aa r X i v : . [ s t a t . M E ] F e b Dynamic Shape Analysis andComparison of Leaf Growth
Stephan HuckemannJune 9, 2018
Abstract
In the statistical analysis of shape a goal beyond the analysis of staticshapes lies in the quantification of ‘same’ deformation of different shapes.Typically, shape spaces are modelled as Riemannian manifolds on whichparallel transport along geodesics naturally qualifies as a measure for the‘similarity’ of deformation. Since these spaces are usually defined as com-binations of Riemannian immersions and submersions, only for few wellfeatured spaces such as spheres or complex projective spaces (which areKendall’s spaces for 2D shapes), parallel transport along geodesics canbe computed explicitly. In this contribution a general numerical methodto compute parallel transport along geodesics when no explicit formulais available is provided. This method is applied to the shape spaces ofclosed 2D contours based on angular direction and to Kendall’s spaces ofshapes of arbitrary dimension. In application to the temporal evolutionof leaf shape over a growing period, one leaf’s shape-growth dynamics canbe applied to another leaf. For a specific poplar tree investigated it isfound that leaves of initially and terminally different shape evolve ratherparallel, i.e. with comparable dynamics.
Key words and phrases: geodesics, parallel–transport, Riemannian im–/submersion,shape analysis, forest biometry, leaf growth
AMS 2000 Subject Classification:
Primary 62H35Secondary 53C22
For more than two millenia, the analysis of form of biological entities has been anenticing object of human occupation. While early work tended to be speculativein nature, the current state of mathematics and computational power allowto develop and simultaneously verify theoretical results, thereby increasinglydriving scientific progress as witnessed today.This present work has been motivated by joint research with the Institutefor Forest Biometry and Informatics at the University of G¨ottingen, to compare1eaf growth dynamics within single specimen, species and taxa for identificationof gene expression. The endeavor is challenging as it touches problems at leastas old as Theophrastus’ (371 – 287 b.C.) famous book on “plant growth”, cf.Theophrastus (1976).The first step of this project is the subject of this work: to develop a frame-work allowing to compare shape dynamics. To this end, we model biologi-cal growth by (generalized) geodesics in shape space. We do so because thegeometry of shape spaces in which travel along geodesics requires no energyseems linked to the physiological reality of growth preferring to minimize en-ergy. This “geodesic hypothesis”, originally stated by Le and Kume (2000))is further supported by earlier research, cf. Huckemann and Ziezold (2006);Hotz et al. (2010). As with the “geodesic hypothesis” one can say that geodesicshape deformation of two different shapes is the “same” if the impetus of thefirst deformation is transplanted to the second with no loss of energy. In the lan-guage of Riemannian geometry this translates to the condition that the initialvelocity of the second geodesics is the parallel transport of the initial velocity ofthe first geodesic. If the deformations are not the same, i.e. the geodesics are notparallel at the first and the second shape, this concept gives a correlation-baseddistance between the deformations.In consequence, the aim of this paper is to provide for parallel transport onshape spacs. Recall that most shape spaces can be viewed as Riemannian im-mersions or submersions or, combinations thereof. Explicit formulae for paralleltransport are only available for special spaces. e.g. for spheres and Kendall’sspaces of planar shapes, cf. Huckemann et al. (2009). In general, parallel trans-port may be difficult to compute and be only available numerically. In thefollowing Section 2 we provide for a general method to compute parallel trans-port on shape spaces. In view of our application the method is illustratedin Section 3 for the spaces of closed 2D contours based on angular directionwith and without specific initial point (cf. Zahn and Roskies (1972) as wellas Klassen et al. (2004)), and in Section 4 for Kendall’s landmark based shapespaces (e.g. Dryden and Mardia (1998)).In Section 5, we compare parallel transport on the spaces of closed contourswith parallel transport on Kendall shape spaces for simple regular polygonal con-figurations. While all sectional curvatures in Kendall’s shape space are boundedfrom below by 1, it turns out that the corresponding subspace of closed contoursis flat.Finally in Section 6, leaf growth of one leaf is transported parallelly to otherleaves and both shape evolutions are compared with one another. For a spe-cific Canadian black poplar investigated we find that leaves with initially andterminally different shapes tend to evolve parallel, in particular so if no shapeanomalies are present. Thus the geodesic hypothesis can be extended to the parallel hypothesis : biological growth of related objects, possibly of initially and terminally differentshape, tends to follow parallel geodesics, Using Euclidean approximations in landmark based shape spaces rather than2eodesics, this hypothesis was originally coined by Morris et al. (2000) who ob-served parallel growth patterns. Readers primarily interested in the applicationcan directly skip to Section 6.
This section begins with a review of basic concepts of Riemannian geometryfound in any standard textbook (specifically Lang (1999) is very appropriate forthe following), in particular formulae relating covariant derivatives of Rieman-nian immersions and submersions. These provide differential equations liftingthe parallel transport on shape space to Euclidean or Hilbert space.For a Riemannian manifold M , possibly of countable dimension denote by h V p , W p i M the Riemannian metric of tangent spaces and by ∇ MV W the covariantderivative of vector-fields. Here V, W ∈ T ( M ) denote vector-fields with values V p , W p in the tangent space T p M of M at p ∈ M . d M ( p, p ′ ) is the inducedmetrical distance on M for p, p ′ ∈ M , V ⊗ W denotes the outer product definedby ( V ⊗ W ) X = h X, V i W . A vector-field W ∈ T ( M ) is parallel along a smoothcurve t → γ ( t ) on M if it satisfies the ordinary differential equation (ODE) ∇ ˙ γ W = 0 . (1)It is well known that there is locally a unique solution W along γ for a giveninitial value. In Euclidean or Hilbert space the left hand side has the simpleform (2).In particular, geodesics are characterized by the fact that their velocity isparallel: ∇ ˙ γ ˙ γ = 0 . The covariant derivative is often called a covariant connection . Indeed, iftwo offsets p, p ′ ∈ M can be joined by a unique geodesic segment of minimallength, their respective tangent spaces are connected via parallel transport (PT). Definition 2.1. w ′ ∈ T p ′ M is the parallel transplant of w ∈ T p M if there are1. a unique unit speed geodesic t → γ ( t ) connecting p = γ (0) with p ′ = γ (cid:0) d M ( p, p ′ ) (cid:1) , and2. a vector field W ∈ T ( M ) parallel along γ with W p = w, W p ′ = w ′ . A sufficient condition for the existence of such a unique connecting geodesicis that M is finite dimensional and p ′ is sufficiently close to p . In case ofinfinite dimension, examples of complete spaces can be constructed which do notfeature minimizing geodesics between arbitrary close points (e.g. Lang (1999,pp.226/7)). For our applications in mind this fact seems less troublesome sinceinfinite dimensional spaces considered here are built from projective limits offinite dimensional spaces.The Euclidean and Hilbert spaces R n (for Hilbert space n = ∞ ) can beidentified with all of their tangent spaces, i.e. h v, w i R n = P ni =1 v i w i and the3ovariant derivative is just the usual multivariate derivative by components, ∇ R n ( v ,...,v n ) ( w , . . . , w n ) = n X i =1 v i (cid:18) ∂w ∂x i , . . . , ∂w n ∂x i (cid:19) . In particular, if v = ˙ x ( t ), i.e. dx i dt = v i we have that ∇ R n ˙ x ( t ) W = ddt W x ( t ) . (2)Thus as desired, parallel transport on Euclidean and Hilbert spaces is givenby affine translations.For short we write W ( t ) for the value of W along a selfunderstood smoothcurve t → γ ( t ) and ˙ W ( t ) := ddt W γ ( t ) in the Euclidean/Hilbert case.A surjective linear mapping f : E → F of topological vector-spaces splitsin F if kern( f ) has a closed complement e F in E such that e F × kern( f ) ∼ = E as topological vector-spaces, in particular, kern( f ) → E → F is a short exactsequence . Another wording is that F splits over E .A smooth mapping Φ : M → N of Riemannian manifolds M and N inducesa differential mapping d Φ p : T p M → T Φ( p ) N of tangent spaces. Φ is calledan immersion if Φ is injective and if every T Φ( p ) N splits over d Φ p T p M ,a submersion if Φ is surjective and if every d Φ p splits in T Φ( p ) N ,an isometry if h V p , W p i M = h d Φ p V p , d Φ p W p i N , ∀ p ∈ M and V, W ∈ T ( M ) . An isometric immersion (submersion) is a
Riemannian immersion (submersion) respectively.
Riemannian Immersions.
If Φ : M → N is a Riemannian immersion thenthe tangent spaces of N split into the tangent spaces of Φ( M ) and its orthogonalcomplements, the normal spaces T Φ( p ) N = T Φ( p ) Φ( M ) ⊕ N Φ( p ) Φ( M ) . As a consequence of the implicit function theorem, every Riemannian immersionΦ : M → N admits locally an implicit representation Ψ : U ∩ N → N Φ( p ) Φ( M )such that Ψ( U ∩ M ) = U ∩ Φ( N ). Here U is a suitable neighborhood of Φ( p )in N . Hence, we have with X, Y ∈ T ( M ) and arbitrary local extensions e X, e Y ∈ T ( U ∩ N ) of d Φ X, d Φ Y ∈ T M that(id T ( N ) − d Ψ) (cid:16) ∇ N e X e Y (cid:17) = d Φ (cid:0) ∇ MX Y (cid:1) . (3)In particular, d Ψ Φ( p ) spans the normal space N Φ( p ) Φ( M ).4 heorem 2.2. Suppose that an embedding id M : M ֒ → R n is a Riemannianimmersion in Euclidean ( n < ∞ ) or Hilbert space ( n = ∞ ), t → γ ( t ) a geodesicin M , { V j ( t ) : j ∈ J } an orthonormal smooth base for N γ ( t ) M and W a vector-field in M . Then W is parallel along γ if and only if it satisfies the lineardifferential equation ˙ W ( t ) = − X j ∈ J ˙ V j ( t ) ⊗ V j ( t ) W ( t ) . Proof.
The assertion is an immediate consequence of (3) and the fact that0 = ddt (cid:10) W ( t ) , V j ( t ) (cid:11) = (cid:10) ˙ W ( t ) , V j ( t ) (cid:11) + (cid:10) W ( t ) , ˙ V j ( t ) (cid:11) for all j ∈ J by hypothesis. Riemannian Submersions
For a Riemannian submersion Φ : M → Q fromthe top space M to the bottom space Q , tangent spaces split as follows: everyfiber Φ − ( q ), q ∈ Q is a submanifold of M that is locally a topological embed-ding. With the vertical space T p Φ − (cid:0) Φ( p ) (cid:1) along the fiber and its orthogonalcomplement, the horizontal space , we have T p M = T p Φ − (cid:0) Φ( p ) (cid:1) ⊕ H p M .
Since H p M ∼ = T Φ( p ) Q , every V ∈ T ( Q ) has a unique horizontal lift e V ∈ H p M characterized by d Φ e V = V . For arbitrary W ∈ T ( M ) denote by W ⊥ : p → W ⊥ p the orthogonal projection to the vertical space.The following Theorem due to O’Neill (1966) (cf. also Lang (1999, p.386))allows to lift bottom space parallel transport to the top space. In addition to(3) this provides the vertical (normal) part as well, which is in general non-zerofor submersions. Theorem 2.3.
Let
Φ : M → Q be a Riemannian submersion and let X, Y ∈ T ( Q ) . Then we have with the Lie bracket [ · , · ] on M that ∇ M e X e Y = ^ ∇ NX Y + 12 [ e X, e Y ] ⊥ . We are now ready for the ODE of parallel transport on a Riemannian im-mersion followed by a Riemannian submersion.
Theorem 2.4.
Suppose that Φ : M ֒ → R n is a Riemannian immersion inEuclidean ( n < ∞ ) or Hilbert space ( n = ∞ ) , Φ : M → Q a Riemanniansubmersion and let W be a vector field on M horizontal along a horizontalgeodesic γ ( t ) on M . Then d Φ W is parallel along Φ ◦ γ ( t ) if and only if ˙ W ( t ) = − X j ∈ J ˙ V j ( t ) ⊗ V j ( t ) W ( t ) − X k ∈ K dω tk (cid:0) ˙ γ ( t ) , W ( t ) (cid:1) . ere, { V j ( t ) : j ∈ J } denote an orthonormal smooth base for the normal space N γ ( t ) M ⊂ R n and dω tk are the exterior derivatives of an orthonormal and smoothbase { U k ( t ) : k ∈ K } of the vertical space T γ ( t ) [ γ ( t )] ⊂ T γ ( t ) M for suitable indexsets J and K .Proof. Suppose that we have a vector field X ∈ T ( Q ) with horizontal lift e X ( t ) =˙ γ ( t ). If Φ ◦ γ is geodesic and d Φ W parallel with horizontal lift W , Theorem2.3 yields ∇ M ˙ γ W = 12 X k ∈ K h [ e X, W ] , U k i ] U k = − X k ∈ K dω k ( ˙ γ, W ) (4)making use of the well known (e.g. Lang (1999, p.126/7)) h [ e X, W ] , U k i = e X h U k , W i − W h U k , e X i − dω k ( e X, W )with the exterior derivative dω k of the one-form ω k dual to U k . On the otherhand, since d Φ : T p M → T p M ⊂ R n is the identity, formula (3) yields ∇ ˙ γ W M = ˙ W ( t ) − X j ∈ J V j ( t ) ⊗ V j ( t ) ˙ W ( t )= ˙ W ( t ) + X j ∈ J ˙ V j ( t ) ⊗ V j ( t ) W ( t ) (5)as in the proof of Theorem 2.2 . Putting together (4) and (5) gives the assertionof the Theorem. We define the two shape spaces of closed 2D constant-speed contours based onangular direction as introduced by Zahn and Roskies (1972) in the geometricformulation of Klassen et al. (2004).Suppose that z : [0 , π ] → C , s z ( s ) is a constant-speed parameterizationof a smooth, closed, curve of length L winding once counterclockwise aroundeach interior point. Let θ ( s ) = arg (cid:0) z ′ ( s ) (cid:1) − arg (cid:0) z ′ (0) (cid:1) − s , with˙ z ( s ) = L π e i (cid:0) θ ( s )+arg( z ′ (0))+ s (cid:1) . (6)Obviously, the Zahn-Roskies shape (ZR-shape) θ is invariant under translation,scaling and rotation z ( s ) → c + λe iψ z ( s ). Moreover, subtracting s (the curvesto be modelled wind once around their interior) norms θ such that it is 2 π -periodic. Vice versa, from every converging Fourier series an a.e. differentiableconstant-speed 2D curve can be reconstructed by integrating (6). This curve is6nique modulo translation, scaling and rotation. Thus a linear subspace of theHilbert space ℓ of Fourier series is the ZR–pre–shape space S ZR := n θ ( s ) = ∞ X n =0 (cid:0) x n cos( ns ) + y n sin( ns ) (cid:1) : k θ k − x = 12 ∞ X n =1 ( x n + y n ) < ∞ x = − ∞ X n =1 x n , y = 0 o . As usual, 2 π h θ, η i := R π θ ( s ) η ( s ) ds and k θ k := h θ, θ i . The tangent spaces T θ S ZR are identified with S ZR ⊂ ℓ . Since the curves in question are closed,we have with the non-linear mappingΨ : ℓ → C θ R π e i (cid:0) θ ( s )+ s (cid:1) dt that the ZR–shape space is the implicit sub-manifoldΣ ZR := { θ ∈ S ZR : Ψ( θ ) = 0 } . Obviously, the ZR-shapes of closed not self-intersecting contours form an opensubset containing the origin, which corresponds to the shape of the circle.Additionally considering closed curves invariant under change of initial point z ( s ) → z ( s + s ) (e.g. amorphous curves with no preassigned initial point) bydefining this action of the unit circle S ∋ s on Σ ZR the invariant ZR–shapespace Σ IZR := (cid:16) Σ ZR \ { } (cid:17) /S (7)is obtained.Since rotation and parameter shift are equivalent for circles, the correspond-ing invariant ZR–shape is thus a singularity of Σ IZR , in fact its only singularity.
Parallel Transport on Σ ZR and Σ IZR
Geodesics on Σ ZR as well as on Σ IZR between two given points can be computed via a technique called geodesic shoot-ing , cf. Miller et al. (2006) as well as Klassen et al. (2004), or much faster viaa variational approach Schmidt et al. (2006). Since Σ ZR ֒ → ℓ is a Rieman-nian immersion with the global implicit definition Ψ = 0 we have that thenormal space in S ZR at θ ∈ Σ ZR is spanned by V ( θ ) = s cos ( θ ( s ) + s ) and V ( θ ) = s sin (cid:0) θ ( s ) + s (cid:1) . Orthogonalization yields the base W := V k V k , W := V − h V , W i W k V − h V , W i W k . As a consequence of Theorem 2.2 we have7 heorem 3.1.
A vector-field W ( t ) in Σ ZR is parallel along a geodesic γ in Σ ZR if and only if it satisfies the linear differential equation ˙ W ( t ) = (8) − (cid:16) ddt W (cid:0) γ ( t ) (cid:1) ⊗ W (cid:0) γ ( t ) (cid:1) + ddt W (cid:0) γ ( t ) (cid:1) ⊗ W (cid:0) γ ( t )) (cid:17) W ( t ) . In practice, (8) can be solved numerically by orthogonally projecting to T γ ( t ) Σ ZR in every iteration step.We now turn to the submersion (7). The vertical space at θ ( s ) = x + ∞ X n =1 (cid:0) x n cos( ns ) + y n sin( ns ) (cid:1) ∈ Σ ZR is spanned (if convergent) by the single vertical unit length direction θ ′ ( s ) k θ ′ ( s ) k = P ∞ n =1 n (cid:0) − x n ∂ y n + y n ∂ x n (cid:1)pP ∞ n =1 n ( x n + y n ) √ . The exterior derivative of its dual is hence dω √ = − P ∞ n =1 ndx n ∧ dy n √ P ∞ n =1 n ( x n + y n ) − P n = n ′ y n x n ′ nn ′ ( n ′ dx n ′ ∧ dx n − ndy n ∧ dy n ′ ) √ P ∞ n =1 n ( x n + y n ) − P n,n ′ nn ′ ( ny n y n ′ + n ′ x n x n ′ ) dy n ∧ dx n ′ √ P ∞ n =1 n ( x n + y n ) . (9)In conjunction with Theorem 2.2, Theorem 2.4 and Theorem 3.1 one obtainsafter a tedious computation Theorem 3.2.
The vector-field W ( t ) = u ( t ) ∂ x + ∞ X n =1 (cid:0) u n ( t ) ∂ x n + v n ( t ) ∂ y n (cid:1) is a horizontal lift to the top space Σ ZR of the bottom space parallel transportalong a geodesic in Σ IZR γ s ( t ) = x ( t ) + ∞ X n =1 (cid:0) x n ( t ) cos( ns ) + y n ( t ) sin( ns ) (cid:1) horizontal in Σ ZR if and only if it satisfies the linear differential equation ˙ W ( t ) = − (cid:16) ddt W (cid:0) γ s ( t ) (cid:1) ⊗ W (cid:0) γ s ( t ) (cid:1) + ddt W (cid:0) γ s ( t ) (cid:1) ⊗ W (cid:0) γ s ( t )) (cid:17) W ( t )+ γ ′ s ( t )2 k γ ′ s ( t ) k (cid:16)(cid:10) γ ′ s ( t ) ⊗ γ ′′ s ( t ) , W ( t ) ⊗ ˙ γ s ( t ) (cid:11) − (cid:10) γ ′ s ( t ) ⊗ ˙ γ s ( t ) , W ( t ) ⊗ γ ′′ s ( t ) (cid:11)(cid:17) − γ ′ s ( t ) k γ ′ s ( t ) k (cid:10) ˙ γ ′ s ( t ) , W ( t ) i . ith the derivatives defined as γ ′ s ( t ) = P ∞ n =1 n (cid:0) − x n ( t ) ∂ y n + y n ( t ) ∂ x n (cid:1) , ˙ γ s ( t ) = ˙ x ( t ) ∂ x + P ∞ n =1 (cid:0) ˙ x n ( t ) ∂ x n + ˙ y n ( t ) ∂ y n (cid:1) , ˙ γ ′ s ( t ) = P ∞ n =1 n (cid:0) − ˙ x n ( t ) ∂ y n + ˙ y n ( t ) ∂ x n (cid:1) ,γ ′′ s ( t ) = − P ∞ n =1 n (cid:0) x n ( t ) ∂ x n + y n ( t ) ∂ y n (cid:1) , if convergent, and the inner product defined by h E i ⊗ E j , E k ⊗ E l i := δ ( i,j ) , ( k,l ) for an orthogonal system E j and index set J ∋ j . In practice, convergence of the series for the derivates is not an issue since asremarked earlier, computations are carried out using only finitely many Fouriercoefficients.
Kendall’s landmark based similarity shape analysis is based on configurations consisting of k ≥ m + 1 labelled vertices in R m called landmarks that do not allcoincide. A configuration x = ( x , . . . , x k ) = ( x ij ) ≤ i ≤ m, ≤ j ≤ k is thus an element of the space M ( m, k ) of matrices with k columns, each an m -dimensional landmark vector. Disregarding center and size, these configurationsare mapped to the pre-shape sphere M = S km := { p ∈ M ( m, k −
1) : k p k = 1 } , where k p k = h p, p i and h p, v i := tr( pv T ) is the standard Euclidean product.This can be done by, say, multiplying by a sub-Helmert matrix, cf. Dryden and Mardia(1998) for a detailed discussion of this and other normalization methods. Thecanonical Riemannian immersion S km ֒ → M ( m, k −
1) comes with a global im-plicit definition Ψ( x ) = k x k − γ on S km relate as ∇ S km ˙ γ ( t ) W ( t ) = ˙ W ( t ) − (cid:10) ˙ W ( t ) , γ ( t ) (cid:11) γ ( t ) (10)As a consequence of (10) and (1), unit-speed geodesics on S km are great circlesof form γ ( t ) = x cos t + v sin t with x, v ∈ S km and h x, v i = 0.In order to filter out rotation information define the regular part ( S km ) ∗ := { x ∈ S kk : rank( x ) > m − } (an open dense subset of S km ) and a smooth and freeaction of SO ( m ) by the usual matrix multiplication ( S km ) ∗ g → ( S km ) ∗ : p gp for g ∈ SO ( m ). The orbit π ( p ) = { gp : g ∈ SO ( m ) } is the Kendall shape of p ∈ S km and the quotient π : ( S km ) ∗ → (Σ km ) ∗ := ( S km ) ∗ /SO ( m ) (11)9s called Kendall’s shape space . Note that projecting from the entire pre-shapesphere S km would have led to a non-manifold quotient ( m ≥ gl ( m ) = o ( m ) ⊕ SM ( m ) of the Lie algebra gl ( m ) = M ( m, m ), the Lie algebra o ( m ) ofskew-symmetric matrices in gl ( m ) and the vector-space of symmetric matrices SM ( m ) in gl ( m ) we have the following orthogonal tangent space decompositionfor x ∈ S km , cf. Kendall et al. (1999, p.109). gl ( m ) = o ( m ) ⊕ SM ( m ) ↓ · x ↑ · x T T x S km ⊕ N x S km = T x π ( x ) ⊕ z }| { H x S km ⊕ N x S km (12)For x ∈ ( S km ) ∗ both mappings are surjective and H x S km = T π ( x ) (Σ km ) ∗ . In orderto compute the horizontal lift of bottom space parallel transport as in Theorem3.2, we need an orthonormal base for the (cid:0) m ( m − / (cid:1) -dimensional verticalspace T x π ( x ) and the exterior derivative of its duals. From (12) we have at oncea (in general not-orthogonal) base { e ij x : 1 ≤ i < j ≤ m } with base system e ij = (cid:16) ε α,β (cid:17) ≤ α,β ≤ m with ε α,β = α = i, β = j − α = j, β = i of o ( m ). From the former obtain an o.g. base system { V ij ( x ) : 1 ≤ i < j ≤ m } of T x π ( x ) through Gram-Schmidt orthogonalization and let ω ij ( x ) be the one-formdual to V ij ( x ) (1 ≤ i < j ≤ m ). For m = 2 there is a single vertical unit-direction V ( x ). For m >
2, however, dω ( x ) is about as complicated as (9), and evenfor m = 3, the other two derivatives and their application to vector-fields resultin expressions too lengthy to be written down, cf. also the rather complicatedexamples in Le (2003). Using a computer algebra program, however, theseexpressions and their respective values can be easily computed by symbolicdifferentiation. Hence, Theorem 2.4 and (10) yield the following Theorem. Thespecial case m = 2 is taken from Huckemann et al. (2009, Theorem A.6), cf.also Le (2003, Theorem 2). Theorem 4.1.
A vector-field W ( t ) is a horizontal lift to the top space of thebottom space (Σ km ) ∗ parallel transport along a geodesic γ ( t ) = x cos t + v sin t horizontal in ( S km ) ∗ if and only if it satisfies the ODE ˙ W ( t ) = D ˙ W ( t ) , γ ( t ) E γ ( t ) − X ≤ i Equidistant deformationalong geodesics in Σ ZR . The bul-let marks the pre-assigned initialpoint. Figure 1(c) depicts the par-allel transplant to σ of the geodesicfrom Figure 1(a) along the geodesicdepicted in Figure 1(b). (a) From rectangle σ to the regu-lar hexagon σ . (b) From rectangle σ to rectangle σ . (c) From σ to the parallel trans-plant of σ from Figure 2(a). Figure 2: Equidistant deformationalong geodesics in Kendall’s land-mark based shape space Σ . Thekinks signify landmarks. Notationas in Figure 1. Denote by Σ ZR k the sub-space of closed curves with k -fold rotational symmetry.The following is an observation of Zahn and Roskies (1972): Theorem 5.1. θ represents a closed curve with k -fold rotational symmetry k > if and only if x n , y n = 0 for all n mod k . k -fold rotational symmetryis again of k -fold rotational symmetry we have at once: Corollary 5.2. For each k = 2 , . . . , Σ ZR k is a flat linear submanifold of Σ ZR and parallel transport on Σ ZR k is affine. Within the subspace Σ ZR of closed curves with two-fold symmetry considera simple example of three shapes: two rectangles σ and σ differing only bytheir initial point and a hexagon σ . The ‘same’ (i.e. parallel) deformation from σ to σ is applied to σ .Figure 1 illustrates that deformation in the geometry of Zahn-Roskies’ shapespace, Figure 2 gives it in the geometry of Kendall’s shape space. Comparing therespective Subfigures (a) and (b) over the different geometries shows that thegeodesic deformation with fixed initial and terminal shape gives almost identicalintermediate shapes.In the respective Subfigures (c), the difference between ‘same’ shape de-formation over the two geometries is hardly notable in the beginning of thedeformation. Near the end, however, it becomes notable: number the kinks(the landmarks in Σ ) counterclockwise from 1 (bullet) to 6 and denote by ( i, j )the line connecting the i -th kink with the j -th kink. Then (2 , 5) remains par-allel to (6 , 1) and (3 , 4) in the Kendall geometry whereas in the Zahn-Roskiesgeometry it turns in direction beyond (2 , 3) and (5 , Let us quickly overview some very recent developments within two millenia ofresearch on plant form. With the application below in mind, we are interestedin a flexible and realistic representation of leaf contour shape. Flexibility in thiscontext means that we are looking for a model in which nature not only choosesvalues of parameters in a pre-defined parameter space but rather the parameterspace itself. We caution that such a non-parametric model may come at thecost that nature’s parameters may not be simply geometrically interpretable Parametric in this sense are the well established models involving allometry,still of interest today: e.g. Gurevitch (1992); Burton (2004), or the superfor-mula of Gielis (2003). Also, models based on landmarks such as Dickinson et al.(1987); Jensen (1990) or Jensen et al. (2002), can be viewed as projecting na-ture to a pre-specified parameter space by leaving out the parts of the contourbetween landmarks. Note that parametric models are highly successful e.g. forplant classification, genetic hybrid identification, cf. Jorgensen and Mauricio(2005), or in the Climate Leaf Analysis Multivariate Program (CLAMP) ofWolfe (1993) which is fundamental to paleoclimate and present day climatereconstruction, cf. Endress et al. (2000).On the other hand, models building on the shape spaces of Zahn and Roskies(cf. Section 3) are non-parametric , even though they are not entirely free of con-straints: in view of landmark-based shape analysis (cf. Section 4), restrictingto unit speed velocities translates into infinitesimally placed landmarks. The12ifferent geometry, however, liberates from the necessity to identify homologouslandmarks, by imposing infinitesimal uniform growth. The latter is certainlydebatable. Curiously, such non-parametric models introduced as eigenshapeanalysis (building on Lohmann (1983)) have initially stirred controversy be-cause parameters were not simply geometrically interpretable and because withlacking initial point, registration was not satisfactory, cf. Rohlf (1986). Whilethe former is precisesly a desired feature, introducing the geometric concept ofthe quotient Σ IZR = (cid:0) Σ ZR \ { } (cid:1) /S by Klassen et al. (2004) settles the latterobjection. It seems, however, that the natural non-Euclidean geometry of Σ ZR is not fully realized in the community, cf. Ray (1992); Krieger et al. (2007);Hearn (2009).For sake of completeness, even though not practicable for our purpose be-cause of high sensitivity to boundary noise, let us briefly mention a third ap-proach of shape modeling based on the leaf’s vein structure. With methods forautomated venation extraction available (cf. Fu and Chi (2006)), although com-putationally much more challenging than contour extraction, Lu et al. (2009)link vein structure to the concept of shape spaces by Blum and Nagel (1978)based on medial skeletons, cf. also Pizer et al. (2003). Undoubtedly, modelingthe vein structure gives deep insight into physiological, hydraulical and biome-chanical aspects of leaf formation. Parameter spaces thus obtained should beclosest to nature in the above sense. Current research in venation patterns,however, shows that leaf shape diversification is still poorly understood (e.g.Niinemets et al. (2007)).Obviously, for our purpose of modeling entire leaf contours while being asnon-parametric as possible, the space Σ ZR suits ideally. For the problem athand there is no need for pre-registration as the leaves in question are naturallyaligned by petiole (the base point where the stalk enters the blade forming themain leaf vein) and apex (the terminal point of the main vein, usually the leaftip) location. One could almost equivalently align by petiole location and theinitial direction of the main leaf vein. be1b8: original be1b8: projection to its geodesic Figure 3: Shape evolution of a black poplar leaf over two weeks. Left: originalcontours. Right: contours obtained from projecting to geodesic evolution. In an application we consider five leaves of a Canadian black poplar tree at an13 e1b9: original be1b9 along geodesic of be1b8be1b3: original be1b3 along geodesic of be1b8be2b4: original be2b4 along geodesic of be1b8be2b1: original be2b1 along geodesic of be1b8 Figure 4: Shape evolution of black poplar leaves over two weeks. Left column:original contours. Right column: contours obtained from traversing the firstcontour along the parallel translate of the geodesic from Figure 3. N + 1)-dimensional subspace of ℓ using Fourier coefficients, ( x , x , y , . . . , x N , y N )with N = 100.Leaf “be1b8” (left image of Figure 3) exhibits the most regular shape. In con-cord with earlier observations of different leaves using landmarks (cf. Hotz et al.(2010)), the temporal evolution occurs almost along the geodesic determined byinitial and terminal shape (as depicted in the right image of Figure 3). Theinitial direction of this geodesic has been parallely transplanted to the initialshapes of leaves “be1b9”, “be1b3”, “be2b4” and “be2b1”. In the right columnsof Figure 4, the shapes along these new geodesics starting at the correspondinginitial shapes have been recorded at the corresponding points in time. The leftcolumns of these figures depict the original temporal shape evolution.The common shape dynamics displayed by the original leaf contours (leftcolumns of Figures 3 and 4) seems two-fold. First, an increase of base angle.Second, different growth ratios are not visible at the apex as its angle remainsnearly unchanged. Individual effects are non-symmetric and non uniform lateralgrowth. Also, leaf ’be1b3’ develops a notch left, slightly below the apex, for leaf’be2b1’ an original notch also left, slightly below the apex attenuates.Obviously (right columns of Figures 3 and 4) leaves “be1b9”, “be1b3’ and‘be2b4” follow rather closely the parallel transplant of the geodesic of leaf“be1b8”. Original non-uniform growth is uniformized and, stronger than origi-nally, apexes acuminate. Even though all of their initial and terminal shapes arequite different, one can say that their temporal evolution is rather similar. Thisseems to be less the case for leaf “be2b1”. Its observed growth tends to eliminateits initial strong dent at north-west-north while along the transplanted geodesic,this dent remains, causing increased distal growth at the tip. One could arguethat in order to restore an original contour defect, natural growth deviates fromits “original” plan. Certainly, such phenomena deserve future research.b be1b8 be1b3 be2b4 be2b1 ρ ( v b , w b ) 0 . 17 0 . 12 0 . 44 0 . µ ( v b , w b ) 0 . 99 0 .