By Rudolf Kingslake

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24) fU (x) = − 2 8π S2 \U Ω × ε (y, ξ) ∂q gs (y, v) dθ|q=0 , y ,ξ ξ,v =q y∈O(ξ) which converges to f (x) as mes U → 0, since y , ξ = 0 on S2 \U. Equation d y − x, ξ = y , ξ ds + y − x, dξ = 0 holds on O which yields y , ξ ds = − y − x, dξ . 25) where the form dϕ |y − x| Ω/ y − x, dξ is equal to the even form on the circle O (y) = ξ ∈ S2 , y − x, ξ = 0 . Orientation in O (y) is defined by the normal covector y − x, dξ . 25), the orientation S2 defined by Ω coincides with the orientation defined by sgn y , ξ ds ∧ dϕ.

Let S be the sphere of all unit quaternions. 5) Factorization Method 49 preserves the norm and orientation, hence, G (q) ∈ SO (3) . It can be written in the form: G (q) s = q02 − |q| 2 s + 2q0 q × s + 2 q,s q, s ∈ S2 . 5) has quadratic components and defines a surjective group homomorphism. It is two-fold, since the quaternions q and −q generate the same orthogonal transform. Let H0 be the vector space over R of all imaginary quaternions. The manifold S+ = {q ∈ S, q0 > 0} is the hemisphere with the boundary S ∩ H0 .

This property is exploited for the efficient reconstruction algorithms by Natterer [84]. 3 The support theorem is due to Helgason (1964) [58]. See further results and a survey in Quinto [121]. 4 An inversion formula for the exponential ray transform was found by Tretiak and Metz [137] for constant attenuation coefficient. Other methods were applied by Kuchment and Shneiberg [73]. The range description was obtained in [3]. An approximate inversion formula for variable attenuation was proposed by Natterer [84].