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Hough Parameter Space Regularisation for Line Detection in 3D

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Hough Parameter Space Regularisation for Line Detection in 3D Hough Parameter Space Regularisation for Line Detection in 3D Manuel Jeltsch1, Christoph Dalitz1 and Regina Pohle-Fr¨ohlich1 1Institute for Pattern Recognition, Niederrhein University of Applied Sciences, Reinarzstr. 49, 47805 Krefeld, Germany [email protected], fchristoph.dalitz, [email protected] Keywords: Hough transform, 3D line detection, ridge detection, laser scan data. Abstract: The Hough transform is a well known technique for detecting lines or other parametric shapes in point clouds. When it is used for finding lines in a 3D-space, an appropriate line representation and quantisation of the parameter space is necessary. In this paper, we address the problem that a straightforward quantisation of the optimal four-parameter representation of a line after Roberts results in an inhomogeneous tessellation of the geometric space that introduces bias with respect to certain line orientations. We present a discretisation of the line directions via tessellation of an icosahedron that overcomes this problem whenever one parameter in the Hough space represents a direction in 3D (e.g. for lines or planes). The new method is applied to the detection of ridges and straight edges in laser scan data of buildings, where it performs better than a straightforward quantisation. 1 INTRODUCTION Chaudhuri, 2015). For line detection in 2D, lines are typically represented with the Hessian nor- mal form, which results in a two dimensional pa- Originally proposed for line detection in a 2D rameter space. In three dimensions, the Hessian space (Hough, 1962), the Hough transform has normal form does not represent lines, but planes, meanwhile become a standard tool for detecting and a straightforward generalisation of the origi- a wide variety of parametric shapes (Mukhopad- nal Hough transform to 3D thus leads to plane de- hyay and Chaudhuri, 2015). Unlike other shape tection with a three dimensional parameter space detection algorithms, the Hough transform does (Ishida et al., 2012). not work on images, but on point clouds. Its ap- plication to images thus requires a preprocessing For line detection in 3D, an appropriate para- step for filtering candidate points. The idea of metric line representation needs to be chosen. the Hough transform is to consider each candi- The text book line representation is the vector ~ ~ date point as a \vote" for all predefined shapes form ~a + tb, where ~a is a point on the line and b to which it might belong. The predefinition of the (with k~bk = 1) is the direction of the line. Even shapes is done by a discretisation of the parame- though this line representation is redundant and ter space. Shapes with many points will then get leads to a five dimensional parameter space, it many votes in the parameter space. has indeed been used for line detection with the Time and space complexity of the Hough Hough transform (Moqiseh and Nayebi, 2008). transform not only depend on the size of the point One way to reduce the space and time complexity cloud, but also on the dimension of the param- is a hierarchical approach that first searches for eter space and the coarseness of the parameter peaks in the slope parameter space, which is only discretisation. For ellipse detection in 2D, e.g., two dimensional, and to further investigate these the parameter space is five dimensional, and there peaks in the intercept parameter space, which is have been a number of suggestions for making two dimensional too (Bhattacharya et al., 2000). this problem more tractable (Mukhopadhyay and A different way to reduce the complexity is to This is a self-archived version of a paper that appeared in the Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP), pp. 345-352 (2016) use a non-redundant line representation, thereby Algorithm 1 Hough transform reducing the number of dimensions. A minimal Input: point cloud X = f~x ; : : : ; ~x g, parameter and optimal representation of a line with four pa- 1 n values Pi = fpi1; : : : ; piNi g for 0 < i ≤ k rameters was given by Roberts (Roberts, 1988; Output: voting array A of size N × ::: × N Schenk, 2004). This representation has already 1 k 1: A(p1; : : : ; pk) 0 for all p1; : : : ; pk been used for needle detection in 3D ultrasonic 2: for ~x 2 X do images (Zhou et al., 2008; Qiu et al., 2013). 3: for p1 2 P1 do In the present paper, we show that a straight- 4: ::: forward discretisation of Roberts' parameter 5: for pk−1 2 Pk−1 do space leads to an inhomogeneous sampling pat- 0 6: p g(~x;p1; : : : ; pk−1) . cf. Eq. (2) tern that favours certain directions. To overcome 0 7: pk nearest neighbour to p from Pk this shortcoming, we suggest a discretisation of 8: A(p1; : : : ; pk) A(p1; : : : ; pk) + 1 two parameters, namely the angles specifying the 9: end for normal vector of Roberts' plane through the ori- 10: ::: gin, by means of a sphere tessellation. As a use 11: end for case, we apply the new method to edge detection 12: end for in laser scan data, where the candidate points 13: return A are prefiltered based on the local curvature in the point cloud. In our experiment, the new approach lead to better results than a straightforward dis- is infinite, but it can be made finite by a discreti- cretisation with the same number of parameter sation of the parameter space: each parameter pi cells. is assumed to be limited to a range of Ni values This paper is organised as follows: in section 2, from a finite set Pi = fpi1; : : : ; piNi g, where the we give a general description of the Hough trans- number Ni of possible parameter values may vary form, introduce Roberts' line representation and from parameter to parameter. our parameter space discretisation. In section 3, Each of the discrete parameter values then we describe how the method can be used to de- represents a cell in parameter space, and a point tect arbitrarily oriented ridges and other straight ~x votes for all cells that contain parameters fulfill- edges in 3D laser scan data. ing Eq. (1). To make the voting process tractable, Eq. (1) must be rewritten as g(~x;p1; : : : ; pk−1) = pk (2) 2 LINE DETECTION IN 3D Then all parameter cells, to which ~x belongs, can be determined in a loop over all parameter values Before we specialise the Hough transform for 3D P1 × ::: × Pk−1 and by computing the cell of the line detection, let us first describe the Hough last parameter through Eq. (2). The resulting transform in its most general case, and then dis- algorithm is listed as Algorithm 1. cuss the problems of 3D line representation and its parameter space discretisation. 2.2 Line parametrisation 2.1 Hough transform For the purpose of algorithmic efficiency it is ad- visable to use a parametrisation that is unique, does not suffer from singularities and is non- Let X = f~x ; : : : ; ~x g be a point cloud in which 1 n redundant. Such a parametrisation for lines in 3D we look for curves defined by k parameters and that uses only four parameters (x0; y0; φ, θ) was described in the parametric form given by Roberts (Roberts, 1988). f(~x;p1; : : : ; pk) = 0 (1) The direction of a line can be specified by the two parameters φ for horizontal orientation (az- For a given list of parameters p1; : : : ; pk, all points imuth) and θ for altitude (elevation) (see Fig. 1). ~x on the curve fulfil Eq. (1). The basic idea of The directional vector ~b is obtained from θ and φ the Hough transform is to consider this equation through not as a condition for ~x, but for the parameters 0 1 0 1 bx cos φ cos θ p1; : : : ; p1: all curves to which a given point be- ~ longs are given by all parameter values that fulfil b = @byA = @sin φ cos θA (3) Eq. (1). Generally, the number of possible curves bz sin θ 346 z line, the parameters x0 and y0 are obtained with: 2 0 bx bxby x = 1 − px − py − bxpz y 1 + bz 1 + bz (4a) ! b2 θ 0 bxby y y = − px + 1 − py − bypz 1 + bz 1 + bz (4b) φ x A point ~p on the line can in turn be obtained (a) with: Figure 1: φ and θ as azimuth(blue) and elevation(red) 0 2 1 0 bxby 1 bx 1 − − 1+b 1+bz z 0 B C 0 B b2 C ~p = x · B − bxby C + y · B1 − y C (5) z @ 1+bz A @ 1+bz A −bx −by To utilise this parametrisation for a 3D Hough transform, a suitable discretisation of the param- b ⃗ y eter space has to be chosen. (x',y') 2.3 Parameter regularisation The discretisation of θ and φ determines the ori- entations that can be detected by the Hough x transform. A na¨ıve approach is the uniform dis- cretisation of the angles θ and φ with constant Figure 2: Line parametrisation by Roberts with x0; y0 step size ∆: (red) and the line's directional vector ~b (blue) θ − θ θ = θ + i · ∆; 0 ≤ i ≤ max min (6a) min ∆ φ − φ φ = φ + j · ∆; 0 ≤ j ≤ max min (6b) min ∆ Since two anti-parallel directional vectors de- The number of resulting direction vectors is scribe the same line, the vectors ~b have to be con- (θ − θ ) · (φ − φ ) fined to a half-space for the representation to be n = max min max min (7) unique. This can be achieved by restricting the uniform ∆ π angle ranges to 0 ≤ θ ≤ 2 and −π < φ ≤ π.
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