Projective Shape Analysis for Spatial Orientation in Virtual Environments

Projective Shape Analysis for Spatial Orientation in Virtual Environments Mihaela Pricop-Jeckstadt1*, Alexander Garthe2, Vic Patrangenaru3, Robert L. Paige4 1*Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering “Traian Lalescu”, National University of Science and Technology Politehnica Bucharest, Splaiul Independent¸ei no. 313, Bucharest, 060042, Romania. 2German Center for Neurodegenerative Diseases, Fetscherstraße 105, Dresden, 01307 , Saxony, Germany. 3Department of Statistics, Florida State University, 222 S Copeland St, Tallahassee, FL 32306, Florida, US. 4Department of Mathematics and Statistics, Missouri S&T University, 300 W 13th St, Rolla, MO 65409, Missouri, US. *Corresponding author(s). E-mail(s): [email protected]; Contributing authors: [email protected]; [email protected];[email protected]; Abstract In this article, we apply extrinsic data analysis to the study of cognitive abilities evaluated based on the learning behaviour in the Dresden Spatial Navigation Task (DSNT) virtual navigational experiment. Our novel mathematical modelling of the spatial orientation and spatial learning is based on recent concepts in objectoriented data analysis like extrinsic mean and extrinsic covariance as well as novel statistical testing methods for random objects on manifolds. Finally, the allocentric orientation patterns in persons exhibiting mild cognitive impairment (MCI) and controls are detected for the first time via projective shape analysis. Keywords: nonparametric statistics, projective geometry, extrinsic data analysis, neuroscience, spatial orientation 1 1 Introduction 55 million people were living with dementia in 2020 and it is a rapidly growing health threat. The number of those living with dementia is projected to almost triple to 135 million by 2050 ([1]). Our reseach brings an important medical and social contribution to healthier aging and to the understanding of some neurological processes fundamental to brain functioning by the interplay between neuropsychology, non-euclidean geometry, statistical methods and applied mathematics (see [2,3]). Moreover, the application of our projective shape approach to spatial orientation go beyond cognitive neuroscience towards automated orientation for cars, drones and satellites as well as robotics. Additionaly, this approach links spatial orientation to quantum mechanics and cosmology via projective geometry (see [4,5]). This setting might allow us to connect loss of time feeling in neurological diseases and loss of spatial orientation, and it might support the hypothesis that the brain functions as a quantum computer ([6,7]). A decline in spatial orientation is one of the earliest symptoms of dementia, as individuals lose the ability to navigate their environment ([8]). Spatial learning - the ability to move effectively from place to place - depends on reference memory, a form of long-term memory that stores stable information over time ([9]). The hippocampus, located in the medial temporal lobe, plays a central role in both long-term memory and spatial navigation ([10]). Seminal theories have proposed that it functions as a “cognitive map,” encoding the layout of the environment ([11,12]). Evidence for this comes from both human neuroimaging, such as the enlarged hippocampal volume observed in London taxi drivers ([13]), and animal models. The Morris Water Maze demonstrated that rats can learn to locate a hidden platform using distal cues, providing a classic paradigm for allocentric navigation ([14,15]). Allocentric strategies, which represent spatial relations independent of the observer’s position, rely heavily on the hippocampus, and patients with mild cognitive impairment (MCI) or Alzheimer’s disease (AD) show marked deficits in such tasks ([16,17]). Because the hippocampus is among the first brain regions affected in AD, impairments in allocentric spatial navigation represent a sensitive cognitive marker for early detection ([18]). A major advance in studying hippocampus-dependent spatial navigation in humans has been the development of virtual reality (VR) techniques. These approaches enable the translation of paradigms such as the Morris Water Maze into computerized 3D environments ([19,20]). In such tasks, participants are placed in a virtual circular pool surrounded by distal cues but without local markers and are instructed to locate a hidden platform using a joystick. VR offers precise experimental control, objective tracking of behavioral responses, and strong ecological validity, as navigation performance in VR correlates well with navigation in real-world settings ([21,22]). The Dresdner Spatial Navigation Task (DSNT), developed at Universit¨atsklinikum Carl Gustav Carus Dresden, is one such paradigm, grounded in the allocentric principles of the Morris Water Maze ([14,23]). In the DSNT, participants must locate a hidden platform within one minute, guided by visual landmarks placed around the pool’s perimeter. Their trajectories are recorded as Cartesian coordinates and reaction times, providing rich quantitative data for analyzing navigational strategies. The task includes multiple trial types: pretests (assessing reflexes and familiarity with the 2 joystick), acquisition trials (assessing learning), and probe trials (assessing memory when the platform is removed). Testing spans two consecutive days: on Day 1, participants complete two pretests, one probe trial, and 12 learning trials with a fixed platform location; on Day 2, two pretests are followed by 12 learning trials in which the platform location is shifted after the fourth trial. Mild cognitive impairment (MCI) is widely regarded as an intermediate stage between healthy aging and dementia, particularly Alzheimer’s disease ([24,25]). In the DSNT dataset, 54 participants were tested: 25 healthy controls, 26 with MCI, and 3 with minor dementia of the Alzheimer type (DAT). All participants also underwent standard neuropsychological assessments. Previous studies suggest that spatial navigation tasks can reveal distinct strategies—reflected in trajectory patterns—that are sensitive to hippocampal integrity and disease status ([8,18]). Building on this evidence, our research examines how trajectory patterns in the DSNT reveal the interplay between spatial learning and spatial long-term memory, as well as landmark-based orientation—specifically, which environmental cues participants rely on when navigating. 2 Projective Shape Data Analysis Projective shape analysis has its origins in the parametric statistics study of projective invariants. In this setting, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations ([26]). A nonparametric methodology, via projective frames in arbitrary dimensions, was developped in the paper [27] on high level image analysis while [28] studies asymptotics of projective invariants and extrinsic and intrinsic sample means on projective manifolds for understanding of 2Dscenes from their digital camera images. For a landmark configuration in R3, ”shape” deals with the residual structure of this configuration when certain transformations are filtered out. The non-shape components consists of size, position and orientation. Proportions, relative angles and the relative arrangement of parts are geometric features that belong to shape. Hence, spatial orientation as understood by the allocentric theory can be better studied in terms of projective shape. Additionally, projective shape is interpreted in terms of axial statistics, allowing a directional interpretation of the estimators. Mathematically, the shape of a configuration consists of its equivalence class under a group of transformations. Here the group action describes the way in which an image is captured. For example, the pinhole camera model assumes that a 2D−image is the result of the projection onto the image plane of an ideal pinhole camera, where the camera aperture is described as a point and no lenses are used to focus light ([29]). If two different images of the same scene are obtained using a pinhole camera, the corresponding transformation between the two images is the composition of two central projections from 3Dto 2D, which is a projective transformation ([30]). The projective shape of an object consists of the geometric information that is invariant under different camera views. Moreover, these images differ only by a projective 3 transformation between themselves and from the original scene, even if the images are taken with different cameras. The pinhole camera model as a projective model can be extended to general dimensions. The homogenous coordinates are unique up to non-zero scalars in the elegant projective geometric framework. Therefore, the homogenous coordinates become points in projective spaces, and the perspective projection landmark model in the DSNT experiment may now be studied in terms of projective shape geometry from 3Dto 3D. 2.1 Projective shape space We review in the following the mathematical results necessary to employ the projective shape analysis ([2]). In projective geometry two non-zero vectors xand yin (m+ 1)- dimensional numerical space Rm+1 are equivalent if they differ by a non-zero scalar multiple, where mis a abstract dimension of the image space. The equivalence class of x∈Rm+1 \ {0}is labeled [x].The set of all such equivalence classes is the projective space P(Rm+1) associated with Rm+1 P(Rm+1) = {[x]; x∈Rm+1 \ {0}}. and is often denoted as RPm. The projective space RPmis topologically an m-dimensional unit sphere Smwith the antipodal points identified, and can be represented as a disjoint union of ordinary and ideal projective points; RPm=              x1 . . . xm 1      ∈RPm         | {z } ordinary points [              x1 . . . xm 0      ∈RPm         | {z } ideal points In this framework for instance, a vanishing point in a three-dimensional image corresponds to an (ideal) point in RP3. We now review concepts of projective transformations and projective shape space. If the calibrations of the cameras in the pinhole camera model are unknown, i.e. if there is no information available on the camera parameters such as focal length, angle between scene and film hyperplane, location of the camera, etc., then an image relies only on information about the scene which is invariant under projective transformations ([30]). Reconstruction of a configuration of points in 3Dfrom two ideal non-calibrated camera images regarded as subsets in projective spaces with unknown camera parameters in absence of occlusions is known as the 3Dreconstruction problem ([31]). This leads to the projective ambiguity of the 3Dreconstruction problem: given two camera images of unknown relative position and internal camera parameters and two matched k−sets of labeled points in the projective space, find all k−sets of points in space as solutions of both 3Dreconstruction problem. It could be proven in [32] that a landmark correspondence is needed to obtain a 3Dreconstruction from any pair of 2D 4 images. This leads to the concept of projective k-ads, as presented below. Ak-ad is an ordered list of klabeled points in Rm; it can be also regarded as a k-ad in RPm, via the standard affine embedding of Rmin RPmgiven by p:Rm→RPm x= (x1, . . . , xm)→p(x)=[x1:· · · :xm: 1] = [(x1...,xm,1)T]. One approach to projective shape analysis ([33]), is based on the idea of a projective frame selected from the points of a finite generic k-ad in mdimensions. Such a representation has the advantage that it associates to the full set of projective invariants ([34]) of such a configuration one point on a projective shape manifold. The projective frame approach can be used to identify the projective shape of a planar curve in the context of a scene that contains four points in general position, that are not necessarily on the curve. The projective frame determined by such control points is used for registration. Ideally two registered images of the same scene should be identical; nevertheless due to registration errors and departure from a planar scene, they are different, and this raises questions about identification of the mean projective shape of a curve, testing for equality of such mean projective shapes of curves, etc ([35]). There are a few problems that arise in such a testing problem from curves in digital images. The identification of the actual curve in image processing, even from less noisy images, is first problem. Secondly, the curve registration problem presents challenges. Thirdly, the classical simple null hypothesis of equality of two mean curves, even when they arise form the same scene, is very likely to be rejected because of inherent registration errors; therefore a neighborhood null hypothesis for functional data is preferred ([36]). Finally once the test is established one has to dwell with intensive computational algorithms involved in functional data analysis. An ordered list of m+ 2 labeled points in RPmis said to form a projective frame (basis) if they span RPm.An ordered list of k≥m+ 2 labeled projective points in RPmare said to be in general position if the first m+ 2 of these points form a projective frame. G(k, m) is the space of all k-ads in general position in RPm. A projective transformation g=gPof RPmis the projective map associated with a nonsingular matrix P∈GL(m+ 1,R) and its action on RPm gP([x]) = g([x1:· · · :xm+1]) = [P(x1· · · xm+1)T] The projective transformations of RPmform a group, denoted by PGL(m). Division by the last coordinate yields a representation of a projective transformation gPin terms of affine coordinates, as v=f(u), with vj=aj m+1 + Σm i=1aj iui am+1 m+1 + Σm i=1am+1 iui,∀j= 1, ..., m where det(P) = det((aj i)i,j=1,...,m+1)) = 0. Two k-ads of points in Rmhave the same the projective shape if they differ by a projective transformation of Rm. 5 The projective shape of a k-ad X= ([x1],...,[xk]) ∈G(k, m) is the orbit of Xunder the action αof PGL(m) on G(k, m), given by α(gP, X) = (gP([x1]), . . . , gP([xk])). Projective shape space PΣk m, is the space of projective shapes of all k-ads in RPmin general position X= ([x1],...,[xk]), such that X= ([x1],...,[xm+2]) is a projective frame : PΣk m=G(k, m)/P GL(m). PΣk mis a manifold, homeomorphic with RPm×RPm× · · · × RPm | {z } qcopies = (RPm)q where q=k−m−2. Projective shape analysis focuses on the properties of a configuration of colinear or coplanar points, as they are seen in a central projection by an external observer and it can be seen as a simplified analysis of vision in absence of occlusions. The image fusing method is based on extrinsic means of projective shapes of configurations of landmarks. Projective shape analysis is identified with multivariate (axial) statistics and it can be interpreted in the context of directional statistics. The projective space RPmis topologically equivalent to a m-dimensional unit sphere Smwith the antipodal points identified. Therefore, PΣk mis a manifold which is homeomorphic with polysphere (Sm)qwith the antipodal points identified. Hence, the extrinsic shape analysis has an elegant interpretation as a directional statistics. 2.2 Extrinsic Statistical Analysis in Projective Shape Spaces The notions of extrinsic mean and extrinsic covariance on manifolds are at the center of our statistical analysis ([2]). Let Qbe a probability measure on (M, ρ), a complete metric space with a manifold structure of dimension m. If j:M → RNis an embedding, a minimizer µEof F(x) = Z||j(x)−j(y)||2Q(dy). is called the extrinsic mean (set) of Q. A point xof RNsuch that there is a unique pin Mfor which ρ0(x, j(M)) = ρ0(x, j(p)) is called j-nonfocal where ρ0is the Euclidean distance in RN.Fcis the set of jnonfocal points. A probability measure Qon Mis said to be j-nonfocal if the mean µof j(Q) is a j-nonfocal point. Aprojection Pj:Fc→j(M) maps any x∈ Fcto the unique ysuch that ρ0(x, j(M)) = ρ0(x, y). The extrinsic sample mean is defined accordingly (see [37]). 6 Theorem 1. Assume Qis a nonfocal probability measure on the manifold Mand X={X1, . . . , Xn}are i.i.d.r.o.’s from Q. (a) If the sample mean j(X)is a j-nonfocal point then the extrinsic sample mean is given by XE=j−1(Pj(j(X))). (b) XEis a strongly consistent estimator of µj,E(Q). Population extrinsic covariance matrix can be defined starting from the covariance of the embedded random object. Let (e1(y), e2(y), . . . , eN(y)) be an adapted frame to the embedding jaround Pj(µ) = j(µE) i.e. it is an orthonormal frame field such that (er(j(p)) = dpj(fr(p)), r = 1, . . . , m, ∀p∈j−1(Uj(µE)) where p→ (f1(p), . . . , fm(p)) is an orthonormal local frame field on an open subset of M. Let tan(v)=(e1(Pj(µ))Tv . . . em(Pj(µ))Tv)Tbe the tangential component of v. The extrinsic covariance matrix of the j-nonfocal distribution Qwith respect to the basis f1(µE), . . . , fm(µE) is defined as the covariance matrix of the random variable tan(Pj(j(X))) with respect to the j−adapted frame. Sample extrinsic covariance matrix is related to the sample covariance matrix of the embedded random object in the usual way. The sample extrinsic covariance matrix of the j-nonfocal distribution Qwith respect to the basis f1(µE), . . . , fm(µE) is defined as Sj,E,n =hXdj(X)Pj(eb)·ea(Pj(j(X)))ia=1,...,m·Sj,n hXdj(X)Pj(eb)·ea(Pj(j(X)))ia=1,...,mT where Sj,n is the sample covariance matrix of j(X). Sj,E,n is a consistent estimator of Σj,E. For the projective space RPm, we will use the Veronese-Whitney embedding jdefined in the following. RPmcan be equivariantly embedded in the space S(m+ 1) of (m+ 1) ×(m+ 1) of symmetric matrices via the Veronese-Whitney (V-W) embedding j:RPm→S(m+ 1), j([x]) = xxT, xTx= 1. The V-W equivariant embedding of the projective shape space PΣk mjk:PΣk m= (RPm)q→(S(m+ 1))qis defined by jk([x1],...,[xq]) = (j([x1]), . . . , j([xq])), where xs∈Rm+1, xT sxs= 1,∀s= 1, . . . , q (see [28,38]). If Yr, r = 1, . . . , n are i.i.d.r.o.’s for which the mean shape µjk exists, it has a multivariate axial representation Yr= ([X1 r],...,[Xq r]),(Xs r)TXs r= 1; s= 1, . . . , q. 7 Let Jsbe the random symmetric matrix given by Js=n−1 n X r=1 Xs r(Xs r)T, s = 1, . . . , q, and let ds(a) and gs(a) be the eigenvalues in increasing order and the corresponding unit eigenvector of Js, a = 1, . . . , m + 1. Then the sample mean projective shape is given by Yjk,n = ([g1(m+ 1)],...,[gq(m+ 1)]). From a general consistency theorem for extrinsic means on manifolds in [37], it follows that the extrinsic sample mean [Y]jk,n is a strongly consistent estimator of µjk. Theorem 2. In the case of the VW embedding jk, the sample extrinsic covariance matrix estimator Sjk,E,n is given by the (mq)×(mq)symmetric matrix Sj,E,n, with the entries in pairs of indices (s, a), s = 1, . . . , q;a= 1, . . . , m, in their lexicographic order given by Sj,E,n(s,a),(t,b)=n−1(ds(m+ 1) −dt(a))−1(dt(m+ 1) −dt(b))−1· n X r=1 (gs(a)TXs r)(gt(b)TXt r)(gs(m+ 1)TXs r)(gt(m+ 1)TXt r). Hotelling’s type statistics will alow us to apply a one-sample statistical test for random objects on projective shape spaces. Let Ds= (gs(1) . . . gs(m)) ∈ M(m+ 1, m;R), s = 1, . . . , q. If µ= ([γ1],...,[γq]),where γs∈Rm+1, γT sγs= 1,for s= 1, . . . , q, we define a Hotelling’s T2-type statistic T(Yjk,n;µ) = n(γT 1D1, . . . , γT qDq)S−1 j,E,n(γT 1D1, . . . , γT qDq)T. Theorem 3. Assume (Yr)r=1,...,n are i.i.d.r.o.’s on (Rpm)q, and Y1is jk-nonfocal, with ΣE>0. Let λs(a)and γs(a)be the eigenvalues in increasing order and corresponding unit eigenvectors of E[Xa i(Xa i)T]. If λs(1) >0, for s= 1, . . . , q, then T(Yjk,n;µjk)converges weakly to χ2 mq. Remark 1. Assume (Yr)r=1,...,n are i.i.d.r.o.’s from a jk-nonfocal probability distribution on (RPm)q, and ΣE>0. An asymptotic (1−α)-confidence region for µjk = [v] is given by Rα(V) = {[v] : T(Yjk,n; [v]) ≤χ2 mq,α}. If the probability measure of Y1has a nonzero absolutely continuous component w.r.t. the volume measure on (RPm)q, then the coverage error of Rα(V)is of order O(n−1). 3 Spatial Orientation in Virtual Environments A major progress in studying hippocampus-dependent spatial navigation in humans is due to the development of virtual reality techniques. This allowed testing navigational skills in various environments by enabling complete control over the complexity 8 of the task and objective recording of behavioural responses. Virtual environments can also be used to investigate spatial cognition since strong correlations have been observed between navigation in the real world and in the virtual reality. In this case, the assumed spatial navigation processing is the allocentric reference frame where the object locations are processed in reference to each other or fixed landmarks, independent of the observer’s position in the environment. The medial temporal lobe, especially the hippocampus, plays an important role in the allocentric spatial navigation, as indicated in many studies and confirmed by the poor performance of patients with AD and MCI in the tests requiring the use of allocentric memory. The Dresdner Spatial Navigation Task belongs to the family of of spatial tests conforming the allocentric principles of Morris Water Maze that is designated as hidden goal or hidden goal tasks. 3.1 The MCI data set The mild cognitive impairment (MCI) is considered to be the link between cognitive changes in the healthy aging persons and those who are predisposed to developing dementia, usually the Alzheimer disease (AD). Hence, the decline in spatial navigation skills can be seen as a cognitive marker in diagnosing AD. The MCI data set, where a group of 54 participants, 25 diagnosed healthy, 26 diagnosed with MCI, and 3 presenting te symptoms of minor dementia of the Alzheimer type (DAT) performed this task, and were additionally subjected to a set of neuropsychological tests. The group consisted of 31 women and 23 men between 51 and 79 years old (mean age 69.1 years). Since the number of DAT patients is not high enough to constitute a separate group, they have been incorporated into the MCI group. The individual planar curves are irregular (the step size around 10ms), and the starting point varies between trials. The final point (aka goal) and the scientific (topological) landmarks are fixed for all participants and all trials. The movement is constraint by the choice of the domain (circular in this case) and presents a drift in the direction of the goal (see Figure 1). When the participants reach the goal, they stop, while some participants can give up the test earlier from other reasons, like lack of concentration or motivation. 3.2 3DExtrinsic Projective Shape Statistical Analysis A data point in (Rm)kcorresponding to the 3D−relative position of the klandmarks with respect to the path position is regarded as a k-ad in RPm, via the standard affine embedding of Rmin RPm. 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Fig. 2: The empirical distribution of the path extrinsic means and the population extrinsic mean for trials 2,5,6,9,11 and for landmark 1 (left:control, right:disease) Fig. 3: The population extrinsic means of the landmark 1,3,11,15,21,71 for trials 1−12 (left:control, right:disease) 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 Fig. 4: The one sample test for equality with the goal projective shape (left:control, right:disease) 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 123456789 11 1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 Fig. 5: The paired sample test for equality of the extrinsic means of the goal projective shape between consecutive trials (left:control, right:disease)