This section introduces relevant geometric definitions used across this chapter. Let η = (x, y, z) ∈ S2 be a unit vector corresponding to the embedding of spherical coordinates in
R3. A pointx∈R3 projects to sphere space as
η= x
kxk (3.1)
§3.3 Geometry 51
The choice of embedded coordinates eliminates coordinate singularities that appear at the poles when using standard (zenith, azimuth) spherical coordinates.
The associated tangent space atηis
TηS2 ={µ∈R3 | hη,µi= 0} (3.2) and is characterized by the projection matrixPη:R3 →TηS2, computed as
Pη= (I−ηη>) (3.3)
In order to apply “linear” filtering to images defined on the sphere, it is natural to consider local representations of the image on a tangent space of the sphere and apply filter kernels directly in tangent coordinates. Moreover, quantities such as image gradient or optical flow are vector fieldsV : S2 → TηS2 defined in tangent space. To implement these operations, we need a set of functions that map points from the sphere to tangent space and vice versa. The termprojection-retraction classis used to denote a pair of such functions. A retraction function is a function that maps points from the tangent space of an arbitrary manifold back to the manifold Absil et al. [2009]. For the particular case of the sphere manifold, there are at least four projection-retraction classes:orthographic, projective, geodesicandchordal.
Letη0 ∈S2denote a reference point in the sphere. Denoteµ(η;η0) :S2 →Tη0S2to be the projection function mapping pointηontoTη0S2, andµ-1(µ;η0) :Tη0S2 → S2to be the retraction function. The four projection-retraction classes are defined as:
(a) Orthographic. (b) Geodesic. (c) Perspective. (d) Chordal.
−150 −100 −50 0 50 100 150 angle (deg.) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k µ k
Orthographic Geodesic Perspective Chordal
−4 −2 0 2 4
0.00 0.05 0.10
(e) Magnitude of projection functions accordint to angular separation between two points on the sphere. Figure 3.5: Projection-retraction classes on the sphere.
1. Orthographic:This class takes the component ofηorthogonal to reference pointη0as the projection vector ontoTη0S2. The projection functionµoth(η;η0)is simply defined
as
µoth(η;η0) =Pη0η (3.4) where Pη0 is the tangent space projection matrix computed in Equation (4.10). For convenience, denote as ν the normalized vector indicating the direction of projection. For a given pointη6=η0, it is computed as
ν(η;η0) = µoth(η;η0) kµoth(η;η0)k
(3.5)
The retraction functionµ-1
oth(µ;η0)takes a pointµ∈Tη0S2and retracts it into the sphere as
µ-1
oth(µ;η0) =µ+η0
p
1−µ>µ (3.6)
2. Geodesic:The geodesic class takes the angular separation betweenηandη0as distance metric between both points in tangent space. The geodesic projection is calculated as
µgeo(η;η0) = arccos (hη,η0i)ν (3.7)
whereν is computed using Equation (3.5). The retraction functionµ-1
geo(µ;η0)is calculated by rotating reference vectorη0 by an
anglekµkaround rotation axis
r= η0×µ
kη0×µk (3.8)
Let R be the rotation matrix computed from r andθ, the geodesic retraction is then defined as
µ-1
geo(µ;η0) =Rη0 (3.9)
3. Perspective: The perspective class resembles a perspective camera model in which the tangent space acts as image plane. The projection functionµper(η;η0)extends the rayη
until it intersects the tangent plane generated byη0, thus satisfying
hαη−η0,η0i= 0 (3.10)
whereαis an unknown scale factor andαη−η0 ∈Tη0S2is the unknown tangent vector. From Equation (3.10),α = 1/(η>η
0), and the perspective projection function is written
as
µper(η;η0) = η
η>η0 −η0 (3.11)
The retraction function is simply calculated as the renormalization of vectorη0 +µ. That is
µ-1
per(µ;η0) =
η0+µ
§3.3 Geometry 53
4. Chordal: This class uses the Euclidian distance between two points in the sphere as distance metric in tangent space. The projection function is
µcho(η;η0) =kη0−ηkν (3.13)
where unit vectorν is calculated using Equation (3.5).
For the retraction function, the approach is the same as for the geodesic case. A rotation matrixRis constructed using rotation axis from Equation (3.8) and rotation angleθ= 2 sin(kµk/2).
µ-1
cho(µ;η0) =Rη0 (3.14)
Figure 3.5e plots the magnitude of projected points µas a function of angle betweenη0 andηfor each projection class. Notice that for small angles, the difference between projection functions is negligible. The choice of a particular projection-retraction class is best considered problem dependent. Thanks to its numerical simplicity, the orthographic class is selected to map points in a small neighborhood aroundη0, and the geodesic class is chosen to deal with points located far away from the reference.
3.3.1 Orthonormal coordinate system
Considering that points µ ∈ TηS2 only have two degrees of freedom, it is convenient to express the 3-component vectors in terms of a 2D coordinate system. LetBη0 ∈R
2×3denote
the orthonormal basis transform that converts a vectorµ ∈ Tη0S2 to 2-component vectors. The namebeta coordinatesis used to refer to the 2D representation of points in tangent space. The beta coordinateβ∈R2of a vectorµ∈Tη0S2is computed as
β:= β1 β2 =Bη0µ (3.15)
Figure 3.6: Orthonormal coordinates in tangent spaceTη0S2. Matrix Bη0 is constructed by taking any normalized vectorν1 ∈ Tη0S
2 using Equation
(3.5) and creating a second vectorν2 =η0×ν1orthogonal to bothη0andν1 Bη0 = ν>1 ν>2 (3.16) The inverse operation, that is, transforming from β to its 3-vector representation µ is
achieved by
µ=B>η
0β (3.17)
In practice, the beta coordinate representation creates a coordinate system in which it is possible to define image operators in terms of convolutions in 2 axis. Since the coordinate system is orthogonal, properties such as separability can be used to speed up computations.