Neighborhood Hypothesis Testing for Mean Change on Infinite. Dimensional Lie Groups and 3D Projective Shape Analysis of. Contours

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1 Neighborhood Hypothesis Testing for Mean Change on Infinite Dimensional Lie Groups and 3D Projective Shape Analysis of Contours Vic Patrangenaru, Mingfei Qiu Department of Statistics, Florida State University October 15, 2014 Abstract A two-sample hypothesis testing method on a finite dimensional Lie group was derived by Crane and Patrangenaru [1] by an extrinsic approach. If the dimension of the data is infinite, the equality of two extrinsic means is highly likely to be rejected. Munk el. al. [2] generalized one- and multi-sample neiborhood hypothesis testing in projective shape analysis, which addressed the infinite dimension problem. In this paper, we develop a general nonparametric hypothesis testing method for estimation of the extrinsic mean change for matched samples on a Hilbert manifold that has an infinite dimensional Lie group structure. keywords extrinsic means, infinite dimensional data, neighborhood hypothesis test, nonparametric bootstrap 1 Introduction With our eyes, all we see are 3D projective shapes. However, based on the current technology level, 2D digital images are still the most frequently used resources to test if two objects are the same, while our brains use 3D. Hartley [3] developed the 3D projective reconstruction technology to retrieve the 3D projective shape of an object from the digital 1

2 images obtaining from two views. Ma el. al. [4] provided the MATLAB code to compute the fundamental matrix, which is the basic tool in investigating the 3D projective shape. The real projective space in m dimensions, RP m is the projective space P (R m+1 ). We are interested in the infinite dimensional data in RP m obtained from the digital images. Nowadays, the dimension of the data obtaining from digital images could be very large, however, the statistical hypothesis testing methods work better for finite dimensional data sets. Munk el. al. [2] pointed out that the well known Hotelling T statistic would break down and the equality for each point of the shape would not hold if the dimension of the data extend to infinity. So they introduced the one- and multi-sample neighborhood hypothesis testing methodology, which utilized neighborhood hypothesis to replace the regular null hypothesis defined by equations; and led to simpler asymptotic analyses. While the mean and covariance used in that paper were the regular ones, Ellingson el. al. [5] extended the one-sample neighborhood hypothesis testing method by employing extrinsic parameters. For two random objects (X, Y ), one can test if X and Y are different by estimate the difference vector D = X Y via one-sampe testing methods. Similarly, in the infinite dimensional case, we define the mean change by the quarternion multiplication, then test the difference by one-sample neighborhood hypothesis testing. Motivated by the pioneering works, this paper develops the neighborhood hypothesis testing method to test the random change on infinite dimensional 3D projective shapes by extrinsic approach. In section 2, we introduce the projective geometry, projective shapes and the extrinsic mean, especially the extrinsic mean defined by Veronese- Whitney embedding (VW embeding). The neighborhood hypothesis testing methodology for 3D projective shapes is derived in section 3. To better analyze the small sample size data set, the nonparametric bootstrapping statistic is also displayed. And we apply this method to two sets of images taken from two green leaves in section 4. 2 Projective geometry projective shape, and the extrinsic mean In this section, we briefly review the 3D projective reconstruction. More details are in Hartley and Zisserman [7], Ma el. al. [4] and Patrangenaru el. al. [8]. We also recall the concept of affine coordinate, homogeneous coordinate, and the projective frame given by Mardia and Patrangenaru [6], Munk el. al. [2] and Patrangenaru el. al. [8]. Moreover, we 2

3 suppose the data obtained from the 3D projective reconstructions is on a manifold, and the extrinsic mean guarantees the mean of the data is also on the manifold, so the data is analyzed by extrinsic approach. We are especially interested in the extrinsic mean on a Hilbert manifold defined by the Veronese-Whitney embedding proposed in Patrangenaru el. al. [5]. 2.1 Projective geometry Shape is the residual structure of a configuration with points in R m, when some transformations of R m are filtered out [6]. In the machine vision area, projective transformation are the only important transformations, given the physical process by which the image data are stored. If two images from the same 2D scene are captured by a pinhole camera, the corresponding transformation between the two images is the composition of two central projections, which is a projective transformation. The affine transformation requires the two central projection being able to approximated by parallel projection (the scene pictured is very far from the camera, or very small). And the similarity transformation has even more restrictions, the parallel projections must be orthogonal projections on the plane of the camera. The real projective space RP m is the set of the axes going through the origin of R m+1. Unlike similarities or affine transformations or R m, projective transformations do not have a group structure under composition, since the domain of definition of a projective transformation depends on the transformation, and the maximal domain of a composition has to be restricted accordingly. To avoid such unwanted situations, rather than considering projective shapes of configurations in R m, one may consider configurations in the real projective space RP m, with the projective general linear group (GL) action. The problem of the reconstruction of a configuration of points in 3D from two ideal uncalibrated camera images with unknown camera parameters, is equivalent to the following: given two camera images RP1 2, RP2 2 of unknown relative position and internal camera parameters and two matching sets of labeled points {p a,1,..., p a,k } RPa 2, a = 1, 2, find all the sets of points in space p 1,..., p k in such that there exist two positions of the planes RP1 2, RP2 2 and internal parameters of the two cameras c a, a = 1, 2 with the property that the c a -image of p j is p a,j, a = 1, 2, j = 1,..., k. 3

4 THEOREM 2.1. In absence of occlusions, any two 3D reconstructed configurations R, R obtained from a pair of 2D matched configurations in uncalibrated cameras images of a 3D configuration C, have the same projective shape. Note that the solution of the reconstruction problem, from a pair of 2D images depends on a landmark correspondence. 2.2 Projective frame If x = (x 1,..., x m+1 ) R m+1 \{0}, then (1) [x] = [x 1 : x 2 :... : x m+1 ] = {λx, λ 0} is a projective point in RP m. Alternatively, a point p RP m is given by the normalized form p = [z 1 : z 2 : : z m+1 ], where m+1 i=1 (zi ) 2 = 1. In particular, a projective line l is a set associated with a vector plane V in R m+1, l = {[x], x V \0}. A number of points in RP m are colinear if they lie on a projective line. The Euclidean space R m can be embedded in RP m, preserving colinearity. Such a standard affine embedding, missing only a hyperplane at infinity, is (2) x = (x 1,..., x m ) [x 1 : : x m : 1]. Then for a point p = [X] = [X 1 : : X m : X m+1 ] RP m, X m+1 0, the affine coordinate (imhomogeneous coordinate) is ( X (3) (x 1, x 2,..., x m 1 ) = X m+1,..., X m ) X m+1. Oppositely, the homogeneous coordinate of p is (X 1,..., X m+1 ), which are defined up to a multiplicative constant only. However, the coordinates of interest in projective shape analysis are neither affine nor homogeneous. We need coordinates that are invariant with respect to the group of projective (general linear) transformationp GL(m). A projective transformation α of RP m is defined in terms of an (m + 1) (m + 1) nonsingular matrix A GL(m + 1, R) by (4) α([x 1 : : x m+1 ]) = [A(x 1,..., x m+1 ) T ]. 4

5 There is an nonsingular matrix A = ((a j i ) i,j=1,..., m+1)) (j denotes for row, i for column), the projective transformation in R m is (5) v j = aj m+1 + m i=1 aj i xi a m+1 m+1 + m, j = 1,..., m. i=1 am+1 i xi The linear span of a subset of RP m is the smallest linear variety containing that subset. Note that k points in RP m with k m + 2 are in general position if their linear span is RP m. A projective frame, also known as projective basis, in RP m is an ordered (m + 2) tuple of points π = (p 1,..., p m+2 ), any m + 1 of which are in general position. For example, in RP 2, 4 points consist a projective frame, each 3 of them could not be colinear; for RP 3 case, 5 points could make a projective frame only when each 4 of them are not coplanar. There is one-to-one correspondence between projective frame and projective transformation. Let (e 1,..., e m+1 ) be the standard basis of R m+1, then the standard projective frame is ([e 1 ],..., [e m+1 ], [e e m+1 ]), where the last point is the unit point. Given a projective frame π = (p 1,..., p m+2 ), there is a unique α P GL(m) with α([e j ]) = p j, j = 1,..., m+1, α([e 1 + +e m+1 ]) = p m+2. Suppose x 1,..., x m+2 are points in general position where x = (x 1,..., x m ) is an arbitrary point in R m, set x = (x 1,..., x m, 1). In the notation, the superscripts are reserved for the components of a point, whereas the subscripts are for the labels of points. (x 1,..., x m+2 ) is the projective frame in R m only if ( x 1,..., x m+2 ) is the projective frame in RP m. Consider the (m + 1) (m + 1) matrix U m = [ x 1,..., x m+1 ], whose jth column is x j = (x j, 1) T, j = 1,..., m + 1. Define an intermediate system of homogeneous coordinates (6) v(x) = Um 1 x, and write v(x) = (v 1 (x),..., v m+1 (x)) T. Set (7) y j (x) = v j (x)/v j (x m+2 ), j = 1,..., m + 1, (8) z j (x) = y j (x)/ y(x), j = 1,..., m + 1, where [z 1 (x) : z 2 (x) : : z m+1 ] is the projective axis of a point x. If z m+1 (x) 0, the affine representative of this point with respect to the last coordinate is (ξ 1 (x),..., ξ m (x)), where (9) ξ j (x) = zj (x) z m+1, j = 1,..., m. (x) 5

6 2.3 Extrinsic mean for infinite dimensional data For the points on a manifold, the extrinsic mean point on the manifold is defined in terms of a least expected Fréchet distance property among all the other points on the manifold (Bhattacharya and Patrangenaru [9], [10]). A finite dimensional manifold is a metric space M with the following property: if x M, then there is some neighborhood U of x and some integer n 0 such that U is homeomorphic to R n [11]. Assume J : M R N is an embedding of the d-dimensional complete manifold M. In [9] Bhattacharya and Patrangenaru defined the extrinsic mean µ J of J-nonfocal random object Y on M by (10) µ J =: J 1 (P J (µ)), where µ = E(J(Y )) is the mean vector of J(Y ) and P J : F c J(M) is the otho-projection on J(M) defined on the complement of the set F of focal points of J(M). If f : M 1 M 2 is a differentiable function defined from the manifold M 1 to the manifold M 2 and x M 1, the differential of the function f at x is labeled D x f. Consequently, the extrinsic covariance matrix of Y was defined by Bhattacharya and Patrangenaru [10] also, with respect to a local frame field y (f 1 (y),..., f d (y)) for which (D y J(f 1 (y),..., D y J(f d (y))) are othonormal vectors in R N. The extrinsic covariance mathrix of Y with respect to (f 1 (µ J ),..., f d (µ J )) is [ d ] (11) Σ E = D µ P J (e b ) e a (P J (µ))e a (P J (µ)) [ d Σ D µ P J (e b ) e a (P J (µ))e a (P J (µ)) a=1 b=1,...,n a=1 b=1,...,n where Σ is the covariance matrix of J(Y ); P J is differentialble at µ; (e 1 (p), e 2 (p).... e N (p)) is a local othoframe field defined on an open neighborhood U R N of P J (M), which is adapted to the embedding J if y J 1 (U), e r (J(y)) = D y J(f r (y)), r = 1,..., d. Since P Σ k m is homeomorphic to (RP m ) q, q = k m 2, and RP m is equivariantly embedded in the space S(m + 1) of (m + 1) (m + 1) symmetric matrices via Veronese-Whitney embedding (VW embedding) J : RP m S(m + 1), ] T, (12) J([x]) = xx T. Ellingson el. al. [5] defined the VW embedding from the Hilbert manifold M to the Hilbert space by tensor product. Suppose j is a one-to-one differentialble function embeds a Hilbert manifold M to a Hilbert Space H 6

7 j : M H, (13) j([x]) = x x, x = 1. We call the extrinsic mean defined by the VW embedding as VW mean. Assume X = [Γ], Γ = 1 is a random object on the Hilbert manifold, the VW mean of X exists if and only if E(Γ Γ) has a simple largest eigenvalue, in which case, the distribution is j-nonfocal. Then the VW mean is µ E = [γ], where γ is the eigenvector corresponds to the largest eigenvalue. 3 Two-sample neighborhood hypothesis testing methodology The test for the mean change in paired samples on a Lie group as well as its bootstrap approach were derived by Crane and Patrangenaru [1] in extrinsic way. Patrangenaru el. al. [12] organized this methodology and proposed the bootstrap confidence region in terms of figures other than numbers. This testing method considers the data to be on the finite dimensional Lie group, leaving the research for the one on the infinite dimensional Lie group to be unexploited. Munk el. al. [2] employed the neighborhood hypothesis for analyzing the data on the Hilbert space, whose dimension is infinite. Taking the data on the infinite dimensional Lie group into consideration, in this paper, we will develop the neighborhood hypothesis testing method for detecting the mean change in the way of extrinsic analysis. 3.1 Two sample test for VW means of 3D prjective shapes A group (G, ) that has in addition an m dimensional manifold structure, such that the group multiplication : G G G, and the inversion I : G G, I(g) = g 1 are differentiable maps between manifolds, is called a Lie group. The projective shape manifold P Σ k 3, k 5, has a Lie group structure, derived from the quaternion multiplication. Recall that if a real number x is identified with (0, 0, 0, x) R 4, and if we label the quadruples (1, 0, 0, 0), (0, 1, 0, 0), respectively (0, 0, 1, 0) by i, j, respectively k, then the multiplication table given by 7

8 i j k i -1 k j j k -1 i k j i -1 where a b product of a on the first column with b on the top row, is listed on the row of a and column of b, extends by linearity to a multiplication of R 4. Note that (R 4, +, ) has a structure of a noncommutative field, the field of quaternions, usually labeled by H. Note that if h, h H, then h h = h h, and the three dimensional sphere inherits a group structure, the group of quaternions of norm one. The identity element is given by 1 (RP 3 ) q = ([0 : 0 : 0 : 1],..., [0 : 0 : 0 : 1]). Assume (X r ) r=1,...,n and (Y r ) r=1,...,n are paired r.o. s on a Lie group (G, ). The change from X to Y was defined to be r. o. C r =: Xr 1 Y r = ([c 1 r],..., [c q r]), (c s r) T c s r = 1, s = 1,..., q, r = 1,..., n. A test for no mean change from X to Y is one for the null hypothesis (14) H 0 : µ E = 1 G, where 1 G is the identity of G, and µ E is the extrinsic mean of C. In this paper, we only taking VW mean into account. Let J s be the r.o. given by (15) J s = n 1 Σ n r=1c s r(c s r) T, s = 1,..., q, where d s (a) and g s (a) are the eigenvalues in increasing order and corresponding unit eigenvectors of J s, a = 1,..., m+ 1, then the sample mean VW projective shape, in the multi-axial representation, is given by (16) C Jk,n = ([g 1 (m + 1)],..., [g q (m + 1)]). Notate 1 G = ([0 : 0 : 0 : 1] T,..., [0 : 0 : 0 : 1] T ) = ([γ 1 ],..., [γ q ]), so we can construct the statistic for the test as (17) T s = T s (C s j,n, [γ s ]) = nγ T s D s G 1 s,nd T s γ s, s = 1,..., q, 8

9 where D s = (g s (1)... g s (m)) s = 1,..., q, and G is the sample covariance matrix (18) G n(s,a),(t,b) = n 1 (d s (m + 1) d s (a)) 1 (d t (m + 1) d t (b)) 1 n (g s (a) T c s r)(g t (b) T c t r)(g s (m + 1) T c s r)(g t (m + 1) T c t r). r=1 The corresponding bootstrap distributions are (19) Ts = T s (C s j, C s j,n) = ng s (m + 1) T DsG 1 s,n Ds T g s (m + 1). Patrangenaru et al. [8] showed that T s has asymptotically a χ 2 m distribution. COROLLARY 3.1. For s = 1,..., q let c s,1 β be the upper 100(1 β)% point of the values of T s given by (19). We set (20) C s,n,β := j 1 (U s,n,β) where (21) U s,n,β = {µ s RP m : T s (y s j,n; µ s ) c s,1 β}. Then (22) Rn,α = q s=1 C s,n, α q with Cs,n,β, U s,n,β given by (20)-(21) is a region of approximately at least 100(1 α)% confidence for µ j k. The coverage error is of order O p (n 2 ). 3.2 Neighborhood hypothesis test for the mean change Ellingson el. al. [5] derived the one-sample neighborhood hypothesis testing methodology. Let M 0 be a compact submanifold of M. Let ϕ 0 : M R be the distance function (23) ϕ 0 (p) = min p 0 M 0 j(p) j(p 0 ) 2, and let M δ 0, B δ 0 be given respectively by (24) M δ 0 = {p M, ϕ 0 (p) < δ 2 }, 9

10 (25) B δ 0 = {p M, ϕ 0 (p) = δ 2 }. The neighborhood null and alternative hypothesis test is (26) H δ : µ E M δ 0 B δ,x 0, A δ : µ E (M δ 0 ) c (B δ,x 0 ) c. For v j(p (H)), tan(v) is the tangential component of v with respect to the basis e a (P j (µ)). Consider the neighborhood hypothesis testing for the particular situation in which the submanifold M 0 consists of a point m 0 on M. Set ϕ 0 = ϕ m0, there is the asymptotic distribution THEOREM 3.1. If M 0 = {m 0 }, the test statistic for the hypotheses specified in (26) has an asymptotically standard normal distribution and is given by: (27) T n = n{ϕ m0 (ˆµ E ) δ 2 }/s n, where (28) s 2 n = 4 ˆν, S E,nˆν and (29) S E,n = 1 n (tanˆ µ n d P j(x) j (j(x i ) j(x) n n )) i=1 (tanˆ µ d j(x) n P j (j(x i ) j(x) n )) is the extrinsic sample covariance operator for {X i } n i=1, and (30) ˆν = (dˆµe,n j) 1 tan j(ˆµe,n )(j(m 0 ) j(ˆµ E,n )). Here tan(v) is the tangential component of v H with respect to the othobasis e a (P j (µ)) a = 1, 2,..., as (31) tan(v) = (e a (P j (µ)) v)e a (P j (µ)). a=1 And for each element of S E,n, we could compute it by (32) S E,n,ab = 1 n (d2 1 d 2 a) 1 (d 2 1 d 2 b) 1 < e a, γ r >< e b, γ r > < e 1, γ r > 2, r=1 10

11 then s n is obtained by (33) s 2 n = 4 S E,n,abˆv aˆv b. a,b=2 By plugging S E,n (by (32)) and s n (by (33)) in T n from (27), one could obtain the rejection criterion for µ 0. In order to take the advantages of this one-sample neighborhood hypothesis testing method, we test if two matched pair r.o. are similar by testing their random change. Similarly to section 3.1, we suppose X and Y are paired r.o. s on the infinite dimensional Lie group, which is also a Hilbert manifold. The change from X to Y was defined to be r. o. C = X 1 Y. Let µ E be the extrinsic mean of C, then M 0 = {m 0 } is the identity {1 G } = {([0 : 0 : 0 : 1] T,..., ([0 : 0 : 0 : 1] T )}. Here, for the statistic T n, (34) s 2 n = 4 ˆν, G nˆν, (35) n G n(s,a)(t,b) = n 1 (d s (m+1) d s (a)) 1 (d t (m+1) d t (b)) 1 (g s (a) T Xr s )(g t (b) T Xr)(g t s (m+1) T Xr s )(g t (m+1) T Xr), t where s, t = 1,..., q; a, b = 1,..., m, and r=1 (36) ˆν = (dˆµe,n j) 1 tan j(ˆµe,n )(j(m 0 ) j(ˆµ E,n )). We perform the confidence region through nonparametric nonpivotal bootstrap (Efron (1979)[13]). The nonparametric bootstrap algorithm for constructing a confidence region for extrinsic mean contour is given by Qiu el. al. [14]. Algorithm 3.1. INPUT x: the normalized coordinates of the contours (x is k n complex matrix ); k: number of matched points on contours ; n: number of contours(columns); N: number of bootstraps OUTPUT CR: confidence region Step 1 Compute extrinsic mean of x For i=1:n X i = x i x i 11

12 End X = sum(x)/n µ V W = eigenvector corresponds to the largest eigenvalue of X Step 2 Bootstrap For j=1:n u 1,..., u n = random integer uniform(1,n) y 1:n = x u1,..., x un For i=1:n End Y i = y i y i Ȳ = sum(y )/n µ BV W =eigenvector corresponds to the largest eigenvalue of Ȳ φ BV W =real part of trace((ȳ X)(Ȳ X) T ) End cutoff=95%quantile of φ CR={µ BV W, µ BV W 0 (φ BV W > cutoff) µ BV W, j = 1,..., N} 4 Testing the mean change of two group of images from leaves There are two leaves from the same tree, named as leaf A and leaf B. 20 pictures for each of them are taking from different directions. We are interested in applying the neighborhood hypothesis testing methodology to those images. It is difficult for people to pair the infinite number landmarks by hand. Ellingson el. al. [5] established the algorithm to automatically pair the landmarks along the contour of an object. Following up their work, we are able to obtain 12

13 Figure 1: Leaf A (top 20 figures) and leaf B (bottom 20 figures). Figure 2: Leaf A contours parameterized (top 20 figures) and leaf B contours parameterized (bottom 20 figures). the 1200 paired landmarks for the contour of each leaf. With these 1200 paired landmarks acquired from each leaf images, we applied the 3D projective reconstruction to each two landmark sets obtained from the same leaf. Taking the computation cost into consideration, only 50 randomly selected landmarks (Figure 4) from the D projected reconstructed landmarks are utilized to make the hypothesis test. And before the test, those 50 landmarks are firstly projective transformed with respect to the projective frame. Under the significant level α = 0.05, the minimum Figure 3: Leaf A 3D reconstruction (top 10 figures) and leaf B 3D reconstruction (bottom 10 figures). 13

14 Figure 4: The 50 randomly selected landmarks (red dots). Figure 5: Significant level α vs δ. δ for the test to be significant is , which is quite large taking into account the diameter of the sample space is no more than 2. Figure 5 illustrates how the value of δ changes associated with the decreasing of α, which implies that if we want the type I error to be small, we have to make a loose null hypothesis. The bootstrap confidence region for the extrinsic mean of the random change of 50 landmarks is displayed in Figure 6. The null hypothesis is that the extrinsic mean of the random change of the two data set is close to the identity ([0 : 0 : 0 : 1] T,..., [0 : 0 : 0 : 1] T ), whose affine coordinate is ((0, 0, 0) T,..., (0, 0, 0) T ). The range of the confidence region is very large, which indicates that the two leaves are different. References [1] M. Crane; V. Patrangenaru. (2011). Random Change on a Lie Group and Mean Glaucomatous Projective Shape Change Detection From Stereo Pair Image. Journal of Multivariate Analysis. 102,

15 Figure 6: Bootstrap confidence region for the extrinsic mean of the random change. [2] Munk, A.; Paige, R.; Pang, J. ; Patrangenaru, V. and Ruymgaart, F. H.(2008). The One and Multisample Problem for Functional Data with Applications to Projective Shape Analysis. J. of Multivariate Anal.. 99, [3] Hartley, R. I. (1994). Projective reconstruction and invariants from multiple images. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 16(10), [4] Ma, Y., Soatto, S., Kosecka, J. and Sastry, S.S. (2006). An Invitation to 3-D Vision, Springer, New York. [5] Ellingson, L., Patrangenaru, V., and Ruymgaart, F. (2013). Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours. Journal of Multivariate Analysis, 122, [6] Mardia, K. V.; Patrangenaru, V. (2005). Directions and projective shapes. The Annals of Statistics, 33, [7] Hartley, R.I. and Zisserman A. (2004). Multiple view Geometry in computer vision,; 2 edition Cambridge University Press. [8] Patrangenaru, V; Liu, X. and Sugathadasa, S. (2010). Nonparametric 3D Projective Shape Estimation from Pairs of 2D Images - I, In Memory of W.P. Dayawansa. Journal of Multivariate Analysis. 101,

16 [9] Bhattacharya, R.; Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Annals of statistics,31, [10] Bhattacharya, R.; Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Annals of statistics,33, [11] Spivak, M. (2005). A comprehensive introduction to differential geometry, Volume I, Third Edition, Publish or perish, Berkeley. [12] Patrangenaru, V., Qiu, M., and Buibas, M. (2014). Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images. Methodology and Computing in Applied Probability, 16(2), [13] Efron, B. (1979) Bootstrap methods: another look at the jackknife. Ann. Statist. 7, [14] Qiu, M., Patrangenaru, V., Ellingson, L. (2014) How far is the Corpus Callosum of an Average Individual from Albert Einstein s? COMPSTAT 2014, Geneva 16