Reconstruction of Three Dimensional Models of Environments with a Mobile Robot

Reconstruction of Three Dimensional Models of Environments with a Mobile Robot
Janelcy Alferes
International Center for Numerical Methods in Engineering Center Technologic CARTIF Technological Park, , Parcel 205 Boecillo, Spain janalf@cartif.es
Abstract - In this paper we present a method for the threedimensional reconstruction of environments, using a mobile platform that has onboard a scanner laser for the acquisition of the geometry of the scene, and a webcam for the registration of the texture of the environment. The used algorithm is based on the localization of the planes that conform the scene, the assignment of the 3D points to the planes for subsequently to apply a triangulation and this way to obtain the surfaces and contours of the model. In the texture the images are assigned to the plane surfaces and by means of a stitching algorithm (image mosaics), the complete scene is compose, in “Vrml” format. To diminish the loss of information of the free forms, and the distortion generated well because the points are considered discarded or because they can not be approximate to planes, we propose a variant in the treatment of the data that consists on projecting the images of the scene and the 3D points on a sphere After to obtain the corresponding planes in angular coordinates (teta, phi), the assignment of texture would be realized on the triangular surfaces obtained of the model points. Index Terms – Scanner laser, image mosaics, 3D models.
Eduardo Zalama and Jaime Gómez
UVA-ETSII University of Valladolid Valladolid, Spain eduzal@eis.uva.es understanding of the model requires of the previous knowledge of the scene architecture. Recently, a method for the environments modeling has been development on Cartif using a laser scanner and a webcam, technique reflected in [14] and [5]. This method causes loss of information relative to the free forms, so that a method for diminish the objects distortion is introduced in this paper, that is to project the 2D and 3D points on a sphere. The paper is organized as follows. First, the acquisition platform and the modeling method employed will be described. Next, the problem is presented and a method to improve the results. Finally, the method is applied on some images to analyze the obtained results and to establish the respective conclusions. II. DESCRIPTION OF THE MOBILE PLATFORM The wheeled mobile robot platform used is showed on the figure 1. It presents a configuration differential. A laser scanner Sick [10] registers the geometry information. The laser, return the distance “r” of the object on the scene and the correspond angle of sweep “ θ ”. The laser takes up to 180o measures in a same plane. To obtain a complete sweeping of the scene, a stepper motor allows that the Sick pans vertically up to 90o. A webcam register texture information of the scene. A servomotor allows that the mounted camera on the sick rotates up to 180o in a same plane, and this way, take the images corresponding to the 3D points taken by the scanner, obtaining a complete sweeping of the scene.
I. INTRODUCTION For many applications, a environment map in 2D is enough. However, there are applications that require of precise and detailed information of the characteristics of the scene. In these cases, the solution that offers a better result is the use of environment maps in 3D. Some of these applications could be: automatic systems of having guided of robots, control of quality, segmentation of images, biomedical applications, design architectural, reconstruction of constructions and in those situations in those that it is required to model an environment with the smallest possible human intervention. The methods based on artificial vision [7],[4],[3],[20], frequently produce ambiguity and errors on the model, because the scene images are not clear. Others methods on study use wheeled mobile robots that has onboard lasers and camera, for register the information [25],[1],[24],[7]. The principal problem is the techniques used for the data treatment and the obtaining of 3D model.In [9], a mobile platform is used with two scanners laser. The accuracy of the data depends of the robot position. In [23] and [22], a platform is used with a scanner 2D and a servomotor. The algorithm requires the information of the scene and it approaches the objects to boxes. In [15] , the authors use the techniques developed in [23], with an additional algorithm based on ICP (iterative closest points) but it doesn't provide great information on the details of the scene. The method used in [16] uses a mixture among RANSAC (Random Sample Consensus) and ICP. The
Fig. 1 Mobile Platform used for the data acquisition
III. ALGORITHM FOR ENVIRONMENT MODEL The stages in the process of scene modelling are shown in the following figure.
Plane ubication Simplification of the planes Extension limitation Ubication of points on the planes Texture addition
Triangles simplification Obtaining Contours
Points triangulation

Fig. 2 Algorithm for to obtain the environment model
A. Geometry Reconstruction The algorithm is based on the approach from the scene to a group of planes [14]. The main objective is to get the normal of the surface that contains each point, denominated “normal local”. The calculation of the normal is associated to the smallest variance in this sample.
P= (x,y,z)
provides the most probable normal in the plane. The other two main variances are associated to the extension of the plane. The normals are simplified until obtaining a simple plane conjunct checking the vicinity, the point occupation and the discarded points. After, the planes are limited for the application of Delaunay triangulation [6], obtaining the surfaces. Finally, the coordinates VRML are created for to build the virtual world. The contours are obtained for elimination of the internal points, maintaining only the edges. With this procedure we obtain a simple representation of de scene for the texture addition. B. Texture Addition 1) Camera Calibration The camera calibration [5] [9] contemplates intrinsic parameters to the system of acquisition system: focal distance (f), displacement of the center of the image (cx, cy), distortion coefficients (k1,k2). And extrinsic parameters, related to the position and orientation of the camera on the world coordinated system: translation Tx , Ty , Tz and rotation angles α, β, γ. The procedure is the following: to determine the positions of 3D reference points); to determine the value of their projections on the image and to look for the correspondence between the 3D points and their projections in the image. We have used the Tsai method [5] because it is precise on the parameter estimation.
n
c = (xc , yc , zc
Distancia entre el pto y el centroide
Banda de Tolerancia
Fig. 3. Plane on the space.
The Hessiana Normal form of the plane shown in the figure, defining the components of the normal vector as n = ( n x , n y , n z ) is:
nx x + ny y + nz z ? dn = 0
(1)
Where d n represents the distance from the plane to the origin of the coordinated system. An adjustment of square minima is used to estimate the position in the space of this plane.
?N ? min ?∑ n x x + n y y + n z z ? d n 2 ? ? i =1 ?
(Xw,Yw,Zw) world coordinantes
Step 1 Transformation of (Xw,Yw,Zw) to ( X ,Y , Z ) c c c
(
)
(2)
( X c ,Yc , Z c ) camera coordinates
Step 2 Pin-Hole Projection perspective
The solution this equation provides a local minimum. It can be written as:
? ?? ??Sxx Sxy Sxz ? ?Sx ? ?nx ? ? ?? ? ? ?? ? 1 ??Sxy S yy S yz ? ? ? (Sx S y Sz )? ?S y ?? ? ?ny ? = 0 ? ? ? ? ?? ? N ? ?? ? ? ?Sz ?? ?nz ? ? ?? ??Sxz S yz Szz ? ?
(xu,yu) image coordinates without distortion
(3)
Step 3 Radial distortion
S x = ∑ xi S xx =
i =1
N
∑ xi
i =1
N
2
S xy =
∑ xi . yi
i =1
N
(xd,yd) image coordinates with distortion
Step 4 Image Acquisition
etc.
This equation can be written in terms of the media and the variance of, dividing (3) among the number of points N:
E
(r,,c) coordenadas en la imagen almacenada en el computador
Fig 4. Camera Calibration
(X
2
.X
)?
E ( X ). n
(4)
σ
(X )
Transformation of ( X W ,YW , Z W ) to ( X ,Y , Z ) :
? XW ? ?X ? ? r1 ?Y ? = R.?Y ? + T ? R = ? r4 W ? ? ? ? ?, ? ZW ? ?Z ? Transformation of X ,Y r7Z ? ? ? ? ?
?T1 ? r2 r3 ? ? ? ? r5 r 6 ? T = ?T2 ? r8 r9 ? ,Y , Z ?https://www.360docs.net/doc/7116771485.html,ing to X? u u u? 3 ? ?
The obtained matrix has been called “Matrix of Space Variance” [14]. It allows to model as white noise the implicit errors in the measurements. The three eigvalues of the matrix will be the main variances. If the cloud of points comes or it belongs to a plane, and the points are distributed homogeneously, the ellipsoid generated by the matrix σ 2 ( X ) is practically a disk, because σ 3 << σ 2 ≤ σ 1 , where
(
)
(
)
(5) the
Pinhole model [14],[9]:
Xu = f Xc Zc , Yu = f Yc Zc
(6)
σ 3 is the smaller eigvalue and correspond to the smaller variance. The eigvector asociated is the normal more
.
Transformation of X u ,Yu , Z u to ( X d ,Yd , Z d ) :
(
)

Xd + dx = Xu
;
Yd + d y = Yu
d y = Yd (k1r 2 + k2r 4 + .....)
2 2
dx =
X d ( k1r 2
+ k2
r = X d 2 + Yd
(
r4
+ .....)
(7)
Imagen A
)
M
Imagen B
Transformation of ( X d ,Yd , Z d ) to (r , c ) .The superior terms in (7) are rejected:
? ? X ? ?X r = r0 + N x ? f c ? = r0 + K1? c ? Z ? ? Zc c? ? ? ? ? ? ; ? ?
M
-1
? m0 M = ? m3 ? ? m6 ?
m1 m4 m7
m2 ? m5 ? ? 1 ? ?
Fig 5. Transformation Image
?Y ? ? Y ? r = c0 + Ny ? f c ? = c0 + K2? c ? ? Z ? ? Zc ? ? c? ? ?
The transformation of a point of coordinated (x,y,w) of the image “A” to a point of coordinated (x',y'w ') is obtained with the following system of 8 equations:
?x ?0 ? ?x ? ?0 ?x ? ?0 ?x ? ?0 ? y 1 0 0 0 ?x* x'' ? y* x'' ? 0 0 x y 1 ?x* y'' ? y* y''? ? y 1 0 0 0 ?x* x'' ? y* x'' ? ? 0 0 x y 1 ?x* y'' ? y* y''? * y 1 0 0 0 ?x* x'' ? y* x'' ? ? 0 0 x y 1 ?x* y'' ? y* y''? y 1 0 0 0 ?x* x'' ? y* x'' ? ? 0 0 x y 1 ?x* y'' ? y* y''? ?
→
(8)
Where N x , N y are the scaling factors. Substituting (5) in (8):
? ? ?n.r ? ?(r 0 r7 + k1r1 ) (r0 r8 + K1r2 ) (r0 r9 + K1r3 ) (r0T3 + K1T1 ) ? ? X w ? ?n.c? = ?(c r + K r ) (c r + K r ) (c r + K r ) (c T + K T )?.?Y ? 2 4 0 8 2 5 0 9 2 6 0 3 2 2 ?? w ? ? ? ? 07 ? ?Z w ? ? ? ? r7 r8 r9 T3 ?n ? ? ? ? 1 ? ? ?
0 ' ?x1' ? ?m ? ? ? ? 1 m? y1' ? ' ? ?x2 ' ?m ? 2 ' ? ? ? 3 ' ?y2 ' ?m ? ?m ? = ?x3' 4? ' ? ? ?y3' ?m ? 5 ' ? ? ? 6 m? ' ?x4 ' ? ?y4 ' ?m ? 7? ? '? ?
→
(11)
Calibration Matrix M
(9)
→ →
b
A
p
Applying minimization, we obtain the calibration matrix M for the N control points. To obtain the control points a grill of squares it has been used. 2) To Eliminate the Perspective It is necessary to apply certain prosecution to the images to carry out the adjustment the images to the model.[21],[13]. By means of the general equation of the plane and their normal, we obtain the reference system fix to the plane in study, that allows to locate the camera in a virtual position front the plane, with the purpose to framing the plane in a perpendicular position and undoing the perspective generated when taking the images. The transformation matrix associated to the camera view for these axes this given for:
?1 ? 0 M pv = R pv .? ?0 ? ?0 ? 0 0 ? x pv ? ? ? ? 1 0 ? y pv ? ? R pv = 0 1 ? z pv ? ? ? ? 0 0 1 ? ?0 0 0 ? ? ? x pv' ? ? ? y pv' ? ? z pv' ? ? ? 1 ?
The search of the 8 parameters, implies to solve the expression p = [ A ]?1 . b using the technical SVD [14]. To implement the local registration of the image, the error (e) to be minimized it is characterized as the sum of the square of the differences of intensity among the overlapped pixels in the images A and B:
∑e
i
2 i
= ∑ [ A' ( x, y ) ? B ( x , y )]
i
2
(12)
Where A’ is the transformed image and B the reference image.To obtain the intensity value of the transformed image A'(x,y), the inverse of M is used, next to a bilineal interpolation. The Hessiana matrix “A” and a gradient vector “b” are calculated to solve the following lineal system, for the m parameters.
( A + λ I )* ?m = b
(13)
[ ]
[ ]
The coefficients akl and bk are normalized by the pixels number (n) of the overlapped region: (10)
akl =
∑ ?mi
i
?e ?ei k ?ml n
?
Whre (x pv , y pv , z pv ) are the coordinates of de camera view point; (x pv ' , y pv ' , z pv ' ) are the coordinates of the virtual camera and R pv the rotational matrix of the platform elements. 3) Image Mosaics Exist alignment data that allow assembling an image with other, when the images has some common region. In the process of determining the appropriate alignment parameters, each image is deformed using a transformation matrix of 8 parameters, applying the optimization algorithm “Levenberg-Marquardt” [21],[13]. For two images A and B, 4 corresponding points are determined, just as it is shown in the figure 5.
,
bk =
∑
i
?ei ?m k n
(14)
The final step to produce a image mosaic is to mix the images using a pondered pyramidal function, according to [1] ? 2 * invX ? ? 2 * invY ? ?.?1 ? 1 ? ? w( x , y ) = ?1 ? 1 ? (15) ? ?? ?
? ancho ? ? alto ?
Where (invX ,invY) are the inverse coordinates of image. that determine the true localizations of each pixel in the original image, without deforming. 3) Detail of Free Forms of the Scene For applications where general information of the scene is required, the developed method is a good solution to obtain the 3D map of the scene. However, if the application requires more detail in the scene, it is necessary to offer

more realism to the scene. Since the used algorithm treats the points assigning them to planes, if exists a sample of points that cannot be assigned to any plane, the algorithm considers them discarded points. In the best case, these points could be assigned some plane but, in any option there is lost information and distortion perspective. Since the camera and the SICK describe spherical movements around a central point, the projection surface that better is adjusted to the system it is a sphere. With the purpose of improving this situation, a variant on the data treatment is proposal that consist on projecting the images and the 3D points, just as it is shown in the figure 13. The optical center is define for the coordinates ( V x ,V y ,V z ) and the horizontal and vertical vision angles ( θ ,φ ) . .
Object surface triangulate Image planes projected on the sphere 3D points projected on the sphere
Fig 9. Projection of the 3D and 2D extinguisher points on the sphere.
4) Mapped Fuction on the sphere. One point on the sphere is defined for the pair (x,y,z) it is showed on figure 10:
Fig 6. View point on a sphere
R
The 3D points of the scene are projected by the camera on a plane image located to “f” distance of the focus. Then, these planes are projected on a sphere [11][26][18] for to obtain the texture of the scene. The figure 4 shows the scheme of projection proposed.
φ
Fig 10. Coordinates of one point on the sphere
θ
Any point on the sphere, is defined for three parameters: the radius (R), the horizontal sweep angle θ and the vertical sweep angle φ . Now, we have must find the relations between these parameters and the image chromatic points 2D. First, we suppose that the focal axe crosses the imagen on θ = 0; φ = 0 . We obtain the planes xy and yz.
Objetc
(x,y,z)
Imge plame (u,v) Focal distance
“f”
Sphere
Fig 7. Projection of the 3D and 2D points on the sphere
Each image will correspond to a certain angle of horizontal and vertical vision, just as it is shown in the figure. Subsequently the 3D points are triangulate [6] for to parameterize the surface on triangles that can be projected on the sphere. This allows to carry out a direct assignment of 2D texture coordinates to the 3D coordinates of the object, minimizing the distortion. The geometric form of the sphere allows to locate the 3D points such that we will have more resolution in the near areas to the center of the sphere, and smaller resolution in the near areas to the ends or “poles” of the sphere. The complete model is obtained when the stitching is carry out on the sphere. As result, we will have a notable improvement on the object perspective. The figure 5 shows the complete scheme of the proposed method for the extinguisher example. We have used the gnomonic projection [8][10], because it has the propriety that the grid straight lines, correspond to the circles on the grid sphere, so there is a direct correspondence between the points.
Fig.11 Plane xy and plane yz.
Using the information of both planes, we obtain a functions conjunct of “mapped spatial” to obtain the coordinates of the resultant image. Since the optical axe can move by means of camera servomotor and the sick motor, we must to add the horizontal ( αh ) and vertical ( αv ), vision angles ( αv ), so, we obtain the following general equations:
? x.sen( αh ) + y .cos( αh ) ? ? R ? ? ? [R ..sen( α v ) + z . cos( α v )]* cos( θ ) ? φ = a tan ? ? R ? ?
θ = a tan?
(16)

It is necessary to apply an interpolation algorithm for to determinate the intensity level on the final image, using one or several pixels from the original image. We have used the bilineal interpolation [27] because it provides good results. The registration of the images corresponds to find the“8” parameters of the transformation, for application of the technical of “Levenberg-Marquardt” previously detailed. For the realization of the stitching, we used the planes θ ? φ obtained of each projection on the sphere.
Vicinity Occupation Plane Simplification Final Result
planes planes Localised planes Discarded points
45 20 14 8556 (22,85%)
11:13min
Fig. 14 Model 3D with texture. Fig 12. Concetric Images. Spherical Mosaic
V.
RESULTADOS EXPERIMENTALES
We have tried the proposed method on real data.. In the following experience we proceed to model a Cartif corridor. For the data acquisition, the robot has been located in two different positions, one to the beginning of the corridor and the other to 4 meters on the axis of the robot's movement. The sweeping is from 0 to 1000 sick steps.. The resolution of stepper motor is fixed in 0,09, so, we have 000*0,9 = 90 sick positions.The obtained points cloud is shown in the figure 13
When applying the described procedure, efficient results are showed for environments formed by plane surfaces. However, when we navigate for the created virtual scene, we observe lost information relative to the free forms on the scene, for example: chairs, doors,, extinguishes, etc. These objects appear embedded in the planes, and there is perspective distortion for different view points. We observe that the view point showed in the figure 15a (view point where the texture was registered), the extinguisher presents few perspective distortion. However, when we watching the object from different view points, we observe that the perspective is distorted, as appears in the figures 15b, 15c.
4
(a) Frontal view point. (b) Left Lateral (c) Right Lateral Fig.15. View points of the extinguisher
Fig. 13. Point Cloud of Cartif Corridor
The following figure shows the initial model obtained after locating the planes of the scene and the surfaces contours. The longitude of the straight line that defines the normal specifies the quantity of points that contains the plane. The table I shows the important information. The figure 14 shows the final model with added texture.
We have tried the proposed method on different images, in order to study the effect of the generated distortion by the geometrics transformations on the sphere. Each point on the image plane is considered a pixel with a position inside the image ((coordinates (y,z)) and a RGB intensity value. We cross the grid discrete ( θ ,φ ) of dimensions associate to the original image size and the function of inverse mapping is applied to obtain the RGB value of the final image, applying bilineal interpolation. The following figure shows the resulting points of an image in coordinates ( θ ,φ ), for different relationships among the radius of the sphere (R) and the size of the plane image (A).
Fig. 13 Surfaces and Contours of the scene. TABLE I PROCESS RESULT
Step Parameter Time
Load file and parameters Initial
Scene points Initial planes Discarded points
37440 239 17805 (47,5%)
1seg 517 seg
a) (R/A=0.5) Fig 16. Plane
θ ?φ .
b) (R/A=1)
There is major distortion on the image extremes because de type of the used transformation. Too, the distortion

increases when the relation (R/A) diminished. We observed better results for sphere radius greater of equal that image width. The following figures show the results of applying the transformations to a group of images of a scene, taken from a same optical center (concentric images).
a) θ = 0o; φ = 0o
c) θ = 30o; φ = 0o
d) θ =45o; φ = 0o
We have present a method for the Environments 3D Modeling, using the data registered by a laser and a camera, both onboard on a wheeled mobile robot. When we realize the described procedure, we observe efficient results for plane surfaces. However, for the free form object, the results show perspective distortion on the object. The reason of this problem is that the used algorithm for the scene construction treats the points assigning them to planes. We have proposed a method for to diminish the perspective distortion relative to the free form object. This one consists on to project the 3D points (these points have the scene geometry information) and the 2D points (these points have the texture information) on a spherical surface. The obtained results allow to study the characteristics of the distortion generated by the used projection, finding distortion in the ends, fact that makes more direct the correspondence with the points 3D of the space. In this respect, there are parameters to consider such as the radius of the sphere and the discretization of the image. As immediate future lines of this work we have the parameterization in triangles of the scene surface, and to obtain, the planes θ ? φ of the 3D points, with the objective of to carry out the correspondence 2D-3D and to assign the texture to the scene. On the other hand, is the study and application of techniques to eliminate those “seams” resultants of the stitching process and this way to obtain realistic result. ACKNOWLEDGMENT Falta REFERENCES Andreozzi, L; Barnobi, L; Giuffrida, A; Santagati, C. (2002). “New Frontiers to Architectural Survey: Laser Scanner 3D”.University of Catania. Axel, W; Amir, M. (2000). Monlouk, “Multisensoric anctive spatial exploration and modeling” in Dynamishe Perzeption: Workshop der GI-Fanchgrupe 1.0.4 Bildverstehen, Ulm. Berlín. Bajcsy, R; Kamberova, G; Nocera, L.(2000) “3D Reconstruction of Enviroments for Virtual Reconstruction”. 4th IEEE Workshop on Applications of Computer Vision. Debevec, P; Taylor, C. (2002) “Modeling and Rendering Architecture from Photographs: A Hibrid Geometry-and Image-based approach”. University of California. Diez, J. (2003). Proyecto Fin de Carrera “Incorporación de Texturas a Modelos 3D navegables del Interior de Edificaciones mediante Robots Móviles dotados de Sistemas de Visión” Universidad de Valladolid, Espa?a. Francés, A; Lambán, L; Rubio, A. (1998). “Diagramas de Voronoy y Topología Digital”. Universidad de Zaragoza. Espa?a. Frintop, S; Rome, E. (2003) “An Attentive Multi-modal Laser Eye”. Fraunhofer Institut fur Autonomr Intelligente Systeme, Germany. GFD Dennou Library. “Map Projections”. González, J. (1999). “Visión por Computador”. Editorial Paraninfo. Espa?a. Howells, L. (2002). “The Design of an Automated Data Analysis System for Robotic Telescopes and its Application to be Stars”. “Introduction to Computer Vision”. CS /ECE 181B. 2003. Itkowitz, B. (1998). “Image Mosaics”. CAS CS585. Lensch, He, Heidrich, Wo. (2001). “A Silhoutte-Based Algorithm for Texture Registration and Stitching”. Graphics Models.
f) θ = -30o; φ = 0o
g) θ = -45o; φ = 0o
Fig. 17 Image Planes projected on the sphere.
The following figure shows the result of applying the Stitching of the obtained planes.
Fig. 18 Stitching of the image planes on the sphere.
a) θ = 0o; φ = 20o
c) θ = 30o; φ = 20o
d) θ = 45o; φ = 20o
[1] [2] [3]
f) θ = -30o; φ = 20o
g) θ = -45o; φ = 20o [4] [5]
Fig. 17 More Image Planes projected on the sphere.
The following figure shows the result of applying the Stitching of the obtained planes
[6] [7] [8] [9] Fig. 18 Stitching of the image planes on the sphere [10] [11] [12] [13]
The following step consists on the projection in the sphere of the scene 3D, for subsequently to carry out the correspondence 2D-3D, step that is in development. CONCLUSIONES

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