TThe Vector Space of Convex Curves: How to Mix Shapes
Dongsung Huh
Gatsby Computational Neuroscience Unit, University College London, London, W1T 4JG, United Kingdom
We present a novel, log-radius profile representation for convex curves and define a new operationfor combining the shape features of curves. Unlike the standard, angle profile-based methods, thisoperation accurately combines the shape features in a visually intuitive manner. This method haveimplications in shape analysis as well as in investigating how the brain perceives and generatescurved shapes and motions.
INTRODUCTION
Understanding how we perceive and understand shapeis a central problem in human and computer vision.Plane curves are the simplest forms that have shape:we can easily recognize objects in cartoons or outlinesof images. Here, we introduce a novel way to representand combine the shape of plane curves that has the twinvirtues of mathematical elegance and intuitive simplicity.
ANGLE PROFILE REPRESENTATION
A plane curve is formally described as a continuousmapping from a closed interval of real numbers to a 2-DEuclidean plane, Γ : I → R , i.e., Γ( s ) = ( x ( s ) , y ( s )) , where s ∈ I is the 1-D coordinate parameterization alongthe curve. If the natural, arc-length coordinate is used(such that ds = (cid:112) dx + dy ), the derivative of the curvewith respect to s yields a unit-length velocity vector: (cid:107) Γ (cid:48) ( s ) (cid:107) = 1. This implies that the curve shape can befully characterized by the velocity vector’s orientationalong the curve, i.e. , the angle profile { θ ( s ) } , since thelength of the vector does not contain any shape informa-tion.The angle profile representation is widely used for iden-tification and categorization of curve shapes, as well asshape synthesis [1]. In particular, one of the most widelyused shape description methods, called Fourier Descrip-tors , analyzes angle profiles of curves as linear combina-tions of their frequency components [2, 3]. However, suchanalysis has serious drawbacks, because the operation forcombining angle profiles does not properly translate tocombination of shape features. (See Appendix A).Here, we introduce a novel, log-radius profile represen-tation that is dual to the angle profile representation.This representation resolves the problems of the angleprofile representation, and provides a new operation forcombining shape features.
LOG-RADIUS PROFILE REPRESENTATION OFCONVEX CURVES
Consider a subset of plane curves with monotonicallyincreasing angle profiles, called convex curves . Such acurve admits an alternative, angle-based parameteriza-tion, Γ( θ ), since any point on the curve can be uniquelyspecified by θ . The angle coordinate, like arc-length, is anatural parameterization of a curve, determined uniquelyby the curve’s geometry. Moreover, it is scale invariant— the coordinate value of a point remains unchangedunder scaling operations on the curve.Differentiating a curve with respect to θ yields a veloc-ity vector whose length is the local radius of curvature: (cid:107) Γ (cid:48) ( θ ) (cid:107) = dsdθ ≡ r ( θ ) > . (1)Then, the shape of a convex curve can be fully charac-terized by the radius profile { r ( θ ) } along the curve, be- = FIG. 1: Examples of elementary shapes shownwith their characteristic frequency ν a r X i v : . [ c s . G R ] J un cause the orientation of the velocity vector only providesredundant information that is already specified by theangle coordinate.To summarize, the radius-profile describes the lengthof the velocity vector as a function of the orientation,whereas the angle-profile describes the orientation of thevelocity vector as a function of the arc-length: They pro-vide complementary ways to represent curves.However, since we will consider scaling operations oncurves, it is more convenient to introduce the log-radiusprofile representation, { l ( θ ) } , defined as l ( θ ) ≡ log r ( θ ) . (2) ELEMENTARY SHAPES
A circle is a simple featureless curve, described by aconstant log-radius of curvature. However, interestingshape features are described by fluctuations of the log-radius profiles.Let us define elementary shapes by sinusoidal log-radius profiles: l ( θ ) = (cid:15) sin( ν ( θ − θ o )) , (3)where ν is the frequency, (cid:15) is the amplitude and θ o is thephase shift, which rotates the shape (Fig. 1).Each elementary shape exhibits a distinctive featurecharacterized by the frequency ν . For example, the elon-gated shape of an elliptic curve is characterized by fre-quency 2, whose log-radius profile oscillates twice perone full rotation of θ , or 2 π radians. At larger integerfrequencies, the shapes resemble rounded regular poly-gons. In general, an elementary shape with a rationalfrequency ν = m/n , where m and n are coprime integers( i.e. no common factors) and m >
1, has a closed shapeof period Θ = 2 πn , and exhibits m degrees of rotationalsymmetry. If m = 1, then the curve does not close andexhibits a translational symmetry. In the zero frequencylimit, the elementary shape approaches a logarithmic spi-ral: l ( θ ) = lim ν → ( a/ν ) sin( νθ ) = aθ .The amplitude (cid:15) modulates the degree of expression ofelementary shapes, while preserving their characteristicfeatures (see Fig 2A). VECTOR SPACE OF CONVEX CURVES
Uniform scaling is the simplest operation on a curve,which scales the overall size of the curve while preserv-ing its shape. In our representation, this corresponds toadding a constant to the log-radius profile.More generally, we define two operations on convexcurves, Γ i : I → R : scalar multiplication (Fig 2A)Γ = a · Γ o ←→ l ( θ ) = a · l o ( θ ) , (4) ======== +++ = 0.3 x 1.6 x 1.6 = 0.5= 0.8 AB FIG. 2: (A) Scalar multiplication of curves, eq (4). Sizeis normalized. (B) Addition operation between two curves,eq (5). This operation accurately combines the shape fea-tures of curves. which amplifies the log-radius profile by a scalar fac-tor ( a ∈ R , ∀ θ ∈ I ), and addition between two curves(Fig 2B) Γ = Γ + Γ ←→ l ( θ ) = l ( θ ) + l ( θ ) . (5)which adds their log-radius profiles in a pointwise manner( ∀ θ ∈ I ).The addition operation can be understood as a general-ized scaling operation, which uses one curve’s log-radiusprofile as the scale factor for modifying the other curve.It reduces to uniform scaling if one of the curves is a cir-cle. The unit circle, l ( θ ) = 0, is the identity element ofthe addition operation.Remarkably, the addition operation exquisitely mergesthe shape features of the added curves in a visually intu-itive manner (See Fig 2B). For examples, adding a shrink-ing spiral progressively decreases the length scale of thecurve, thereby “spiralizing” its shape, whereas addingan ellipse elongates the curve in one direction and com-presses it in the perpendicular direction, thereby “ellip-tizing” the curve. Thus, adding a spiral and an ellipseproduces an elliptic spiral, which combines the shapefeatures of both curves. This accurate combination ofshape features owes to the angle coordinate representa-tion, which is invariant under scaling operations.The scalar multiplication and addition operations de-fine a vector space over the set of convex curves, whichis spanned by the basis set of elementary shape curveseq (3); that is, any curve in this space can be representedas a linear combination of elementary curves. Moreover,an inner-product between curves can be defined as, (cid:104) Γ , Γ (cid:105) ≡ (cid:90) θ ∈ I l ( θ ) l ( θ ) dθ, (6)which induces a norm (cid:107) Γ (cid:107) ≡ (cid:112) (cid:104) Γ , Γ (cid:105) . Thus, convexcurves with finite norm ( (cid:107) Γ (cid:107) < ∞ ) form a Hilbert space,isomorphic to the space of square-integrable functions, L . DISCUSSION
We presented a novel, log-radius profile representationfor describing convex shapes, which is dual to the stan-dard, angle profile-based representation and offers a com-plementary view of curve shapes. The angle profile repre-sentation is closely related to “bending” operations: thesimplest curve is a straight line, which can be bent intovarious curves. In contrast, the log-radius profile repre-sentation is closely related to “scaling” operations: thesimplest curve is a circle, which can be non-uniformlyscaled into other curve shapes.The log-radius profile representation resolves the afore-mentioned problems of the angle profile representation(See Appendix A): The elementary curves preserve their characteristic shape features over all range of ampli-tude, and the addition operation accurately combinesthe shape features. Therefore, Fourier transform of log-radius profiles indeed properly analyzes the curve shapesinto visually meaningful shape features.Recent applications of this method have revealed sur-prising details of regularities in kinematics of curved handmovements [4], as well as in speed perception of curvedmotion [5]. It may also be useful in investigating theshape perception process in vision.Here, we considered representation of 1-D convexcurves in 2-D space. This result can be generalized tohigher dimensional, convex surfaces. The angle coordi-nate then becomes related to normal vector to the surfaceand inverse Gauss map.
Acknowledgement
I thank Terrence J. Sejnowski for helpful comments.This research was supported by Gatsby Charitable Foun-dation.
Appendix A: Fourier Descriptors
The Fourier Descriptor method describes the angleprofile of a curve as a linear combination of frequencycomponents θ ( s ) = s Θ + ∞ (cid:88) k =1 a k cos( ks Θ) + b k sin( ks Θ) , (A1)where s ∈ [0 ,
1] is the normalized arc-length coordinateand Θ determines the total number of turns (2 π for sim-ple closed curves) [3]. The coefficients a k , b k are calledFourier descriptors.However, such approach has serious drawbacks. First,the curve shape described by a single frequency compo-nent θ k ( s ) = s Θ + a k cos( ks Θ) (A2)exhibits a wildly varying shape as the coefficient a k in-creases (Fig A1, top), thus failing to represent a unique,consistent shape feature. Secondly, in the angle profilerepresentation, linear addition of the frequency compo-nents does not properly combine their shape features, buttends to produce rather deformed shapes (Fig A1, bot-tom). Therefore, decomposing an angle profile into thefrequency components via Fourier Transform does notproperly translate to a meaningful analysis of the curve’sshape into basic shape features. = +
21 4 a = k = FIG. A1: Problems of angle profile representation. Top: Asingle frequency component shape described by eq (A2). Bot-tom: Addition operation in angle profile representation. [1] Dengsheng Zhang and Guojun Lu, “Review of shape repre-sentation and description techniques,” Pattern recognition , 1–19 (2004).[2] RL Cosgriff, “Identification of shape,” Ohio State Univ.Res. Foundation, Columbus, OH, Tech. Rep. ASTIA AD , 792 (1960).[3] Charles T Zahn and Ralph Z Roskies, “Fourier descriptorsfor plane closed curves,” Computers, IEEE Transactionson100