Currents and K-functions for Fiber Point Processes
Pernille EH. Hansen, Rasmus Waagepetersen, Anne Marie Svane, Jon Sporring, Hans JT. Stephensen, Stine Hasselholt, Stefan Sommer
CCurrents and K -functions for Fiber PointProcesses Pernille EH. Hansen − − − , RasmusWaagepetersen − − − , Anne Marie Svane − − − ,Jon Sporring − − − , Hans JT. Stephensen − − − Stine Hasselholt − − − , and Stefan Sommer − − − Department of Computer Science, University of Copenhagen, Copenhagen,Denmark { pehh, sporring, hast, sommer } @di.ku.dk Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark { rw, annemarie } @math.aau.dk Stereology and Microscopy, Aarhus University, Aarhus, Denmark [email protected]
Abstract.
Analysis of images of sets of fibers such as myelin sheaths orskeletal muscles must account for both the spatial distribution of fibersand differences in fiber shape. This necessitates a combination of pointprocess and shape analysis methodology. In this paper, we develop a K -function for shape-valued point processes by embedding shapes ascurrents, thus equipping the point process domain with metric structureinherited from a reproducing kernel Hilbert space. We extend Ripley’s K -function which measures deviations from spatial homogeneity of pointprocesses to fiber data. The paper provides a theoretical account of thestatistical foundation of the K -function and its extension to fiber data,and we test the developed K -function on simulated as well as real datasets. This includes a fiber data set consisting of myelin sheaths, visu-alizing the spatial and fiber shape behavior of myelin configurations atdifferent debts. Keywords: point processes, shape analysis, K -function, fibers, myelinsheaths. We present a generalization of Ripley’s K -function for shape-valued point pro-cesses, in particular, for point processes where each observation is a curve in R , a fiber. Fiber structures appear naturally in the human body, for examplein tracts in the central nervous system and in skeletal muscles. The introduced K -function captures both spatial and shape clustering or repulsion, thus pro-viding a powerful descriptive statistic for analysis of medical image of sets offiber or more general shape data. As an example, Fig. 1 displays myelin sheathsin four configurations from different debts in a mouse brain. We develop themethodology to quantify the visually apparent differences in both spatial andshape distribution of the fibers. a r X i v : . [ s t a t . M E ] F e b PEH. Hansen et al.
Ripley’s K -function [6] is a well-known tool for analyzing second order momentstructure of point processes [1] providing a measure of deviance from completespatial randomness in point sets. For a stationary point process, K ( t ) gives theexpected number of points within distance t from a typical point. An estimatorof Ripley’s K -function for a point set { p i } ni =1 inside an observation window W is, ˆ K ( t ) = 1 n ˆ λ (cid:88) i (cid:54) = j p i , p j ) < t ] (1)where ˆ λ = n | W | is the sample intensity, | W | is the volume of the observationwindow, and 1 is the indicator function. By comparing ˆ K ( t ) with the K -functioncorresponding to complete spatial randomness, we can measure the deviationfrom spatial homogeneity. Smaller values of ˆ K ( t ) indicate clustering whereas thepoints tend to repel each other for greater values.Generalizations of Ripley’s K -function have previously been considered forcurve pieces in [3] and several approaches were presented in [7] for space curves.In this paper, we present a K -function inspired by the currents approach from[7] and provide the theoretical account for the statistical foundation.The challenges when generalizing the K -function to shape-valued point pro-cesses arise in defining a distance measure on the shape space and determininga meaningful descriptive quantity that is well-defined and that we are able toestimate. In this paper, we provide a well-defined K -function for point processesin general metric spaces and use the embedding of shapes as currents to obtaina distance measure of shapes. We construct the following two-parameter K -function for a curve-valued pointprocess X ˆ K ( t, s ) = 1 | W | λ (cid:88) γ ∈ X : c ( γ ) ∈ W (cid:88) γ (cid:48) ∈ X \{ γ } (cid:107) c ( γ ) − c ( γ (cid:48) ) (cid:107) ≤ t, d m ( γ, γ (cid:48) ) ≤ s ] (2)for t, s >
0, where c ( γ ) is the center point of the curve γ , d m the minimalcurrents distance with respect to translation, and λ is the spatial intensity ofthe center points. By introducing a second distance parameter, we are able toseparate the spatial distance of the curves from the difference in shape, allowingus to measure both spatial and shape homogeneity.The paper thereby presents the following contributions:1. A K -function for shape-valued point processes along with a theoretical ac-count for the statistical foundation.2. We suggest a certain fiber process which we argue corresponds to completerandomness of points, an analogue of the Poisson process.3. An application of the K -function to several generated data set and a realdata set of myelin sheaths. urrents and K -functions for Fiber Point Processes 3ST01 ST06 ST17 ST20 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) . . . . . . . . K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) . . . . . . . . K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 Fig. 1: K -functions for samples of myelin sheaths. Each column corresponds tomeasured on a data set. (Top row): the centerlines of the myelin sheathed ax-ons. (Middle row): The K -function for fixed values of t . (Bottom row): The K -function for fixed values of s . We model a random collection of shapes as a point process on the space ofshapes. Shape spaces are usually defined as the space of embeddings B e ( M , R d )of a manifold M into R d [2]. For example, the space of closed curves in R is B e ( S , R ), where S denotes the 1-sphere, and the space of fibers is B e ( I, R )for some real interval I .Formally, a point process X on a metric space S is a measurable map fromsome probability space ( Ω, F , P ) into the space of locally finite subsets of S .Thus, for each ω ∈ Ω , X ( ω ) ⊆ S , and for every compact Borel set B ⊆ S , X ( ω ) ∩ B is a finite set. Measurability of X means that all sets of the form { ω ∈ Ω | X ( ω ) ∩ B ) = m } , where m ∈ N and B ⊆ S is a Borel set, must bemeasurable.There are different ways to endow B e ( M , R d ) with a metric [5]. In this paper,we consider the representation of shapes as currents embedded in the dual space PEH. Hansen et al. of a reproducing kernel Hilbert space (RKHS). Thus, the RKHS metric inducesa metric for our shape space. This can be combined with the Euclidean metric on R d to obtain a suitable metric on B e ( M , R d ). This approach is very useful dueto its generality and computability, as it requires very little information aboutthe shape. Shapes are usually more difficult to work with than points, as they usuallycannot be captured in any finite dimensional vector space. An approach alreadyconsidered for anatomical structures [4] [8] is embedding shapes as currents.We will give a brief introduction to this setup and refer to [4] for a detaileddescription. We can characterize a piece-wise smooth curve γ ∈ B e ( I, R d ) bycomputing its path-integral of all vector fields wV γ ( w ) = (cid:90) γ w ( x ) t τ ( x ) dλ ( x ) , (3)where τ ( x ) is the unit tangent of γ at x and λ is the length measure on the curve.Likewise, an oriented hypersurface S embedded in R d can be characterized byits flux integral of vector fields wV S ( w ) = (cid:90) S w ( x ) t n ( x ) dλ ( x ) , (4)where n ( x ) is the unit normal at x and λ is the surface area measure on S . Theseare both examples of representing shapes as currents, i.e. as elements in the dualspace of the space of vector fields on R d . Formally, the space of m -currents C m is the dual space of the space C ( R d , ( Λ m R d ) ∗ ) of differential m -forms.It is not only curves and hypersurfaces that can be represented as currents.Let M be an oriented rectifiable sub-manifold of dimension m in R d with pos-itively oriented basis of the tangent space u ( x ) , ..., u m ( x ) for all x ∈ M . Thesub-manifold M can be embedded into the space of m -currents as the current T M ( w ) = (cid:90) M I ( x ) w ( x ) (cid:16) u ( x ) ∧ ... ∧ u m | u ( x ) ∧ ... ∧ u m | (cid:17) dλ ( x ) (5)where w ∈ C ( R d , ( Λ m R d ) ∗ ) is an m -differential form and I : T → R is ascalar function satisfying (cid:82) T | I ( x ) | dλ ( x ) < ∞ [4]. Since shapes are embeddedsub-manifolds, this means that shapes can be embedded into C m . The space of m -currents C m is continuously embedded into the dual space of areproducing kernel Hilbert space (RKHS) H with arbitrary kernel K H : R d × R d → R d × d [4]. It follows from Riesz representation theorem that v ∈ H can urrents and K -functions for Fiber Point Processes 5 be embedded in the dual space H ∗ as the functional L H ( v ) ∈ H ∗ defined by L H ( v )( w ) = (cid:104) v, w (cid:105) H for w ∈ H .Elements v ( y ) = K H ( x, y ) α form a basis for H where x, α ∈ R d , and basiselements in H are lifted to basis elements in H ∗ as δ αx := L H ( v ) which are calledthe Dirac delta currents. The element V γ from (3) can be written in terms of thebasis elements δ τ γ ( x i ) x i where τ γ ( x i ) are the unit tangent vectors of γ at x i . Thismeans that the curve γ is embedded into H ∗ as the 1-current V γ ( w ) = (cid:90) γ w ( x ) t τ γ ( x ) dλ ( x ) = (cid:90) γ δ τ γ ( x ) x ( w ) dλ ( x ) (6)where λ is the length measure on the curve. Furthermore, it is approximatedby the Riemann sum of Dirac delta currents V γ ( w ) ≈ ˜ V γ ( w ) = (cid:80) i δ τ ( x i ) ∆x i x i ( w )where x i are sampled points along the curve according to λ . The dual space H ∗ inherits the inner product from the inner product on the RKHS via the inversemapping L − H , so that the inner product for two curves γ and γ in H ∗ is (cid:104) V γ , V γ (cid:105) H ∗ = (cid:90) γ (cid:90) γ τ tγ ( x ) K H ( x, y ) τ γ ( y ) dλ γ ( x ) dλ γ ( y ) . (7)Writing || V γ || H ∗ = (cid:104) V γ , V γ (cid:105) H ∗ , we finally arrive at the currents distance of twocurves γ and γ d c ( V γ , V γ ) = || V γ − V γ || H ∗ = (cid:16) || V γ || H ∗ + || V γ || H ∗ − (cid:104) V γ , V γ (cid:105) H ∗ (cid:17) / . (8)In practice, we usually don’t know the orientation of the curves, thus we chooseto consider the minimal distance between them, d ( V γ , V γ ) = min { d c ( V γ , V γ ) , d c ( V γ , V − γ ) } (9)where − γ denotes the curve with opposite orientation of γ . If the orientationof the data is important, this step may be omitted. From (6) we see that K H serves as a weight of the inner product between τ ( x i ) and τ ( y j ) depending onthe positions x i and y j . To illustrate the distance metric, consider the generalized Gaussian kernel K pσ ( x, y ) = exp (cid:16) −| x − y | p σ p (cid:17) Id , (10)where σ, p ∈ (0 , ∞ ], and consider two lines of equal length parametrized by l u ( t ) = x u + ut, l v ( t ) = x v + vt where x u , x v , u, v ∈ R d and 0 < t ≤ T ∈ R . Forvery short lines far from each other, i.e., T / | x u − x v | →
0, we have, d ( ˜ V l u , ˜ V l v ) T → d − (cid:0) −| x u − x v | p σ p (cid:1) d , (11) PEH. Hansen et al. where d = u t u + v t v and d = max ( u t v, − u t v ). Since the d and d are con-stants, and since the exponential and the square root functions are both mono-tonic, then in the limit, d ( ˜ V l u , ˜ V l v ) /T is one-to-one with | x u − x v | p which isone-to-one with | x u − x v | . Thus, for very short lines, d/T is one-to-one with theeuclidean distance between the points x u , and x v . Further, in the limit p → ∞ we have, d ( ˜ V l u , ˜ V l v ) T → d − d , when | x u − x v | < σ,d − (cid:0) − (cid:1) d , when | x u − x v | = σ,d , otherwise . (12)Thus, for very short lines and very large exponents, ( d − d /T ) / (2 d ) convergesto a unit step function in | x u − x v | where the step is at σ . K -function Let S be the image of the embedding of B e ( I, R d ) into C . For brevity, γ isidentified with its representation in C . We model a random collection of curvesas a point process X in S .Let c : S → R d be a center function on the space of fibers in R d that associatesa center point to each fiber. A center function should be translation covariant inthe sense that c ( γ + x ) = c ( γ ) + x for all x ∈ R d . It could be the center of massor the midpoint of the curve with respect to curve length. Let S c denote thespace of centered fibers wrt. c , i.e., those γ ∈ S for which c ( γ ) = 0. We define γ c := γ − c ( γ ) ∈ S c to be the centering of γ .For Borel sets B , A ⊂ R d and B , A ⊂ S c , define the first moment measure µ ( B × B ) = E (cid:88) γ ∈ X c ( γ ) ∈ B , γ c ∈ B ]and the second moment measure α (( A × A ) × ( B × B )) = E (cid:54) = (cid:88) γ,γ (cid:48) ∈ X c ( γ ) ∈ A , γ c ∈ A ]1[ c ( γ (cid:48) ) ∈ B , γ (cid:48) c ∈ B ] . We assume that µ is translation invariant in its first argument, i.e. µ ( B × B ) = µ (( B + h ) × B )for any h ∈ R d . This is for instance the case if the distribution of X is invariantunder translations. This implies that µ ( · × B ) is proportional to the Lebesguemeasure for all B . Thus we can write µ ( B × B ) = | B | ν ( B ) urrents and K -functions for Fiber Point Processes 7 for some measure ν ( · ) on S c . Note that the total measure ν ( S c ) is the spatialintensity of the center points, i.e. the expected number of center points in a unitvolume window. In applications, this will typically be finite. In this case, we maynormalize ν to obtain a probability measure which could be interpreted as thedistribution of a single centered fiber.We define the reduced Campbell measure C ! ( A × A × F ) = E (cid:88) γ ∈ X c ( γ ) ∈ A , γ c ∈ A , X \ { γ } ∈ F ]where the ”!” represents the removal of the point γ from X . By disintegration, C ! ( A × A × F ) = (cid:90) A × A P ! c,γ c ( F ) µ (d( c, γ c )) . By the standard proof, we get for any measurable function h : R d × S c × N → [0 , ∞ ) E (cid:88) γ ∈ X h ( c ( γ ) , γ c , X \ { γ } ) = (cid:90) R d × S c E ! c,γ c h ( c, γ c , X ) µ (d( c, γ c )) . (13)In particular, α (( A × A ) × ( B × B )) = (cid:90) A × A E ! c,γ c (cid:88) γ (cid:48) ∈ X c ( γ (cid:48) ) ∈ B , γ (cid:48) c ∈ B ] µ (d( c, γ c )) . (14)Assume also that α is invariant under joint translation of the arguments A , B .Then K c,γ c ( B × B ) := E ! c,γ c (cid:88) γ (cid:48) ∈ X c ( γ (cid:48) ) ∈ B , γ (cid:48) c ∈ B ] (15)= E !0 ,γ (cid:88) γ (cid:48) ∈ X c ( γ (cid:48) ) ∈ B − c, γ (cid:48) c ∈ B ] (16)= K ,γ (( B − c ) × B ) . (17)Assume that also E ! c,γ c h ( c, γ c , X ) does not depend on c , which is true if the distri-bution of X is invariant over translations. Then, using (13) and the factorizationof µ , E (cid:88) γ ∈ X c ( γ ) ∈ W ] h ( c ( γ ) , γ c , X \ { γ } ) = (cid:90) W × S c E ! c,γ c h ( c, γ c , X ) µ (d( c, γ c ))= (cid:90) W × S c E !0 ,γ h (0 , γ , X ) µ (d( c, γ )) = | W | (cid:90) S c E !0 ,γ h (0 , γ , X ) ν (d γ ) (18)From this it follows that E h = 1 | W | (cid:88) γ ∈ X c ( γ ) ∈ W ] h ( c ( γ ) , γ c , X \ { γ } ) PEH. Hansen et al. is an unbiased estimator of (cid:82) S c E !0 ,γ h (0 , γ , X ) ν (d γ ) . Furthermore, if ν ( S c ) isfinite, (cid:90) S c E !0 ,γ h (0 , γ , X ) ν (d γ ) = ν ( S c ) E ˜ ν E !0 ,Γ h (0 , Γ , X )where Γ is a random centered fiber with distribution ˜ ν ( · ) = ν ( · ) /ν ( S c ) and E ˜ ν is expectation with respect to this distribution of Γ . K -function for fibers In order to define a K -function, we must make an appropriate choice of h . Aseemingly natural choice for h that coincides with [7], is h ( c, γ c , X ) = (cid:88) γ (cid:48) ∈ X d ( γ c + c, γ (cid:48) ) ≤ t ] = (cid:88) γ (cid:48) ∈ X d ( γ, γ (cid:48) ) ≤ t ] . However this choice allows the K -function to be a.s. infinite, due to the fact that d ( γ, γ (cid:48) ) ≤ (cid:112) || γ || H ∗ + || γ (cid:48) || H ∗ ). If every curve in X has || γ || H ∗ ≤ M , e.g. ifthe length of fibers is bounded, then choosing t ≥ M results in any fiber in X having infinitely many neighbors within distance t .A solution is to separate the spatial distance of the curves from the differ-ence in shape by introducing another radius parameter for the distance betweencenter points. Accounting for spatial distance with this parameter, we chooseto minimize the influence of spatial distance by measuring the currents distancebetween the centered curves. Thus, we choose h as h ( c, γ c , X ) = (cid:88) γ (cid:48) ∈ X \{ γ } (cid:107) c ( γ ) − c ( γ (cid:48) ) (cid:107) ≤ t, d ( γ c , γ (cid:48) c ) ≤ s ] (19)for s, t >
0, where || c ( γ ) − c ( γ (cid:48) ) || is the usual distance in R d between centerpoints. Thus we define the empirical K -function for t, s > K ( t, s ) = 1 | W | ν ( S c ) (cid:88) γ ∈ X : c ( γ ) ∈ W,γ (cid:48) ∈ X \{ γ } (cid:107) c ( γ ) − c ( γ (cid:48) ) (cid:107) ≤ t, d ( γ, γ (cid:48) ) ≤ s ] . (20)Since ν ( S c ) is the intensity of fiber centers, it is estimated by N/ | W | where N isthe observed number of centers c ( γ ) inside W . The K -function is the expectationof the empirical K -function K ( t, s ) = E ˆ K ( t, s ) = E ˜ ν E !0 ,Γ h (0 , Γ , X ) = E ˜ ν K ,Γ ( B (0 , t ) × B c ( Γ , s ))where B ( x, r ) = { y : || x − y || ≤ r } and B c ( γ, s ) = { γ (cid:48) : d ( γ, γ (cid:48) ) ≤ s } . K -function for general shapes The currents metric and the K -function easily extends to shape-valued pointprocesses with values in B e ( M , Ω ) for more general manifolds M and Ω ⊂ R d . urrents and K -functions for Fiber Point Processes 9 Shapes A , B ∈ B e ( M , R d ) are embedded as m -currents V A and V B as in (5).Since C m is continuously embedded into the dual RKHS H ∗ , we get the distancemeasure d c between shapes.If c : B e ( M , Ω ) → R d is a center function, then we can generalize the K -function to a point process X with values in B e ( M , Ω ). Identifying elements of B e ( M , Ω ) with their embedding in H ∗ , we can write the same K -functionˆ K ( t, s ) = 1 | W | λ (cid:88) U∈ X : c ( U ) ∈ W, U (cid:48) ∈ X \{U} (cid:107) c ( U ) − c ( U (cid:48) ) (cid:107) ≤ t, d m ( U , U (cid:48) ) ≤ s ] (21)for t, s >
0, where d m is constructed as in Section 3 . λ is the spatialintensity of the center points. To obtain a measure of spatial homogeneity, Ripley’s K -function for points isusually compared with the K -function for a Poisson process, K P ( t ) = vol( B d ( t )),corresponding to complete spatial randomness. We are now in a more compli-cated situation where the K -function has two parameters and we do not have anotion of complete randomness of fibers. The aim of the experiments on gener-ated data sets is to analyze the behavior of the K -function on different types ofdistributions and suggest a fiber process that corresponds to complete random-ness. This will serve as a way to compare the results in 4.2. The four generated data sets X , X , X and X each contain 500 fibers withcurve length l = 40 and center points in [0 , and is visualized in the first rowof Fig. 2. Each data set is created by sampling center points from a distributionon R and fibers from a distribution on S , that is then translated by the centerpoints. For the first three data sets, the center points are generated by a Poissonprocess and the fibers are uniformly rotated lines in X , uniformly rotated spiralsin X and Brownian motions in X . The data set X has clustered center pointsand within each cluster the fibers are slightly perturbed lines.To avoid most edge effects, we choose the window W ≈ [13 , ⊂ R for thecalculation of the K -function. Furthermore, we choose a Gaussian kernel K σ asin (10) with p = 2 and σ = . Finally, c is defined to be the mass center of thecurve.The first row of Fig. 2 shows the generated data sets and the respective K -functions are visualized the second and third row. In the second row, s (cid:55)→ K ( t, s )is plotted for fixed values of t . For example, the graphs with t = 50 show theexpected numbers of fibers within currents distance s , where the distance ofcenter points are 50 at most. Lastly in the third row, t (cid:55)→ K ( t, s ) is plotted forfixed values of s . Similarly, the graphs with s = 70 show the expected numbers −
25 0 25 50 75 100 125 −
25 0 25 50 75100125 − − s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 s : ( d ( γ,γ ) ) K ( s , t ) t=90t=70t=50t=30t=10 t : ( || c ( γ ) − c ( γ ) || ) K ( s , t ) s=90s=70s=50s=30s=10 Fig. 2: K -function on the generated data, where each column corresponds to onedata set. (Top row): The data sets X , X , X and X described in 4.1. (Middlerow): The K -function of the data set above for fixed values of t . (Bottom row):The K -function of the data set above for fixed values of s .of fibers with center point distance t when the currents distances are 70 at most.Thus, the graphs in the second row capture the fiber shape difference of eachdata set whereas the graphs in the third row capture spatial difference.It is distribution X that we consider to a natural suggestion for a uniformrandomness distribution of fibers. This is because Brownian motions are well-known for modelling randomness, thus representing shape randomness. And bytranslating these Brownian motion with a Poisson process, we argue that thisdistribution is a good choice.Considering only the data sets with uniformly distributed center points, i.e.,the first three columns of Fig. 2, we see a big difference in the second row ofplots. This indicates that the K -function is sensitive to the change in shape. The K -function for the Brownian motions captures much more mass for smaller radiicompared to the lines, with the spirals being somewhere in-between. The plotsin the third row are very much as expected, since we generated the center pointsfrom a Poisson process. Finally, the second row plot for X indicate a slight shapeclustering when compared to the uniformly rotated lines. This makes sense, sinceeach cluster is directed differently. urrents and K -functions for Fiber Point Processes 11 Myelin surrounds the nerve cell axons and is an example of a fiber structurein the brain. Based on 3D reconstructions from the region motor cortex of themouse brain, centre lines were generated in the myelin sheaths. The data setsST01, ST06, ST17 and ST20 displayed in the first row of Fig. 1 represent themyelin sheaths from four samples at different debts.For real shape-valued data sets, it very common that only parts of the shapesare observed. This is the case for many fiber data sets as well. This fact isimportant to have in mind when choosing c , since we should have a clear ideaof when c ( γ ) is observed, in order to get an unbiased estimate.Since myelin sheaths tend to be quite long, we chose to divide the fibers oflength greater than 40 into several fibers segments of length 40. This has thebenefits, that the mass center is an appropriate choice for c and that the resultsare comparable with the results of Fig. 2, since the curves are of similar length.The results of the estimated K-function on the four data sets ST01, ST06,ST17 and ST20 are visualized in Fig. 1, where s (cid:55)→ K ( t, s ) is plotted in thesecond row for fixed values of t and t (cid:55)→ K ( t, s ) is plotted in the third row forfixed values of s . The plots in the second row showing the fiber shape are verysimilar, resembling the fiber distribution of X . We notice a slight difference inST20, where the graphs have a more pronounced cut off. When noticing thescale of the y -axis, we see that the expected number of neighbor fibers varysignificantly between the data sets.The third row plots indicate that the center point distributions of each dataset is similar to the center point distribution of X , X and X , which we gen-erated from a Poisson process. For ST17, we notice a slight clustering of centerpoints for t ∈ [10 , t = 30, indi-cating that the neighbors for fibers i ST20 are of more similar shape than theothers. References
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