A CLASS OF CIRCULAR AND ELLIPTICAL RADON TRANSFORMS 3 In [4], the authors provided an exact inversion formula for CRT from radially partial data in the circular geometry of data acquisition. The current paper builds up on the results and techniques established in [4] and presents some further results both for CRT and ERT. We prove that such transforms can be uniquely inverted from radially incomplete data, that is, from data where t is limited to a small subset of R+. Our inversion formulas recover the image function f in annular regions defined by the smallest and largest available values of t. The results hold for cases when f is supported inside C (e.g. in mammography [13, 14]), outside of C (e.g. in intravascular imaging [6]), or simultaneously both inside and outside (e.g. in radar imaging [3]). The rest of the paper is organized as follows. In Section 2, we introduce the notations and definitions. In Section 3, we present the main results in the form of three theorems. The proofs of these theorems are provided in Section 4. Section 5 includes additional remarks and acknowledgments. 2. Notation Let us consider a generalized Radon transform integrating a function f(x) of two variables along ellipses with the foci located on the circle C(0,R) centered at 0 of radius R and 2a units apart (see Figure 1). We denote the fixed difference between the polar angles of the two foci by 2α, where α (0,π/2) and define a = R sin α, b = R cos α. We parameterize the location of the foci by γT (φ) = R (cos(φ α), sin(φ α)), γR(φ) = R (cos(φ + α), sin(φ + α)) for φ [0, 2π]. Thus the foci move on the circle and are always 2a units apart. For φ [0, 2π] and ρ 0, let E(ρ, φ) = {x R2 : |x γT (φ)| + |x γR(φ)| = 2 ρ2 + a2}. Note that the center of the ellipse E(ρ, φ) is (b cos φ, b sin φ) and ρ is the minor semiaxis of E(ρ, φ). Consider a compactly supported function f(r, θ), where (r, θ) denote the polar coordinates in the plane. The elliptical Radon transform (ERT) of f on the ellipse parameterized by (ρ, φ) is denoted by REf(ρ, φ) = E(ρ,φ) f(r, θ) ds, where ds is the arc-length measure on the ellipse. If we take a = α = 0, then the foci coincide and the ellipses E(ρ, φ) become circles C(ρ, φ) = {x R2 : |x γT (φ)| = ρ}. The resulting circular Radon transform (CRT) of f on the circle parameterized by (ρ, φ) is denoted by RCf(ρ, φ) = C(ρ,φ) f(r, θ) ds, where ds is the arc-length measure on the circle. In the rest of the text, we will use the notation g(ρ, φ) to denote either REf(ρ, φ), or (when working with CRT) RCf(ρ, φ).
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