Varieties, Parameters, and Moduli

Bernd SturmfelsUC Berkeley

Algebraic VisionTU Berlin, October 8-9, 2015

Quartic CurvesVarieties are zero sets of polynomials. Homogeneous polynomialsspecify projective varieties in Pn. Points can be real, complex, ....

My example: Plane quartic curves

HYPERBOLIC POLYNOMIALS, INTERLACERS, AND SUMS OFSQUARES

MARIO KUMMER, DANIEL PLAUMANN, AND CYNTHIA VINZANT

Abstract. Hyperbolic polynomials are real polynomials whose real hypersurfaces are max-imally nested ovaloids, the innermost of which is convex. These polynomials appear in manyareas of mathematics, including optimization, combinatorics and differential equations. Herewe investigate the special connection between a hyperbolic polynomial and the set of poly-nomials that interlace it. This set of interlacers is a convex cone, which we write as a linearslice of the cone of nonnegative polynomials. In particular, this allows us to realize any hy-perbolicity cone as a slice of the cone of nonnegative polynomials. Using a sums of squaresrelaxation, we then approximate a hyperbolicity cone by the projection of a spectrahedron.A multiaffine example coming from the Vámos matroid shows that this relaxation is notalways exact. Using this theory, we characterize the real stable multiaffine polynomials thathave a definite determinantal representation and construct one when it exists.

1. Introduction

A homogeneous polynomial f 2 R[x] of degree d in variables x = (x1, . . . , xn) is calledhyperbolic with respect to a point e 2 Rn if f(e) 6= 0 and for every a 2 Rn, all roots of theunivariate polynomial f(te + a) 2 R[t] are real. Its hyperbolicity cone, denoted C(f, e) isthe connected component of e in Rn\VR(f) and can also be defined as

C(f, e) = {a 2 Rn : f(te � a) 6= 0 when t 0}.

As shown in Gårding [6], C(f, e) is an open convex cone and f is hyperbolic with respectto any point contained in it. Hyperbolicity is reflected in the topology of the real projectivevariety VR(f) in Pn�1(R). If VR(f) is smooth, then f is hyperbolic if and only if VR(f)consists of bd

2c nested ovaloids, and a pseudo-hyperplane if d is odd (see [8, Thm. 5.2]).In[121]:= f = ContourPlot3D@2 x^4 + y^4 + z^4 - 3 y^2 x^2 - 3 z^2 x^2 + y^2 z^2 ã 0 ,8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4<, Mesh Ø None,ContourStyle Ø 8 Blue, Opacity@.5D<, Boxed Ø False, Axes Ø None, PlotPoints Ø 100D

Out[121]=

In[137]:= p1 = ContourPlot3D@x ã 1.6, 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4<,Mesh Ø None, ContourStyle Ø 8Magenta<, Boxed Ø False, Axes Ø NoneD;

p2 = ContourPlot3D@z ã 1.5, 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4<,Mesh Ø None, ContourStyle Ø 8Green<, Boxed Ø False, Axes Ø NoneD;

Show@8f, p1<DShow@8f, p2<DShow@8f, p1, p2<D

Figure 1. A quartic hyperbolic hypersurface and two of its affine slices.

A hyperbolic program, introduced and developed by Güler [7], Renegar [14] and others,is the problem of maximizing a linear function over an affine section of the convex cone

2010 Mathematics Subject Classification. Primary: 14P99, Secondary: 05E99, 11E25, 52A20, 90C22.1

arX

iv:1

212.

6696

v2 [

mat

h.A

G]

3 N

ov 2

013

a0x4 + a1x

3y + a2x3z + a3x

2y2 + a4x2yz + a5x

2z2 + a6xy3 + a7xy

2z+a8xyz

2 + a9xz3 + a10y

4 + a11y3z + a12y

2z2 + a13yz3 + a14z

4 = 0

Quartic CurvesComputers like to think about equations:

a0x4 + a1x

3y + a2x3z + a3x

2y2 + a4x2yz + a5x

2z2 + a6xy3 + a7xy

2z+a8xyz

2 + a9xz3 + a10y

4 + a11y3z + a12y

2z2 + a13yz3 + a14z

4 = 0

Humans like to think about pictures:

29/08/15 09:08Triple_torus_illustration.png 2.204×1.550 pixels

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The Three Levels

Level 1: Variety Baby BearLevel 2: Parameter Space Papa BearLevel 3: Moduli Space Mama Bear

Level 1: Quartic curve dimension 2Level 2: Space of coefficients dimension 14Level 3: Moduli space of plane quartics dimension 6

Level 1: Coordinates on the curve are (x : y : z)Level 2: Coordinates on P14 are (a0 : a1 : a2 : · · · : a14)Level 3: Coordinates on P14//PGL(3) via geometric invariant theory

Algebraic geometers prefer the 6-dimensional moduli space M3.

It maps birationally onto P14//PGL(3) and its concrete models.

Quiz: How would YOU choose six moduli coordinates?

Level 2: Cameras

Most people think Mathematicians think29/08/15 11:0272157624493316109_model_huge_674cedbc7f.jpg 507×457 pixels

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Level 2: Cameras

A camera is a 3⇥4-matrix A of rank 3. It defines a linearprojection P3 99K P2. The focal point is f = ker(A).

29/08/15 11:0272157624493316109_model_huge_674cedbc7f.jpg 507×457 pixels

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24? ? ? ?? ? ? ?? ? ? ?

35

The space of n cameras (A1, A2, . . . , An)A camera is a 3×4-matrix Ai of rank 3. It defines a linearprojection P3 99K P2. The focal point is fi = ker(Ai ).

A configuration of n cameras is a tuple A = (A1,A2, . . . ,An). Thespace of such tuples is our parameter space. It has dimension 12n.

A calibrated camera has the form [R | t ] where R ∈ SO(3), t ∈ R3.

Level 1: Pictures

Multiview geometry or epipolar geometry is foundationalfor computer vision. It concerns the space of pictures ofthree-dimensional objects seen from n ≥ 2 cameras.

[R. Hartley and A. Zisserman: Multiple View Geometryin Computer Vision, Cambridge University Press, 2003]

Fix a configuration of cameras A = (A1,A2, . . . ,An)whose foci f1, f2, . . . , fn are in general position in P3.

Taking n pictures means applying the rational map

φA : P3 99K (P2)n, x 7→ (A1x,A2x, . . . ,Anx).

The closure of the image of φA is an algebraic variety,denoted VA and called the multiview variety of thetuple A = (A1,A2, . . . ,An) of 3× 4-matrices.

Two CamerasA1 =

[? ? ? ?? ? ? ?? ? ? ?

]A2 =

[� � � �� � � �� � � �

]

The multiview variety VA is the hypersurface in P2 × P2 defined by

f (x , y , z) = det

? ? ? ? x1 0? ? ? ? y1 0? ? ? ? z1 0

� � � � 0 x2� � � � 0 y2� � � � 0 z2

This polynomial is bilinear:

f (x , y , z) =[x1 y1 z1

]·

� � �� � �� � �

·

x2y2z2

This 3×3-matrix is the fundamental matrix of the two cameras.

What is its rank? How to write � in terms of ? and � ?

Two Cameras

Bernd likes formulas.Michael likes drawings.

RIGID MULTIVIEW VARIETIES

MICHAEL JOSWIG, JOE KILEEL, BERND STURMFELS, AND ANDRE WAGNER

Abstract. The multiview variety from computer vision is generalizedto images by n cameras of points linked by a distance constraint. Theresulting five-dimensional variety lives in a product of 2n projectiveplanes. We determine the defining polynomial equations, and we exploregeneralizations of this variety to scenarios of interest in applications.

1. Introduction

The emerging field of Algebraic Vision is concerned with interactions be-tween computer vision and algebraic geometry. A central role in this en-deavor is played by projective varieties that arise in multiview geometry [5].

The set-up is as follows: A camera is a linear map from the three-dimensional projective space P3 to the projective plane P2, both over R.We represent n cameras by matrices A1, A2, . . . , An 2 R3⇥4 of rank 3. Thekernel of Aj is the focal point fj 2 P3. Each image point uj 2 P2 of cameraAj has a line through fj as its fiber in P3. This is the back-projected line.

We assume throughout that the focal points of the n cameras are ingeneral position, i.e. all distinct, no three on a line, and no four on a plane.Let �jk denote the line in P3 spanned by the focal points fj and fk. Thisis the baseline of the camera pair Aj , Ak. The image of the focal point fj

in the image plane P2 of the camera Ak is the epipole ek j . Note that thebaseline �jk is the back-projected line of ek j with respect to Aj and alsothe back-projected line of ej k with respect to Ak. See Figure 1 for a sketch.

u2

f2f1

u1

X

e2 1e1 2

Figure 1. Two-view Geometry

Fix a point X in P3 which is not on the baseline �jk, and let uj and uk bethe images of X under Aj and Ak. Since X is not on the baseline, neither

1

Algebraic geometry is the art of mixing formulas with drawings.

Can you see the fundamental matrix?

Multiview Variety

TheoremThe multiview variety VA ⊂ (P2)n is 3-dimensional. Its ideal isminimally generated by

(n2

)bilinear and

(n3

)trilinear polynomials.

For generic A, using lexicographic monomial order,the reduced Grobner basis has

(n2

)+ 3(n3

)+(n4

)elements.

[Aholt-St-Thomas, Canadian J. Math, 2013]

n = 3: The coefficients of the (unique?) trilinear generator

2∑

i , j , k=0

tijk · xiyjzk

form a 3×3×3-tensor (tijk), known as the trifocal tensor.They are certain 4×4-minors of the 4×9-matrix

[AT1 |AT

2 |AT3

].

Rigid Multiview Variety

Take pictures of two points X ,Y ∈ P3 that are linked by a distanceconstraint. Let V (Q) be the hypersurface in P3 × P3 defined by

Q = (X0Y3−Y0X3)2 +(X1Y3−Y1X3)2 +(X2Y3−Y2X3)2−X 23Y

23 .

The 5-dimensional rigid multiview variety is the image of the map

ψA : V (Q) ↪→ P3 × P3 99K VA × VA ⊂ (P2)n × (P2)n,(X ,Y ) 7→

((A1X , . . .AnX ), (A1Y , . . .AnY )

)

[Joswig-Kileel-St-Wagner, 2015]30/08/15 08:057.Tipps-wie-du-den-perfekten-Elevator-Pitch-durchführst.png 2.774×985 pixels

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“We determine defining polynomial equations and exploregeneralizations to scenarios of interest in applications.”

The Three Levels

Level 1: Variety Baby BearLevel 2: Parameter Space Papa BearLevel 3: Moduli Space Mama Bear

Level 1: Multiview Variety dimension 3 (or 5)

Level 2: Space of Tuples of Cameras dimension 12nLevel 3: Compactified Camera Space Γn dimension 11n − 15

Level 1: Homogeneous coordinates on P2 × P2 × · · · × P2

Level 2: Coordinates are matrix entries of A1,A2, . . . ,An

Level 3: Which coordinates? Can geometric invariant theory help?

[AST 2012]: Γn is the main components in the multigraded Hilbertscheme of all ideals with the same Hilbert function Nn → N as VA.

Level 3: Trifocal Tensors

The trifocal variety T consists of all trifocal tensors (tijk)in P26 = P(C3⊗3⊗3). It has dimension 18 = 11 · 3− 15.

There is a birational isomorphism

Γ3 99K T .

The tijk serve as moduli coordinates.

2 CHRIS AHOLTA, LUKE OEDINGB

Figure 1. The trifocal tensor as a map P2 ⇥ P2 ! P2

l1

l 2

l3

l 1,2

f1

f2

f3

2

3

1

The image in camera i of a point x 2 P3 is determined by intersectingthe line hfi, xi with the plane ⇡i. Now consider lines li ✓ ⇡i for i = 1, 2.The planes hf1, l1i and hf2, l2i generically intersect in a line l1,2 ✓ P3,and the plane hf3, l1,2i generically intersects ⇡3 in a line l3. See Figure 1.

We have described for a su�ciently general camera configuration, amap

P2 ⇥ P2 ! P2,

given by (l1, l2) 7! l3. This map must come from a bilinear map

C3 ⇥ C3 ! C3.

To help avoid ambiguity, fix A, B, C ' C3 so that this map is nowA⇥B ! C. This bilinear map is equivalently a tensor T 2 A⇤⌦B⇤⌦C,called a trifocal tensor because of its derivation. For more details, see[HZ03, Chapter 15],[AT10], [Hey00], [PF98].

One way to connect these two seemingly di↵erent algebraic and geo-metric constructions is via the following construction, which shows (inan invariant way) how the parametrization using special minors of a4 ⇥ 9 matrix gives rise to a tensor parametrization. This also relatesto the compactified camera space considered in [AST11].

The row space of�AT

1 | AT2 | AT

3

�determines a point in the Grass-

mannian Gr(4, 9). Set U1, U2, U3 respectively as the 3-dimensional col-umn spaces of the blocked matrix

�AT

1 | AT2 | AT

3

�. The direct sum

W = U1 � U2 � U3 is 9-dimensional, and we can view the matrix

TheoremThe 18-dim’l variety T has degree 297.Its ideal is generated by 10 cubics, 81 quintics and 1980 sextics.

[Aholt-Oeding, Math. of Computation, 2014]

Level 3: Quadrifocal Variety

Let n = 4. There are 81 = 34 ways of picking one row each fromthe matrices A1,A2,A3,A4. The resulting 4×4-determinants arethe entries in the quadrifocal tensor (qijkl) of format 3×3×3×3.

The quadrifocal variety Q is the subvariety of P80 consisting of allquadrifocal tensors (qijkl). It has dimension 29 = 11 · 4− 15.

There is a birational isomorphism

Γ4 99K Q.

The qijkl serve as moduli coordinates.

It is believed that the ideal of Q is generatedby 600 cubics, 1377 quintics and 37586 sextics. [Oeding 2015]

Restricting to any 2×2×2×2-block gives relations amongprincipal minors of a 4×4-matrix. [Lin-St, J. Algebra, 2009]

Calibrated LociOur moduli spaces can be restricted to calibrated cameras

Ai = [Ri | ti ], where Ri ∈ SO(3), ti ∈ R3.

Key object: the compactified calibrated camera space Γcaln ⊂ Γn

For n = 2 this is the 5-dim’l variety of essential matrices

Γcal2 = {3×3-matrices with σ1=σ2>σ3=0} ⊂ Γ2 = V (det) ⊂ P8

[Demazure 1988]: defined by 10 cubic equations[Agrawal-Lee-St-Thomas 2014]: intersection with linear subspaces[Ottaviani 2015]: representation over C as 4×4-matrices of rank 2

[Kileel 2015]: For n = 3, we get the 11-dimensional calibratedtrifocal variety T cal in P26. It has degree 4912 and is cut outset-theoretically by equations of degree 12 and 45, derived from

Theorem (Kileel)

A trifocal tensor is calibrated if and only if all three fundamentalmatrices are essential. The matrices entries have degrees 4, 4, 15.

Conclusion

The three levels from

appear naturally in

2 CHRIS AHOLTA, LUKE OEDINGB

Figure 1. The trifocal tensor as a map P2 ⇥ P2 ! P2

l1

l 2

l3

l 1,2

f1

f2

f3

2

3

1

The image in camera i of a point x 2 P3 is determined by intersectingthe line hfi, xi with the plane ⇡i. Now consider lines li ✓ ⇡i for i = 1, 2.The planes hf1, l1i and hf2, l2i generically intersect in a line l1,2 ✓ P3,and the plane hf3, l1,2i generically intersects ⇡3 in a line l3. See Figure 1.

We have described for a su�ciently general camera configuration, amap

P2 ⇥ P2 ! P2,

given by (l1, l2) 7! l3. This map must come from a bilinear map

C3 ⇥ C3 ! C3.

To help avoid ambiguity, fix A, B, C ' C3 so that this map is nowA⇥B ! C. This bilinear map is equivalently a tensor T 2 A⇤⌦B⇤⌦C,called a trifocal tensor because of its derivation. For more details, see[HZ03, Chapter 15],[AT10], [Hey00], [PF98].

One way to connect these two seemingly di↵erent algebraic and geo-metric constructions is via the following construction, which shows (inan invariant way) how the parametrization using special minors of a4 ⇥ 9 matrix gives rise to a tensor parametrization. This also relatesto the compactified camera space considered in [AST11].

The row space of�AT

1 | AT2 | AT

3

�determines a point in the Grass-

mannian Gr(4, 9). Set U1, U2, U3 respectively as the 3-dimensional col-umn spaces of the blocked matrix

�AT

1 | AT2 | AT

3

�. The direct sum

W = U1 � U2 � U3 is 9-dimensional, and we can view the matrix

Can you see the trifocal tensor?