Conditions for Segmentation of Motion with Affine Fundamental Matrix

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2 Conditions for Segmentation of Motion with Affine Fundamental Matrix Shafriza Nisha Basah 1, Reza Hoseinnezhad 2, and Alireza Bab-Hadiashar 1 Faculty of Engineering and Industrial Sciences, Swinburne University of Technology 1, Melbourne School of Engineering, The University of Melbourne 2, Victoria, Australia sbasah@swin.edu.au,rezah@unimelb.edu.au,abab-hadiashar@swin.edu.au Abstract. Various computer vision applications involve recovery and estimation of multiple motions from images of dynamic scenes. The exact nature of objects motions and the camera parameters are often not known a priori and therefore, the most general motion model (the fundamental matrix) is applied. Although the estimation of a fundamental matrix and its use for motion segmentation are well understood, the conditions governing the feasibility of segmentation for different types of motions are yet to be discovered. In this paper, we study the feasibility of separating a motion (of a rigid 3D object) with affine fundamental matrix in a dynamic scene from another similar motion (unwanted motion). We show that successful segmentation of the target motion depends on the difference between rotation angles and translational directions, the location of points belonging to the unwanted motion, the magnitude of the unwanted translation viewed by a particular camera and the level of noise. Extensive set of controlled experiments using synthetic images were conducted to show the validity of the proposed constraints. The similarity between the experimental results and the theoretical analysis verifies the conditions for segmentation of motion with affine fundamental matrix. These results are important for practitioners designing solutions for computer vision problems. 1 Introduction Recovering structure-and-motion (SaM) from images of dynamic scenes is an indispensable part of many computer vision applications ranging from local navigation of a mobile robot to image rendering in multimedia applications. The main problem in SaM recovery is that the exact nature of objects motions and the camera parameters are often not known a priori. Thus, any motion of 3D object needs to be modelled in the form of a fundamental matrix [1] (if all moving points are in the same plane, the motion can be modelled as a homography). Motion estimation and segmentation based on the fundamental matrix are well understood and solved in the established work summarised in (Chapters.9 12,[2]). Soon after that work, researchers resumed to the more challenging multibody structure-and-motion (termed MSaM by Schindler and Suter

3 2 Shafriza Nisha Basah, Reza Hoseinnezhad, and Alireza Bab-Hadiashar in [3]) where multiple objects in motions need to be concurrently estimated and segmented. However, the conditions governing the feasibility of segmentation involving MSaM for different types of motions are yet to be established. These conditions are important as they provide information on the limits of current MSaM methods and would provide useful guidelines for practitioners designing solutions for computer vision problems. Well known examples of previous works in motion segmentation using the fundamental matrix are by Torr [4], Vidal et.al [5] and Schindler and Suter [3]. Torr uses the fundamental matrix to estimate an object in motion and cluster it in a probabilistic framework using the Expectation Maximisation algorithm [4]. Vidal.et.al propose to estimate the number of moving objects in motion and cluster those motions using the multibody fundamental matrix; the generalization of the epipolar constraint and the fundamental matrix of multiple motions [5]. Schindler and Suter have implemented the geometric model selection to replace degenerate motion in a dynamic scene using multibody fundamental matrix [3]. The focus of this paper is to study the feasibility of the detection and segmentation of an unknown motion (of a rigid 3D object) with affine fundamental matrix in a dynamic scene (using images taken by an uncalibrated camera). Motion with affine fundamental matrix consists of rotation around Z axis of the image plane and translation parallel to the image plane (no translation in Z direction) [1]. This work is an extension to the study of conditions for motionbackground segmentation and segmentation of translational motions in [6, 7]. In Section 2, we derive the conditions for the detection and segmentation of motion with affine fundamental matrix and provide quantitative measures for detection using theoretical analysis. Section 3 details the Monte Carlo experiments using synthetic images conducted to verify the theoretical analysis and the proposed conditions for successful segmentation of motion with affine fundamental matrix. Section 4 concludes the paper. 2 Segmentation of Motion with Affine Fundamental Matrix Consider n = n i + n o feature points [X i, Y i, ] belonging to two 3D objects undergoing motion-a and motion-b (i =, 1...n i denote the points belonging to motion-a and i = n i + 1, n i n denote points belonging to motion-b). Each motion consists of rotations around Z axis followed by a non-zero translation in the X Y plane denoted by θ a and T a for motion-a and θ b and T b for motion-b where T a = [T xa, T ya, T za ] and T b = [T xb, T yb, T zb ] with T za = T zb =. The location [X i, Y i, ] before and after the translations are visible in the image plane and are denoted by m 1i = [x 1i, y 1i ] and m 2i = [x 2i, y 2i ]. All points in the image plane are perturbed by measurement noise e assumed to be independent and identically distributed (i.i.d) with Gaussian distribution: x 1i = x 1i + e 1 ix, y 1i = y 1i + e 1 iy, x 2i = x 2i + e 2 ix, and y 2i = y 2i + e 2 iy, (1)

4 Conditions for Segmentation of Motion with Affine Fundamental Matrix 3 where e 1 ix, e1 iy, e2 ix and e2 iy N(, σ2 n ) and σ n is the unknown scale of noise. The underlined variables denote the true or noise-free locations of points in image plane. The relationship between all noise-free matching points in the image plane and world coordinate points as viewed by a camera (with focal length f and principle points [P x,p y ]) are: x 1i = fx i + P x, x 2i = x 1i cosθ y 1i sin θ + ft x + P x, y 1i = fy i + P y, y 2i = x 1i sin θ + y 1i cosθ + ft y + P y, (2) where P x = P x (1 cosθ)+p y sin θ and P y = P y (1 cosθ) P x sin θ. The symbols θ, T x and T y in (2) denote the motion parameters where θ = θ a, T x = T xa and T y = T ya for motion-a and θ = θ b, T x = T xb and T y = T yb for motionb respectively. We aim to segment the matching points belonging to motion-a from the mixture of matching points belonging to motion-a and motion-b in two images, thus the points undergoing motion-a are considered the inliers and the points undergoing motion-b would be the outliers or the unwanted motion. The fundamental matrix of motion-a is computed using: F = A T [T] xra 1, (3) where R is the rotation matrix of the motion and [T] x is the skew symmetric matrix of translation T [8, 2,9]. For motion-a (θ = θ a, T x = T xa and T y = T ya ), equation (3) yields: F a = 1 T ya T xa, (4) f T xa sin θ a T ya cosθ a T ya sin θ a + T xa cosθ a Q where Q = (T ya cosθ a T xa sin θ a T ya )P x + (T xa T ya sin θ a T xa cosθ a )P y. We assume that, a perfect estimator provides the true fundamental matrix given in (4). If F a is known, the Sampson distances can be computed using [1,1,11]: d i = m 2i Fm 1i [ ( / x 1i) 2 + ( / y 1i) 2 + ( / x 2i) 2 + ( / y 2]. (5) 2i) m 2i Fm 1i Substitution of real plus noise forms in (1) and F a in (4) into equation (5) yields: d i =(2(T 2 ya + T 2 xa)) 1 2 [(x1i + e 1 ix)(t xa sinθ a T ya cosθ a ) + (y 1i + e 1 iy) (T ya sin θ a + T xa cosθ a ) + (x 2i + e 2 ix)t ya (y 2i + e 2 iy)t xa + Q]. (6) For points undergoing motion-a (i =, 1...n i ), the above expression without noise terms equals zero because the true F a is used to compute d i s. Thus, equation (6) can be simplified to: d i =(2(T 2 ya + T 2 xa )) 1 2 [e 1 ix (T xa sin θ a T ya cosθ a ) + e 1 iy (T ya sin θ a + T xa cosθ a ) + e 2 ix T ya e 2 iy T xa]. (7)

5 4 Shafriza Nisha Basah, Reza Hoseinnezhad, and Alireza Bab-Hadiashar Distances d i s of the points belonging to motion-a in (7) turn out to be a linear combination of the i.i.d. noises therefore, they are also normally distributed with zero mean. The variance of d i s (i =, 1...n i ) also equals σ 2 n as the numerator and denominator cancel each other. Therefore, the distribution of d i s of motiona are the same as the noise e which are N(, σ 2 n). The points belonging to motion-a are to be separated from the points belonging to motion-b. The distances of points belonging to motion-b (i = n i + 1, n i n) with respect to F a are calculated using (6) in which [x 2i,y 2i ] are replaced with the terms in (1) for motion-b, yields: d i =(2(T 2 ya + T 2 xa )) 1 2 [x1i (T xa sin θ a T ya cosθ a ) + y 1i (T ya sinθ a + T xa cosθ a ) + x 2i T ya y 2i T xa + Q] + e, (8) where e N(, σ 2 n ) based on the distribution of d i s in (7). By combining equation (8) with the image-world points relationship for motion-b in (2), and expressing it in term of directions of translation (φ a for T a and φ b for T b ) and magnitude of T b, we obtain (after manipulations using several trigonometric identities): d i = 2sin θ 2 ( x 1i cosθ + ý 1i sin Θ) + K sin φ + e, (9) where x 1i = x 1i P x, ý 1i = y 1i P y, θ = θ a θ b, φ = φ a φ b, Θ = φ a θa+θ b 2 and K = f T b 2. By using the harmonic addition theorem [12], equation (9) is simplified to: where G i = d i = G i sin θ 2 cos Θ i + K sin φ + e, (1) 2( x 2 1i + ý 2 ) and Θ 1i i = Θ + tan 1 ( x 1i /ý 1i ) + β (the value of β = if x 1i or β = π if x 1i < ). Since the distribution of d i s of motion-a is always N(, σ 2 n) (according to equation (7)), the feasibility of identification and segmentation of points belonging to motion-a using a robust estimator depends on the distribution of d i s of motion-b. If the population of d i s from points belonging to motion-b overlap with population of d i s from points belonging to motion-a, they would not be separable. However if both populations (d i s from motion-a and b) do not overlap, the population of d i s of each motion will be separable. Thus, a robust estimator should be able to correctly identify and segment the points belonging to motion-a. In order to ensure that both populations do not overlap, the following conditions must be satisfied 1 : d i 5σ n or d i 5σ n when i = n i + 1, n i n, (11) where d i s are the noise-free d i s given by all terms in equation (8) except for the noise term e. If the conditions in (11) is satisfied 99.4% of points belonging 1 Since the d i s of motion-a is distributed according to N(, σ 2 n) and d i s of motion-b are also perturbed by measurement noise of N(, σ 2 n), from probability theory if the maximum or minimum value of d i s of motion-b are at least 5σ n away from the mean of d i s of motion-a, then only about.6% of d i s of each population would overlap.

6 Conditions for Segmentation of Motion with Affine Fundamental Matrix 5 to motion-a will be correctly segmented. Since the term G i s in (1) are always positive, the range of the term G i cos Θ i in (1) are: Ĝ G i cos Θ i Ĝ for all i s, (12) where Ĝ is the maximum values of G i s depending on the locations of the objects undergoing motion-b. Thus, the range of d i s from equations (1) and (12) are: θ Ĝsin 2 + K θ sin φ d Ĝsin i 2 + K sin φ. (13) Combining the inequalities in (11) and (13), the conditions for successful segmentation of points belonging to motion-a are expressed as: Ĝsin θ 2 + K θ sin φ 5σ n or Ĝsin 2 + K sin φ 5σ n. (14) Solving for θ, the inequalities in (14) are expressed as: θ 2 5σ sin 1 n ± K sin φ Ĝ θ 2 5σ sin 1 n ± K sin φ Ĝ for θ, for θ <. In most computer vision problems, the distance between the camera and the object in motion is roughly known. Therefore, the term in equation (15) can be expressed in term of average distance Z between camera and objects in motions or Z. We assume that an accurate estimate for F a is obtained by minimising the cost function of a robust estimator. Having F a, the distances (d i s) of all matching points can be computed. Then d 2 i s for all points are used as residuals for segmentation to identify and segment points belonging to motion-a using a robust estimator. Here, we use the Modified Selective Statistical Estimator (MSSE) [13] as it has been shown to outperform other robust estimation techniques in term of its consistency [14]. In MSSE, the residuals are sorted in an ascending order and the scale estimate given by the smallest kth distances is calculated using [13] for a particular value of k: (15) σ 2 k = k i=1 d2 i k 1. (16) While incrementing k, the MSSE algorithm is terminated when d k+1 is larger than 2.5 times the scale estimate given by the smallest k distances: d 2 k+1 > 2.52 σ 2 k. (17) With the above threshold, at least 99.4% of the inliers will be segmented if there are normally distributed [13].

7 6 Shafriza Nisha Basah, Reza Hoseinnezhad, and Alireza Bab-Hadiashar From our analysis, the separability of motion with affine fundamental matrix depends on the difference between rotation angles and translational directions ( θ and φ), the location of points belonging to motion-b (Ĝ), the magnitude of T b viewed by a particular camera ( K Z ) and the level of noise (σ n ) presented in equation (15). We verified these conditions using Monte Carlo experiments and the results are presented in the next section. The correctness of these conditions are verified by studying the variance of the result of the Monte Carlo experiments. The conditions for segmentation for more general motions (including T z and rotation around other axes) are too complex to be derived theoretically. However, the derived condition for segmentation of motion with affine fundamental matrix is served as the basis of approximation for more general motions when T z and rotation around other axes are very small or close to zero. 3 Monte Carlo Experiments The Monte Carlo experiments with synthetic images have two parts. The first part was conducted to verify the conditions for segmentation in (15) for separating motion-a from motion-b. The second part of the experiments was designed to examine how the conditions change when the inlier ratio ɛ was varied. In each iteration in the Monte Carlo experiments, 2 pairs of points in the world coordinate according to motion-a were mixed with the pairs of matching points according to motion-b (the number of matching points belonging to motion-b depends on the inlier ratio ɛ). All X and Y coordinates of the matching points were randomly generated while Z coordinates were uniformly distributed according to U( Z σ Z, Z+σ Z ) where Z = 1m and σz Z = 1%. Then, all matching points were projected to two images using a synthetic camera (with f = 73 pixels, [P x, P y ]=[32,24] and image size of pixels). Random noise with the distribution of N(, σn) 2 was added to all points. We assumed that the image points ( x 1i s and ý 1i s) belonging to each motion (motion-a and b) were clustered together, since in many computer vision applications the objects in motions are rigid. The points belonging to motion-b were assumed to be within 2% 2% width and length of the image size and according to Ĝ, while the points belonging to motion-a could be anywhere in the image plane (since its d i s will always be N(, σn) 2 according to (7)). The segmentation was performed using MSSE with d 2 i s (calculated based on the true F a) as the segmentation residuals. The ratio of the number of segmented inliers over the actual inliers ζ was calculated and recorded. Each experiment consists of 1 experimental iterations and the mean and standard deviation of 1 ζ s were recorded (denoted by ζ and σζ). These experiments were then repeated for various ɛ. In the first part of the experiments, we consider two scenarios; Scenario-I with parameters K Z =5 (corresponding to T b = 1m), σ n =.5 and Ĝ =.75G max while Scenario-II with parameters K Z =4 (corresponding to T b =.8m), σ n = 1 and Ĝ =.75G max 2. The conditions for segmentation for Scenario-I and II 2 The term G max is the value of G i when [ x 1i,ý 1i ] are maximum and in this case [32,24] according to a camera with image size with [P x, P y]=[32,24].

8 Conditions for Segmentation of Motion with Affine Fundamental Matrix 7 were generated from (15) and shown in Fig.2(c)-(d) and Fig.2(e)-(f) respectively, where the shaded area denote the area where motion-a will be successfully segmented from motion-b (in this area, both population of d i s will not overlap). This analysis were also performed in five different cases with ɛ = 3%. The motion parameters were selected from the magnitudes of θ s and φ s from the shaded region (Case-1 and 2) and unshaded region (Case-3, 4 and 5) in Fig.2(c)-(d). In all cases, the histogram of d i s of all image points were plotted and ζ s were recorded. For five instances of the data samples generated in Case 1 to 5 for Scenario-I (in Fig.2(c)-(d)), the histogram of d i s for all points are plotted in Fig.1. These figures show that in Case-1 ( θ = 1 and φ = 1 ) and Case-2 ( θ = 5 and φ = 3 ), the points belonging to motion-a were correctly segmented, denoted by ζ =.98 for both cases (ζ = 1 indicates perfect segmentation). Successful segmentation was expected as the population of d i s of points belonging to motion-a and motion-b were not overlapping, thus there were separable as shown in Fig.1(a) and 1(b). As the magnitudes of θ and φ were selected to be outside the shaded region in Fig.2(c)-(d) in Case-3 ( θ = 2 and φ = 1 ) and Case-4 ( θ = 1 and φ = 4 ), the points belonging to motion-a were incorrectly segmented denoted by ζ = 1.92 and The failure is due to the overlap of the population of d i s of both motions and thus there were very little distinction between them as shown in Fig.1(c) and 1(d). However, in Case-5 ( θ = 1 and φ = 4 ) it was observed that the points belonging to motion-a was correctly segmented (ζ =.99 and non overlapping d i s in Fig.1(d)) even though the magnitudes of θ and φ were the same as in Case-4. These experiment results were consistent with the conditions for segmentation derived in equation (15), where population of d i s of both motions are not overlap in the shaded region of Fig.2(c)-(d). Thus the points belonging to motion-a will be correctly segmented when the magnitudes of θ and φ are in this region. However, when the magnitudes of θ and φ are outside the shaded region in Fig.2(c)-(d), the are no guarantee that points belonging to motion-a will be correctly segmented since there is a chance that the population of d i s of both motions would overlap. In the second part of the experiments, the effect of varying ɛ from 3% to 8% to the conditions for segmentation of Scenario-I and II (the shaded regions) in Fig.2(c) to (e) were examined. The mean and sigma of 1 ζ s (denoted as ζ and σζ) were recorded for each pair of θ s and φ s in the experiments. Both θ and φ in the experiments were varied from to 9 with the increment of 2.5. Fig.2(a) and 2(b) show ζ and σζ versus θ and φ for Scenario-I. It was observed that for small θ and φ (both < 5 ), points from motion-a were mixed with points from motion-b and segmented (ζ > 1). In such cases, an inaccurate inlier-outlier dichotomy would result in an incorrect motion estimation and segmentation. As both φ and θ increased to 9, the magnitudes of ζ s approaching.99 and σζ s reduced to around.2. From Fig.2(a) and 2(b), there are areas when ζ =.99 and σζ <.1 Generally smaller Ĝ means that the points moved by motion-b were closer to [Px, Py] of the image.

9 8 Shafriza Nisha Basah, Reza Hoseinnezhad, and Alireza Bab-Hadiashar Frequency di (a) Case 1, ζ =.98 Frequency di (b) Case 2, ζ =.98 Frequency di (c) Case 3, ζ = Frequency 15 1 Frequency di (d) Case 4, ζ = di (e) Case 5, ζ =.99 Fig.1. Histogram for d i s for all points for Case-1 to 5 in Scenario-I. (when θ between to 12 and φ from 6 to 9 ) indicating correct and consistent segmentation of motion-a. Then the magnitudes of θ s and φ s when ζ.994 and σζ.1 were extracted from Fig.2(a) and 2(b) and compared to the analytical conditions for segmentation as shown in Fig.2(c). The magnitudes of ζ.994 and σζ.1 were selected to make sure that the segmentation of motion-a was correct and consistent for all iterations (1 iterations for each θ and φ) in the experiments 3. The extracted θ s and φ s when ζ.994 and σζ.1 for different inlier ratio ɛ and Scenario-II are shown in Fig.2(c) to 2(f). When the value of ɛ is increased from 3% to 8%, the region where the magnitudes of ζ.994 and σζ.1 are slightly expanded as shown in Fig.2(d) and 2(f). In addition, we observed that when θ > 2 for Scenario-I and θ > 45 for Scenario-II, the points belonging to motiona were also correctly and consistently segmented ( ζ.994 and σζ.1). This is because, when ɛ was higher (more points of motion-a than motion-b) and high values on θ, the density of d i s for points belonging to motion-b were not widely spread. Thus, the likelihood of both populations of d i s to overlap decreased. Hence, expanding the region for correct and consistent segmentation of points belonging to motion-a. The similarity between the experimental and analytical results for the magnitude of θ s and φ s to achieve correct and consistent segmentation ( ζ.994, σζ.1 and both population of d i s do not overlap) in Fig.2(c) to 2(f) verifies the segmentation analysis in Section.2. 3 From probability theory, the standard deviation of a uniformly distributed variables B s is according to Bmax B min 12, where B max and B min are the maximum and minimum values of B s [15]. Thus, if ζ s was uniformly distributed with σζ =.1, the values of ζ s were between ζ ±.17.

10 Conditions for Segmentation of Motion with Affine Fundamental Matrix 9 ζ σζ (a) (b) 8 Analytical result Monte Carlo experiments 8 Analytical result Monte Carlo experiments (c) Scenario-I, ɛ = 3% (d) Scenario-I, ɛ = 8% 8 Analytical result Monte Carlo experiments 8 Analytical result Monte Carlo experiments (e) Scenario-II, ɛ = 3% (f) Scenario-II, ɛ = 8% Fig.2. ζ and σζ versus θ and φ for Scenario-I when ɛ = 3% in (a) and (b). The analytical and the extracted magnitudes of θ s and φ s when motion-a will be correctly and consistently segmented in (c)-(d) for Scenario-I and (e)-(f) for Scenario- II (the boundary of the region where ζ.994 and σζ.1 from Monte Carlo experiments were plotted instead of the shaded region for illustration purposes). 4 Conclusions The conditions for segmentation of motions with affine fundamental matrix were proposed in terms of the difference between rotation angles ( θ) and translational directions ( φ), the location of points belonging to the unwanted motion (Ĝ), the magnitude of the unwanted translation viewed by a particular camera ( K Z ) and the level of noise (σ n ). If this conditions are satisfied, it is guaranteed that the population of distances belonging to multiple affine motions do not overlap, and therefore the target motion can be successfully segmented. The proposed conditions were both studied by theoretical analysis and verified by

11 1 Shafriza Nisha Basah, Reza Hoseinnezhad, and Alireza Bab-Hadiashar Monte Carlo experiments with synthetic images. The performance of these conditions for various inlier ratio was also examined. The magnitudes of θ s and φ s (for particular values of K Z, Ĝ and σ n) for correct and consistent segmentation did not changed significantly when the inlier ratio was varied. However when the inlier ratio was high (ɛ = 8%), the segmentation was also successful when the magnitude of θ was large ( θ > 2 for Scenario-I and θ > 45 for Scenario-II). This is explained by the fact that, less contamination of the unwanted motion and high magnitude of θ reduced the likelihood for both populations of distances to overlap resulting in the successful segmentation of the target object. References 1. Torr, P.H.S., Zisserman, A., Maybank, S.J.: Robust detection of degenerate configurations while estimating the fundamental matrix. Vision Computing and Image Understanding 71 (1998) Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Second edn. Cambridge University Press, Cambridge, UK (23) 3. Schindler, K., Suter, D.: Two-view multibody structure-and-motion with outliers through model selection. IEEE Transactions on Pattern Analysis and Machine Intelligence 28 (26) Torr, P.H.S.: Motion Segmentation and Outlier Detection. Phd thesis, Department of Engineering Science, University of Oxford (1995) 5. Vidal, R., Ma, Y., Soatto, S., Sastry, S.: Two-view multibody structure from motion. International Journal of Computer Vision 68 (26) Basah, S.N., Hoseinnezhad, R., Bab-Hadiashar, A.: Limits of motion-background segmentation using fundamental matrix estimation. DICTA 8 (28) Basah, S.N., Bab-Hadiashar, A., Hoseinnezhad, R.: Conditions for segmentation of 2d translations of 3d objects. Image Analysis and Processing - ICIAP 29. LNCS 5716 (29) Armangu, X., Salvi, J.: Overall view regarding fundamental matrix estimation. Image and Vision Computing 21 (23) Zhang, Z.: Determining the epipolar geometry and its uncertainty: A review. International Journal of Computer Vision 27(2) (1998) Torr, P.H.S., Murray, D.W.: The development and comparison of robust methodsfor estimating the fundamental matrix. International Journal of Computer Vision 24 (1997) Weng, J., Huang, T., Ahuja, N.: Motion and structure from two perspective views: Algorithms, error analysis, and error estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989) Weisstein, E.W.: Harmonic addition theorem. (From MathWorld-A Wolfram Web Resource Bab-Hadiashar, A., Suter, D.: Robust segmentation of visual data using ranked unbiased scale estimate. Robotica 17 (1999) Hoseinnezhad, R., Bab-Hadiashar, A.: Consistency of robust estimators in multistructural visual data segmentation. Pattern Recognition 4 (27) Evans, M., Hastings, N., Peacock, B.: Statistical Distributions. Third edn. Wiley (2)