The Journal of Geometric Analysis Volume 14, Number 2, 2004 Novikov’s Inversion Formula for the Attenuated Radon Transform—A New Approach By Jan Boman and Jan-Olov Strömberg ABSTRACT. We study the inversion of weighted Radon transforms in two dimensions, R ρ f (L) =  L f(·) ρ(L, ·)ds , where the weight function ρ(L,x), L a line and x ∈ L, has a special form. It was an important breakthrough when R.G. Novikov recently gave an explicit formula for the inverse of R ρ when ρ has the form (1.2); in this case R ρ is called the attenuated Radon transform. Here we prove similar results for a somewhat larger class of ρ using completely different and quite elementary methods. 1. Introduction In so-called Emission Computed Tomography one is led to the inversion of the generalized Radon transform in two dimensions: R ρ f(θ,p) =  x·θ =p f(x)ρ(x,θ)ds, (θ,p) ∈ T × R , (1.1) where f is a continuous function with compact support, ds is arc length measure on the line x · θ = p, T denotes the set of unit vectors in R 2 , and ρ(x,θ) is a weight function defined by ρ(x,θ) = exp  ∞ 0 µ ( x + tθ ⊥ ) dt  . (1.2) Here, µ(x) is a given function with compact support and the operator ⊥ is defined by (cos α, sin α) ⊥ = (− sin α, cos α). Physically, f(x) is a radiation intensity to be determined, −µ(x) is a known attenuation coefficient, and R ρ f(θ,p) is the measured radiation flux coming out of the body along the (oriented) line x · θ = p. The transform R ρ with ρ given by (1.2) is called the attenuated Radon transform. If µ = 0, then R ρ reduces to the classical Radon transform. If µ is constant on the support of f , an inversion formula for R ρ has been known for a long time. For about 20 years it has been an open problem whether R ρ is always injective if ρ(x,θ) has the form (1.2). On the other hand, it is known that there exist smooth and positive ρ(x,θ)—not of the kind (1.2)—such that the transform C ∞ 0  f  → R ρ f is not injective, [2]. Math Subject Classifications. Primary: 44A12, 92C55. Key Words and Phrases. Attenuated Radon transform, generalized Radon transform, weighted Radon transform. © 2004 The Journal of Geometric Analysis ISSN 1050-6926