
home use and a lower budget (Matter and Form 2014). What has not been discussed in
depth is the other side of the spectrum. What will we print? The models and data must
come from somewhere. As the demand and ability to print 3D models increases, the
supply of models must also follow according to basic economic theory. When this
happens, the market will desire a method for gathering 3D model data that is accessible
on an amateur scale to as many individuals as possible, at a low cost.
Methods such as laser scanning are not new, the first triangulation laser scanning
technology was developed already in 1978 (Mayer 1999). However, these methods were
and are still inaccessible to the mainstream public, at least to some degree, as the cost of
acquiring the technology or the knowledge needed to operate it is simply too high for
the average user. A relatively cheap alternative to laser scanning is stereo
photogrammetry. With the help of specific software and the advanced triangulation
algorithms that are available today, this method can be used, similarly to a 3D scanner,
to reconstruct and digitally model real life physical objects from just a set of ordinary
photos. These methods now open up new possibilities when it comes to creating and
sharing 3D models on a much larger scale, by amateur users, as they become more
accessible from an economic and technological viewpoint to the general public.
This is a very important area as these technologies are on the brink of becoming
mainstream and integrated to our everyday lives. 3D printing and 3D reconstruction
might be as common as sharing a file over the internet is today, in just a couple of years.
It is important to analyze these different methods, where they are in their current state
and possibly draw conclusions on what still needs to and can be improved in the future.
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1.1 Goal of the thesis The goal of this thesis is to take a closer look at 3D reconstruction of objects, by using
photogrammetry software as an alternative to laser scanning. Furthermore, to
investigate if this is a viable alternative for mainstream users and available to generate
models of a comparable quality to that of a reconstructed model from a laser scanner.
The main focus will be on comparing the models reconstructed from the two methods
mentioned above (reconstruction with laser scanning and reconstruction with stereo
photogrammetry), as described in Figure 1. Specifically, I will investigate the deviations
of the surfaces between the two methods while trying to assess the quality of the
models, by comparing them to each other numerically.
Figure 1 – Overview of reconstruction methods
1.2 Research questions • Can photogrammetry yield similar results to 3D laser scanning when used in an
amateur home setting for small objects?
o Numerical comparison of deviation between the reconstructed surfaces
• How do these two methods (laser scanning and stereo photogrammetry)
reconstructing models compare to each other?
o Strengths and weaknesses
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1.3 Limitations The comparison is limited to two photogrammetric methods and the laser scanning is
used just as reference measurement. The quality of the scanned models and accuaracy
for the specific scanner has already been carried out in another publication (Spytkowska
2008). Ideally a set of models from several laser scanners would be used as to be able to
generalize the results from laser scanning and identify similarities. In addition, only two
photogrammetric software applications will be used to reconstruct 3D models, which
could also limits the conclusions drawn from comparing the two techniques and
identifying general strengths and weaknesses.
Furthermore, the quality of the reconstructed objects will be assessed subjectively, due
to the nature of such qualitative evaluations. The limitation is the fact that there is no
digital original object to compare and quantify the deviation from the digital
reconstructions, as in most cases the originals are small physical objects.
The reconstructed models will be compared to a digital representation of the original, in
this case, a 3D scanned version of the original object, which in turn will be treated as
reference for deviation measurements. This will provide quantifiable results of deviation
between the two methods, 3D scanning and photogrammetry, but there is no way to
compare these to that of the original object.
One of the big limitations of this thesis is that in order to stay true to the “amateur
mainstream user” perspective a number of commercial software applications have been
used in parts of the process. This generates some “black box” parts where the
transparency of the process is limited, as we only know what we put in and what comes
out. Assumptions are made in these cases based upon general procedures and
algorithms within the field but complete certainty can’t be achieved. It has been made
clear which these parts are and when assumptions are made they are clearly indicated.
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2 Background 2.1 Related research Laser scanning has historically been used for scanning small objects in a controlled
environment, where there is a possibility of scanning the object in 360 degrees angle.
This is due to the fact that 3D scanners often have a small field of view and larger objects
often have to be scanned in several iterations and then combined together in order to
reproduce and complete the model (Chen et al. 2000; Seitz & Curless 2006). However
there are some exceptions with laser scanners such as LIDARs which are laser scanners
optimized for capturing large geographic areas and similar scenarios. On the other hand,
photogrammetry has mostly been used in these types of situations, when there is a need
to scan large-‐scale areas, such as air photography, cartography, mapping of
archaeological sites and other situations with large objects, when there is no need to
capture small details but rather focus on extracting measurements etc. Photogrammetry
is rapidly becoming more and more common, for reconstructing high detailed 3D
models (Blizard 2014; Poznanski 2014). This can be seen especially on the Internet,
where its accessibility to the mainstream public has inspired many hobbyists and so
called DIY (do it yourself) enthusiasts (Blizard 2014) but also in the entertainment
industry such as games (Poznanski 2014) and movies (Wolff 2004). Previously
photogrammetry has also been used in combination with laser scans as a compliment to
provide accurate textures for the model that were generated by laser scanning.
Attempts to reconstruct digital 3D models with the help of photogrammetry were
already done in 1984 (Benard 1984) and there are even some cases where the results of
the reconstructions have been compared to laser scanning techniques (Baltsavias 1999;
Fassi & Fregonese 2013). However most of them are, as mentioned earlier, focused on
archaeological, architectural or geo-‐data scenarios where the objects are usually very
large in scale. This thesis focuses on the perspective of smaller sized objects, up to one
cubic meter, and as some earlier research suggests (Baltsavias 1999) this area might be
more challenging for photogrammetric reconstruction of models. In addition, 3D
scanners have largely dominated the digital reconstruction of small objects as they are
usually built for this purpose whereas reconstruction with the help of photogrammetry
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has not been applied in these scenarios as often and if so only partly or complimentary
for smaller objects.
2.2 Triangulation Triangulation is a central concept for both of the techniques applied in this thesis, as
both photogrammetry and most laser scanning methods are based around this principle.
It is a method used to calculate the position of points in 3D space. It is used in a wide
range of applications and scenarios such as navigation, astronomy and many more due
to its broad and dynamic origin. Triangulation works by mathematically intersecting
converging lines in space from at least two known points to that of the investigated
point. By measuring the difference in the angle to the investigated point the precise
location of the point in 3D space can be determined. (Spytkowska 2008)
2.3 3D laser scanning 2.3.1 How laser triangulation sensors work
A triangulating laser scanner simplified consists of two components, a transmitter and a
receiver. The transmitter usually consists of a laser diode that projects a ray of light on
the object. A charged coupled device (CCD) sensor detects the reflection and due to
displacements in the object the angle of reflection varies depending on form and
distance. Thus the difference can be measured due to the principles of triangulation.
From this data, a point cloud is generated based on each specific measurement, each
point with a specific distance from the scanner. This measurement is carried out
thousands of times to generate a large point cloud representing the object. This discrete
point data cloud can then be interpolated, usually with the help of some complimentary
software, to create a 3D surface that consists of not just points but polygons.
(Spytkowska 2008; Dold & Brenner 2006)
2.4 Photogrammetry Photogrammetry is the science of acquiring measurements and 3D coordinates from
photographs. It can be compared to the reverse engineering of photographs as it aims to
turn 2D information into 3D information, the opposite of a camera (Figure 2).
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Photogrammetry is often applied in topography scenarios like satellite and aerial
photography but also in close range scenarios, such as 3D reconstruction.
Figure 2 – Input and output of optical data capturing methods
2.4.1 The basics of Photogrammetry
The core principle for photogrammetry is triangulation. Due to several photographs (at
least two or more) with overlapping information, rays, as they are called, can be
calculated from the camera position to the object. By calculating the difference in angle
between the different rays the distance and position of the camera can be established.
This is also exactly the way our eyes work when estimating depth. In photogrammetry
the difference in the (x,y)-‐plane is used to triangulate the measured point. Furthermore
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photogrammetry applies this principle to multiple points at a time with theoretically no
limit to the number of points measured at the same time (Slama et al. 1980).
To be able to compare the 3D laser scanning method to the photogrammetric approach
we have to go from physical 3D domain to digital 3D domain as seen in Figure 2.
However strictly photogrammetry consists of 2D input in the form of photos, thus we
have to include the photography part from Figure 2 to achieve the complete flow of
physical 3D domain to digital 3D domain.
As mentioned before (Figure 2) photogrammetry is in a way the reversed process of
photography. Unfortunately the photographic process is not perfect as information is
lost when taking a photo, if it was perfect, as in no information lost, just two photos
would be more than enough to recreate the 3D scene. So to compensate for this missing
information several photographs (absolute minimum of two) have to be used to aid the
calculation. The coordinates acquired from these calculations are the final result from
photogrammetry, this can be presented in the form of a point cloud or some other data
set that then usually is used further to extrapolate a 3D surface (Greve 1996; Schenk
2005).
For best results the images used as input for photogrammetry a few parameters are
important. First is the focus of the camera, since photogrammetry uses pixel for points in
triangulation blurry images are very tricky for pinpointing the position of features and
other elements that are out of focus. Furthermore the resolution is a very important
parameter much due to the same reason mentioned earlier meaning more pixels equal
more accurate calculations. Consistent lightning is also a factor that helps
photogrammetry in identifying various elements in the photograph. Varying light casts
different shadows from picture to picture this can have a destructive effect, as the same
feature can have varying intensity and color depending on the image analyzed at the
moment (Greve 1996; Schenk 2005).
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3 Method 3.1 Literature study To see what has been done in the field of 3D reconstruction with the use of
photogrammetry a literature study has been done. Google Scholar and KTH Royal
Institute of Technology’s “Primo” service have been used as primary knowledge wells
for academic research within the area. Furthermore high detail 3D reconstruction from
photogrammetry is a relatively new concept relevant information has been found on
blogs and other technical news themed websites. Google search and Wikipedia have
been used as a compliment or in conjunction to the named sources above.
3.2 Quantitative evaluation Quantitative methods are applied in this report to measure the difference in distance
between points on several reconstructed 3D surfaces. This is done with the help of
software that generates points on the reference surface and then tries to match them
with the test surface. Quantitative distance data is received for each of these points.
Unfortunately only the disparity between two reconstructed objects can be compared
quantitatively and not between the reconstructed objects and the original. This is due to
the fact that the original is not in the digital domain and any “conversion” from physical
to digital is simply another form of reconstruction method. This leaves the “quality”
parameter, from original to reconstruction, unquantifiable and a subjective matter. What
can be compared are the disparities of the surfaces from the reconstructed models,
which is also the main goal in this thesis, to find out if the methods are comparable and
yield similar results in a non professional environment.
3.3 Qualitative observations Furthermore the resulting reconstructed models will be assessed from a qualitative
approach with the aim to give some way of categorizing the perceived quality and
similarity of the digital reconstructed object compared to the physical original. This
assessment is purely subjective and may vary depending on the individual. This
observation is carried out with the purpose of another input into the main research
question, whether or not the tested methods yield comparable results. A method might
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give relatively small quantifiable differences but the discrepancy in perceived quality
between the original object and the reconstructed model may be very large.
4 Evaluation setup The photogrammetric evaluation process consists of two major parts, the first is the
reconstruction of the physical object in 3D domain – this will be called data generation.
The second part consists of analyzing the reconstructed objects and comparing them to
each other. This is where the quantified data is extracted from the objects – this will be
called data comparison.
As the software applications used in this thesis are commercial, the specific algorithms
and techniques used are not available to the public domain. Therefore a general
approach to reconstruction with photogrammetry and laser scanning has been
described and for the specific software that follows all information that is available for
each application will be presented and assumptions based on what is known.
4.1 Data generation -‐ Photogrammetric approach 4.1.1 Photo environment setup and camera parameters The data generation environment was set up in a home setting with varying lightning
conditions as to stay in line with the thesis main goal of analyzing the possibility for
mainstream public to apply this technique in a non professional context. A tripod was
used to aid the stability and positioning of the camera and the lightning consisted of
mixed indoor light. The camera was placed as close as possible to the object for the
majority of the picture to cover most of the image area to preserve as much detail as
possible.
The camera used was a FUJIFILM X100S. The pictures taken had the aperture value of 16
as to maximize the depth of field for maximum amount of the element to be in focus.
Depending on the lightning registered by the camera different shutter speeds were
applied although most of the taken pictures had a shutter speed of ¼ seconds. The
pictures were saved digitally in .jpg format with a resolution of 4896 x 3264 pixels.
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As mentioned before resolution and focus are two very important parameters for the
reconstruction process and these have been taken into consideration in this setup.
However consistent lightning is also an important factor for best results from
photogrammetric reconstruction but there has been no effort in this setup to minimize
this. This is to mimic home conditions of mainstream users to stay in line with the thesis
goal and as in a home environment there is usually no way to achieve evenly lit objects
such as with studio lightning conditions. However it is assumed that a good depth of
field and focus can be achieved in a home environment.
4.1.2 Photogrammetric processing: Agisoft’s Photoscan
The procedure of photograph processing and 3D model construction is described with
the following four stages according to the PhotoScan manual (Agisoft LLC 2011).
4.1.2.1 Camera alignment The software searches for common points in the collection of photographs. To be able to
match them, the software also calculates the position and orientation of the camera for
each picture. It is very probable that this is carried out with some form of triangulation
described earlier. The result of this process is a sparse point cloud where several points
have been identified and matched over the different camera positions. It is important to
have in mind that several points can be calculated and matched per camera location (a
single photo). However this sparse point cloud is not what the reconstruction of the 3D
model is based on unless explicitly specified by the user. The information that is used
further in the process of reconstruction is mainly the set of camera positions gathered in
this stage, presumably with the intention of using them as a starting point for further
triangulation of points.
4.1.2.2 Dense point cloud
The generation of a dense point cloud is done with the help of the estimated camera
positions and the respective matching photographs. Unfortunately no more information
is available on this stage. It is likely that a more complex triangulation process is used in
this stage to generate a dense point cloud with the help of the initially calculated camera
positions.
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4.1.2.3 Mesh construction The third stage consists of constructing a 3D polygonal mesh representing the object
surface based on the dense point cloud.
In most cases two algorithmic methods are available that the software can apply for 3D
mesh generation:
• Height field – Optimized for modeling of planar surfaces, such as terrains or bas-‐
reliefs. For aerial photography processing, it requires a lower amount of memory
and allows for larger data sets processing.
• Arbitrary -‐ For closed objects, such as statues, buildings, etc. It doesn't make any
assumptions on the type of the object modeled, which comes at a cost of higher
memory consumption.
In this thesis the arbitrary algorithm method was used for all cases of photogrammetric
reconstruction processing.
Once the mesh is constructed, the user has the ability to somewhat edit it. Non-‐complex
corrections such as mesh decimation (simplification), removal of detached components
and the closing of holes can automatically be performed by the software in the mesh
generation process.
Furthermore the final stage of the model construction is the application of textures, as
this is not relevant in this thesis it will be ignored.
4.1.3 Photogrammetric processing: Autodesk’s 123D Catch Unfortunately very little information can be found about 123D Catch’s reconstruction
process. Presumably the process is somewhat similar to that of PhotoScan. It can also be
assumed that again triangulation is a key algorithm and that it is based on characteristic
points identified in the set of pictures.
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The practical process is very similar to PhotoScan’s. The user takes a set of overlapping
photos of the object from different angles. The difference is the processing is done via
the software applications server. The user uploads the photoset to the server and after a
while, depending on image size and quantity the server returns a reconstructed 3D
model.
This is both an advantage and a flaw comparing it to PhotoScan. By automating the
process only the photographs have to be provided, this can be helpful in the aspect of a
mainstream user and it can be assumed that the server has better computational power
than if the reconstruction would be performed locally. However this limits the amount of
influence the user has on the reconstruction such as key parameters for reconstruction
and insight into the process and therefore also the end result.
4.2 Data generation -‐ laser scanning approach 4.2.1 Environment and setup The scans were carried out in Gliwice, Poland at the Institute of Theoretical and Applied
Informatics (part of the Polish Academy of Sciences) in one of their offices. The laser
scanner was a Konica Minolta Vivid 9i and it was connected to a mechanically rotatable
platform, which together with the scanner were operated through the complimentary
native software for the scanner.
For the scanner to encompass the whole object several scans had to be carried out from
different angles, this was controlled with the rotatable platform. The angle of rotation
per scan was 30° up to a complete circle of 360° resulting in 12 overlapping scans.
4.2.2 Konica Minolta Vivid 9i The scanner used for the conducted tests is a Konica Minolta Vivid 9i. The Vivid 9i is a
scanner created for small to medium sized objects which it can capture with high detail.
This is a triangulating laser scanner with a charged coupled device (CCD) receiver like
the ones described in chapter 2.3.1.
4.2.3 Model reconstruction from point cloud
When the laser scanner has completed its registration of points on the object, each of the
point clouds are interpolated individually to create a part of the objects surface. Theses
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pieces have to be aggregated into a single surface to complete the process. This was
done by aligning each individual piece in respect to the others and merging them into a
single shape to construct a 3D model. The software application used for this was
Geomagic Qualify which is used to measure, align and compare fabricated parts to their
digital blueprint in the production industry.
4.3 Data comparison 4.3.1 Alignment
Alignment is a very important part of the measurement process, as the objects need to
be identically aligned for the distance results to be accurate. This is in turn a problem as
the objects vary in shape and form, it becomes a subjective judgement if the objects are
aligned properly. First n-‐point alignment is used for a rough alignment of the objects.
This is based on a physical person identifying common points on both objects and
marking these. The minimum for this alignment is 3 points but can be as many as one
would like, therefore the name n-‐point alignment. When the points are identified the
objects are matched up so the points on both objects align with eachother.
After this rough alignment a global registration algorithm is used. Randomly selected
points on the reference surface are used to reposition the test object to minimize the
overall distance measured from the points. This somewhat eliminates the subjective
factor from the alignment process but the first alignment step is still based on human
perception.
4.3.2 Measurement The laser-‐scanned surface is the reference surface for our comparison. The other
surfaces reconstructed with the help of the photogrammetric applications are the test
surfaces. For each comparison an amount of randomly selected points on the reference
surface, presumably depending on the number of vertices, are used to measure the
distance to the corresponding point on the test surface for each individual point with the
help of Geomagic Qualify. This in turn generates a data set of point pairs with their
measured distance that is used as the data for the quantifiable comparison of the
methods.
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Due to the surfaces being different in form and quality it is not always possible to map a
point on the reference surface to a corresponding location on the test surface. In these
cases the difference can’t be measured between the two points. However it is fair to
assume that the deviation in the current point in this case is larger or similar to the
maximum deviation of the measured points.
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5 Results In this section there are four cases presented with varying objects in complexity and size
to represent a wide spectrum of smaller objects that may be found in a home
environment for amateur 3D reconstruction. For each case a view of the original and all
reconstructions is shown with a description of each surface. Furthermore color maps
are presented from four different perspectives of the deviation between the compared
surfaces. The deviation is always measured in millimeters and the legend of the color
map shows that green areas are the ones closest to zero in deviation. Further towards
red are larger positive deviations and towards blue larger negative deviations.
5.1 Case 1 – Quadric object This first case is a type of calibration case different to the other test cases that follow in
this section. The original object here is a digital 3D model of a simple quadric surface
(A). The digital original was then reproduced into the physical domain (scale 1:1) with
the help of a 3D printer (B) to then again get reconstructed to digital domain (C&D) with
the help of the photogrammetric methods presented earlier. The physical object is small
and measures 100x95x45 mm (height x width x depth).
The interesting thing here is that we have a digital original giving us some way of
comparing the deviation between the original and the reconstructed models. This is not
possible in the other cases, as the original object is physical.
Figure 3 -‐ A) Digital original 3D model. B) Physical 3D-‐print based on A. C) Reconstructed
digital model with PhotoScan based on B. D) Reconstructed digital model with 123D Catch based
on B.
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5.1.1 3D print reconstructed with PhotoScan
What is presented here are four different perspectives (top, bottom, isometric and side
view), of the deviation between the original (A) and the reconstruction with the help of
PhotoScan (C).
Figure 4 – Case 1, 3D scan compared to reconstruction with PhotoScan
The numerical difference between the digital original and the reconstructed model is
very small. As shown in the full table of points in appendix chapter 9.1.1, about 96% of
the points are placed in the interval of -‐0,1386 to 0,304 mm. So the numerical difference
over 96% of the surface is less than half of a millimeter.
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
1.1312 -1.0498 0.1594 / -0.0785 0.1
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5.1.2 3D print reconstructed with 123D Catch
The same perspectives as in the previous comparison are also displayed here. This is the
comparison of the digital original (A) and the reconstructed model with the help of 123D
Catch (D).
Figure 5 – Case 1, 3D scan compared to reconstruction with 123D Catch
The results are similar to the reconstruction with PhotoScan (C) and again the numerical
difference here is small, although a bit larger than the results from PhotoScan. What we
can see here is also that the differences gravitate a bit more towards a negative distance
difference whereas when reconstructing with PhotoScan we see the differences
gravitating towards a positive difference.
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
0.845 -2.395 0.1825 / -0.2531 0.2282
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5.2 Case 2 – Angel figure In this case the original object is a physical figure of an angel, which consists of a mix of
small soft and hard features with an overall complex surface. The object is small and
measures 80x50x40 mm (height x width x depth). The comparison is conducted
between the photogrammetricly reconstructed models (C&D) from the two different
methods and compared to that of a reconstructed digital model from a 3D scanner (B).
Figure 6 -‐ A) Physical original. B) Digital 3D-‐model reconstructed with the Vivid 9i based on A. C) Reconstructed digital model with PhotoScan based on A. D) Reconstructed digital model with 123D Catch based on A
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5.2.1 3D model reconstructed with PhotoScan
Here we see four different perspectives (front, left, back and right), of the deviation
between the 3D-‐scanned reconstruction (B) and the reconstruction with the help of
PhotoScan (C).
The color map does not classify the grey areas in this comparison. This is due to a too
large discrepancy between the two compared surfaces in the specific area and the
deviation measurement algorithm cannot identify the corresponding point on the other
surface. Simply put a point on surface C does not match any point on surface B and
therefore the difference cannot be measured.
Figure 7 -‐ Case 2, 3D scan compared to reconstructed model with PhotoScan
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
8.5393 -9.6250 0.6051 / -0.6442 1.0395
20
The reconstructed model is very noisy and distorted, even so we see that the numerical
difference over most areas of the surface is still small, only 0.5 mm. With that said it is
important to acknowledge the grey areas where points were unable to match we can
assume the difference is greater than +/-‐5 mm.
5.2.2 3D model reconstructed with 123D Catch The same perspectives (front, left, back and right), of the deviation between the 3D-‐
scanned reconstruction (B) and the reconstruction with the help of 123D Catch (D). As
previously the color map does not classify the grey areas due to the same reasons in
5.2.1.
Figure 8 -‐ Case 2, 3D scan compared to reconstructed model with 123D Catch
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
3.7093 -3.6926 0.6848 / -0.4766 0.8400
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Although both photogrammetric reconstructions contain artifacts and some noise the
key thing to notice here is that there are larger areas on the test surface (D) which can’t
be matched to the reference surface (B) than in the previous comparison, C vs B. This
has a significant impact on the key values like average deviation as pieces of the dataset
are missing.
5.3 Case 3 – Monkey figure As in the previous case the original object here is a physical figure of three monkeys,
which consist of mostly small sharp features. The overall surface of the object is very
complex with a high amount of detail. The object measures 70x105x50 mm (height x
width x depth). A comparison is performed between the photogrammetricly
reconstructed models from the two different methods (C&D) and compared to that of a
reconstructed digital model from a 3D scanner (B).
Figure 9 -‐ A) Physical original object. B) Digital 3D-‐model reconstructed with the Vivid 9i based on A, C) Reconstructed digital model with PhotoScan based on A. D) Reconstructed digital model with 123D Catch based on A.
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5.3.1 3D model reconstructed with PhotoScan Four perspectives (front, left, back and right), of the deviation between the 3D-‐scanned
reconstruction (B) and the reconstruction with the help of PhotoScan (C) are shown
here.
Figure 10 – Case 3, 3D scan compared to reconstructed model with PhotoScan
This statue has a similar size comparable to Case 2 yet the discrepancies here are
significantly smaller. The deviation is also very evenly spread over the whole object
suggesting there is little to no noise.
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
3.6367 -3.0904 0.1769 / -0.1312 0.1854
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5.3.2 3D model reconstructed with 123D Catch
The same perspectives (front, left, back and right). The deviation here is larger than in
the comparison of B vs C and the deviation is not as evenly spread out as in 5.3.1.
Figure 11 – Case 3, 3D scan compared to reconstructed model with 123D Catch
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
6.1359 -7.7362 0.8444 / -1.3935 1.4147
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5.4 Case 4 – Wooden cat This case is unique in the sense that the object here is significantly larger than the
previous cases. The object measures 410x150x50 mm (height x width x depth). As
before in the previous cases the original object here is a physical figure of cat carved in
wood that consists mostly of large soft features and is quite simplistic in shape. The
comparison is conducted between the photogrammetricly reconstructed models from
the two different methods (C&D) and compared to that of a reconstructed digital model
from a 3D scanner (B).
Figure 12 -‐ A) Physical original 3D model. B) Digital 3D-‐model reconstructed with the Vivid 9i
based on A. C) Reconstructed digital model with PhotoScan based on A. D) Reconstructed digital
model with 123D Catch based on A.
5.4.1 3D model reconstructed with PhotoScan
We can see from the color map that a lot of noise has distorted the model quite heavily
in some areas yet still most of the surface only has a small deviation of about 1mm which
may seem like a lot compared to the earlier cases but taking into account the size of the
object it is actually a very small deviation.
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Figure 13 – Case 4, 3D scan compared to reconstructed model with PhotoScan
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
20.3581 -20.3864 2.0123 / -1.7814 3.2267
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5.4.2 3D model reconstructed with 123D Catch
This reconstructed model is not as noisy as the previous case and the discrepancies are
much more evenly grouped. Again even though some areas are quite deviant most of the
surface has a very small deviation of 0.5 mm.
Figure 14 – Case 4, 3D scan compared to reconstructed model with 123D Catch
Deviation (mm) Max. Upper Deviation
Max. Lower Deviation
Average Deviation
Standard Deviation
10.6311 -7.8892 0.6068 / -0.9564 1.1290
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6 Analysis and discussion These comparisons are carried out between the 3D laser scan of the object and the
photogrammetric reconstruction so small numerical differences do not always
correspond to an ideal reconstruction as errors might very well exist in the laser scan of
the object (Spytkowska 2008). This is not a point that has or will be explored further in
this thesis, as the goal is only to compare the methods with each other and not with the
physical original on a quantifiable level. As explained earlier this is not possible in most
cases and neither the goal of this thesis. Nevertheless it is important to keep that in mind
when analyzing these results. We cannot assume the digital laser scan of the object is
ideal but it is the reference for comparison of the photogrammetric reconstruction
methods.
6.1 Comparing the results of the photogrammetric
reconstructions and the laser scanned reconstructions Although the numerical differences in all these cases are by comparison relatively small
excluding some minor areas it is important to note the difference in perceived quality of
the objects. Looking at the results, the conclusion that lower average deviation also
correlates to a better-‐perceived quality can be said to be true. Furthermore we can see
that the more evenly the deviation is spread out over the surface the better the
perceived quality of the reconstruction. Even if large areas are quite off, the key factor
affecting the perceived quality is noise, and large areas with similar deviation often
suggest small amounts of noise or at least a uniform distortion. This of course assumes
that the 3D laser scanner has reconstructed the physical original in a satisfying manner,
which is assumed here as discussed earlier, since it is these two surfaces we are
comparing and not the original. These conclusions were expected but cannot be said to
always be true since this is a subjective judgment and the perception of quality might
vary for each individual.
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6.2 Strengths and weaknesses of the photogrammetric
reconstruction software applications It is also interesting to try and analyze the strengths and weaknesses of the algorithms
of the two photogrammetric software applications in their ability to reconstruct the
surface of the object. In general we can see that the algorithms struggle when faced with
areas and whole surfaces that lack large or multiple changes, in other words smooth soft
features and same colored surfaces as seen in case 2 and 4. The type of change whether
it is surface based or color based seems to be of less importance. As we can see in case 1,
the simple and smooth surface of the quadric is compensated by the complex pattern
painted on the surface. Furthermore surface differences like the fur of the monkeys in
case 3 also result in the same type of change such as color changes due to lightning
giving the features a shadow and thus a different intensity. Therefore changes in either
surface or color result in a similar difference on a photograph and it can be assumed that
this is what the algorithms use to identify points of measurement for the triangulation as
mentioned in chapter 2. This property will be named Δ (Delta); objects with a high
amount of Δ produce better results when reconstructed.
The cat, case 4, is a good example where we see quite small Δ, as changes in color and
surface over large areas contribute to confusion for the photogrammetric applications as
they most likely have no way of identifying the points on the surface and therefore
struggle aggregating the orientation of the respective images.
We can also come to the conclusion here that Δ is not the only factor for good
photogrammetric reconstructions but also the density of the change. If we look at case 2,
the angel figure has some areas with soft round features (low Δ) and some areas like the
hair with high Δ. The areas with high Δ have been reconstructed quite well however the
lack of surface nearby with high amounts of Δ create a challenge for the reconstruction
algorithm and the result is quite poor. The conclusion that can be drawn from this is that
even if an object has a high amount of Δ it has to be spread out over the whole surface
for a good reconstruction, concentrated amounts will only give a good result in that
specific area. The density of Δ will be called ρ (rho).
29
It is also clear that there is a difference in the algorithms between the two applications.
PhotoScan’s reconstruction generates a substantial amount of noise but still manages a
quite small overall numerical deviation over the whole surface, suggesting a more
intense frequency of points triangulated or a large difference in the interpolation
between points. In the case of 123D’s reconstruction the points calculated are probably
fewer, this contributes to, in most cases, that the perceived quality is much higher due to
the lower amount of noise and it still captures the overall features. However it’s
important to note that in the best reconstruction, case 3, PhotoScans point frequency
generates a more accurate reconstruction by capturing far more detail than 123D Catch
thanks to the larger amount of points calculated, although as mentioned above, this is
not always an advantage as errors can become plentiful.
Another interesting difference between the algorithms is that PhotoScans deviation is
usually positive, differences between the reconstructed surface point outwards from the
object however 123Ds deviation is often negative, pointing inwards from the surface.
This results in PhotoScans reconstructions generally ending up larger in size than their
equivalent laser scan and 123D reconstruction ending up smaller than their counterpart.
This is presumably a side effect of each applications calculation algorithm and it is hard
to know if it is related in any way to the perceived quality of the object although it can
have some impact when trying to recreate objects made to scale.
6.3 The photogrammetric reconstruction process as a whole Photogrammetry, as can probably derived from just the name, is highly dependent on
the input data, in other words the photographs quality. This is not the same subjective
qualities of the human perception of a “nice” picture, but rather good quality as in
maximum preservation of unaltered visual information as described in chapter 2.4.1 and
4.4.1. This is the main bottleneck for this technology and as photographs with altered or
low amount of visual information also give bad results when processed by the
reconstruction algorithm. Bad input can be somewhat mitigated with the help of high
amounts of Δ and ρ on the object but only to some extent. This thesis focus was more on
the results of photogrammetric reconstruction and comparing those to that of
reconstruction by 3D laser scanning. Even though the results are fairly impressing in
some cases, not only numerically but especially in perceived quality, such as case 3, it is
30
fair to say that more work and time should have been focused on the process of
generating the input data, as this generates the biggest differences in quality in the
reconstruction process. Unfortunately this is very time consuming and somewhat
beyond the goal of this thesis as we focus on the whole process rather than the details of
each part, but nevertheless this is the most important and the most obvious conclusion
that can be drawn from these cases.
The dependency on input data goes so far that even lack of data is better than low
quality data. What this means is that photographs with low quality visual information
such as a blurry or heavily shadowed photographs do more harm than good. This is due
to the need to aggregate information from several photographs during the
reconstruction algorithm. Visual information of poor quality can confuse the algorithm
during the aggregation process. In the attempt to identify expected features or rather
points, based on assumptions from earlier presumably proper visual information, they
cant be recognized in the following picture as they differ in some form, even though it is
to the human eye the same point. This in the worst-‐case scenario leads to earlier proper
visual information getting discarded and also becoming useless.
Photogrammetry is all about good input data, and as mentioned before – if a photograph
was an ideal medium for information and no information was lost in the transformation
from three dimensions into two, the reconstruction process for photogrammetry
wouldn’t need more than a single photograph – unfortunately this is not the case.
6.4 Can photogrammetry yield similar results to 3D laser
scanning when used in an amateur home setting for
smaller objects? The short and simple answer to this question is yes. In case three with the monkey
statues we see a near ideal reconstruction, not only in numerical terms but also in
perceived visual quality of the object. However there are three main factors along the
way that have a significant impact on the results of the reconstruction. The quality of the
visual information preserved in the photograph, focus and lightning are two big
components in maximizing captured visual data. Secondly, the features of the object
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either form, color and intensity – Δ and the density of these changes – ρ, enabling the
reconstruction algorithm to establish a clear and robust set of points to use for
triangulation calculations. The third and final piece is the algorithm used for
reconstruction and calculation of the points, the transformation from two-‐dimensional
coordinates into three-‐dimensional ones. This is also where some questions are still
unanswered as the algorithms of these commercial software applications are not public
domain and have not been investigated in depth in this thesis. These components need
to be, not just average, but of good quality for the reconstruction to yield similar results
to that of a good 3D laser scanner.
7 Conclusion This thesis has described the background and general technology behind both methods,
3D scanning with laser and photogrammetry, as means of reproducing physical objects
in digital three-‐dimensional space. The focus has been to compare if, with the help of
photogrammetry software in a non-‐professional setting, it is possible to get results of
similar quality to that of a laser scanning approach which has been the preferred
method for some time when reconstructing smaller objects. This has been tested with
four cases of objects of varying surface features and size, these four cases are not enough
to draw generalized conclusions but still give a good spread of data as to see where
these methods strengths and weaknesses lie and what parameters affect this type of
reconstruction.
The results from the cases indicate that it is indeed possible with the help of
photogrammetry to reconstruct a model of very similar quality to that of one
reconstructed with a laser scanner. However it is also obvious that these good results
are not consistent and need not only good external conditions but also rely on the shape
and features of the objects surface. Objects with unique non-‐repeatable variations are
the ones that produce the best results when reconstructed. It does not seem to matter
whether these variations are physical, color or intensity based, the key result is
differentiated points or pixels on the resulting photograph that provide the algorithm
with highly varying points used to identify and match these in the following
photographs.
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It is fair to say that amateur photogrammetry can yield very good results when used for
digital reconstruction of 3D objects although as mentioned before these results of good
quality are not very consistent and require a certain amount of variables. If these
prerequisites are not matched the results in most cases are very underwhelming
compared to those of a laser scanner. However strictly numerically the results are not
very far off and it is debatable if the methods produce comparable results but when we
weigh in the perceived quality of the object the results are usually underwhelming in an
amateur setting.
7.1 Future research Three-‐dimensional reconstruction from photogrammetry is still relatively new
compared to other reconstruction techniques and it is interesting to see what can be
done to improve these methods in the future, as there is certainly potential for these
methods. For further research on this topic it would be interesting to look at
experiments with more focus on optimizing the process of photography, as this is the
core of photogrammetric reconstruction – the input data.
Another interesting angle of approach would be to compare amateur laser scanning with
the results of amateur photogrammetry. As in this thesis the focus has been on amateur
photogrammetry results compared to professional laser scanning it might be an unfair
comparison of the two techniques. Furthermore there seems to be a drive to produce
more and more equipment for 3D reconstruction on an amateur level as demand for
both 3D printers and 3D scanners rise.
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9 Appendix 9.1 Case 1 – deviation results 9.1.1 Photoscan model compared to 3D scanned model
9.1.2 123D Catch model compared to 3D scanned model
Deviation interval (mm) # Points % amount -‐0.8782 to -‐0.4990 1616 1.4827 -‐0.4990 to -‐0.1198 66074 60.6217 -‐0.1198 to 0.1198 23582 21.6361 0.1198 to 0.4990 16476 15.1164 0.4990 to 0.8782 1245 1.1423
Deviation interval (mm) # Points % amount -‐1.1312 to -‐0.9658 7 0.0027 -‐0.9658 to -‐0.8003 8 0.0031 -‐0.8003 to -‐0.6349 6 0.0024 -‐0.6349 to -‐0.4695 45 0.0176 -‐0.4695 to -‐0.304 526 0.2061 -‐0.304 to -‐0.1386 3268 1.2804 -‐0.1386 to 0.1386 100728 39.4664 0.1386 to 0.304 144364 56.5634 0.304 to 0.4695 5774 2.2623 0.4695 to 0.6349 448 0.1755 0.6349 to 0.8003 47 0.0184 0.8003 to 0.9658 1 0.0004 0.9658 to 1.1312 2 0.0008
36
9.2 Case 2 – deviation results 9.2.1 Photoscan model compared to 3D scanned model
Deviation interval (mm) # Points % amount -‐5.0000 to -‐4.2469 175 0.2408 -‐4.2469 to -‐3.4937 231 0.3179 -‐3.4937 to -‐2.7406 232 0.3193 -‐2.7406 to -‐1.9875 772 1.0624 -‐1.9875 to -‐1.2344 3677 5.0600 -‐1.2344 to -‐0.4813 7630 10.4998 -‐0.4813 to 0.4813 45926 63.1998 0.4813 to 1.2344 8499 11.6957 1.2344 to 1.9875 2549 3.5077 1.9875 to 2.7406 1352 1.8605 2.7406 to 3.4938 721 0.9922 3.4938 to 4.2469 408 0.5615 4.2469 to 5.0000 156 0.2147
9.2.2 123D Catch model compared to 3D scanned model
Deviation interval (mm) # Points % amount -‐4.2042 to -‐3.4084 25 0.0602 -‐3.4084 to -‐2.6127 41 0.0987 -‐2.6127 to -‐1.8169 89 0.2143 -‐1.8169 to -‐1.0211 1310 3.1544 -‐1.0211 to -‐0.2253 9811 23.6245 -‐0.2253 to 0.2253 11712 28.2020 0.2253 to 1.0211 13912 33.4995 1.0211 to 1.8169 2411 5.8056 1.8169 to 2.6127 1205 2.9016 2.6127 to 3.4084 769 1.8517 3.4084 to 4.2042 244 0.5875
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9.3 Case 3 – deviation results 9.3.1 Photoscan model compared to 3D scanned model Deviation interval (mm) # Points % amount -‐3.6367 to -‐3.0609 1 0.0006 -‐3.0609 to -‐2.4851 2 0.0012 -‐2.4851 to -‐1.9093 0 0.0000 -‐1.9093 to-‐1.3334 1 0.0006 -‐1.3334 to -‐0.7576 3 0.0018 -‐0.7576 to -‐0.1818 16122 9.4107 -‐0.1818 to 0.1818 107687 62.8591 0.1818 to 0.7576 47352 27.6403 0.7576 to 1.3334 143 0.0835 1.3334 to 1.9093 2 0.0012 1.9093 to 3.0609 0 0.0000 3.0609 to 3.6367 2 0.0012
9.3.2 123D Catch model compared to 3D scanned model
Deviation interval (mm) # Points % amount -‐7.7362 to -‐6.5113 2 0.0023 -‐6.5113 to-‐5.2864 0 0.0000 -‐5.2864 to -‐4.0615 2653 2.9958 -‐4.0615 to -‐2.8366 6492 7.3308 -‐2.8366 to -‐1.6117 12105 13.6690 -‐1.6117 to -‐0.3868 37165 41.9668 -‐0.3868 to 0.3868 15998 18.0650 0.3868 to 1.6117 10853 12.2552 1.6117 to 2.8366 2943 3.3232 2.8366 to 4.0615 294 0.3320 4.0615 to 5.2864 51 0.0576 5.2864 to 6.5113 2 0.0023 6.5113 to 7.7362 0 0.0000
38
9.4 Case 4 – deviation results 9.4.1 Photoscan model compared to 3D scanned model
Deviation interval (mm) # Points % amount -‐20.3864 to -‐17.1585 280 0.1521 -‐17.1585 to -‐13.9307 1247 0.6774 -‐13.9307 to -‐10.7028 1687 0.9164 -‐10.7028 to -‐7.4750 2759 1.4988 -‐7.4750 to -‐4.2472 5773 3.1361 -‐4.2472 to -‐1.0193 46904 25.4802 -‐1.0193 to 1.0193 100389 54.5355 1.0193 to 4.2472 16875 9.1672 4.2472 to 7.4750 4868 2.6445 7.4750 to 10.7028 1761 0.9566 10.7028 to 13.9307 826 0.4487 13.9307 to 17.1585 528 0.2868 17.1585 to 20.3864 183 0.0994
9.4.2 123D Catch model compared to 3D scanned model
Deviation interval (mm) # Points % amount -‐8.9478 to -‐7.2646 5 0.0074 -‐7.2646 to -‐5.5813 35 0.0518 -‐5.5813 to -‐3.8981 233 0.3450 -‐3.8981 to -‐2.2148 4101 6.0715 -‐2.2148 to -‐0.5316 19616 29.0414 -‐0.5316 to 0.5316 33827 50.0807 0.5316 to 2.2148 8675 12.8433 2.2148 to 3.8981 872 1.2910 3.8981 to 5.5813 137 0.2028 5.5813 to 7.2646 29 0.0429 7.2646 to 8.9478 10 0.0148 8.9478 to 10.6311 4 0.0059