The Sphere Method for the Inverse Problems of the Radon Transforms
- PDF / 221,417 Bytes
- 11 Pages / 612 x 792 pts (letter) Page_size
- 79 Downloads / 166 Views
DOI: 10.1007/s13226-020-0449-x
THE SPHERE METHOD FOR THE INVERSE PROBLEMS OF THE RADON TRANSFORMS Yufeng Yu School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041000, China e-mail: [email protected] (Received 5 November 2018; after final revision 20 May 2019; accepted 29 May 2019) Using a different method - the sphere method, which is based on the technique of changing the integral on a plane into the integral on a hemisphere, we give some concise inverse formulas of the Radon transforms of functions with support in a cone with vertex at the origin and flare angle less than π/2, or with compact support. These formulas are easy for the computer to operate and thus can be applied in the imaging techniques of computerized tomography. Key words : Radon transform; inverse problems; the sphere method. 2010 Mathematics Subject Classification : 42B99, 65R10.
1. I NTRODUCTION The hyperplane in Rn , with the normal vector θ ∈ Sn−1 and the distance |t|(t ∈ R) from the origin, is define by H(θ, t) = {x ∈ Rn : x · θ = t},
(1)
where · denotes the standard inner product in Rn . The Radon transform [1, 3-5] of a function f on Rn is defined by Z Rf (θ, t) = H(θ,t)
f (y)dyH ,
where (θ, t) ∈ Sn−1 × R and dyH is the Lebesgue measure on H(θ, t).
(2)
1054
YUFENG YU
The Radon transform has many applications, covering from mathematical theories to practical areas, for example, the integral geometry [2, 3] and the remarkable computerized tomography [1, 2, 5, 6]. In computerized tomography, a variety of inverse formulas of the Radon transforms are applied for the image reconstructing. There have been several classical methods for the inverse problems of the Radon transforms, for example, the method of mean value operators [10, 12], the method of Riesz potentials [6, 10], the convolution-backprojection [1, 2, 5, 6, 9, 11] and continuous ridgelet transforms [11]. Next, we give an overview of the three kinds of methods. For the method of mean value operators [10], as the preliminaries, the spherical mean of function f on Rn is defined by Z
1
(Mt f ) (x) =
σn−1
f (x + tθ)dθ f or x ∈ Rn , t > 0, Sn−1
where σn−1 is the Legesgue measure on Sn−1 . Then by the following operator [10] Z Z ³ ´ k ϕ˘r (x) = ϕ γR + x + rγen dγ = ϕ (γτr + x) dγ SO(n)
SO(n)
for x ∈ Rn and r ≥ 0, the k-plane transform (1 ≤ k < n) [10] of f , denoted by fˆ, can be represented by the spherical mean Mt f of function f . If k = n−1, then the k-plane transform becomes the Radon transform. Finally, function f can be reconstructed from Mt f by the properties of the spherical mean of functions, where we recommend the readers refer to Corollary 5.3 and Theorem 5.4 in [10], and Theorem 3.4 and 3.5 in [12] for details. The main idea of the method of Riesz potentials [6, 10] is as follows. First by the dual Radon transform R∗ [5, 6, 10], defined by Z ∗
R ϕ(x) =
ϕ(θ, x · θ)dθ, Sn−1
the Radon transform Rf of function f can be transformed to its Riesz potential I α f [13]. Then by the inversion methods of Riesz potentials [6, 10, 13], one
Data Loading...
