Image Reconstruction for Fan Beam X-Ray

By Anna Johnson,2015-03-15 06:37
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Image Reconstruction for Fan Beam X-Ray

    Image Reconstruction for Fan Beam X-Ray

    hy Using General Hankel Tomograp

    Transform Pair (2)

    Zhao Shuang-Ren

    Doubletask, Toronto, Canada

    Abstract: A new transform pair has been introduced for the

    fan-beam image reconstruction[10]. This new integral transform

    plays the same role as the Hankel ( Fourier Bessel) transform in the

    parallel beam case and includes the Hankel transform as a special

    case. In this paper we develop this method for the equal-angle fan

    beam geometry. We call this new pair the General Hankel transform

which gives transform and inverse transform from an object

function f(r,?? to fan beam projections R?(?). One member of the

    pair can be used to calculate projection data for simulation; the

    other can be used as an algorithm for fan beam reconstruction

without interpolation in polar coordinates. Compared with other

    algorithms for fan beam reconstruction the general Hankel

transform method is especially useful when f(r,??! the image to be

reconstructed, has only low frequencies in the;? angle direction and

only few projections are available.

    1. Introduction

    The problem of reconstructing an object from a set of its

    projections arises in among other fields, computer-aided tomography

    CAT, radio astronomy, electron microscopy, and spotlight-mode

    synthetic aperture radar. Traditional algorithms of reconstruction for

    parallel or fan beam projections is the convolution backprojection

    method[1-6], the Fourier method[7], the Hankel transform method [8].

    The convolution-backprojection method is fast and widely used. The

    Fourier method is faster than the convolution-backprojection method, but

    the reconstruction quality is poor because interpolation in the Fourier

    domain introduces additional artifacts[9]. For a special case where the

object to be reconstructed has no high frequencies along the;? angle, the

    Hankel transform method can be used and good quality of reconstruction

    can be achieved by few projections. However, the Hankel method is only

    used for parallel beam projections. Ref.[10] has introduced a new integral

    transform pair which extended the Hankel method for the fan beam

equal-space case. This new integral transform pair called General

    Hankel transform pair plays exactly the same role in the fan-beam case as

    the Hankel transform pair in the parallel-beam case. In this paper we

    developed this method for the fan beam equal-angle geometry.

    2. Two integral formulas and General Hankel

    Transform pair for fan beam tomography

    The spirit of the Hankel transform method for parallel tomography

is first to obtain the integral transform from the object function f(r,?) to

the projection P?(u) and the inverse transform from the projection P?(u)

    to the object function f(r,?), then expands these into Fourier series. We


follow this spirit to obtain the general Hankel Transform pair which can

be applied to fan beam geometry.

From Ref. [11] the method of back projection for the case of equi-

angle detectors is available

2; 1;?m (1) ? ? f(r,??( ? L2