In this section, we briefly review several elementary facts from group theory and differential geometry. For a more detailed and rigorous treatment, we refer the reader to [32, 38, 104, 1, 68].
A d-dimensional manifold M can be informally defined as a set M that is locally home- omorphic to the Euclidean space Rd. The tangent space TxM at a point x ∈ M is the
vector space consisting of all the tangents of all smooth curves in M passing through x. A Riemannian manifold is a manifold whose tangent spaces are equipped with a smoothly varying inner product, which is called aRiemannian metric. We use the notation g(ξ, ζ) to denote the inner product of two elements ξ, ζ ∈TxM(where the point xwill be clear from
the context). The metric naturally induces a norm kξk=. pg(ξ, ξ).
A geodesic curve on M is the generalization of a straight line (that is, a curve with zero acceleration). We denote as γx,ξ(t) the geodesic emanating from x in the direction of ξ ∈ TxM. Theexponential map expx :TxM → Mis defined asexpxξ
.
=γx,ξ(1). Thelogarithm
map logx :M →TxMis the inverse of the exponential map and is generally defined only in
a neighborhood ofx. Where defined, we have the identityd(x, y) =klogx(y)k, whered(x, y)
is the Riemannian distance of x, y induced by the metric.
LetF :M → N be a smooth map between two manifolds MandN. The linear mapping
is called thedifferential ofF atx. For any curveγ(t) on Mwe have
DF(γ(t))[ ˙γ(t)] = d
dtF(γ(t)). (2.53)
Furthermore, given a real-valued function f :M →R, the Riemannian gradient gradf(x)
of f at a pointx∈ Mis the unique element ofTxM satisfying
g(gradf(x), ξ) =Df(x)[ξ], (2.54)
for allξ ∈TxM. The Riemannian Hessian is the self-adjoint linear map
Hessf(x) :TxM →TxM, ξ7→Hessf(x)[ξ], satisfying g(ξ,Hessf(x)[ξ]) = d 2 dt2f(γx,ξ(t)) t=0 , (2.55) for allξ ∈TxM.
LetM,N be manifolds such thatN ⊂ M. IfN has the subspace topology inherited from
M, thenN is called anembedded submanifold ofMandMis termed theemdedding space. Note that given a Riemannian metric on M, its restriction to N induces a Riemmanian metric onN.
A group(G,·)is a setGalong with a binary operation·:G×G→Gsatisfying the axioms of closure, associativity, existence of an identity element e∈Gand existence of inverse for each element in the group. ALie group is a group that is also a manifold. IfG is a group and Mis a set, a left action of G onM is a map G× M → M, written as (g, p)7→ g·p, satisfying g1 ·(g2·p) = (g1g2)·p, for all g1, g2 ∈ M, p ∈ M and e·p = p for all p ∈ M. The action is said continuous if the corresponding map is continuous, and it is said free if
g·p=pfor somep∈M implies thatg=e. A group action induces an equivalence relation
Let Mbe a manifold equipped with an equivalence relation ∼. The equivalence class of a pointx∈ Mis denoted by[x] ={y∈ M:y∼x}. Thequotient space M =M/∼is the set of all equivalence classes andM is termed thetotal space or ambient space. The canonical projection is the map π : M → M defined by π(x) = [x]. The quotient space is called a quotient manifold if the canonical projection is a submersion, that is the differential ofπ at every point is surjective. If the quotient space is a manifold anddim(M/∼)<dim(M), then each equivalence class π−1(π(x)) , x ∈ M, is an embedded submanifold of M. Consider any x ∈ M and let x ∈ π−1(x) ⊆ M. The vertical space Vx = Tx(π−1(x)) at x is the
tangent space to the equivalence class π−1(x). The horizontal space Hx is the orthogonal complement ofVx inTxM, that is,
Vx⊕ Hx =TxM. (2.56)
Given any and ξ ∈ TxM, there exists exactly one horizontal lift ξx ∈ Hx satisfying Dπ(x)[ξ] =ξ.
In the context of this work, we will frequently use the Lie group of three dimensional rotations
SO(3) ={R∈R3×3:RTR=I,det(R) = 1}. (2.57)
The tangent space at a pointR∈SO(3)is given by
TRSO(3) ={RΩ : Ω∈so(3)}, (2.58)
whereso(3)denotes the vector space of3×3skew-symmetric matrices. GivenR, Q∈SO(3)
and ξ∈TRSO(3), the exponential and the logarithm maps are given by
expR(ξ) =RexpI(RTξ), (2.59)
where expI and logI denote the exponential and the logarithm map at the identity which coincide with the matrix exponential and logarithm. For the group of rotations there are explicit formulas, e.g. Rodrigues’ formula, for the matrix exponential and the matrix loga- rithm [84]. Before defining the metric of manifold of rotations, we need to introduce some notation,. Thehat operator ∧:R3 →so(3)is defined as
b u=. 0 −u3 u2 u3 0 −u1 −u2 u1 0 , (2.61)
whereu= (u1, u2, u3)T. If u, v∈R3 and×denotes the cross product of vectors inR3, then
u×v = uvb . The inverse map of hat operator is the vee operator ∨ : so(3) → R
3. The
standard metric ofSO(3)at a pointR∈SO(3)is given by
g(ξ1, ξ2) = 1 2tr(ξ T 1ξ2) = 1 2tr(Ω T 1Ω2) =ω1Tω2, (2.62) whereξi =RΩi ∈TRSO(3) andω1= Ω∨1.
For modeling the translational part of the trifocal tensor, we use Kendall’s shape space [61]. Following the Kendall’s notation, we define
S32 ={X∈R2
×2 :kXk
F = 1}, (2.63)
as the space of triangles in 2-D. The tangent space at a pointX∈S3 2 is
TXS32 ={ξ∈R2
×2 : tr(XTξ) = 0}=X⊥
, (2.64)
and the Riemannian metric is the usual Euclidean inner product. We also introduce the space
which is the space of non-degenerate triangles. Finally, the exponential and the logarithm maps can be computed as
expX(ξ) = cos(kξk)X+sin(kξk)
kξk ξ, (2.66) logX(Y) = arccos(tr(X TY)) p 1−tr(XTY)2(Y −Xtr(X TY)), (2.67) for X, Y ∈S3 2 andξ ∈TXS32.