Recognizing three-dimensional objects on the basis ...

Recognizing three-dimensional objects on the basis of range-images Daniel Kocielinski∗

Stanislaw Ambroszkiewicz∗

∗ Institute †

of Computer Science, Polish Academy of Sciences, Warsaw, Poland Institute of Computer Science, University of Podlasie, Siedlce, Poland

Abstract— The motivation of presented work comes from our considerations towards recognizing objects by a mobile robot which is equipped with a three-dimensional laser scanner. Our research has been aimed in the direction of three-dimensional modeling of environment for application in robotics. The method to accomplish this objective is to use the shape of objects perceived as their particular property (perhaps the most important in the recognizing process). We have developed an original approach to identifying shapes of objects on the basis of rangeimages. We adopted the assumption that the shape of an object is characterized by two graphs. The first one (region adjacency graph) represents the relationships of regions contiguity; it may be perceived as ”a higher level of description of the shape of the object surface”. The second graph (boundary segment adjacency graph) represents the boundaries of regions, their segments and contiguities; this may be seen as ”low level of the shape description”. The table M (a set of points in e3 space, which is specified on the basis of range-image) is created by scanning fragments of surfaces of some objects within the surroundings of mobile robot. In our research, we assume that the goal of computing table M is to recognize the shape of the surfaces which are approximated by M . Any triangulation is possible by connecting each three adjacent elements of M , and thus a surface consisting of all the triangular elements (triangles) can be created. A method of triangulation is needed which takes into consideration local characteristics of a surface. An appropriate triangulation is particularly important at the stage of searching for homogeneous regions. Our approach to the objects shape recognition (using rangeimages) is based on segmentation of uniform regions with the same characteristics which are discrete approximations of the cohesive fragments of the object surface. At the same time, this approach is a practical realization of the proposed definition of shape. According to this, the shape of an object consists of a set of homogeneous regions, each having certain geometrical properties which result from assumed homogeneity criteria. They are in particular: flatness, convexity, concavity. We have realized that thus defined homogeneous regions are elementary properties of object shapes existing in a real environment.

I. T HE OBJECTIVE AND SCOPE OF THE RESEARCH Our research has been aimed towards three-dimensional modeling of environment for application in robotics. We have reduced this problem to the problem of recognizing objects by a mobile robot which is equipped with a three-dimensional laser scanner. As we introduce further, the method to accomplish this objective is to use the shape of objects perceived as their particular property (perhaps the most important in the recognizing process). The results of our research clearly indicate that recognizing objects of any shapes is possible by

use of a three-dimensional rangefinder. We have developed an original approach to identifying shapes of objects on the basis of range-images. The guiding idea for our research is constructing a semantic map of an environment on the basis of a sequence of measurements (given in the form of range-images) which are effected by a mobile robot, equipped with a laser rangefinder. We have developed our own method for segmentation of the rangeimages on the basis of a definition of shape of an object. Also, we have developed the concept of data structures, which can both allow for the modeling of any shapes of complete objects, and are the result of a segmentation process of range-images (representing the views of objects for points at which the measurements are made). We adopted the assumption that the shape of an object is characterized by two graphs. The first one represents the relationships of regions contiguity; it may be perceived as ”a higher level of description of the shape of the object surface”. The second graph represents the boundaries of regions, their segments and contiguities; this may be seen as ”low level of the shape description”. By means of these graphs, it is possible to construct topological-metric models of object shapes. II. T HE CONCEPT OF SHAPE The shape of a physical object seems to be its most important property on the basis of which it can be identified. The concept of shape remains a challenge for researchers (see review by Zhang & Lu [1]). The classic definitions of shape rely largely on visual data, whereas our proposed approach is based on the idea that the shape is an abstraction and conceptualization of data which are analogous to the tactile stimuli as they are perceived by humans. The idea is based on the following intuition: ”Close your eyes and touch the object with your fingers. Assuming that the surface of the object is smooth, feel and identify convex, concave and flat regions on it as well as their variations (for example concave-convex ones etc.). Notice places where there are the corners, saddle points, edges, and recognize the two-dimensional shape of these edges (i.e. whether they are straight lines, convex or concave curves) and their relation to identified regions of the surface. Next, determine edges of these regions and divide them into parts (segments) corresponding to previously identified edges.” Elements of shape of the objects, intuitively identified in this way, can be represented in the form of two graphs.

The first of these, well-known from literature, is a region adjacency graph where the graph nodes correspond to the recognized regions, while each of its edges represents a shared segment of the edge of two adjacent regions which are adjacent to each other in this segment. In the second graph, which is called a boundary segment adjacency graph, the nodes correspond to the boundary segments, whereas each edge of the graph (between two segments) represents their contiguity (i.e. indicates whether the edges of the particular region have a shared point). The structure of these graphs, enriched with information on the type of identified regions and the types of boundary segments, creates the proposed formalization of the concept of ”shape”. We want to draw attention to the fact that the thus defined concept is invariant with regard to any object transformation which retains this structure. Rotation, scaling, and general affinitive transformations all belong to this type of transformations. In search of a good definition of the concept of shape we have directed our attention towards mathematics, and in particular: topology, differential geometry and the theory of manifolds. The explicit goal of our research was to develop efficient computational methods that could be used for modeling and recognition of three-dimensional shapes of objects, scanned by 3D laser rangefinders and represented as rangeimages. III. G ENERAL APPROACH TO SEGMENTATION OF THE RANGE - IMAGES A range-image is obtained by use of a three-dimensional laser scanner. It is a table of points in 3D space which are given as triplets of numbers (x, y, z) which are corresponding to the coordinates of the points on the scanned surface. Points neighbouring in the table are also neighbouring on the surface of the scanned object. Such a range-image usually corresponds to a view of an object ”as perceived by an observer staying in a certain definite point”. The approach to the problem of extraction of the objects properties by use of range-images is similar to the approaches used in processing visual images. However, there are some essential differences resulting from the nature of the data. The process of segmentation of an object is classically defined in a following way: ”Let R mean all the area of a range-image. The segmentation may be treated as a process of a dividing the area R into n smaller regions R1 , R2 , · · · , Rn in such a way that: Pn 1) i=1 Ri = R 2) Ri is a cohesive area, i = 1, 2, · · · , n 3) Ri ∪ Rj = ∅ for all such i, j that i 6= j 4) P (Ri ) = TRUE, for i = 1, 2, · · · , n and 5) P (Ri ∩ Rj ) = FALSE, for i 6= j where P (Ri ) is a predicate.” The segmentation of the range-images is very often done by means of region expansion and edge detection. The technique of region expansion was used in our own (briefly described later) approach to the problem of surface segmentation.

IV. P ROPOSED APPROACH TO SEGMENTATION The approach to the objects shape recognition (using rangeimages) which we propose is based on segmentation of uniform regions with the same characteristics which are discrete approximations of the cohesive fragments of the object surface. At the same time, this approach is a practical realization of the proposed definition of shape. According to this, the shape of an object consists of a set of homogeneous regions, each having certain geometrical properties which result from assumed homogeneity criteria. They are in particular: flatness, convexity, concavity. But there are no major limitations for the creation of defining homogeneity criteria. We have realized that thus defined homogeneous regions are elementary properties of object shapes existing in a real environment. We have noticed that the surfaces of complex irregular shapes are usually various combinations and intersections of regions of elementary characteristics. For example, in this approach, a cylinder surface is an intersection of flat and convex regions. Thus, a complex region (of irregular shape) is composed of a set of homogeneous regions which are intersecting and adjoining according to some fixed rules. In our research we have focused on defining the homogeneity criteria and on the rules of constructing complex regions. This approach enables us to characterize shapes of real objects and is a foundation for their recognition based on range-images. V. S EGMENTATION OF REGIONS IN RANGE - IMAGE Segmentation of flat regions alone is often used in robotics, e.g. in the subjects related to simultaneous creation of a map and of a localization (so-called SLAM 3D) [2, 3, 4, 5]. On the other hand, simultaneous segmentation of flat, convex and concave (and even convex-concave) regions is more complicated but also often applied in computer processing of range-images. More such techniques are described in literature. These kinds of segmentation are usually based on the concept of a local curvature of surface at point P . Two main curvatures (the greatest and the smallest) are the basis for defining the average and Gauss curvatures. Depending on the signs of theirs values, P. Besl [6] defined eight fundamental types of surface. The method of computing these curvatures for a discrete rangeimage (see [7]) consists in computing the local coordinate system at first, where one axis coincides with the normal vector for the (approximating) surface in this point P . Bicubic surface, approximating some neighborhood of this point, is usually defined as a second- or third-degree multi-dimensional function (in the form z = f (x, y)) which comes as near as possible to the points of the neighborhood under consideration. The definition of shape we propose involves in fact many more (in theory, indeed, a limitless number of) types of surfaces. For this reason, the concepts of main curvatures, average and Gauss are insufficient. Instead, we propose a somewhat different definition of local curvatures which allow us to describe surface in more general ways. We use the ”slope” concept in our work because of some differences in a classic approach to curvature.

Formally, a range-image (as a result of scanning the surface of objects) can be represented by a table M of mM × nM dimensions. The elements of this table (points in 3D space) are marked as M (i, j); where i = 1, 2, · · · , mM and j = 1, 2, · · · , nM . The direct table neighbours of any element (point) M (i, j) (where 1 < i < mM and 1 < j < nM ), i.e. elements: M (i−1, j−1), M (i−1, j), M (i−1, j+1), M (i, j+ 1), M (i + 1, j + 1), M (i + 1, j), M (i + 1, j − 1), M (i, j − 1), correspond to adjacent (in relation to M (i, j)) points in a Euclid E 3 space. This property is used, in particular, in our definition of slope. A. The directional slopes at a point For a given point M (i, j) = P (where l < i < mM − l + 1 and l < j < nM − l + 1) is defined some neighborhood of this point having a radius l as a table (window) Ol (P ), which dimensions are l × l. This table is a sub-table of M containing (in addition to M (i, j)) only those elements of M (i0 , j 0 ), for which |i − i0 | < l + 1 and |j − j 0 | < l + 1. Accordingly, the neighborhood O1 (P ) contains — in addition to element P — also the direct neighbors of point P . Then, in particular, the point P corresponds to the element O1 (P )(2, 2). For a point P (not lying at the edge of table M ) the points belonging to its direct neighborhood O1 (P ) are enumerated as follows: Pi,j = O1 (P )(i, j), where 1 ≤ i ≤ 3 and 1 ≤ j ≤ 3). According to this numbering the point under consideration P = P2,2 . For each point Pi,j 6= P2,2 (adjacent in relation to P ) we define the slope at point P , where the direction is determined by point Pi,j under consideration. In the first step, we calculate an additional point P 0 . The manner of its calculating is specified below. In the second step, we calculate a radius R of the circle which is tangential to the base Pi,j P 0 of triangle 4(P Pi,j P 0 ) and also to extension of its other two edges (i.e. P Pi,j and P P 0 ) (see figure 1). It is important to emphasize that the coordinates of points in the table M are calculated according to the fixed system of coordinates for which the location of the observer O = (xO , yO , zO ) is determined — where in relation to this observer the measurements are performed for table M . Let the distance between points O and P be signified by |OP |, with that between points O and Ph being |OPh |; moreover point Ph arises as a result of the intersection of the height of triangle 4(P Pi,j P 0 )) (this height being projected from the point P ) with segment Pi,j P 0 . Then the slope is defined as follows: 0

po(P, 4(P Pi,j P )) =



1 R

− R1

if |OP| ≤ |OPh | otherwise

(1)

Therefore, the absolute value of slope equals the inverse of the radius R (according to the figure 1). For a point P = P2,2 , each of its eight neighboring points Pi,j (where Pi,j 6= P2,2 ) is taken as a directional point to calculate a slope. Selection of a point P 0 (in relation to a directional Pi,j ) are done in the following steps: 1) Definition of points Pa and Pb — directly adjacent to the point Pi,j under consideration, different from P2,2

Fig. 1. A slope as a reversing of a radius R of a circle which is tangential to the base Pi,j P 0 .

and, at the same time, belonging to the neighborhood O1 (P ). For example, for P1,1 such points are P2,1 = Pa and P1,2 = Pb . 2) Determination of a point Pp , which is the point of intersection of a straight line t (passing through the points Pa and Pb ) with a line l (passing through the directional point Pi,j under consideration and which is the two-secant angle 6 (Pa Pi,j Pb )). Referring to the example, the point Pp lies on a straight line passing through the points Pa = P2,1 and Pb = P1,2 ; also this point lies on two-secant angle 6 (P2,1 P1,1 P1,2 ). 3) Determination of a plane s which passes through the points P2,2 , Pi,j and Pp . The plane s should be perpendicular, to the extent possible, to the surface of the scanned object at point P = P 2, 2. 4) Determination of a point Pl as the intersection of the plane s with one of the edges Pc Pd of the neighborhood O1 (P ) under consideration; while Pc and Pd are adjacent to each other and are directional in relation to P2,2 — different from Pi,j under consideration. The appropriate point Pl is the sought point P 0 . In the example presented, sought point P 0 is determined by the intersection of the plane P2,2 P1,1 Pp with one of the segments: P1,2 P1,3 , P1,3 P2,3 , P2,3 P3,3 , P3,3 P3,2 , P3,2 P3,1 , P3,1 P2,1 . Definition 5.1: Directional Characteristics of the slope of a point P is defined as a series of point(P ) = (p1 , p2 , · · · , p8 )

(2)

of its eight directional slopes (p1 is the slope towards a neighbor P1,1 , then p2 is the slope towards P1,2 , next p3 is the slope in direction of P1,3 and so on, clockwise). It should be noted that this description takes into account

only the direct neighbors of a point P , so the points in the neighborhood O1 (P ). If the neighborhood of point P were to be larger, it could increase the number of directional slopes. VI. T RIANGULAR SLOPES AND TRIANGULATION OF A SURFACE

The table M is created by scanning fragments of surfaces of some objects within the surroundings. In our research, we assume that the goal of computing table M is to recognize the shape of the surfaces which are approximated by M . In computer graphics, a triangulation of surfaces is often used, based on discreet points. When the table of points M is given, any triangulation is possible by connecting each three adjacent elements of M , and thus a surface consisting of all the triangular elements (triangles) can be created. However, such triangulation is not unambiguous and may not always give the correct result due to accidental choice of the triangles. For this reason, a method of triangulation is needed which takes into consideration local characteristics of a surface. An appropriate triangulation is particularly important at the stage of searching for homogeneous regions. In our work, we have proposed a method of triangularition of a set of points (describing surfaces) which rests on the homogeneity criterium. To define an appropriate method of triangulation, we have adopted the following assumptions. For each element M (i, j) such that i < mM and j < nM , a group of four points is defined in a table M marked by Q(M (i, j)) = (U1 , U2 , U3 , U4 ), where U1 = M (i, j), U2 = M (i, j + 1), U3 = M (i + 1, j), while U4 = M (i + 1, j + 1). Each group of four points can be divided into two triangles with a shared edge (see image 2); there are two methods to do this: • left-slopingly, then a group of four points Q(P1 ) is divided by a diagonal from the top-left corner to bottomright one, that is by a segment U1 U4 ; now then, two triangles are created: 4(U1 U4 U2 ) and 4(U1 U4 U3 ); • right-slopingly, then the group of four points is divided by a diagonal from the top-right corner to the bottom-left one by segment U2 U3 ; now then, the triangles 4(U2 U3 U1 ) and 4(U2 U3 U4 ) are created. To triangulate the table M it is necessary to divide each group of four points Q(P ) where P is an element of table M which does not belong either to its last row or last column. To determine the correct division of the group of four points Q(P ) we introduce the concept of a triangular slope which is calculated for triangles sharing an edge (or an apex). We also use it to define characteristics of a local surface. The figure 2 features the groups of four points, each of them divided by left- or right-sloping diagonals, and illustrates a possible triangulation. In determining slope between adjacent triangles we need to highlight that they can be adjacent (as elements of table M ) by their hypotenuse or short edges. It is important to emphasize that the terms ”hypotenuse” and ”short side” of a triangle are not used in the geometrical sense but the one resulting from the table-neighborhood of the points. If both triangles are ”spread” on the same group of four points Q(P )

Fig. 2. The groups of four points on the basis of which triangulation is done.

then they are adjacent to each other by hypotenuses, otherwise, i.e. if they are ”spread” on the adjoining groups of four points Q(P1 ) and Q(P2 ) which have two shared points then they are adjacent to each other by short sides of triangles. Apart from the neighbouring by edges, neighbouring by common apexes of the triangles is taken into account. Depending on the type of the neighbouring, an appropriate way of counting the values of slope between the two triangles under consideration is determined. We have proposed a method of counting the triangular edge and apex slopes which characterizes correctly enough the curvatures between each pair of the adjacent triangles. These slopes are the basis for defining the best triangulation and segmentation of homogeneous regions. A. Triangular edge slopes To define the triangular edge and apex slopes, we employed the classic definition of curvature of a given point in E 3 space which (due to the difference of approach) we hereinafter term a directional point slope (see V-A). It is calculated at each point P of table M in relation to each of eight points (neighboring with P , which are marked as Pi,j ). Let us consider any group of four points Q(P ) of the table M , where a point P does not belong to the last column or the last row of table M . For a pair of triangles (41 , 42 ) for the left-sloping and right-sloping division the slope for this pair is marked by: triangle(41 , 42 ) These two slopes (right- and left-sloping) are linked to each other and are calculated as the point slopes (see Def. 1). And so for the first pair it is: triangle(4(U1 U4 U2 ), 4(U1 U4 U3 )) = po(S, 4(U2 SU3 )) (3) for some point of S, lying on the left-sloping diagonal, which will be designated below. And for the second pair of the triangles it is: triangle(4(U2 U3 U1 ), 4(U2 U3 U4 )) = po(T, 4(U1 T U4 )) (4) for some point T , lying on the right-sloping diagonal.

On figure 3 we have marked points S and T , adopting the notation: U1 = P 1, U2 = P 2, U3 = P 4 and U4 = P 5. Thus, the triangular slopes (between the two pairs of triangles) are the point slopes calculated at certain points S and T , lying respectively on the left and right diagonals of the group of four points Q(P ). The manner of their designation is as follows: through points U1 and U4 is drawn a line k and through points U2 and U3 is drawn the line l. The section with the smallest distance between the lines k and l determines (as its two ends of) two appropriate points: S and T , where the first of them lies on the left-sloping diagonal and the second on the right-sloping diagonal. The choice of these points in particular can be intuitively explained in the manner that, while they must be aligned with the corresponding diagonals, they lie at the same time as near as possible to the center of gravity of the entire group of four points. The slope value at the point S (with two additional points U2 slope U3 ) sets the slope between a pair of triangles 4(U1 U4 U2 ) and 4(U1 U4 U3 ). In contrast, the slope at the point T (with two additional points U1 slope U4 ), sets the slope between 4(U2 U3 U1 ) and 4(U2 U3 U4 ). If one of them is negative in value then the other is positive, and vice versa. In other words, if the first pair of triangles creates convexity in relation to the observer’s point O, then the second pair creates concavity. The exception is the situation when the four points of the group Q(P ) lie together on one plane; then the two triangular slope values are close to zero value. It is important that when comparing the distance of points S and T to the point of observer O it is possible to determine whether the combinations formed by such pairs of triangles are a convexity or a concavity. For a left-sloping pair, the convexity will be when point S will be nearer to the observer O than the T , while for a right-sloping pair the situation is reversed.

short side. Two additional points Q and R are determined, with this determining going as follows: first, two apexes P5 and P8 are taken, forming a common edge for both triangles, and next, a line k is drawn through these apexes. A second line L is drawn through the two opposite apexes P7 and P6 in relation to a section P5 P8 ; on figure 3 this line is marked as a doted line. Then, the segment is calculated having the smallest distance between the lines k i l so defined. The ends of this segment define points of Q and R, and they lie on lines k and l, respectively. The slope value at the point Q, where the directional points are apexes P7 slope P6 , determines the slope between the pair of triangles under consideration, 4(P5 P8 P7 ) and 4(P5 P8 P6 ), and is defined as: triangle(4(P5 P8 P7 ), 4(P5 P8 P6 )) = po(Q, 4(P7 QP6 )) (5) Comparing the distance between each other points Q and R and observer O, the sign of the slope is determined, i.e. its convexity or concavity. B. Triangular apex slopes For each triangle 4(U1 U2 U3 ) and, being adjacent to it by the apex Ui (i = 1, 2, 3), triangle 4(Ui W Z), the slope between them (termed the triangle apex slope) is determined as follows. For the purposes of illustration, it can be assumed that in figure 3 U1 = P 4, U2 = P 5, U3 = P 7 and W = P 2, Z = P 6; where Ui = P 5. First, is designated the plane hs such as minimizes the sum of distances to four other apexes of the pair of triangles under consideration, i.e. points W , Z and Uj , Uk ; where Uj and Uk are two different apexes of triangle 4(U1 U2 U3 ), which different from Ui . Then is designated the plane hp , being perpendicular to the plane hs , and intersecting W Z and Uj Uk segments at points K and L, respectively. At the same time, the condition that point K lie as near as possible to the center of W Z segment and that point L lie as near as possible to the center of Uj Uk segment is fulfilled. The points K and L thereby designated are directional points, relative to which the point slope value is determined at apex Ui . The apex slope for triangles 4(Ui W Z) and 4(U1 U2 U3 ), via a common adjacent apex Ui , is defined in such a way as: triangle(4(U1 U2 U3 ), 4(Ui W Z)) = po(Ui , 4(Ui LK)) (6) VII. S EGMENTATION OF HOMOGENEOUS REGIONS

Fig. 3.

The method of calculating triangular edge slope

As for the above pairs of triangles (adjacent to each other through the hypotenuses — diagonals of groups of four points), the slopes are calculated for such pairs of them as are adjacent to each other by vertical or horizontal short sides. Consider, as in figure 3, two triangles 4(P5 P8 P7 ) and 4(P5 P8 P6 ); where their common edge, i.e. P5 P8 , is a vertical

The overall concept for the segmentation of the rangeimage being proposed by us in our research is comparatively simple and employs a specific method of triangulation and also definitions of triangular slopes. This segmentation consists of isolating sets of triangles forming a cohesive homogeneous region by attaching themselves to further triangles the mutual position of which in 3D space matches the defined homogeneity criterion. The fundamental step of identifying

a homogeneous region is the application of the specified homogeneity criterion selected to the set of separate triangles currently isolated (constituting a search region) and to a new triangle being adjacent to the region currently located. In each such a step a pair of the triangles is considered, where the first of these already belongs to this set, and the second triangle (adjacent to the first one) is checked for the possibility of its being attached to the region being searched. In the simplest version of the segmentation method being proposed by us, we have assumed that the homogeneity criterion is determined by a range of numbers (for example (a, b)) acceptable slope values. In this approach, all values of the slopes which correspond to adjacent pairs of triangles (belonging to the homogeneous region being searched) will be located in a given range (defined by the homogeneity criterion selected). For recognition of the flat regions we have adopted a particular form of the homogeneity criterion. A flat region is taken to be a cohesive set of triangles, the apexes of which lie within a small (predetermined) distance from the plane approximating them. A range of the values of slopes can be reduced to the following three groups with a fixed ε > 0 (by definition near to 0): (−∞, −ε], (−ε, ε), [ε, +∞). The first of them signifies concavity, the second flatness, and a third convexity. VIII. J OINING OF HOMOGENEOUS REGIONS After the recognizing of homogeneous regions based on a range-image (which is the result of scanning the real environment) the size of homogeneous regions are usually small, and their count is significant. The reason is that the real object surfaces are usually heterogeneous in nature. So what is required is a way of aggregating these regions. We can not ignore the fact that the surfaces of objects can be divided not only on the convex, concave, and flat surfaces. For example, in a lot of situations any area in one direction is convex, and in second one is flat, for example lateral surface of cylinder. In this case, the result of a segmentation usually is a series of parallel convex regions in one direction, and a series of parallel flat regions in the other one. We suppose that there may be many ways to defining homogeneous regions. As a such regions may be assume even rough surfaces, if only there are the certain regularities for them, i.e. the relationship between their triangular components. This is possible by using particular criteria for the definition of homogeneity. On those grounds, it is necessary to join groups of homogeneous regions into complex regions which characterize the larger surfaces. This is possible by determining their common parts. To achieve this goal it is necessary to make use of the fact that the triangles often simultaneously belong to several homogeneous regions, extending in different directions. These triangles constitute elements of homogeneous regions such as enable those regions to be joined to each other through them. At the same time, it is important to emphasize that these regions can differ from each other with regard to any properties (homogeneity criteria on the basis of which they are

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