Modeling geometric-optical illusions: A variational approach
aa r X i v : . [ q - b i o . N C ] A p r Modeling geometric–optical illusions:A variational approach
Werner Ehm and Jiˇr´ı Wackermann
Institute for Frontier Areas of Psychology and Mental Health,Freiburg, Germany { ehm, jw } @igpp.de Abstract — Visual distortions of perceived lengths, angles, or forms, are generally knownas “geometric–optical illusions” (
GOI ). In the present paper we focus on a class of
GOI swhere the distortion of a straight line segment (the “target” stimulus) is induced by anarray of non-intersecting curvilinear elements (“context” stimulus). Assuming local target–context interactions in a vector field representation of the context, we propose to model theperceptual distortion of the target as the solution to a minimization problem in the calculusof variations. We discuss properties of the solutions and reproduction of the respective formof the perceptual distortion for several types of contexts. Moreover, we draw a connectionbetween the interactionist model of
GOI s and Riemannian geometry: the context stimulus isunderstood as perturbing the geometry of the visual field from which the illusory distortionnaturally arises. The approach is illustrated by data from a psychophysical experiment withnine subjects and six different contexts.
Keywords: calculus of variations, geodesic, geometric–optical illusions, Hering type illu-sions, Riemannian geometry, vector field, visual perception “Geometric-optical illusions” (
GOI ) is a covering term for a broad class of phenomena, wherevisual perception of lengths, angles, areas or forms in a figure (e. g. a simple line drawing) isaltered by other components of the figure. These phenomena demonstrate, generally, the de-pendence of a percept on its context, and allow to study the structural principles underlying theorganization of visual percepts, or “laws of seeing” [24]. Since their discovery [26, 27],
GOI shave been the subject of intensive experimental research (for comprehensive reviews see [7]and [29]), but they are still far from being well understood. The variety of proposed explana-tions ranges from physiological theories, based on mutual interactions between elements of theneural substrate (e. g., retina or primary cortical areas) [3, 6, 36], to purely mentalist theories,interpreting the
GOI s as results of “unconscious inferences” [15] or inappropriately applied cog-nitive strategies [14]. However, no unitary theory of the
GOI s has been established until presentdays, and it is even doubtful whether such a unified explanatory theory is conceivable [7].In the present paper we study a well-defined class of
GOI s that are reducible to a com-mon generating principle. The emphasis is not on the human vision system or on psycho-logical factors, nor will physiological or psychological “mechanisms” be proposed; we aimat a representative-descriptive rather than explanatory-causal theory. Specifically, we focuson a class of
GOI s in which perception of a target element—usually a segment of a straightline—appears distorted when presented with an array of (curvi)linear elements, in the follow-ing called the context . An example for such target–context interactions first reported by Hering1 ) b) c)d) e) f)
Figure 1: Examples of geometric–optical illusions. Upper row: a) Classic form of Hering’s [16] illusion and b) itsmodification due to Thi´ery and Wundt [38]; c) Illusory bending of straight lines in a flat, non-perspectival context.Lower row: d,e) Distortions of square shape in two different contexts [11, 28]; f) Trapezoid deformation of squareshape similar to e) in a different context, obtained by permutation of quadrants of pattern d) [33]. [16] is the illusory curvature of straight lines over which an array of concurrent lines is su-perposed (Fig. 1a). Since then a great number of
GOI s have been constructed, discovered, orre-discovered on the same principle [38, 11, 28] (Figs. 1b,d,e).These phenomena—hereafter called illusions of “Hering type”—are of particular interestfor several reasons. First, they depend on local interactions between the target and the contextelements, as is evidenced by variant figures in which parts of the context pattern are deleted [33].Next, they demonstrably do not depend on a “scenic” impression induced by the context patterns(Fig. 1e,f). Finally, they all exhibit angular expansion at the target–context intersections: theillusory distortion of the target always acts to enlarge the acute angles at the intersection points(Fig. 1, passim). This effect, also dubbed “regression to right angles,” seems to be constitutivefor the class of
GOI s of our interest [4, 19] as well as in other types of
GOI s [20, 21].These observations set up the framework for our modeling approach [10]. Starting with aminimal set of assumptions plus the fact that the straight line is the shortest path connectingtwo points, we propose a variational principle for the perception of a linear target, draw aconnection to Riemannian geometry, and show that approximate solutions of the respectivevariational problem reproduce the perceptual distortions of the target (Sections 2 to 4). Further,we report on a related psycho–physical pilot experiment using six different context patterns(Section 5). Finally, we discuss achievements and limitations of this work (Section 6).The main text covers the basic approach along with the applied methods and the results. Allmathematical details, derivations, and proofs are given in the Appendix.2
Variational problem for Hering type illusions
Our focus in this paper is on the case where the target is a straight line, and the context con-sists of a family of (generally curved) lines that intersect the target but not each other. We willconceive of the context lines as the stream lines of a planar flow given by a continuously differ-entiable vector field v defined on some region Ξ ⊆ R containing the target in its interior. Toany point ξ ∈ Ξ is attached a vector, v ( ξ ) , indicating the “velocity” of the flow at the point ξ . Inview of the purely geometric character of the context it is natural to assume that | v ( ξ ) | = 1 forall ξ ∈ Ξ . Here | a | = p h a, a i and h a, b i = a b + a b denote the Euclidean norm (length) andinner product, respectively, of vectors a, b ∈ R . In geometrical terms, the normalized innerproduct h a, b i / | a || b | gives the cosine of the angle between a and b , which we denote as ∠ ( a, b ) .In graphical presentations of GOI s only a finite sample of context curves is displayed. Thecomplete set of context lines, which form a continuum in the plane, may then be conceived ofas continuously interpolating the sample. The target is here assumed to be the straight line, τ ,connecting two given endpoints τ , τ ∈ R . In illusions of Hering type τ is not perceivedas a straight line: it appears slightly curved. The basic idea of our approach is to model thedeviating percept as a perturbation of τ that is characterized by a minimum principle. Settingup the principle involves three components:(a) the local interactions hypothesis: the context v “acts” only along candidate paths, in thevicinity of the target;(b) the angular expansion hypothesis (“regression to right angles”), based on the phenomenol-ogy of GOI s (cf. Introduction);(c) the fact that the straight line is the shortest path between two points.Observing (b) and (c) we then posit the principle that, given the context vector field v , thestraight line target τ is distorted so that (i) the stream lines of v (the context lines) are intersected“as orthogonally as possible”, and (ii) the distorted line is as short as possible. This can be formulated mathematically as an optimization problem under side conditions.Since there is no a priori criterion suggesting length or orthogonality as the primary or theside condition, we propose to optimize a weighted mixture of the two terms. Specifically, weconsider the following
Variational problem [ VP ]: Given the vector field v , t , t ∈ R such that t < t , endpoints τ , τ ∈ R , and some number α ≥ , minimize the functional J ( x ) = Z t t | ˙ x ( t ) | dt + α Z t t h ˙ x ( t ) , v ( x ( t )) i | ˙ x ( t ) | dt (1) over the set X of all twice continuously differentiable planar curves x ≡ { x ( t ) , t ∈ [ t , t ] } with given endpoints x ( t ) = τ and x ( t ) = τ such that x ( t ) ∈ Ξ and | ˙ x ( t ) | > for every t . For clarity it has to be emphasized that the vector field v just serves us to represent the context; it is neitherrelated to any kind of field theory or perceptive field, nor to the receptive field of the retina. Likewise, there is nosupposition as to where the percept is located “materially.” Finally, the term “flow” is used only metaphorically; itshall not convey any idea of motion. Subscripts may index objects (such as τ , τ ) as well as the components of a vector (such that y = ( y , y ) if y ∈ R is a row vector). The appropriate interpretation will always be evident from the context.
3n this setting, X comprises the possible candidates for the actual percept. The first termin (1), R t t | ˙ x ( t ) | dt = R t t | dx ( t ) | , represents the length of x . (The superscript dot denotes thederivative w. r. t. the parameter t .) The second term accounts for the context–target interaction:its integrand is essentially the square of the cosine of the angle subtended by v and the curve x atthe point x ( t ) , hence it measures the deflection from orthogonality along the curve. The divisionby | ˙ x ( t ) | is to make the right-hand side of (1) invariant under reparameterizations of the “time”parameter t , so that it depends only on the trace of the curve x , and not on its parameterization.By its coordinate-free formulation, the problem is also invariant under translations and rotations.The number α ≥ , finally, accounts for the strength of the illusory effect. Obviously, for α = 0 only the length term is being minimized, and the solution of the problem reduces to the straightline between τ and τ , that is, to τ . Since the actual percept deviates only slightly from thestraight line target, one may anticipate that α should be small. Remark 1
The above minimum principle is distantly related to Fermat’s principle, which char-acterizes the path of a light ray through an inhomogeneous medium. Indeed, on rewriting thefunctional (1) in the form x R t t F ( x ( t ) , ˙ x ( t )) dt with integrand F ( x ( t ) , ˙ x ( t )) = | ˙ x ( t ) | + α h ˙ x ( t ) , v ( x ( t )) i / | ˙ x ( t ) | = | ˙ x ( t ) | (cid:18) α h ˙ x ( t ) , v ( x ( t )) i | ˙ x ( t ) | (cid:19) , (2)one sees that the variational problem amounts to minimizing the functional x R t t n ( t ) | dx ( t ) | where n ( t ) = 1 + α h ˙ x ( t ) , v ( x ( t )) i / | ˙ x ( t ) | = 1 + α cos ∠ ( ˙ x ( t ) , v ( x ( t ))) is the “refraction index”—which in our case depends not only on the “medium” (here: thecontext) as traversed by the path, via v ( x ( t )) , but also on the tangents to the path, ˙ x ( t ) . Let us first introduce some notation. With any curve x ∈ X we associate two more curves ρ ≡ ρ x , ρ ⊥ ≡ ρ ⊥ x , called the Frenet 2-gon: for every t , ρ ( t ) = ˙ x ( t ) / | ˙ x ( t ) | denotes thetangent direction vector; ρ ⊥ ( t ) denotes the unit (normal) vector obtained when rotating ρ ( t ) counterclockwise by ◦ , making { ρ ( t ) , ρ ⊥ ( t ) } a positively oriented orthonormal basis of R .Concerning the context, we write v ′ ( ξ ) for the total derivative of v at the point ξ ∈ Ξ , whichis a linear mapping from R into itself; v ′ ( ξ ) ∗ denotes its adjoint. In standard coordinates, v ′ ( ξ ) is given by the × matrix of partial derivatives of v at ξ (“Jacobian”), with entries ∂ j v k ( ξ ) , j, k ∈ { , } , where quite generally, ∂ j stands for the partial derivative w. r. t. the j -thargument. Finally, ω ( ξ ) = ∂ v ( ξ ) − ∂ v ( ξ ) denotes the rotation of v at the point ξ . For simplicity, we henceforth assume Ξ = R . A functional of the form y R t t F ( y ( t ) , ˙ y ( t )) dt is invariant under reparameterization if the integrand is in the sense that F ( u, cv ) = c F ( u, v ) for all c > and arguments u, v [8]. .1 Euler-Lagrange equation We apply the apparatus of the calculus of variations [8]. Generally, a curve x ∈ X at whicha functional of the form x R t t F ( x ( t ) , ˙ x ( t )) dt attains a minimum necessarily satisfies the Euler-Lagrange equation ddt ∇ ˙ x F ( x ( t ) , ˙ x ( t )) − ∇ x F ( x ( t ) , ˙ x ( t )) = 0 (for all t ). (3)Here ∇ x F, ∇ ˙ x F denote the partial gradients of F with respect to the (vector) arguments x, ˙ x ,respectively. In our special case where F is given by (2), the Euler-Lagrange equation becomes (cid:0) − α h ρ, v ( x ) i (cid:1) ˙ ρ = − α (cid:20) [ v ( x ) −h ρ, v ( x ) i ρ ] ddt h ρ, v ( x ) i + h ρ, v ( x ) i (cid:0) v ′ ( x ) − v ′ ( x ) ∗ (cid:1) ˙ x (cid:21) , where for compactness of notation we omitted the parameter t . This system of two nonlin-ear, second-order differential equations reduces in fact to one single equation that concerns the“normal” component orthogonal to the solution curve. Proposition 1
For α < the normal component of the Euler-Lagrange equation is given by h ˙ ρ, ρ ⊥ i = − α | ˙ x | h v ( x ) , ρ ⊥ ih ρ, v ′ ( x ) ρ i + h v ( x ) , ρ i ω ( x )1 − α h v ( x ) , ρ i + 2 α h v ( x ) , ρ ⊥ i . (4)In the simplest special case of a constant vector field one would expect that the straight line τ should result as the unique solution to (4). Indeed, since v ′ = 0 in this case, the right-handside of (4) vanishes, which implies h ˙ ρ, ρ ⊥ i = 0 , hence ˙ ρ = 0 , meaning that the direction vector ρ does not change along x . Consequently, x is a straight line, and since its endpoints are fixedat those of τ it follows that x = τ . The very formulation of the variational problem VP and its resemblance to Fermat’s principle(cf. Remark 1) suggest to look for a strictly geometrical interpretation. Such an interpretationcan indeed be given for a slight modification of VP .Intuitively, the medium, here represented by the context, perturbs the flat Euclidean geom-etry so that the shortest path between two points is curved rather than straight. Mathematically,such a non-Euclidean Riemannian geometry [23] requires specifying a metric tensor G on somedifferentiable manifold by means of which the length of a parameterized curve x ( t ) , t ≤ t ≤ t in the manifold is, invariantly under reparameterization, expressed as L G ( x ) = Z t t p h ˙ x ( t ) , G ( x ( t )) ˙ x ( t ) i dt. (5)The metric tensor attaches to each point ξ of the manifold a positive definite symmetric matrix G = G ( ξ ) = ( g j,k ( ξ )) j,k =1 , ξ . The usual Euclidean geometry corresponds to the special case G = I , the × identity matrix. A curve z is (a segment of) a geodesic (in the given geometry)if for any two t < t the functional x → L G ( x ) in (5) attains its minimum among all smoothcurves x with the same endpoints z ( t ) , z ( t ) as z at the curve x = z .Here, the manifold will be identified with the drawing plane R . The metric is, for given α ≥ and vector field v , defined by G ≡ G ( ξ ) = I + 2 α v ( ξ ) ⊗ v ( ξ ) ( ξ ∈ R ) , (6)with entries g jj = 1 + 2 α v j ( ξ ) , g jk = 2 α v j ( ξ ) v k ( ξ ) ( j = k ) . The rationale for this choice isstraightforward: the root of the quadratic form h ˙ x ( t ) , G ( x ( t )) ˙ x ( t ) i approximates the function F from (2) to the first order in α . It may thus be expected that minimization of the criteria (1)and (5) should yield similar solutions. Precise statements are given in the next subsection.Hereafter we will refer to the problem of minimizing the functional L G : X → R as GP . Instating the following necessary condition, and further below, we shall use subscripts α wheneverwe want to emphasize that some quantity depends on the parameter α figuring in GP (or VP ). Proposition 2
Let α ≥ . A curve γ α ∈ X that is a solution to GP (i.e., a geodesic) satisfiesthe (Euler-Lagrange) equation ¨ x = − α | ˙ x | (cid:16) t α ( x ) ρ + n α ( x ) ρ ⊥ (cid:17) , (7) where for general x ∈ X t α ( x ) = 11 + 2 α (cid:16) h v ( x ) , ρ ih ρ, v ′ ( x ) ρ i − α h v ( x ) , ρ i h v ( x ) , ρ ⊥ i ω ( x ) (cid:17) , (8) n α ( x ) = 11 + 2 α (cid:16) h v ( x ) , ρ ⊥ ih ρ, v ′ ( x ) ρ i + h v ( x ) , ρ i ω ( x ) { α h v ( x ) , ρ i } (cid:17) . (9)The system (7) is split into its tangential and normal components by forming inner productswith ρ and ρ ⊥ , respectively. In the latter case this gives after division by | ˙ x | (and noting that h ¨ x/ | ˙ x | , ρ ⊥ i = h ˙ ρ, ρ ⊥ i ) the equation h ˙ ρ, ρ ⊥ i = − α | ˙ x | n α ( x ) , which may be compared to (4): for small α , the right-hand sides of the two equations are almostthe same. The purpose of this subsection is to derive approximations to the geodesic γ α which are thenused to define what we call the shape of the perceptual distortion. That shape, denoted σ , isuniquely given by the target τ and the vector field v ; in particular, knowing the parameter α isnot required for determining σ . Using the shape as the fundamental link, we then clarify theconnection between the variational problems VP and GP .6n explicit expression for the geodesic γ α (solution to (7)) generally is not available; how-ever, for small α it can be approximated by means of a rapidly converging iterative procedure.Given α ≥ and x ∈ X we set S α ( x, ˙ x ) = − | ˙ x | (cid:16) t α ( x ) ρ + n α ( x ) ρ ⊥ (cid:17) , (10)which considered as a function of t ∈ [ t , t ] represents a curve in R . The following is essen-tially the Picard-Lindel¨of scheme for the iterative solution of an ordinary differential equation(system). Let a sequence of curves x α,n ∈ X be defined as follows. One starts with x α, = τ ,the target line, which we take to be parameterized as τ ( t ) = τ + T − ( t − t ) ( τ − τ ) , T = t − t ; for n = 1 , , . . . , ˙ x α,n +1 ( t ) = b α,n + α Z tt S α ( x α,n , ˙ x α,n )( u ) du, (11) x α,n +1 ( t ) = τ + Z tt ˙ x α,n +1 ( u ) du. (12)The side condition x α,n +1 ( t ) = τ , i.e. x α,n +1 ∈ X , is achieved by putting b α,n = T − (cid:18) τ − τ − α Z t t Z tt S α ( x α,n , ˙ x α,n )( u ) du dt (cid:19) . (13) Proposition 3
Suppose that the mapping ξ v ( ξ ) is twice continuously differentiable in aneighborhood of the target τ . Then there is α ∗ ∈ (0 , and a constant C such that the followingholds: for every ≤ α ≤ α ∗ there exists a solution γ α ∈ X to eq. (7) such that || x α,n − γ α || ∞ = sup t ≤ t ≤ t | x α,n ( t ) − γ α ( t ) | ≤ Cα n +1 ( n = 0 , , , . . . ) . (14)This means that for each sufficiently small α the sequence x α,n converges exponentiallyfast to a geodesic, γ α . By (14), already the first iteration, x α, , equals γ α up to terms of order O ( α ) , which will prove sufficiently accurate for our purposes. On the other hand, γ α (or x α, )differs from τ by terms of order O ( α ) , which suggests an ansatz γ α . = τ + ασ wherein σ would represent the limit as α → of the rescaled deflection ( γ α − τ ) /α of γ α from τ . Assuch, σ describes the approximative shape of this deflection. The scheme (11) to (13) suggeststhat σ should be given by the conditions ¨ σ = S ( τ, ˙ τ ) and σ ( t ) = σ ( t ) = 0 via a twofoldintegration, σ ( t ) = Z tt Z st S ( τ, ˙ τ )( r ) dr ds − T − ( t − t ) Z t t Z st S ( τ, ˙ τ )( r ) dr ds. (15)For a more explicit description, note first that the 2-gon for the straight line τ is constant along τ . We denote the corresponding pair of orthonormal vectors as ρ , ρ ⊥ ; thus ρ = ( τ − τ ) /ℓ Thus far, α was a fixed parameter, assumed “small.” In the following we conceive of α as an ‘order parameter’indexing a family of problems VP α , GP α , to be studied asymptotically as α → . ℓ = | τ − τ | the length of τ , and ˙ τ = T − ℓρ . Observing (10), (8) and (9) one then findsthat ¨ σ = S ( τ, ˙ τ ) = − ℓ/T ) (cid:16) t ( τ ) ρ + n ( τ ) ρ ⊥ (cid:17) (16) = − ℓ/T ) (cid:18) [ h v ( τ ) , ρ ih ρ , v ′ ( τ ) ρ i ] ρ + [ h v ( τ ) , ρ ⊥ ih ρ , v ′ ( τ ) ρ i + h v ( τ ) , ρ i ω ( τ )] ρ ⊥ (cid:19) . The approximation b x α = τ + ασ will represent our final guess (“prediction”) for the (biased)percept of the target. Let us say that a certain curve η is the approximate shape of the deflectionsof a family of curves y α ∈ X ( α > from the target, or briefly, the shape (of y α ), if || y α − τ − αη || ∞ = O ( α ) as α → . For example, the shape of b x α is σ (trivially). Proposition 4
Under the conditions of Proposition 3 the following holds. || γ α − b x α || ∞ = O ( α ) ( α → . (17) Moreover, curves y α ∈ X ( α > with shape η satisfy the Euler-Lagrange equation (4) up toterms of the order O ( α ) as α → if and only if h η, ρ ⊥ i = h σ, ρ ⊥ i . The first statement implies that the geodesics γ α as well as the approximations x α,n , n ≥ all share the same shape as b x α , namely σ . (This follows from (17) and (14), which togethergive || x α,n − b x α || ∞ = O ( α ) .) In particular, b x α approximates the solution γ α to eq. (7) tothe first order in α (i.e., up to terms of order O ( α ) ). On the other hand, b x α also represents afirst-order approximate solution to eq. (4) since it trivially satisfies the if-condition in the secondstatement. In fact, any first-order approximate solution to (4) necessarily has, to first order, thesame lateral deflection from the target as b x α , in that the normal components of their respectiveshapes are identical.The important conclusion here is that the same curve, b x α , represents an approximate solu-tion, accurate up to terms of order O ( α ) , to both problems VP and GP simultaneously. Hence,the phenomenologically motivated and the geometrical approaches leading to the variationalproblems VP and GP , respectively, yield, to first order, identical predictions for the shape of theperceptual distortion. In that sense, the two approaches are equivalent. Remark 2
Since h ¨ b x α , ρ ⊥ i /α = h ¨ σ, ρ ⊥ i = − ℓ/T ) n ( τ ) , by (16), the sign of n ( τ ) = h v ( τ ) , ρ ⊥ ih ρ , v ′ ( τ ) ρ i + h v ( τ ) , ρ i ω ( τ ) determines whether b x α , when traveled through from τ to τ , is bending to the left-hand side(sign n ( τ ) = − ) or to the right-hand side (sign n ( τ ) = +1 ), respectively. Therefore, thequalitative winding behavior of b x α can be read off already from that sign; knowing the shape σ completely is not required for this purpose. The observant reader will notice that we consider approximations at two different levels: the level of solutioncurves in case of problem GP (first statement), and the level of Euler-Lagrange equations in case of problem VP (second statement). The latter transition frees us from having to refer to ‘solutions to eq. (4)’ the existence of whichis unclear in case of problem VP (other than with GP ). σ depends only on the known quantities v ( τ ) and τ , making b x α easily calcu-lable for any trial parameter α . This allows for a straightforward implementation of the methodof compensatory measurement in our experimental study described in Section 5. Here we introduce three families of context curves forming the streamlines of the vector field v . We represent such a family by means of a real-valued smooth function c ( u, θ ) dependingon two real arguments u, θ such that c is strictly increasing in θ for each fixed u . The contextcurves u C θ ( u ) = ( u, c ( u, θ )) then do not intersect for different θ s, and we may assumethat for every point ξ = ( ξ , ξ ) in some region Ξ ⊆ R there exists θ = ϑ ( ξ , ξ ) suchthat c ( ξ , ϑ ( ξ , ξ )) = ξ . Within this setting, one can calculate the crucial quantities t α , n α explicitly in terms of partial derivatives of c .Of particular interest is the behavior of the normal and tangential components, n α and t α ,along the target. Suppose that τ is the horizontal line segment between τ = ( − ℓ/ , and τ = ( ℓ/ ,
0) ( ℓ > , parameterized by t ≡ u ≡ ξ ∈ [ − ℓ/ , ℓ/ . Then for α = 0 we have n ( τ ) ≡ n = (cid:2) ∂ c (cid:0) − ( ∂ c ) (cid:1) + ∂ c · ( ∂ c ) /∂ c (cid:3) / (cid:2) ∂ c ) (cid:3) , (18) t ( τ ) ≡ t = − ∂ c (cid:0) ∂ c − ∂ c · ∂ c/∂ c (cid:1) / (cid:2) ∂ c ) (cid:3) , (19)wherein the partial derivatives are evaluated at the arguments ( ξ , ϑ ( ξ , . (Note that one has c ( ξ , ϑ ( ξ , by the definition of ϑ , so the parameter θ for which the curve C θ crossesthe target at the point ( ξ , is θ = ϑ ( ξ , .) We only give the expressions for the primarilyimportant quantity n , which describes the lateral deflection of the percept from the target.The following three types of functions c will be considered. Type 1 (vertical shifts): c ( u, θ ) = q ( u ) + θ for some given function q . Then ∂ c = q ′ , ∂ c = 1 , ∂ c = q ′′ , ∂ c = 0 (primes here denote derivatives w. r. t. u ), whence n = q ′′ (cid:0) − q ′ (cid:1) / (cid:0) q ′ (cid:1) . If q is even and convex then n > at least in the central part of τ , since q ′ (0) = 0 . Thus in viewof Remark 2, our principle predicts that the curve appears concave there (bending downwardaway from the origin. This fits with the perceived curvatures in Figs. 3a, 3b, 3d, as well asFig. 1d (lower edge) or Fig. 1c (upper line). Of course, by symmetry the converse holds if q isconcave instead of convex; see Figs. 3c, 3e, 3f, 1d (upper edge), 1c (lower line). Type 2 (dilation): c is of the form c ( u, θ ) = θq ( u ) − a with a constant a > and a function q satisfying q ( u ) > q (0) > , < | u | ≤ ℓ . Here ∂ c = θq ′ , ∂ c = q, ∂ c = θq ′′ , ∂ c = q ′ For notational convenience coordinate vectors are written as row vectors.
9) b) −0.5 0 0.5−0.200.20.4 −0.5 0 0.50.000.010.02 prediction1st iterate2nd iterate3rd iterate
Figure 2: a) Illustration of context-induced effects: Straight line target τ (red line) embedded in a type 2 context(array of curves). The dashed line above τ represents our prediction for the average observer’s percept, the lower(dashed) line is counterbalanced so as to be perceived as straight by the average observer. See text for detailedexplanation. — b) Comparison of various approximations. Shown are: (i) the prediction b x ¯ α , and the first threeiterations converging to the geodesic γ ¯ α ; (ii) the analogous curves when the tangential component is ignored. Seetext. Note that up to scaling all curves share approximately the same form, namely that of the shape of the perceptualdistortion, σ . with θ = ϑ ( ξ , ξ ) = ( a + ξ ) /q ( ξ ) (and ξ = u ). Hence along τ , where ξ = 0 or θ = a/q , n = (cid:2) (cid:0) − ( aq ′ /q ) (cid:1) aq ′′ /q + a ( q ′ /q ) (cid:3) / (cid:2) aq ′ /q ) (cid:3) . Again, n is positive (negative) near the origin if q ′ (0) = 0 and q is convex (concave) there-abouts. The conclusion in regard to the perceived curvature thus is the same as for type 1. Type 3 (segments of concentric circles): c ( u, θ ) = √ θ − u − a, | u | ≤ θ where a is a positiveconstant and θ > a . The curves C θ represent concentric upper half circles intersecting the x -axis at the points ±√ θ − a . Observing θ = ϑ ( ξ , ξ ) = p ξ + ( a + ξ ) one finds that for ξ = 0 the numerator of n in (18) equals the constant − /a , and n = − a − (cid:0) ξ /a ) (cid:1) − . In particular, n < , and the principle predicts that the perceived curve should be convex(bending upward away from the origin), in agreement with, e.g., Figs. 1d, 1f, upper edge.It should be noted that the conclusions regarding curvature are similar for the various contexttypes. This suggests a simple rule of thumb: the percept tends to be bent in the opposite directionas the context curves. See e.g. Fig. 2a, 2b, where the curvatures of the pattern and the shape σ differ between the center and the margins, in opposite ways.Let us discuss this example in more detail. The context in Fig. 2a is of type 2, with q ( u ) =1 + sin ( πu ) , | u | ≤ / , a = 0 . , and twelve θ ’s equally spaced between 0.1 and 0.3. Thetwo lines above and below the target τ are determined as b x ± = τ ± ¯ ασ , respectively, where σ isthe shape computed numerically via (15), (16) using (18), (19); and ¯ α = 0 . is the average α value (across trials and participants) obtained in the experiment described in Section 5. Note thatthe target appears slightly bent upward in the middle, and this effect is roughly doubled when b x + , which is our prediction for the average observer’s percept, is drawn within the same context.10onversely, subtracting the distortion as in b x − removes the perceived curvature for the averageobserver. This ‘compensation principle’ is used in the implementation of our experiment.Fig. 2b shows the prediction b x + along with the first three iterates x ¯ α,n , n = 1 , , com-puted via (10) to (13), which approximate the exact geodesic γ ¯ α . Also plotted are four curvesobtained in the very same way except that the term S ¯ α ( x, ˙ x ) throughout is replaced by the term − | ˙ x | n ( x ) ρ ⊥ ignoring the tangential component. These eight curves come in two groups offour curves each which within groups are almost identical. The lower quadruple consists of b x + , x ¯ α, , and their counterparts computed with − | ˙ x | n ( x ) ρ ⊥ instead of S ¯ α ( x, ˙ x ) ; the upperquadruple consists of the respective second and third iterates. Similar results were found for allcases considered. In Section 3.2, we attached to each context a metric tensor G via the associated vector field.An intrinsic property of the geometry induced by G is the Gaussian curvature, K . This is ascalar quantity that describes how, and how strongly, the corresponding manifold (here R ) isdeformed at each of its points [23]. An approximation to K valid for our setting is K = 2 α C + O ( α ) , where C = v ∂ ω − v ∂ ω − ω (20)depends on the rotation ω of the respective vector field v in the first place.For contexts of type 1 and 2, C (hence K ) turns out to vary across the manifold, and toassume positive as well as negative values. The geometry thus does not reduce to one of theclassical non-Euclidean geometries (elliptic, hyperbolic, etc.) where K is constant. This isdifferent with contexts of type 3: here C can be shown to vanish identically, which implies a flat(essentially Euclidean) geometry. Our approach predicts the shape of the perceived distortion of the target as given by the expres-sion σ introduced in Section 3.3. The magnitude of the distortion is determined by the parameter α , which has to be estimated empirically. For that purpose we carried out an illustrative experi-ment, using the method of compensatory measurement : a line distorted in the opposite directionis presented to the observer, who adjusts α until a straight line is perceived. The rationale forthis procedure is clear: If τ is (approximately) perceived as τ + ασ , then for small α , τ − ασ will (approximately) be perceived as τ .The stimuli were constructed for six different contexts, as shown in Fig. 3. Each stimulusconsisted of a sequence of 21 Encapsulated Postscript pictures (‘frames’), displaying a constantarray of context curves, drawn black on a white background, with superimposed curved linesof the form τ − α k σ ( k = 1 , . . . , , the α k s being equally spaced in the (sufficiently large)interval [ − . , . . The target τ was always a horizontal straight line segment, drawn in redfor easier visual identification. The 21 frames belonging to a single stimulus were combined to In the experiment, only one line was shown at a time (together with the context). The ensuing perceptual biasmay well be different from the one seen in Fig. 2a; and of course, it may differ between observers. ) b)c) d)e) f) Figure 3: Six context patterns used in the reported experiment. Classification according to Section 4: a–c): type 1;d, e): type 2; f): type 3. Shown are stimuli for α = 0 , i. e., superimposed red lines are exactly straight lines. a single multipage PDF file, which was displayed on a LCD monitor watched binocularly froma distance of 100 cm. The observers’ task was to scroll through the sequence of frames and toindicate that one where the red line appeared to them as most similar to a straight line. Eachtrial thus resulted in an estimate b α of the model parameter α .Nine observers participated in the experiment. Each participant was presented stimuli of sixdifferent classes (contexts), in a randomized order, and six trials were done with each stimulusclass. The study thus yielded a total of 9 (observers) × × α . The complete data set is presented in Fig. 4. All α -estimates are positive, inaccordance with the predicted direction of the distortion. Despite interindividual differences in Two
PDF files were prepared for each context, with the frames sequence in the ‘forward’ order α , . . . , α , andin the ‘backward’ order α , . . . , α . These two versions were used alternately in each experimental session (seebelow). The case α = 0 (exactly straight line) was never contained in the sequence. Three of the six trials were run with the ‘forward’ and three with the ‘backward’ frames sequence to avoidpossible directional bias in the observer’s response. .00.10.20.3 a a b c d e f a a b c d e f a a b c d e f a a b c d e f a a b c d e f a a b c d e f a a b c d e f a a b c d e f a a b c d e f Figure 4: Results of the experiment described in Section 5. Each of the nine panels displays the complete data setfrom an individual participant (36 trials). Abscissæ: context patterns labeled as in Figure 2; ordinates: estimates of α ; cross marks: single-trial estimates; filled circles (connected): arithmetic means. susceptibility to the illusory effect, the pattern of the α s (disregarding the average magnitude) isremarkably similar across subjects.For further analysis, we attempted to decompose the responses into an individual factor anda factor depending on the context. Let α ( i, c ) denote the average α -estimate across trialsreported by observer i for context c . If the α ( i, c ) are proportional to the product of anindividual factor, η ( i ) , times a context-dependent factor, κ ( c ) , then dividing these factors outrenders the thus normalized responses α ( i, c ) / ( η ( i ) κ ( c )) ≡ e α ( i, c ) constant. In that (ideal)case one can argue that those two factors fully “explain” the (systematic) variation in the data.The goal thus is to find subject- and context-dependent factors reducing the variation in the e α ( i, c ) as far as possible.A natural choice for η ( i ) is the average of the α ( i, c ) across contexts, η ( i ) = α ( i, · ) . Suitablecandidates for the factor κ ( c ) could be various geometrical quantities related to, for example,the number and angles of the context-target intersections, or the curvature of the context lines. The strength-of-effect parameter α may reflect a variety of factors, including factors that depend on σ . a ( no r m a li z ed ) a b c d e fa) 012 a ( no r m a li z ed ) a b c d e fb) Figure 5: Individual (grey, dotted lines) and group (black, solid line) response profiles. a) First step: responses α ( i, c ) normalized intra-individually; coefficient of variation = 0.48. b) Second step: profiles resulting from Step 1further divided by context-dependent factors κ ( c ) ; coefficient of variation = 0.18. Details are provided in Section 5. The best among those considered was κ = s T Z t t h ˙ σ ( t ) , ρ ⊥ i dt , where “best” means the following. Let individually normalized response profiles be defined as π ( i, c ) = α ( i, c ) /η ( i ) , and let π ( c ) = π ( · , c ) denote their average across observers (consideredas functions of c , each). The above κ was best in the sense that division by this term maximallyreduced the coefficient of variation, namely from CV = 0.48 for the profile π ( c ) to CV = 0.18for the profile π ( c ) /κ ( c ) additionally normalized by κ ( c ) . The individual profiles π ( i, c ) alongwith the group mean π ( c ) are shown in Fig. 5a. Similarly, Fig. 5b presents the respective profilesadditionally normalized by context, π ( i, c ) /κ ( c ) = e α ( i, c ) and π ( c ) /κ ( c ) = e α ( · , c ) . The present paper marks but one step in our approach to the study of visual field geometry. Ad-mittedly, the approach presented here has certain limitations. Some of these limitations follownaturally from our decision for a “phenomenological,” i. e. purely descriptive theory of the
GOI phenomena [33], disregarding possibly underlying neurophysiological or neuropsychologicalmechanisms. Other limitations reflect the momentary state of development of the theory, andwill hopefully be overcome at later stages:1. Modeling the context by a continuous vector field relies upon a convenient, yet unrealisticidealization; in reality, the context always consists of an array of finitely many distinct curves.To what extent this idealization is justifiable remains an open question. Noteworthily, κ depends only on the component of σ orthogonal to τ , so that κ represents a kind of “energy”contained in the lateral deflection of the percept from the target. One might hope that a variable density of the target–context intersection points could be mimicked by admittingnonlinear parameterization of the context curves, e. g., by working with functions ( u, θ ) c ( u, φ ( θ )) where φ depends nonlinearly on θ . It turns out, however, that the terms n ( τ ) , t ( τ ) , hence the shape of the distortion, areinvariant under such reparameterizations.
14. Optical properties (color, background brightness, figure/background contrast, etc.) of thestimulus as well as its global geometric properties (relative size in the visual field, orientationw. r. t. gravicentric coordinates, etc.) have no representation in the present approach. One mayexpect that these properties do not affect the form but only the magnitude of the perceptualdistortion, and can thus be accounted for by the “illusion strength” parameter α . Experimentalstudies must decide which parameters of the stimulus may enter the model via parameter α .3. The assumption of local interactions [Section 2, sub (a)] implies that “holistic” proper-ties of the context pattern (symmetry, presence or absence of “focal” points, etc.) are plainlyignored. While we feel that the global, “scenic” appearance of the context has been over-interpreted in some explanatory approaches, e. g. [32, 14, 9], we cannot a priori exclude thatsuch holistic properties may play a modulating rˆole. These aspects, as well as those mentionedabove sub 2, call for more experimental research.4. The variational approach with fixed endpoints imposes a severe restriction on admissiblepercepts. For example, the present framework does not allow to treat the well-known Z ¨ollnerillusion [40], where the target lines are perceived as tilted, but preserve their straight line ap-pearance. Moreover, extension from straight lines to targets of simple geometric forms—e. g.distortions of circles to oval shapes in Hering-like contexts [11, 28]—is certainly desirable.More generally, one may think of targets representing geodesics in some Riemannian basis ge-ometry that is perturbed by the context similarly as the Euclidean metric tensor I is perturbedby the term α v ⊗ v in (6).A demarcation of our approach against so-called “field theories” of GOI s [7, pp. 167–170]appears necessary. In our approach, the vector field is a convenient mathematical representationof the context pattern. By contrast, some researchers think of a vector field induced in the neuralsubstrate by the context part of the stimulus as a physical entity. This idea, originating in earlytheories of psychophysical isomorphism [22], inspired some modeling/explanatory approaches[5, 28, 12, 25] that remained mostly on a qualitative or semi-quantitative level.Closer in spirit to the present approach is the work of Hoffman [17, 18] and Smith [30]wherein, too, the“realist” concept of a (neuro)physical field was abandoned in favor of a purelyformal, mathematical treatment. These developments based on vector fields and Lie derivativesrepresent a line of research parallel to ours: they, too, assume a local interactions and angularexpansion hypothesis, and yield a prediction for the perceptual distortion of a form similar toours. However, neither did these works make use of the calculus of variations for the derivationof the distortion, nor did they establish a connection with Riemannian geometry.A Riemannian geometry for visual perception was in fact derived by Zhang and Wu [39]by elegant considerations of perceptual coherency of a visual object under rigid translations.Zhang and Wu build on the image intensity function and properties of motion detectors, andderive an affine connection depending on derivatives of the image function as the fundamentalconstituent of the geometry. Their concepts are not easily seen to be applicable to the presentsetting, however, which deals exclusively with static percepts. One difference concerns theGaussian curvature, K , implied by the respective geometries. In [39], K ≡ always, makingthe geometry flat or pseudo-Euclidean, whereas in our approach K may assume positive andnegative values across the manifold, and may also vanish identically, depending on the context;cf. Section 4.1. Furthermore, Zhang and Wu’s geodesics are “perceptually straight,” whereasours are not, being the curved percept of a “physically straight” line.15ummarizing, we believe that our approach, in spite of its limitations discussed above, hasits undisputable merits and potential for further developments:1. Approximate solutions to the variational problem introduced in Section 2 yield phe-nomenologically correct predictions for the perceptual distortion once the only free parameterof the model, α , has been determined (experimentally). This is evidenced by the fact that prop-erly counter-distorted targets appear, in a given context, as straight lines without perceivableresidual distortion. A more thorough validation of the predicted shape of the distortion wouldcertainly be desirable, but this is beyond the scope of the present work.2. The experimental data reported in Section 5 show a remarkable stability of the illusoryeffect across participants and context types. This finding supports the notion that GOI s arenot mere failures of the visual system, but that they reflect intrinsic principles of the structuralorganization of visual percepts [24]. The method of the reported experiment can be used tostudy dependence of the “illusion strength” parameter α on various properties of the stimulus.3. Perhaps the most important feature of the mathematical model is its explicitly geometricalinterpretation, which allows us to characterize the percept of a straight line as a geodesic in anappropriate, context-dependent Riemannian geometry (Section 3.2). The analogy between thetheory of a (world-)space metric, dependent on the mass distribution, and a possible theory ofvisual space metric, dependent on the perceptual content of the visual field, has been noticedby several authors [35, 37, 34]. An important contribution here is the work of Zhang andWu [39] who studied the perceptual coherence of a visual object under rigid motions. Here wedemonstrate for the purely static case of geometric-optical illusions how the presence of contextelements in the visual field perturbs its (initially Euclidean) geometry, as reflected by the metrictensor (6), and how the illusory distortion naturally arises from the perturbed geometry. Acknowledgments
The authors are grateful to Steffen Heinze for his suggestions concerning variational calculusand Riemannian geometry, and for valuable remarks on an earlier draft of the paper. Thanks arealso due to the Editor and three reviewers for their constructive criticism and recommendationsthat helped to improve the paper. Variational calculus is a widely used tool in vision science, computer vision (e.g., [13, 2]), and elsewhere. Itsuse in, and specific application to, the current setting seem to be new, however. ppendix A1. The Euler-Lagrange equations for problems VP and GP Proof of Proposition 1.
Using the notation introduced in Section 3 we calculate ∇ ˙ x F = ˙ x | ˙ x | + 2 α h ˙ x, v ( x ) i| ˙ x | v ( x ) − α h ˙ x, v ( x ) i | ˙ x | ˙ x = ρ + 2 α h ρ, v ( x ) i v ( x ) − α h ρ, v ( x ) i ρ,ddt ∇ ˙ x F = (cid:0) − α h ρ, v ( x ) i (cid:1) ˙ ρ − α h ρ, v ( x ) i ρ ddt h ρ, v ( x ) i + 2 α v ( x ) ddt h ρ, v ( x ) i + 2 α h ρ, v ( x ) i v ′ ( x ) ˙ x = (cid:0) − α h ρ, v ( x ) i (cid:1) ˙ ρ + 2 α h ρ, v ( x ) i v ′ ( x ) ˙ x + 2 α [ v ( x ) − h ρ, v ( x ) i ρ ] ddt h ρ, v ( x ) i , ∇ x F = 2 α h ˙ x, v ( x ) i| ˙ x | v ′ ( x ) ∗ ˙ x = 2 α h ρ, v ( x ) i v ′ ( x ) ∗ ˙ x, so the general Euler-Lagrange equation (3) assumes the form (cid:0) − α h ρ, v ( x ) i (cid:1) ˙ ρ (21) = − α (cid:26) [ v ( x ) −h ρ, v ( x ) i ρ ] ddt h ρ, v ( x ) i + h ρ, v ( x ) i (cid:2) v ′ ( x ) − v ′ ( x ) ∗ (cid:3) ˙ x (cid:27) . Initially, (21) is a system of two nonlinear, second-order differential equations. However,both sides of (21) are in fact orthogonal to ρ for all t ∈ [ t , t ] , meaning that the tangentialcomponent is trivial and only the component orthogonal to it matters. To see this, note that ˙ ρ = | ˙ x | − ¨ x − | ˙ x | − h ¨ x, ˙ x i ˙ x , whence h ˙ ρ, ρ i = 0 ; moreover, h v ( x ) − h ρ, v ( x ) i ρ, ρ i = 0 as well as (cid:10)(cid:0) v ′ ( x ) − v ′ ( x ) ∗ (cid:1) ˙ x, ρ (cid:11) = | ˙ x | (cid:0)(cid:10) v ′ ( x ) ρ, ρ (cid:11) − (cid:10) ρ, v ′ ( x ) ρ (cid:11)(cid:1) = 0 . Thus effectively, the system (21) reduces to one equation. Now ˙ ρ = h ˙ ρ, ρ ⊥ i ρ ⊥ (22)(since h ˙ ρ, ρ i = 0 ), so forming the inner product of (21) with ρ ⊥ we get the relevant part of theEuler-Lagrange equation (system), h ˙ ρ, ρ ⊥ i (cid:0) − α h ρ, v ( x ) i (cid:1) + 2 α h v ( x ) , ρ ⊥ i ddt h ρ, v ( x ) i + 2 α h ρ, v ( x ) i D(cid:0) v ′ ( x ) − v ′ ( x ) ∗ (cid:1) ˙ x, ρ ⊥ E = h ˙ ρ, ρ ⊥ i (cid:16) − α h v ( x ) , ρ i + 2 α h v ( x ) , ρ ⊥ i (cid:17) + 2 α | ˙ x | (cid:16) h v ( x ) , ρ ⊥ ih v ′ ( x ) ρ, ρ i + h v ( x ) , ρ i ( ∂ v ( x ) − ∂ v ( x )) (cid:17) , (23)17here the second equality follows via (22) from ddt h ρ, v ( x ) i = h ˙ ρ, v ( x ) i + h ρ, v ′ ( x ) ˙ x i = h ˙ ρ, ρ ⊥ ih v ( x ) , ρ ⊥ i + | ˙ x |h v ′ ( x ) ρ, ρ i and (cid:0) v ′ ( x ) − v ′ ( x ) ∗ (cid:1) ˙ x = (cid:18) ∂ v ( x ) − ∂ v ( x ) ∂ v ( x ) − ∂ v ( x ) 0 (cid:19) (cid:18) ˙ x ˙ x (cid:19) = ( ∂ v ( x ) − ∂ v ( x )) (cid:18) − ˙ x ˙ x (cid:19) = ( ∂ v ( x ) − ∂ v ( x )) | ˙ x | ρ ⊥ . (24)The form (4) of the Euler-Lagrange equation then follows on dividing (23) by the expression − α h v ( x ) , ρ i + 2 α h v ( x ) , ρ ⊥ i (which is strictly positive because α < ) and rearranging. (cid:3) Proof of Proposition 2.
Let us first state the Euler-Lagrange equation for the modified func-tional x R t t h ˙ x ( t ) , G ( x ( t )) ˙ x ( t ) i dt ; it is x + 2 α v ddt h ˙ x, v i + 2 α h ˙ x, v i ( v ′ − v ′∗ ) ˙ x. (25)(Here and in the following we suppress the argument x of v and v ′ , for compactness of notation.)Forming the inner product with ˙ x gives h ˙ x, ¨ x i + 2 α h ˙ x, v i ddt h ˙ x, v i + 2 α h ˙ x, v i h ˙ x, ( v ′ − v ′∗ ) ˙ x i (26) = ddt (cid:18) | ˙ x | + α h ˙ x, v i (cid:19) = 12 ddt h ˙ x ( t ) , G ( x ( t )) ˙ x ( t ) i . Thus h ˙ x ( t ) , G ( x ( t )) ˙ x ( t ) i is constant as a function of t , or a “first integral”, with the conse-quence that the Euler-Lagrange equation for the modified functional amounts to the same as theEuler-Lagrange equation for the original functional x L G ( x ) . Let us proceed with derivingthe former equation.With ddt h ˙ x, v i = h ¨ x, v i + h ˙ x, v ′ ˙ x i and (24), equation (25) can be written as G ¨ x + 2 α h ˙ x, v ′ ˙ x i v + 2 α | ˙ x | h ˙ x, v i ( ∂ v − ∂ v ) ρ ⊥ , (27)where again G = I + 2 α v ⊗ v, with inverse G − = I − α α v ⊗ v. Hence G − v = v/ (1 + 2 α ) , G − ρ ⊥ = ρ ⊥ − α α h v, ρ ⊥ i v,
18o on writing v = h v, ρ i ρ + h v, ρ ⊥ i ρ ⊥ and recalling the notation ω = ∂ v − ∂ v , we can state(27) as a differential equation in explicit form, − ¨ x = 2 α | ˙ x | α (cid:20) h ρ, v ′ ρ i (cid:16) h v, ρ i ρ + h v, ρ ⊥ i ρ ⊥ (cid:17) + h v, ρ i ω (cid:16) ρ ⊥ (1 + 2 α ) − α h v, ρ ⊥ i {h v, ρ i ρ + h v, ρ ⊥ i ρ ⊥ } (cid:17) (cid:21) = 2 α | ˙ x | α (cid:20) (cid:16) h ρ, v ′ ρ ih v, ρ i − α ω h v, ρ i h v, ρ ⊥ i (cid:17) ρ + (cid:16) h ρ, v ′ ρ i h v, ρ ⊥ i + ω h v, ρ i { α h v, ρ i } (cid:17) ρ ⊥ (cid:21) . (28)The proof of Proposition 2 is complete. (cid:3) A2. Approximations
Proof of Proposition 3.
Equation (7) can be written as a first-order differential equation bymeans of the common recipe of enlarging the “state space,” from curves x to pairs of curves x, ˙ x . The iteration (11), (12) then becomes the well-known Picard-Lindel¨of scheme, exceptthat here we do not have an initial value problem for x, ˙ x ; rather, the two endpoints of x arefixed. There is only one obstacle for a straightforward application of the classical proof: oneneeds an a priori estimate for the distance of the iterates from the target, which has to remainbounded. Once this is achieved, it is a standard exercise to establish the boundedness andLipschitz conditions necessary for an application of the Banach fixed point theorem.We leave that aside and concentrate on the a priori estimate. Let J α,n = Z t t | ˙ x α,n ( u ) | du, and suppose initially that sup ξ || v ′ ( ξ ) || = M < ∞ , the norm being declared as || A || = P j,k | a j,k | for matrices A = ( a j,k ) . Putting ρ = ( τ − τ ) /ℓ we have by (11) and (13) ˙ x α,n +1 = ( ℓ/T ) ρ + αU α ( x α,n , ˙ x α,n ) (29)where U α ( x, ˙ x )( t ) = Z tt S α ( x, ˙ x )( u ) du − T Z t t Z tt S α ( x, ˙ x )( u ) du dt. From the straightforward bound | S α ( x, ˙ x ) | ≤ | ˙ x | || v ′ ( x ) || (30)one readily gets the estimate || U α ( x α,n , ˙ x α,n ) || ∞ ≤ M J α,n . (31)19hus by (29) J α,n +1 = Z t t (cid:20) ( ℓ/T ) + 2 α ( ℓ/T ) h ρ , U α ( x, ˙ x )( t ) i + α | U α ( x, ˙ x )( t ) | (cid:21) dt ≤ ℓ /T + 2 αℓ M J α,n + T ( α M J α,n ) = ℓ /T [1 + α M J α,n
T /ℓ ] , or with K α,n +1 = J α,n +1 T /ℓ and A α = α M ℓ , K α,n +1 ≤ (cid:0) α M K α,n ( ℓ /T ) T /ℓ (cid:1) = (1 + A α K α,n ) . Since J α, = R t t | ˙ τ ( u ) | du = ℓ /T , the starting value is K α, = 1 .We therefore have to study a “sub-recursion” of the form x n +1 ≤ ϕ ( x n ) , ϕ ( x ) = (1 + ax ) , x = 1 , where a is a positive constant. For a < / the function ϕ has two fixed points, z ± ( a ) = (cid:0) − a ± √ − a (cid:1) / (2 a ) . As a ↓ the smaller fixed point remains bounded; in fact, ≤ z − ( a ) ≤ for all a ∈ [0 , / .We now proceed by induction. We have x n ≤ z − ( a ) for n = 0 , so suppose this holds forsome n ≥ . Then ϕ ( x n ) ≤ ϕ ( z − ( a )) by the monotonicity of ϕ , and hence x n +1 ≤ ϕ ( x n ) ≤ ϕ ( z − ( a )) = z − ( a ) , as claimed.The conclusion for our initial problem is that if we choose α ∗ < (48 M ℓ ) − then sup n ≥ , α ≤ α ∗ J α,n ≤ ℓ /T. (32)As a consequence one has by (29) and (31) the uniform bound || ˙ x α,n +1 || ∞ ≤ ℓ/T + α M ℓ /T = ℓ/T (1 + α M ℓ ) ≤ ℓ/T ; (33)moreover, differentiating (29) and using (30) and (33) gives || ¨ x α,n +1 || ∞ = α || S α ( x α,n , ˙ x α,n ) || ∞ ≤ α M ( ℓ/T ) , whence Z t t | ¨ x α,n +1 ( u ) | du ≤ α M ℓ /T ≤ ℓ/ (2 T ) (34)(for every n ≥ and α ≤ α ∗ ). The required a priori estimates are now obtained from (34) bysetting y = τ, x = x α,n in the following lemma, the easy proof of which is omitted. Lemma 1
Suppose x, y ∈ X are such that R t t | ¨ x ( u ) − ¨ y ( u ) | du ≤ B . Then || x − y || ∞ ≤ BT and || ˙ x − ˙ y || ∞ ≤ B. ¨ τ = 0 we may set B = α M ℓ /T and conclude that || x α,n − τ || ∞ ≤ α M ℓ ≤ ℓ, || ˙ x α,n − ˙ τ || ∞ ≤ α M ℓ /T ≤ ℓ/T (35)for n ≥ , α ≤ α ∗ . These estimates were derived under the assumption that || v ′ ( ξ ) || is globallybounded. However, by (35) and because α ∗ may be arbitrarily small, it suffices that || v ′ ( ξ ) || islocally bounded in the vicinity of τ , which it is. This concludes the crucial part of the proof. (cid:3) Proof of Proposition 4.
By definition we have ¨ b x α − ¨ x α, = α [ S ( τ, ˙ τ ) − S α ( τ, ˙ τ )] , whence itreadily follows that Z t t | ¨ b x α ( u ) − ¨ x α, ( u ) | du = O ( α ) (as α → ). Therefore || b x α − x α, || ∞ = O ( α ) , by Lemma 1, so applying (14) with n = 1 completes the proof of (17).As for the second assertion, let us consider curves y α ∈ X with shape η (i.e., of the form y α = τ + αη + O ( α ) ). Let ρ α ≡ ρ y α = ˙ y α / | ˙ y α | . Straightforward expansions give ρ α = ρ + α ( T /ℓ ) h ˙ η, ρ ⊥ i ρ ⊥ + O ( α ) , ˙ ρ α = α ( T /ℓ ) h ¨ η, ρ ⊥ i ρ ⊥ + O ( α ) . The approximation ρ ⊥ α = ρ ⊥ − α ( T /ℓ ) h ˙ η, ρ ⊥ i ρ + O ( α ) is readily verified on noting that | ρ ⊥ − α ( T /ℓ ) h ˙ η, ρ ⊥ i ρ | = 1 + O ( α ) and (cid:28) ρ + α ( T /ℓ ) h ˙ η, ρ ⊥ i ρ ⊥ , ρ ⊥ − α ( T /ℓ ) h ˙ η, ρ ⊥ i ρ (cid:29) = O ( α ) . Thus for curves y α with shape η one has h ˙ ρ α , ρ ⊥ α i = α ( T /ℓ ) h ¨ η, ρ ⊥ i + O ( α ) . (36)On the other hand, since y α = τ + O ( α ) as α → , hence | ˙ y α | = ℓ/T + O ( α ) , the right-handside of (4) evaluated at y α behaves as − α ( ℓ/T ) [ h v ( τ ) , ρ ⊥ ih ρ , v ′ ( τ ) ρ i + h v ( τ ) , ρ i ω ( τ )] + O ( α ) = − α ( ℓ/T ) n ( τ ) + O ( α ) . The comparison with (36) shows that y α satisfies the Euler-Lagrange equation (4) up to termsof order O ( α ) if and only if h ¨ η, ρ ⊥ i = − ℓ/T ) n ( τ ) . By (16), this condition is equivalent to h ¨ η, ρ ⊥ i = h ¨ σ, ρ ⊥ i , and hence, by the boundary condi-tions, also equivalent to h η, ρ ⊥ i = h σ, ρ ⊥ i . (cid:3)
3. Context representations
Consider two functions c and ϑ as introduced in Section 4, along with the family of contextcurves u → C θ ( u ) = ( u, c ( u, θ )) parameterized by θ . The tangent direction of such a curve atthe point C θ ( u ) is given by the unit vector v ( C θ ( u )) = ( 1 , ∂ c ( u, θ ) ) p ∂ c ( u, θ )) . Since by assumption there exists for every ξ = ( ξ , ξ ) in some planar region Ξ a parameter θ = ϑ ( ξ , ξ ) such that c ( ξ , ϑ ( ξ , ξ )) = ξ , the points C θ ( u ) = ( u, c ( u, θ )) fill the region Ξ and we get a vector field v on Ξ by setting v ( ξ ) = ( 1 , ∂ c ( ξ , ϑ ( ξ , ξ )) ) p ∂ c ( ξ , ϑ ( ξ , ξ )) ) . (37)Toward calculating the Jacobian of v note first that because of ξ = c ( ξ , ϑ ( ξ , ξ )) we have ∂∂ξ ξ = ∂ c · ∂ ϑ, ∂∂ξ ξ = ∂ c + ∂ c · ∂ ϑ, hence ∂ ϑ = − ∂ c∂ c , ∂ ϑ = 1 ∂ c , where here and below it is understood that ∂ k ϑ and ∂ k c are evaluated at the arguments ξ , ξ and ξ , ϑ ( ξ , ξ ) , respectively. Similarly, ∂∂ξ ∂ c = ∂ c + ∂ c · ∂ ϑ = ∂ c − ∂ c · ∂ c∂ c , ∂∂ξ ∂ c = ∂ c · ∂ ϑ = ∂ c∂ c . With p = 1 + ( ∂ c ) the two components of v can be written as v = p − / , v = p − / ∂ c ,respectively. We have ∂ v = ∂∂ξ p − / = − p − / ∂ c (cid:18) ∂ c − ∂ c · ∂ c∂ c (cid:19) ,∂ v = ∂∂ξ p − / = − p − / ∂ c · ∂ c∂ c , furthermore ∂ v = p − / (cid:18) ∂ c − ∂ c · ∂ c∂ c (cid:19) − p − / ( ∂ c ) (cid:18) ∂ c − ∂ c · ∂ c∂ c (cid:19) = p − / (cid:18) ∂ c − ∂ c · ∂ c∂ c (cid:19) , and similarly ∂ v = p − / ∂ c∂ c . v ′ ( ξ ) = p − / − ∂ c (cid:16) ∂ c − ∂ c · ∂ c∂ c (cid:17) − ∂ c · ∂ c∂ c ∂ c − ∂ c · ∂ c∂ c ∂ c∂ c , (38)and its rotation is ω ( ξ ) = p − / ∂ c (39)(with the convention that all those partial derivatives are evaluated at ξ , ϑ ( ξ , ξ ) ).Given a planar curve x with associated 2-gon ρ, ρ ⊥ , one readily derives explicit expressionsfor the crucial quantities t α ( x ) , n α ( x ) from (38) and (39). For example, if as in Section 4we take x = τ where τ ( t ) = ( t, , t ∈ [ − ℓ/ , ℓ/ , the 2-gon is constant along τ , with ρ = (1 , , ρ ⊥ = (0 , , and we get for α = 0 t ( τ ) = − ∂ c (cid:16) ∂ c − ∂ c · ∂ c∂ c (cid:17) (1 + ( ∂ c ) ) , n ( τ ) = ∂ c − ( ∂ c ) (cid:16) ∂ c − ∂ c · ∂ c∂ c (cid:17) (1 + ( ∂ c ) ) . The latter expression gives (18).
A4. Gaussian curvature
Derivation of the approximation (20) to the Gaussian curvature.
It is a consequence of Gauss’s theorema egregium that the Gaussian curvature K of the Riemannian geometry induced by themetric tensor G can be expressed in terms of G itself [23]. A formula convenient for our purposeis [31, p. 114] K = − √ g (cid:26) ∂ (cid:18) ∂ E − ∂ F √ g (cid:19) + ∂ (cid:18) ∂ G − ∂ F √ g (cid:19)(cid:27) − g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ∂ E ∂ EF ∂ F ∂ FG ∂ G ∂ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (40)Here we have switched to the classical notation E ≡ g = 1+2 αv , F ≡ g = g = 2 αv v , G ≡ g = 1+2 αv , g ≡ g g − g , thereby accepting a change in the meaning of the symbol G , which will not cause confusion.Now g = 1 + 2 α is constant (since || v || = 1) . Therefore, √ g times the term in curly bracketson the right-hand side of (40) is equal to ∂ E + ∂ G − ∂ F = 2 α (cid:8) ∂ v + ∂ v − ∂ ( v v ) (cid:9) = 4 α (cid:8) ( ∂ v ) + v ∂ v + ( ∂ v ) + v ∂ v − ∂ v ∂ v − v ∂ v − ∂ v ∂ v − v ∂ v (cid:9) = 4 α (cid:8) ( ∂ v − ∂ v ) + ∂ v ∂ v − ∂ v ∂ v + v ∂ ( ∂ v − ∂ v ) − v ∂ ( ∂ v − ∂ v ) (cid:9) = 4 α (cid:8) ω − v ∂ ω + v ∂ ω − ( ∂ v ∂ v − ∂ v ∂ v ) (cid:9) , ) b)c) d)e) Figure 6: Gaussian curvature of the Riemannian manifold associated with the respective context. Regions with(approximately) positive or negative curvature ( C > or C < ) are marked red and blue, respectively. Labels a) toe) refer to the same contexts as shown in Fig. 3. For details see Section 4.1. with ω the rotation of v . The last term in brackets, ∂ v ∂ v − ∂ v ∂ v , equals the determinantof the Jacobian matrix v ′ . This determinant vanishes because v ′ is rank-deficient due to theconstraint v + v = 1 . (This can be seen also from (38).) The determinant in the last term of(40) is of the order O ( α ) . Therefore, putting everything together one finds that K = − α α ) (cid:0) ω − v ∂ ω + v ∂ ω (cid:1) + O ( α )= 2 α (cid:0) v ∂ ω − v ∂ ω − ω (cid:1) + O ( α ) , which is (20). (cid:3) Explicit expressions for the quantity C = v ∂ ω − v ∂ ω − ω can be derived from (39)for each of the three context types. In Fig. 6, parts of the respective context are color-markeddepending on the sign of C . The vertical stripes in the panels a and c reflect the independence of24 on its second argument for contexts of type 1: C = C ( ξ ) in this case. Such simplification doesnot occur with type 2 contexts (panels d, e). By contrast, C = 0 everywhere for contexts of type3 (not shown). This could be verified using (39) and the particular form of the function c in thiscase. It is more illuminating to recall that the context curves are (segments of) concentric circles,which by translation invariance can be assumed to be centered at the origin. The correspondingvector field then is v ( ξ ) = ( ξ , − ξ ) / p ξ + ξ , from which C = 0 easily follows. References [1] Akhiezer, N. I. (1962).
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