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Recognizing Algebraic Surfaces from their Outlines

D.A. FORSYTHdaf@cs.berkeley.edu

Computer Science Division, University of California at Berkeley,Berkeley, CA 94720

Received ??. Revised ??.

Abstract.

The outline in a single picture of a generic algebraic surface of degree three or greater completelydetermines the projective geometry of the surface. The result holds for a generic perspective view ofa generic algebraic surface, where the camera calibration parameters and the focal point are unknown.Known camera calibration appears not to reduce the projective ambiguity. The result is constructive.

Keywords: Recognition, Computer Vision, Algebraic Surfaces, Invariant Theory, Outlines.

1. Introduction

Outlines, the points in an image where a surfaceturns away from the camera, are a potentially im-portant source of information about the objectsin a scene. Typically, image edges appear at mostoutline points, and image edges can be computedreasonably reliably. This potential has not beenrealised in the case of curved surfaces, because thecomplicated relationship between the outline of acurved surface and the surface makes outlines hardto interpret. This paper shows that, although therelationship between surface and outline is com-plicated, for a large class of surfaces the outlineis sufficiently highly structured to determine thesurface’s projective geometry from a single view.

1.1. Recognising curved surfaces

There have been many approaches to recoveringshape information for curved surfaces from im-ages, including attempts to extend line labellingto curved shapes (e.g. Freeman and Shapira,1978; Malik, 1987), the development of constraint-based systems (e.g. Brooks, 1983), the study ofhow the topology of a surface’s outline changesas it is viewed from different points, formalisedinto a structure known as an aspect graph (forexample, Koenderink, 1987; Koenderink, 1992;

Plantinga and Dyer, 1987; Ponce and Kriegman,1992; Rieger, 1992) and studies of the relation-ship between the differential geometry of the out-line and that of the surface, for single images (e.g.Koenderink and Van Doorn, 1979; Marr, 1982;Koenderink, 1984) as well as for motion sequences(e.g. Giblin and Weiss, 1986; Cipolla and Zisser-man, 1992).Each approach has characteristic disadvan-

tages: extensions to line labelling and aspectgraphs can be extremely complicated for even sim-ple curved surfaces (examples in Petitjean et al.,1992; Ponce and Kriegman, 1992; Rieger, 1992);constraint-based systems must search a model-base, and studies of differential properties seldomyield sufficient information to identify a surface.Recently, there have been attempts to representthe system of outlines of a curved surface as alinear combination of outlines (see, for example,Ullman and Basri, 1991). This approach is repre-sented as providing an approximation sufficientlyaccurate for some purposes, although it cannotcapture all the complexities of the outline. Thereare two main difficulties: it is hard to specify cor-respondences between outline points, so that thelinear combination is ill-defined; and, because thescheme is based on a local approximation, it can-not capture the global interactions between a sur-face and a focal point that produce the outline.However, the approximation can yield plausible

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outlines when views are taken from similar view-ing positions.Recovering surface geometry from a single out-

line is intractable if the surface is constrained onlyto be smooth or piecewise smooth, because sig-nificant changes can be made to the surface ge-ometry without affecting the outline from a givenviewpoint. As a result, an important part of theproblem involves constructing as large a class ofsurfaces as possible that can either be directlyrecognised or usefully constrained, from their out-line alone. In this context, studies have focusedon rotationally symmetric surfaces (for example,Dhome et al., 1990; Forsyth et al., 1992) andstraight homogenous generalised cylinders (for ex-ample, the work of Binford et al., 1989; Ponce,1989; Ulupinar and Nevatia, 1990 and 1991; Zer-roug and Nevatia, in press)Recent work of Ponce and Kriegman (in Ponce

and Kriegman, 1989, 1990a, 1990b; Ponce et al.,1991; Ponce and Kriegman, 1992) focussed at-tention on algebraic surfaces. Algebraic surfaces,which consist of all the points in space where asingle polynomial vanishes, have numerous advan-tages as objects of study:• Many man-made surfaces are made up of“patches” of algebraic surface, as most popu-lar CAD/CAM surfaces are algebraic.

• The geometry of an algebraic surface is deter-mined by a relatively small number of param-eters (the coefficients of the polynomial thatgives the surface). At the same time, the sur-faces have a rich and useful geometry.

• Algebraic surfaces have important “rigidity”properties. For example, one cannot add a lo-cal bump to an algebraic surface and obtain an-other algebraic surface; the whole surface mustbe deformed instead.Ponce and Kriegman showed that elimination

theory can be used to predict the outline of an al-gebraic surface viewed from an arbitrary viewingposition. For a given surface a viewing position isthen chosen using an iterative technique, to givea curve most like the curve observed. The objectis then recognized by searching a database, andselecting the member giving the best fit to theobserved outline. This work shows that outlinecurves strongly constrain the viewed surface, buthas the disadvantage that it cannot recover sur-

face parameters without solving an optimizationproblem, so that for a big model base, each modelmay have to be tested in turn against the imageoutline. Furthermore, camera parameters must beknown to predict the outline correctly.

1.2. Indexing for recognition

A number of recent papers have shown how in-dexing can be used to avoid searching a modelbase (e.g.Lamdan et al., 1988; Forsyth et al., 1991;Rothwell et al., 1991; Wayner, 1991; Stein andMedioni, 1992). Objects are indexed by comput-ing descriptions that are unaffected by the posi-tion and intrinsic parameters of the camera, andthat differ from object to object. These descrip-tions, often known as indexing functions, have thesame value for any view of a given object, and socan be used to index into a model base withoutsearch.In a typical system that works for plane ob-

jects, projective invariants1 are computed for arange of geometric primitives in the image. Ifthe values of these invariants match the valuesof the invariants for a known model, we havegood evidence that the image features are withina camera transformation of the model features.As a result, these invariants index into a modelbase directly. Object models consist of a set of in-variant values and are therefore relatively sparse,meaning that hypothesis verification is requiredto confirm a model match. However, no search-ing of the model base is required because thehypothesised object’s identity is determined bythe invariant descriptors measured. Systems ofthis sort have been demonstrated for plane ob-jects in a number of papers (Forsyth et al., 1991),(Lamdan et al., 1988), (Rothwell et al., 1992),(Taubin and Cooper, 1992), (Weiss, 1988),(Zisserman et al., 1992). These systems are at-tractive because, in the ideal case, an object de-scription is computed from the image and iden-tifies the object, without requiring that a modelbase be searched. As a result, systems with rela-tively large model bases can be constructed2 .In the case of plane objects, indexing func-

tions are easy to compute, because a view ofa plane curve from an arbitrary focal point iswithin a projective transformation of the origi-

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nal curve. Constructing indexing functions forthree dimensional objects is challenging, becausea change in viewing position can lead to a pro-found change in the geometry of the outline.Furthermore, any indexing function should beboth invariant and computable from outline in-formation alone. Indexing functions with theseproperties have been demonstrated for polyhedra(Rothwell et al., 1993), and for rotationally sym-metric surfaces (Forsyth et al., 1992).This paper shows that such indexing functions

can be computed for algebraic surfaces viewed inperspective using an uncalibrated camera, by es-tablishing:

Theorem 1 The equation of its outline in a per-spective image completely determines the projec-tive geometry of an algebraic surface of degree 2or greater, for a generic view of a generic alge-braic surface.

Here generic means “almost every”; precisely,the generic algebraic surfaces are all algebraic sur-faces except those whose coefficients satisfy a non-trivial system of algebraic relations, to be de-termined later. At this point, we assume thatthe surface is smooth and irreducible. The re-sult is similar to that independently obtained by(D’Almeida, 1992), who demonstrated necessaryand sufficient conditions for a curve to be an out-line, but did not show that an outline determinesa surface. Note that the theorem is trival for sur-faces of degree two, as generic surfaces of degree 2are all projectively equivalent. The main result isgiven by the following two properties of outlines:• The contour generator of a generic algebraicsurface is determined as a space curve uniquely(up to a projectivity of space), by a generic pro-jection of the curve on to a plane. In particu-lar, the outline of an algebraic surface in a givenview contains sufficient information to computeboth the contour generator of that surface andthe focal point through which the contour gen-erator was formed, together in some arbitraryprojective frame.

• Given the contour generator of a generic sur-face viewed from a generic focal point, and thatfocal point, the surface can be uniquely deter-mined.

2. The outline of an algebraic surface

Throughout the paper, we assume an idealisedpinhole camera. These cameras possess a focalpoint and an image plane, which is modelled asthe projective plane. Points in space appear inthe image as the intersection between the imageplane and a line through the focal point and thepoint in space. An orthographic view occurs whenthe focal point is “at infinity”.It is easy to see that if the focal point is fixed

and the image plane is moved, the resulting dis-tortion of the image is a collineation, a one-to-onemap of the projective plane to itself that takesstraight lines to straight lines. In what follows, itis assumed that neither the position of the imageplane with respect to the focal point nor the sizeand aspect ratio of the pixels on the camera planeis known, so that the image presented to the algo-rithm is within some arbitrary collineation of the“correct” image. In this model, the image planemakes no contribution to the geometry, and itsposition in space is ignored.The outline of a surface is a plane curve in the

image, which itself is the projection of a spacecurve, known as a contour generator3. The con-tour generator is given by those points on the sur-face where the surface turns away from the imageplane; formally, the line through the focal pointto the surface is tangent to the surface. As a re-sult, at an outline point, if the relevant surfacepatch is visible, nearby pixels in the image willsee vastly different points on the surface, and sooutline points usually have sharp changes in im-age brightness associated with them. Figure 1 il-lustrates these concepts.We shall study generic algebraic surfaces in

projective 3-space, otherwise written as P 3. Apoint in P 3 is given by four homogenous coor-dinates, where two sets of homogenous coordi-nates refer to the same point if they are withina scalar multiple of one another. Appendix 1in (Mundy and Zisserman, 1992b) contains exam-ples and discussions of the practical applications ofhomogenous coordinates. Projective three-spaceis similar to the three dimensional space in whichwe live, but contains an extra plane, consisting ofof infinitely distant points. In P 3, an algebraicsurface is given by the vanishing of a single ho-mogenous polynomial in these four coordinates.

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Fig. 1. The outline and contour generator of a curved object, viewed from a perspective camera.

We will assume that the camera has an infinite

film plane as well, so that the image plane can be

modelled by P 2, the projective plane. The pointsin space that project to points “at infinity” in the

camera film plane lie on the plane parallel to theimage plane and passing through the focal point.

We use the following notation:• (u0, u1, u2, u3) are the coordinates of a point inP 3.

• (x0, x1, x2) are the coordinates of a point in P 2,the image plane.

• (f0, f1, f2, f3) is the camera focal point.• S(u0, u1, u2, u3) is the homogenous polynomialthat vanishes on the surface.

• d is the degree of S.Since the surface is generic, it is irreducible;

that is, S(u0 , u1, u2, u3) does not factor. The

contour generator lies on the surface, and so

S(u0, u1, u2, u3) = 0 on the contour generator.The plane tangent to the surface at a point on

the contour generator must pass through the focalpoint, by the definition of the contour generator.As a result, the expression

f0∂S

∂u0+ f1

∂S

∂u1+ f2

∂S

∂u2+ f3

∂S

∂u3

vanishes on the contour generator. This expres-sion will be called T for short in what follows. Weimmediately have that:• The contour generator is an algebraic spacecurve given by the vanishing of just twopolynomials4, S and T , and so is a completeintersection. Furthermore, for a generic choiceof the focal point, T has degree d − 1, and sothe contour generator has degree d(d− 1). Thesurface given by T = 0 is known as the firstpolar of S.

• The family of contour generators on a surfaceis a family of curves linearly parametrised byfocal points alone. Such families are known

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to algebraic geometers as linear systems, andare widely studied. The study of such systemsmakes general statements about contour gener-ators possible. For example, a generic contourgenerator on a generic surface is smooth, byBertini’s theorem5 . In fact, the contour gener-ator is a plane section of the dual of the surface,where the sectioning plane depends on the focalpoint chosen. The simplicity of the system ofcontour generators stands in stark contrast tothe complexity displayed by the family of out-lines, as the focal point changes, informationconventionally captured by an aspect graph.

• Contour generators are projectively covariant;that is, for a surface S, viewed from a focalpoint f , with contour generator C, if P is an ar-bitrary projectivity of space, then P (C) is thecontour generator of P (S) viewed from P (f).This is because the contour generator is definedby tangency and incidence conditions alone.It is important to note that the treatment that

follows assumes that the complex points of boththe algebraic curve and the algebraic surface aremeaningful. For example, when a count is given ofthe number of singular points on the outline of agiven type, that count includes the complex singu-larities. It is conceivable that an algebraic surfacecould have an outline that consisted entirely ofcomplex points; a natural example is the outlineof a sphere viewed from a focal point lying insidethe sphere. In this case, while in principle the out-line constrains the surface just as effectively as ifit had a large collection of real points, in practicethe outline is difficult to observe.Another effect that can make the outline diffi-

cult to observe is self-occlusion, where sections ofthe outline are occluded by the surface and so are,in practice, invisible. Self occlusion is difficult tostudy given the methods here, which emphasizethe global structure of curves and surfaces, andso cannot predict which section of an outline willbe occluded. However, the outline of a genericalgebraic surface is an irreducible algebraic curve(by genericity), meaning that a finite collection ofpoints determines the complete geometry of theoutline. This property means that partial occlu-sion is irrelevant for the cases studied here.This is why the statement of the theorem em-

phasizes the equation of the outline. In practice,

if the outline has sufficient visible real points thata fitting process can determine its equation fromthe real points alone, its complex points and sin-gularities follow. In principle, a fitting processshould be robust to occlusions, as for irreduciblealgebraic curves (the genericity assumptions as-sure that the outlines covered in this paper are ir-reducible), only a finite number of points is neces-sary to determine the equation of the curve. Thismeans that, to apply the result, we must assumethat the view yields enough real points on the out-line to determine its equation; this is not a partic-ularly strong restriction in principle.

2.1. The singularities of the outline

Since the contour generator of a non-singular al-gebraic surface viewed through a generic focalpoint is smooth, the singularities of the outlinemust be a result of the projection from the con-tour generator to the outline. These singulari-ties are the key to obtaining the contour gener-ator from the outline; fortunately, they are highlystructured. Generically there are only cusps andnodes. The following results have been knownsince at least the late 19th century; see, for ex-ample, (Basset, 1910).

Cusps A cusp in the outline is a local event onthe contour generator, so that cusps are relativelyeasily studied; a cusp occurs when the contourgenerator is tangent to the ray through the fo-cal point; see, for example, (Koenderink, 1992) forthis widely known result.

Lemma 1 Cusps in the outline are the projectionsof points on the contour generator where the sec-ond polar of the surface through the focal pointvanishes; accordingly, there are d(d − 1)(d − 2)cusps in the outline of a surface of degree d.

Proof: If p is a point on the contour generatorthat projects to a cusp on the outline, and f isthe focal point, then the line pf is tangent to thecontour generator at p. For any point q on thecontour generator, the line tangent to the contourgenerator at q is given by the intersection of theplane tangent to the surface at q and the planetangent to the first polar at q. Because the line

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tangent at p passes through f , the plane tangentto the first polar at p must pass through f as well.Recall that the surface was written as S and thefirst polar was written as T . We have that theexpression:

f0∂T

∂u0+ f1

∂T

∂u1+ f2

∂T

∂u2+ f3

∂T

∂u3

must vanish at p. This expression, which is thefirst polar of T through f , is also known as thesecond polar of S through f , and has degree d−2if d is the degree of S; call this expression P forconvenience.In turn, if S, T and P vanish at a point p, then

the point is on the surface and on the contour gen-erator by definition; furthermore, the plane tan-gent to the surface at p passes through f , andthe plane tangent to T at p passes through f , sotheir intersection, which is tangent to the contourgenerator, passes through f . Thus, the contourgenerator cusps at exactly those points where S,T and P vanish; by Bezout’s theorem there ared(d−1)(d−2) such points, and so the outline hasd(d− 1)(d− 2) cusps.

Double points Double points (nodes) on theoutline occur when a line through the focal pointis tangent to S in two distinct points, and so areglobal events; determining the number of doublepoints on the outline requires more complex rea-soning.

Lemma 2 There are 12d(d− 1)(d− 2)(d− 3) dou-

ble points on the outline of an algebraic surface ofdegree d.

Proof: The contour generator is a complete in-tersection, and so its genus is given by the formula

gcg =1

2d(d− 1)(2d− 5) + 1

where d is the degree of the surface (cf Hartshorne,1977, p. 188, ex 8.4g). Project the contour gen-erator into the image through the focal point; theresulting curve is birational to the contour gener-ator, and so has the same genus. The singularitiesare stable (by the generic choice of surface and fo-

cal point), so the genus-degree formula for planecurves yields that

gcg = (d2 − d− 1)(d2 − d− 2)− (nc + nd)

where nc is the number of cusps and nd is thenumber of double points. Rearranging the formulaand substituting the above result on the numberof cusps yields nd =

12d(d − 1)(d − 2)(d − 3).

2.2. Global properties of the singularities of theoutline

A property of the outline that will prove impor-tant later is that its singularities lie on the inter-section of two plane curves, whose degree (whichis relatively low for the number of points) canbe determined using elimination theory. Thesecurves can be studied, without loss of gener-ality, by assuming that the focal point is thepoint (0, 0, 0, 1). This makes computing theoutline relatively simple; a point (u0, u1, u2, u3)projects through (0, 0, 0, 1) to (u0, u1, u2), becauseif (u0, u1, u2) is fixed and the fourth coordinatevaries, the locus of points obtained is a line, lim-iting to the origin as u3 becomes large. Thus,(u0, u1, u2) yield the line, and u3 is a coordinatealong the line.The equation of a surface S of degree d can be

rewritten as:

S(u0 , u1, u2, u3) = H0(u0, u1, u2)ud3+

H1(u0, u1, u2)ud−13 + ...+Hd(u0, u1, u2)

where Hi(u0, u1, u2) is homogenous of degree i inu0, u1, and u2. The focal point is (0, 0, 0, 1), sothat the first polar through the focal point is:

∂S

∂u3= dH0(u0, u1, u2)u

d−13 +

(d− 1)H1(u0, u1, u2)ud−23 + ...Hd−1(u0, u1, u2)

and this vanishes on the contour generatortoo. Now the outline consists of those points(u0, u1, u2) where both equations vanish; theequation of the outline is therefore obtained byeliminating u3 between S and

∂S∂u3. The singular-

ities of the outline all have multiplicity two, andare those points (u0, u1, u2) where S and

∂S∂u3have

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two common roots in u3. The equations yieldingthese points can be obtained using a techniquefrom (Salmon).Consider two polynomials in u3,

F (u3) =

i=d∑

i=0

ad−iui3

and

G(u3) =

i=d−1∑

i=0

bd−1−iui3

If F and G have two common roots, then theremust be some

M(u3) =

i=d−3∑

i=0

Ad−3−iui3

and

N(u3) =i=d−2∑

i=0

A2d−4−iui3

such that FM + GN = 0 identically. M is theproduct of all the factors of G that do not ap-pear in F , and N is the product of all the factorsof F that do not appear in G. The polynomialFM +GN has degree 2d−3, and there are 2d−3unknown Ai, and 2d−2 monomials in u3. We canconstruct a 2d− 3 by 2d− 2 matrix C, such thatFM +GN = atCu, where a is the vector

(A0, A1, ..., Ad−3, Ad−2, Ad−1, ..., A2d−5, A2d−4)t

u is the vector

(u2d−33 , u2d−23 , ..., u23, u3, 1)

and C is the 2d−3 by 2d−2 matrix whose entriesare shown in figure 2.Since, for an appropriate choice of Ai, FM +

GN = 0 identically (i.e. all the coefficients van-ish), there is some choice of a such that atC = 0.Hence, the 2d − 3 by 2d − 3 minors of C mustvanish.In our case, aj = Hj, and bj = (d− j)Hj . The

method of construction of the matrix ensures thatthe minors are homogenous; the degree of a minorin (u0, u1, u2) can be determined by computing thedegree of a typical monomial in the minor. Such

a monomial can be obtained by striking one col-umn of C, and multiplying 2d− 3 elements fromthe remaining square matrix using each row andeach column only once. The resulting monomialwill have the form HaHbHc..., and its degree isthe sum of the subscripts. This process shows thedegrees of the minors are:

(d− 1)(d− 2), (d− 1)(d− 2) + 1,

(d− 1)(d− 2) + 2, .., (d− 2)(d− 1) + 2d− 3

At a singularity of the outline, these minors mustall vanish, so that there exists a family of curves,which intersect at most in points, of these degrees,which pass through the singular points. In par-ticular, the singularities must lie on (though notnecessarily exhaust) the intersection of a curveof degree (d − 1)(d − 2) with a curve of degree(d− 1)(d− 2)+ 1, where these curves do not havea common component.This means that the singularities are strongly

constrained. A curve of degree s has (1/2)(s +1)(s+2) coefficients, meaning that (1/2)(s+1)(s+2)−1 general points uniquely specify such a curve.There are in total (1/2)(d2− 2d)(d2− 2d+1) sin-gularities; if these were in general position, thelowest degree curve that would pass through all ofthem would have degree (d− 1)2.The matrixC yields a great deal of information

about the structure of the problem. Write

C = (c0, c1, c2, ..., c2d−3)

where the ci are column vectors. Let

Cl = (c0, c1, c2, ..., c2d−4)

By inspecting the diagonal elements, it can be seenthat the determinant of Cl, which is square, hasdegree (d−1)(d−2). LetD = Adjoint(Cl) (wherethe adjoint is the transpose of the matrix of co-factors), and let

Cr = (c0, c1, ..., c2d−6, c2d−5, c2d−3)

Inspecting the diagonal elements shows thatDet(Cr) has degree (d − 1)(d − 2) + 1. Write Pfor Det(Cl) and Q for Det(Cr). Now both P andQ are 2d− 3 by 2d− 3 minors of C, and so mustvanish on all the singular points of the outline.SinceCu is a vector of polynomials, all of which

vanish at every point on the contour generator,

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a0 a1 a2 .. .. .. ad 0 0 0 .. 00 a0 a1 a2 .. .. .. ad 0 0 .. 00 0 a0 a1 a2 .. .. .. ad 0 .. 0.. (d− 2 rows)0 0 0 .. .. a0 a1 a2 .. .. .. adb0 b1 .. .. .. bd−3 bd−2 bd−1 0 0 .. 00 b0 b1 .. .. .. bd−3 bd−2 bd−1 0 .. 0.. (d− 1 rows)0 0 0 0 .. b0 b1 .. .. bd−3 bd−2 bd−1

Fig. 2. The entries in the 2d− 3 by 2d− 2 matrix C, whose minors are the polynomials that vanish on the singularities ofthe outline.

DCu must also consist of a vector of polynomi-als, each of which vanishes at every point on thecontour generator. In particular, the last row ofDC has the form:

(0, 0, 0, ..., 0,P,Q)

and so the last element of DCu is the equation

Pu3 +Q

which must vanish at every point on the contourgenerator. This equation cannot vanish trivially;that is, at “almost every” point on the contourgenerator, both P and Q are non-zero, by the fol-lowing argument:both P and Q are homogenous polynomi-als in the variables u0, u1 and u2 and sothey vanish on a cone passing through thepoint (0, 0, 0, 1). If P and Q were to vanishon the entire contour generator, this conewould contain the contour generator, andso P and Q would have to vanish on theprojection of the contour generator throughthe point (0, 0, 0, 1) to any plane. How-ever, a projection of the contour generatorto a plane through (0, 0, 0, 1) must (by thegeneric choice of surface) be irreducible andhave degree d(d − 1); neither P nor Q canvanish at every point of an irreducible curveof this degree, because their degrees are toolow.

This means that, if P and Q can be determinedfrom the image, the contour generator can be re-constructed from the outline, because the “miss-ing” homogenous coordinate of the contour gen-erator, u3 (which can loosely be thought of as“depth”) can be determined as

u3 =−Q

P

This expression, though not strictly a function, ismeaningful, because the degree of Q is one largerthan the degree of P; as a result, if (u0, u1, u2)were to be scaled by λ, the expression for u3 wouldbe scaled by λ too.In fact, P and Q can be determined from im-

age information alone, up to an ambiguity whichis a subgroup of the projective group; P is theonly polynomial of degree (d− 1)(d− 2) that van-ishes on all the singularities of the outline, andhence can be determined from image informationup to scale. In turn, Q is a polynomial of degree(d − 1)(d − 2) + 1 that vanishes on all the singu-larities of the outline. There is a four dimensionalspace of such polynomials; section 3.1 shows thatthe ambiguity arising from choosing one of thesepolynomials to act as Q arbitrarily is just a pro-jective transformation of the contour generator.The constraints that singularities lie on curves

of particular degrees determine the family ofcurves that are generic outlines of smooth sur-faces, according to an result of (D’Almeida, 1992)which states that:Theorem: (D’Almeida) Let Γ be a planecurve of degree n(n− 1), n ≥ 3. The neces-sary and sufficient condition that there existsa smooth surface S ⊂ P 3 and a generic pointp of P 3 such that Γ is the curve of ramifi-cation of the projection of S through p, is asfollows:The curve Γ has d = n(n−1)(n−2)(n−3)/2ordinary double points, k = n(n − 1)(n− 2)cusps and no other singularities. There aretwo curves µ0 and µ1 of degrees n

2 − 3n+2

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and n2− 3n+3 respectively, without a com-mon component, that pass through the sin-gular points of Γ. The minimal degree of aplane curve containing the singular points ofΓ is n2 − 3n+ 2

Note that the “curve of ramification” is equivalentto our outline. Any errors in translation are mine.

2.3. Further global properties of the outline

The study of outlines is quite rich in curious ge-ometric properties; in particular, the form of ageneric outline is strongly constrained, and theprojective invariants of a generic outline must sat-isfy constraints. For example, note that theremust exist a frame in which the outline of a cu-bic surface has the form C2 −Q3 = 0, where Q isquadratic and C is cubic.This can be shown by representing the equation

of surface as

H0(u0, u1, u2)u33 +H1(u0, u1, u2)u

23+

H2(u0, u1, u2)u3 +H3(u0, u1, u2)

By choice of frame, the focal point can begiven coordinates (0, 0, 0, 1) and we can ensureH1(u0, u1, u2) = 0 identically; divide by H0(which is a constant), to get the form

u33 +H2(u0, u1, u2)u3 +H3(u0, u1, u2)

The polar through the focal point is now

T (u0, u1, u2, u3) = 3u23 +H2(u0, u1, u2)

The resultant with respect to u3 has degree six,and consists of terms formed from H3 (degree 3)and H2 (degree 2), and so must have the formC2 −Q3 = 0, for an appropriate choice of C andQ.Similar statements are possible about the out-

lines of surfaces of higher degree, but the form ofthe constraint becomes more complex; a possiblebenefit of such a result includes controlling thecomplexity of the fitting problem - most algebraiccurves are not outlines.

3. Obtaining the contour generator fromthe outline

Determinining the contour generator from theoutline requires knowledge of the “depth” to thecontour generator at each point of the outline.The last sections indicated how this depth is tobe found, by showing an expression that gives thehomogenous coordinate u3 as a rational functionof the other three homogenous coordinates on theoutline. In particular, this rational function canbe determined from the singularities of the out-line using the property that both numerator anddenominator vanish on the singularities of the out-line. This means that the expression for u3 isundetermined at these points. Surprisingly, thisis a useful property, because it makes it possi-ble to obtain a non-singular space curve from asingular plane curve. In particular, the processsketched above for determining the contour gen-erator from the outline is widespread in algebraicgeometry, and is known as “blowing up.” This sec-tion provides some simpler examples of blowing upto demonstrate how the process can “undo” sin-gularities; it then shows that the reconstruction ofthe contour generator is correct by showing thatit is the only possible such reconstruction, up to aprojective transformation of space. This latter re-sult requires some complicated machinery, whichis briefly introduced.

3.1. Blowing up

The outline has only cusps and double points assingularities, by the assumption that both surfaceand viewing position are generic. This means thatthere is no need to blow up more complex singu-larities. In the case of blowing up cusps or doublepoints, the central issue is a depth function thatcan be evaluated along the plane span of the curve,giving the coordinates of the space curve in affinecoordinates as

(x, y,f(x, y)

g(x, y))

or in homogenous coordinates as

(x0, x1, x2,F (x0, x1, x2)

G(x0, x1, x2))

10 ??

This is equivalent to:

(G(x0, x1, x2)x0, G(x0, x1, x2)x1,

G(x0, x1, x2)x2, F (x0, x1, x2))

Clearly, in the case of homogenous coordinatesthe degree of G is one less than the degree ofF . The depth function must have two values ateach double point (these are given as limits as apoint on the curve approaches the double point),so as to construct two points at different depthsin space that correspond to the single point in theimage. As the following two examples show, this isachieved by having the depth function undefined(0/0) at the singularities, with appropriate limit-ing properties close to the singularities; in the caseof homogenous coordinates, all four homogenouscoordinates vanish simultaneously, again with ap-propriate limiting properties.

Example: blowing up a double point inthe affine plane: Consider the curve given byy3 − x2 + y2 = 0, which has a double point at theorigin where the curve crosses itself transversally.The curve can be parametrised as

(x, y) = (t3 − t, t2 − 1)

where t is some complex parameter. The curvepasses through the double point when t = 1 ort = −1.The function

f(x, y) = (x, y, x/y)

which takes a point in the plane to a point inspace, is undefined at the origin; furthermore,

limt→0f(t cos θ, t sin θ)

depends on θ, so that when the function is appliedto a curve approaching the origin, the value of thez-coordinate depends on the direction of the ap-proach.In particular, applying this function to the

curve under consideration produces

(x, y, z) = (t3 − t2, t2 − 1, t)

less the points t = 1 and t = −1, where thefunction is not defined. However, at these pointsthe space curve has meaningful limits, which are

(0, 0, 1) and (0, 0,−1). By attaching these limitpoints we obtain a smooth space curve from a sin-gular plane curve.

Example: blowing up a cusp in the projec-tive plane: In the projective plane, points aregiven by three homogenous coordinates. In thiscase, a polynomial cannot be a function, becausescaling each homogenous coordinate changes thevalue of the polynomial without changing thepoint at which the function is defined. Thus, func-tions are given by ratios of homogenous polynomi-als of the same degree in homogenous coordinates.In fact, a function that maps a curve in the pro-jective plane to a curve in projective space canbe given as four homogenous polynomials of thesame degree in the homogenous coordinates of theplane; in this form, each polynomial represents ahomogenous coordinate in space.Consider the curve given by x30 − x2x

21 = 0

in the projective plane; this curve has a cusp at(0, 0, 1), and can be parametrised as (r2s, r3, s3),where (r, s) are the homogenous coordinates of apoint on the projective line. Consider the follow-ing function from the projective plane to projec-tive three-space:

f(x0, x1, x2) = (x20, x1x0, x1x2, x0x2)

Applied to the curve, this function yields the para-metric space curve given in homogenous coordi-nates by:

(r4s2, r5s, r3s3, r2s4)

which is equivalent to that given in homogenouscoordinates by:

(r2s, r3, rs2, s3)

This curve is a twisted cubic - this is perhaps eas-iest to see by dividing by the fourth coordinate,writing r/s = t and ignoring the point at infin-ity, giving the curve in affine (non-homogenous)coordinates as (t2, t3, t); this space curve has nosingularities.

Blowing up the outline The key to blowingup a curve with double points and cusps, as theexamples have shown, is to obtain a depth func-tion that goes to 0/0 at the double points andcusps of the curve. For the outline of an algebraic

?? 11

Fig. 3. Four frames from a fly-by, showing a plane curve with a double point and its blow-up. The blown-up curve isa non-singular space curve, shown here lying above the plane curve. It projects to the plane curve under orthographicprojection in this case.

curve in affine coordinates, such a function is eas-ily available. Recall from section 2.2 that thereexist two equations P of degree (d − 1)(d − 2),and Q of degree (d− 1)(d− 2)+1 (with no factorin common with P), both of which vanish on thesingularities of the outline. These equations yieldthe necessary depth function.Because the reconstruction is proceeding up to

a projective ambiguity, it is possible to choose afocal point; choose this focal point to be (0, 0, 0, 1),to simplify the working. Now the contour gener-ator is some curve (u0(t), u1(t), u2(t), u3(t)), andthe outline in the image plane consists of the curve(x0(t), x1(t), x2(t)) = (u0(t), u1(t), u2(t)). Recon-structing the contour generator consists, in effect,of supplying the missing u3(t). However, from theprevious section, Pu3+Q = 0 on the contour gen-erator, and P and Q are expressions in u0, u1, andu2 alone, which can be determined from the imageinformation, so that u3 can be determined at eachpoint on the curve.In particular, given an outline in the projective

plane, apply the map

(x0, x1, x2)→ (x0P, x1P, x2P,Q)

taking every point on the outline to a point inspace. For convenience, call this map the “liftingmap”. At the singular points of the outline, thelifting map degenerates (as both P and Q vanishat these points, the image of these points in themap given is (0, 0, 0, 0), which is not a meaningfulpoint in projective space). The result of the fol-lowing section shows that the closure (required tofill in the missing points where the map degener-ates) of the image of the outline in the lifting mapmust be the contour generator.

The lifting map has further useful properties;in particular, it has the property that

πoLift = Identity

where π is projection through the point (0, 0, 0, 1).In coordinates, drop the fourth homogenous coor-dinate, so that

πoLift : (x0, x1, x2)→ (x0P, x1P, x2P)

Since we are working in homogenous coordinates,(x0P, x1P, x2P) is equivalent to (x0, x1, x2). Thismeans that, if Lift takes the outline to the con-tour generator, it does so with a notion of theappropriate focal point through which to projectthe contour generator back on to the outline - theparticular lift constructed presumes that the fo-cal point is (0, 0, 0, 1), which can be done withoutloss of generality by choice of coordinates. Anyother particular focal point can be chosen as well,though the form of the resulting lift is slightlymore complicated; the important thing is that,once the lifting process has been applied, both thecontour generator and the focal point are avail-able, in a single coordinate system. This data issufficient to determine the surface.The lifting map contains an intrinsic projec-

tive ambiguity, because Q cannot be determineduniquely. There are sufficient singularities for Pto be known up to a scale - which is not a source ofambiguity, because we are working in homogenouscoordinates - but there is a four-dimensional spaceof curves of degree (d−1)(d−2)+1 that vanish onthe singularities, spanned by (Q, x0P, x1P, x2P).An element of this space is given byQa = a0x0P+a1x1P + a2x2P + a3Q Now if the lifting map usesQa instead of Q, the resulting curve is:

12 ??

(x0P, x1P, x2P,Qa) = (x0P, x1P, x2P,Q)M

where M is the matrix:

1 0 0 a00 1 0 a10 0 1 a20 0 0 a3

This is clearly just a projective transformation, aslong as a3 �= 0. Since both P and the whole spaceof possible Qa’s can be determined from the out-line, satisfying the requirement that a3 �= 0 simplyinvolves choosing a Qa that does not share a factorwith P, which is easily done.

3.2. Uniqueness of the lift

The sections above have shown constructively thatthe outline can be lifted to yield the contour gener-ator, and have demonstrated a lifting process thatmust yield the contour generator from the outline.It is also possible to show that this is, up to pro-jective transformations, the only process that willdo so; the proof is not novel, and requires a cer-tain amount of technical algebraic geometry, butis included here for completeness. Space does notallow a comprehensive introduction to the mate-rial required, but subsection 3 introduces the gen-eral approach, and sketches the direction that themathematics in subsection 3 takes, as the form ofargument used represents a powerful tool for solv-ing questions about space curves. The reader is re-ferred to (Hartshorne, 1977), which is difficult butcomprehensive, or to (Gomez-mont, 1989), whichis much more approachable but less wide-ranging.The reader willing to accept that the lifting pro-cess in the previous section yields the contour gen-erator may wish to skip both sections, or read onlysubsection 3.

Thrust of the mathematics The central ques-tion is: given a projection of an abstract alge-braic curve satisfying particular constraints, inhow many projectively different ways could thatcurve be embedded in space, consistent with theimage data? The result that will appear is thatthere is a natural choice of depth function to ob-tain the contour generator from the outline. This

result is a statement about the possible embed-dings of a curve in space that are consistent withthe image data.Embeddings of curves are generally attacked

through a technical device called a line bundle,which consists of a collection of sets made up ofthe cartesian product of an open set on the curveand an affine line. These sets are pasted togetherin a precise way using transition functions. Transi-tion functions are associated with the intersectionof two of these sets; their domain is the open set onthe curve, and their range is the line. Transitionfunctions allow studies of sections of line bundles,which associate points on the line with points onthe curve. Formally, a section is a map from thecurve to the line bundle, so that the projection ofthe map onto the first factor is the identity; thismeans that, in some coordinate system, the maphas the form f : p → (p, q), where p is a pointon the curve and q is a point on the line. Wheretwo sets intersect, there are two ways of writingeach point on the curve and each point on theline - one set of coordinates for each set. Transi-tion functions define the correspondence betweenpoints on the line in the coordinates associatedwith the first set, and those in the coordinates as-sociated with the second set. In fact, the choiceof transition functions yields the bundle.

The result is an object that locally looks likea piece of curve crossed with the affine line (c.f.the vector bundles of differential geometry). Linebundles in algebraic geometry have more rigidityproperties than the bundles of differential geome-try, for two reasons. Firstly, the topology used todefine open sets is the Zariski topology, where allalgebraic sets are closed; this means that an openset on a curve consists of the whole curve, lesssome finite number of points. Secondly, the bun-dles under consideration are typically holomorphic- this means that the transition functions are an-alytic in their domain.The following example, which is a modified ver-

sion of example 4.7 in (Gomez-mont, 1989), dis-plays a family of line bundles over the projectiveline. The first open set on the projective line willconsist of the points given in coordinates as s, fors some complex number (henceforth, the complexnumbers will be denoted by C); call this set U .This is the projective line less one point, the point

?? 13

at infinity. The second open set will consist of thepoints given in coordinates as t, for t some com-plex parameter; call this set V . Again, this is theprojective line less a point (which would be the ori-gin in s coordinates). Define the change of coordi-nates in crossing from U to V by t = 1/s. The twosets, pasted together in this way, give the wholeof the projective line; figure 4 illustrates how thesets are assembled together to yield a line.The line bundle will consist of the sets U × C

and V ×C, pasted together in an appropriate way.The set U

⋂V consists of the whole line, less two

points. There must be two transition functions:fV U , which takes coordinates on the line in U ’sframe to those in V ’s frame, and fUV , which takescoordinates on the line in V ’s frame to those in U ’sframe. Consider a point p in U

⋂V . Write:

• pU for the coordinates of p in U ’s frame;• qU for the coordinate in U ’s frame of a pointon the line C;

• pV for the coordinates of p in V ’s frame;• qV for the coordinate in V ’s frame of the pointon the line C that would be written qU in U ’sframe.

Then the pair (pU , qU) corresponds to the pair(pV , qV ) = (pV , fV U (pU )qU). Clearly, we havethat fUV fV U = 1, and that neither function van-ishes on U

⋂V . We can now define a family of line

bundles by the transition functions fV U = s−n =

tn and fUV = sn = t−n. These functions spec-

ify how the coordinates of a holomorphic sectionchange as we move from U to V and back.In U , a holomorphic section of this bundle must

have the form (s, σ(s)), where σ is a holomorphicfunction on C. As a result, σ has a representationof the form

i=∞∑

i=0

aisi

on U . In U⋂V , this section must also have the

representation (in the coordinate t on V )

tni=∞∑

i=0

ait−i =

i=∞∑

i=0

aitn−i

and this expression must also be holomorphic. Inturn, this means that for n < 0, there are no holo-morphic sections. For n ≥ 0, there are holomor-

phic sections which have the form (s,∑i=ni=0 ais

i)(for any choice of ai) in U and in V , the form

(t,∑i=ni=0 ais

n−i). Hence, for a given n, there isan n+1 dimensional vector space of holomorphicsections, corresponding to the choice of ai.Taken on its own, a section of a line bundle has

no interest for us here; however, a ratio of holo-morphic sections of a line bundle is a meromorphic(rational, with poles) function on the curve. Thus,in the example above, with n = 3, there is a fourdimensional vector space of sections. The foursections given in U coordinates by (s, 1), (s, s),(s, s2), (s, s3) can be thought of as a map, appliedto the curve, taking it to the points given in ho-mogenous coordinates in space as:

(1, s, s2, s3)

These four distinct holomorphic sections of thisline bundle map the line to the twisted cubic inprojective space, less one point; this missing pointis the image of the point at infinity, and can beobtained by evaluating these sections at the onepoint in V that does not lie in U

⋂V . In general,

four distinct holomorphic sections of a line bun-dle on a curve represent a map taking the curveto a curve in P 3, and the resulting space curveis algebraic. Note that sections can be added toone another or multiplied by constants, so thatone usually considers the linear span of a set ofsections. The attractive features of line bundlesas tools are illustrated by our example:• there are “few” holomorphic sections;• it is very often possible to tell “how many”holomorphic sections there are;

• a bundle that has n independent holomorphicsections represents a map taking the curve to acurve in P (n−1).

As a result of these properties, line bundles are acentral tool in studying embeddings of algebraiccurves.It can be seen from the example that different

line bundles represent embeddings with differentproperties; we shall be concerned with a line bun-dle often represented as OC(1), where C is thecontour generator. In the case of the projectiveline given above, this would be the bundle thatwould result for n = 1. For a general plane curveC, a general section of OC(1) would vanish ei-ther on a set of points where a line intersects the

14 ??

Fig. 4. Pasting together two affine lines to get a projective line. The first copy of the affine line parametrizes all the pointson the projective line save the point at infinity; the second copy parametrizes the point at infinity, but lacks the origin.These two copies intersect almost everywhere; they are pasted together by specifying how the coordinate of a given pointon one set relates to the coordinate of the equivalent point on the other.

curve, or on a set of points that are functionallyequivalent to a linear section. In this case, func-tional equivalence means that the points are givenby the vanishing of (p1π)/p2, where p1 and p2 arehomogenous polynomials of the same degree, π isthe equation of a line, and the expression (p1π)/p2has no poles, so that the zeros of p2 must all lieon zeros of p1π. Clearly, if p1 is the equation ofsome arbitrary line and p2 = π, this condition issatisfied. For some curves, there are other casesthat will satisfy this condition. For example, if Cis the outline of a surface, then p1 = Q, p2 = Pπwill also satisfy this condition, where P, Q arethe equations vanishing on the singularities anddefined above. This follows because the expres-sion Pu3 + Q, which was shown above to vanishon the contour generator, demonstrates that allthe zeros of P that lie on the contour generatorcoincide with zeros of Q.For a space curve, a choice of four linearly in-

dependent sections of the bundle OC(1) gives anembedding of the curve in space; in particular,this bundle admits four sections that can be rep-resented in coordinates as (u0, u1, u2, u3) (whichbasically just embeds the curve where it is inspace). Four linearly independent sections chosenfrom the linear span of this family would yield an

embedding of the curve that is projectively equiv-alent to the original curve. More interestingly,three linearly independent sections chosen fromthe linear span of this family would represent aprojection of this curve onto a plane through somefocal point; to recover the space curve, one wouldneed to determine a fourth section in the fam-ily generated as the span of (u0, u1, u2, u3) (whichwould generate our “depth function”). Of course,if OC(1) admits more than this four dimensionalvector space of sections, the problem is hopeless,as it would not be possible to determine whetherthe fourth section chosen actually lies in the spanof (u0, u1, u2, u3), and so one could not know with-out other sources of information whether the em-bedding chosen corresponded to the correct one.The crucial fact is that OC(1) has only a four di-mensional family of sections forC a contour gener-ator (in fact, for C a complete intersection). Thismeans that the fourth section can be determinedfrom a projection of the curve up to at worst aprojective ambiguity, so that the contour genera-tor can be recovered from the outline.

Mathematical details At issue is OC(1), forC the contour generator; if H0(C,OC(1)), whichis the space of sections of OC(1), is isomorphic to

?? 15

H0(P 3,OP3(1)), then H0(C,OC(1)) has dimen-

sion four. Since the outline is birational to thecontour generator, OO(1) is the same as OC(1),where O represents the outline; three linearly in-dependent sections of OO(1) are known (in coor-dinates, (x0, x1, x2)). If a fourth can then be de-termined, then O can be embedded in space usingthese four sections, and the result must be projec-tively equivalent to C.

Lemma 3 Given an algebraic curve C, whichis a complete intersection in P 3 and is nota plane curve, H0(C,OC(1)) is isomorphic toH0(P 3,OP3(1)).

Proof: I am indebted to Prof. O. DeBarre, ofthe Mathematics Department, University of Iowa,for pointing out this lemma, and showing me howit could be proven. This proof largely followshis; errors or inaccuracies are of my own addition.Note that a similar fact appears as an exercise in(Hartshorne, 1977) (p. 188, ex. 8.4).Consider the following exact sequence of

sheaves associated with the curve:

0→ I → OP3 → OC → 0

where the symbols have their usual meaning asin, for example, (Hartshorne, 1977). Taking theassociated cohomology sequence, and twisting by1, we obtain the following long exact sequence:

0→ H0(P 3, I(1))→ H0(P 3,OP3(1))→

H0(C,OC(1))→ H1(P 3, I(1))→ . . .

H0(P 3, I(1)) represents those homogenous linearexpressions that vanish on the curve, and must beempty because the curve does not lie in any plane.H0(P 3,OP3(1)) represents the hyperplanes in P

3

and H0(C,OC(1)) represents the space of sectionsof the line bundle given by a hyperplane section ofC. If we can prove that H1(P 3, I(1)) is empty, wehave that H0(C,OC(1)) is isomorphic to the sys-tem of hyperplanes in P 3, and so that the sectionsof this bundle form a four-dimensional space.The curve is a complete intersection, given

(say) by p = 0, q = 0, for polynomials p and q.As a result, we have the following free resolutionof its ideal:

0→ R→ R⊕ R→ I → 0

where R is the graded ring of homogenous polyno-mials in four variables over the complex numbers,and I is the curve’s ideal. In this sequence, theinjection R → R⊕ R is given by f :→ (−pf, qf),and the surjection R⊕R→ I is given by (a, b) :→qa+ pb. Keeping track of the grading, we find:

0→ R(1−m− n)→

R(1−m)⊕ R(1− n)→ I(1)→ 0

This free resolution yields the exact sequenceof line bundles:

0→ OP3 (1−m− n)→

OP3 (1−m) ⊕OP3 (1− n)→ I(1)→ 0

Taking the associated cohomology sequence, andrecalling the standard result (for example, inHartshorne, 1977, p. 225) that

Hi(P n,OPn(j)) = 0

for 0 < i < n and for all j ∈ Z gives thatH1(P 3, I(1)) is empty, and so we have:

0→ H0(P 3,OP3 (1))→ H0(C,OC(1))→ 0

that is, the two are isomorphic.

Lemma 4 The expression

Q

P

where Q, P are the polynomials, given in sec-tion 2.2 that vanish on all the singular points ofthe outline O, represents in coordinates an ele-ment of H0(O,OO(1)), and hence an element ofH0(C,OC(1)), for C the contour generator.

Proof: We have shown above that

Pu3 +Q = 0

on the contour generator; this is sufficient.

Summary Given the outline O of a surface, thespace curve C given by applying the map

(x0, x1, x2)→ (x0P, x1P, x2P,Q)

where P and Q are polynomials that can be deter-mined by an overconstrained fitting process fromthe singularities of the outline, is the contour gen-erator of the surface when it is viewed from the

16 ??

point (0, 0, 0, 1). Applying this map to a largenumber of points on the outline yields a set ofpoints lying on the contour generator. As a re-sult, the equations that vanish on the contour gen-erator can be determined using a fitting process.Amongst this collection of equations lies the equa-tion of a surface, projectively equivalent to theoriginal surface.

4. Obtaining the surface from the contourgenerator

The previous sections showed that it is possible totake the image outline of an algebraic surface andobtain a point in space and a space curve, whichare respectively the focal point and the contourgenerator that gave rise to the outline, and are inthe same coordinate frame - that is, the outline isobtained by projecting the reconstructed contourgenerator through the reconstructed focal point.The contour generator and focal point resultingfrom this reconstruction are projectively equiva-lent to the original contour generator and focalpoint.Once the contour generator through a particu-

lar focal point is known, it is a relatively simplematter to obtain the surface, because of the strongrelationship between the polynomials that vanishon the contour generator. There is one equationof degree d− 1 that vanishes on the contour gen-erator; if there were more, its degree would be(d − 1)2 or less. This equation is the first polarof the surface through (0, 0, 0, 1); the coefficientsof this equation can be determined from a set ofpoints on the contour generator by a fitting pro-cess. Call this equation Tm.There is a five-dimensional linear space of equa-

tions of degree d that vanish on the contour gen-erator, and this space can be determined by a fit-ting process. The equations lie in the linear spacespanned by (u0T, u1T, u2T, u3T, S). The fittingprocess will yield a basis for this space; call theelements of this basis (B0, B1, B2, B3, B4). Since

Tm =∂S

∂u3

and S lies in the span of this basis, it follows that

S =i=4∑

i=0

νiBi (1)

for some set of constants νi and that

Tm =

i=4∑

i=0

νi∂Bi

∂u3(2)

Clearly, this equation is true in coefficients. Thecoefficients of Tm are known, as are the coefficientsof Bi, and hence those of their partial derivatives.Since Tm must have at least 10 coefficients for theproblem to be interesting (S must have degree 3 orgreater for the result to be non-trivial), the termsνi can be determined from equation 2. Once νi areknown, S can be reconstructed from equation 1.There must be at least one solution, because thecurve is known to be a contour generator. In thegeneral case, this is the only solution.

Lemma 5 For S a generic surface viewed from agiven focal point f, there is no other surface S′,such that the contour generator of S′ viewed fromf is the same curve as the contour generator of Sviewed from f.

Proof: The process that forms the contour gen-erator is covariant. It is therefore sufficient todemonstrate that this lemma holds for a partic-ular focal point. This focal point can be chosen tobe (0, 0, 0, 1).If there are two different surfaces, whose equa-

tions are S and S′, which have the same contourgenerator when viewed through this focal point,the tangency relation that defines this contourgenerator must be the same for both surfaces, asthe contour generator has degree d(d− 1), and soonly one form of degree d − 1 can vanish on it.This can be written as:

∂S′

∂u3= λ0

∂S

∂u3

where λ0 is an unknown constant to allow for scal-ing the equations (which does not affect the geom-etry of the underlying curve).Furthermore, we have that the linear system of

five degree d forms that vanishes on the contourgenerator is the same for each surface. Thus, inparticular, there are constants λi such that

?? 17

S′ = λ1u0∂S

∂u3+ λ2u1

∂S

∂u3+ λ3u2

∂S

∂u3+

λ4u3∂S

∂u3+ λ5S

As a result, we can write:

∂S′

∂u3= λ1u0

∂2S

∂u3∂u3+ λ2u1

∂2S

∂u3∂u3+

λ3u2∂2S

∂u3∂u3+ λ4u3

∂2S

∂u3∂u3+ (λ5 + λ4)

∂S

∂u3

This can be rewritten as:

λ0∂S

∂u3= λ1u0

∂2S

∂u3∂u3+ λ2u1

∂2S

∂u3∂u3+

λ3u2∂2S

∂u3∂u3+ λ4u3

∂2S

∂u3∂u3+ (λ5 + λ4)

∂S

∂u3

By rearranging terms, and setting µ0 = λ1, µ1 =λ2, µ2 = λ3, µ3 = λ4, µ4 = λ5 + λ4 − λ0, weobtain:

µ0u0∂2S

∂u3∂u3+ µ1u1

∂2S

∂u3∂u3+ µ2u2

∂2S

∂u3∂u3+

µ3u3∂2S

∂u3∂u3+ µ4

∂S

∂u3= 0

For the case that S′ = λ0S, we must have thatλ1 = 0, λ2 = 0, λ3 = 0, λ4 = 0, and λ5 = λ0, sothat all the µi must vanish, and the equation istrivially true. If we have S′, S where S′ �= λ0S,then there must be some solution for the aboveequation where not all µi vanish. This yields anoverdetermined system of equations in the coef-ficients of S, where the µi are unknown. Forthese equations to be satisfied, the determinantsof the coefficient matrices, which are easily shownto be non-trivial expressions in the coefficients ofS alone, must vanish. In turn, these determi-nants represent constraints that the coefficients ofS must satisfy, and so S is not a general surface.

5. Geometric ambiguities

The discussion above assumed abstract projec-tion. Because the focal point for the reconstruc-tion and the sections of the line bundle used tolift the outline were chosen arbitrarily, it is notsurprising that the best possible reconstruction is

up to a projective transformation. However, thisleaves a substantial ambiguity in the surface’s ge-ometry. It is often the case that the internal pa-rameters of a camera are fully or partially known,and one might hope that a better reconstruction ispossible in this case. Surprisingly, unless a model-base is available, a better reconstruction appearsimpossible.Consider a calibrated camera, where, without

loss of generality, the focal point lies at (0, 0, 0, 1).The outline of an object is formed by a cone ofrays through this point, and tangent to the objectitself. The intrinsic ambiguity of the reconstruc-tion process must include all transformations ofthe object that fix this cone of rays and the fo-cal point - this is the group of dilations of space,written as:

1 0 0 00 1 0 00 0 1 0a b c d

where d �= 0. If there is no modelbase, then thegeometry of the surface observed must be given asa set of invariants to some transformation group.In particular, in most conceivable applications thedescription must be invariant to Euclidean trans-formations of space. It is easily verified6 thatthe smallest subgroup of the projective group thatcontains both the Euclidean group and the dila-tions is the projective group itself. This meansthat to describe algebraic surfaces by invariantsusing only outline information and without refer-ence to a modelbase, one must use projective in-variants, whether the camera is calibrated or not.However, a modelbase changes the ambiguity

substantially. If, for example, there is a dis-crete modelbase with a small number of mod-els, it is straightforward to extend the consis-tency approach of (Forsyth et al., 1994) to yield aEuclidean reconstruction of a system of surfaces,though the study of ambiguities in the reconstruc-tion appears to become difficult. It is not knownwhether these ambiguities allow reconstructionswhen there are parametrised families of models.

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6. Discussion

There is now a constructive path from observa-tions of outline points to the full projective geom-

etry of the surface, which goes as follows:• Fit an algebraic curve that is an outline to theobservations - this process will also yield the

degree of the surface (by a search over degreesd(d− 1) for increasing d, if necessary).

• Determine the singularities of this fitted curve(using the coefficients of the curve).

• Compute the coefficients of the polynomialsthat would blow up these singularities, as de-scribed above in section 3.1.

• Use these coefficients to form a map from theplane to space (section 3.1). Apply this map to

a large number of points on the outline, yield-ing a collection of points in space.

• Determine the unique surface of degree d − 1passing through these points in space.

• Determine the five-parameter family of surfacesof degree d passing through these points in

space.• Determine the νi of the previous section, usingthe methods given there.

• These νi can be substituted into the equationsabove, to give the coefficients of the surface.

Although a simple implementation that suc-cessfully identifies cubic surfaces from synthetic

outline data, exists, there are real difficulties inconstructing an implementation of this approach

that works in a practical vision system:• Computing the outline from image data re-quires fitting high degree algebraic curves to

edge points. The degree goes up as the squareof the degree of the surface.

• Computing the contour generator from the out-line is tricky, as it requires finding singularities

in the outline. Unfortunately, a small changein the coefficients of the outline can lead to

substantial errors in the computed singulari-ties, both in location and in multiplicity. Such

errors are guaranteed by the fact that we areusing a fitted curve. Furthermore, the singu-larities must have special properties, for the

curve to be an outline at all. This has ad-vantages and disadvantages: the curve can be

chosen from a smaller, more specialised class of

curves, which may make fitting more robust; atthe same time, the curve produced by a generalfitter cannot, in general, even be an outline.

• In practice, determining the surface from a sys-tem of points on the contour generator involvesa process of fitting algebraic surfaces to pointsin space, and has the associated instabilities.Considerable precision in the points is required;in the experiments on synthetic data, this couldbe supplied, but it is doubtful whether suchprecision is available in practical situations.

Despite its present impracticality, this result isvaluable, primarily because it shows that shapefrom outline is possible in the context of avery large and interesting range of surfaces, andthereby opens several promising avenues of re-search:• It is hard to be a contour generator. In thecase of algebraic surfaces, “most” curves arenot contour generators, because either their de-gree, their genus, or the number and type oftheir singularities is wrong. There is good rea-son to believe that a similar result must holdfor surfaces drawn from a “small” parametrisedfamily of smooth surfaces, because the rangeof contour generators for a given surface is sosmall, although the mechanisms of proof andof computation may be more complex. This isthe subject of active ongoing research.

• It is an example of a recognition algorithmthat recognises an object drawn from a large,parametrised world (generic algebraic surfacesof degree three or greater) without searching amodel-base. If this algorithm is presented withsuch a surface, it can (in principle) immediatelydescribe the surface up to the intrinsic ambigui-ties of the viewing geometry. Given the way thealgorithm is framed, verification appears to beeither extremely difficult or impossible, and sothe role of the model-base becomes uncertain.

• It suggests that the global properties of systemsof contour generators are important objects ofstudy. Compare the simple, neat structure ofthe family of contour generators on a projectivealgebraic surface with the extraordinary com-plexity of its aspect graph; in this case, theaspect graph is, in principle, redundant, be-cause a single outline contains sufficient infor-mation to determine the entire surface. Fur-

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thermore, in the case of projective algebraicsurfaces, the system of contour generators isone case of a well understood class of objects:a linear system of curves on a surface. A studyof this system might yield a much more practi-cal algorithm for recognising a surface from twoor three uncalibrated views, with an unknowntransformation between the views, by exploit-ing the fact that each outline represents a curvedrawn from a “small” (three-dimensional) lin-ear system of curves.

• It opens a number of curious geometric ques-tions; for example, what is the relationship be-tween the six cusps on the outline of a cubicsurface, and the surface? The contour genera-tor is obtained from the outline by blowing upthese six cusps on the outline; but, by stan-dard results, if we were to extend this blowingup process to the whole plane, we would obtaina surface passing through the contour genera-tor, and the degree of this surface could not begreater than three. Note that this is not a gen-eral cubic surface, because the six points blownup are not in general position, but appears tobe a surface bearing some substantial relation-ship to the original cubic surface.

Determining a class of surfaces that is plasticenough to be useful for modelling a wide rangeof real objects, yet rigid enough to allow strongstatements about the shape of a particular sur-face from a single outline, is the central issue instudying shape from contour. We have shown thatalgebraic surfaces represent one extreme; a verylarge class of surface that is so rigid that an out-line determines a surface. There is room for muchfuture work.

Acknowledgements

Olivier De Barre, of the University of Iowa Math-ematics Department pointed lemma 3 out to me,and showed me how it could be proven. Thiswork has benefited from conversations with TomBuchanan (who referred me to D’Almeida’s work),Olivier Faugeras, Margaret Fleck, Peter Giblin,Richard Hartley, Steve Maybank, Joe Mundy,John Oliensis, Charlie Rothwell, Richard Weissand Andrew Zisserman. Anonymous referees pro-vided extensive and extremely helpful comments.

This work was supported in part by a grant fromUnited States Air Force Office of Scientific Re-search AFOSR-91-0361, in part by the NationalScience Foundation under award no. IRI-92-09729, in part by Magdalen College, Oxford, inpart by the University of Iowa, and in part by aNational Science Foundation Young InvestigatorAward with matching funds from GE, Tektronix,Rockwell and Eugene Rikel.

Notes

1. A clear introduction to applying invari-ant theory in computer vision appears in(Mundy and Zisserman, 1992a).

2. Current systems using indexing functions have model-bases containing of the order of thirty objects.

3. There are a number of widely used terms for bothcurves, and no standard terminology has yet emerged.

4. Some algebraic curves in space must be given by morethan two equations - see, for example (Ohm, 1980)

5. A generic element of a linear system is smooth awayfrom its base points; see, say, Griffiths and Harris,1986). Note that for a smooth surface, the system ofcontour generators has no base points, but if the sur-face is singular, the singularities of the surface are basepoints, and the contour generator may be singular.

6. The most practical technique is simply to form the com-mutators for the Lie algebra of the group containingboth Euclidean transformations and dilations, as de-scribed in (Olver, 1986), and then note that the span ofthe set of commutators and generators is the Lie algebraof the projective group.

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