KT-1 Signature for Circular Cone-Beam Tomography

Cone-beam X-ray tomography is used widely to scan simultaneously several 2D slices of a 3D object. Feldkamp et al. proposed a circular cone-beam reconstruction algorithm (abbreviated as FDK) for this purpose and it is the most popular reconstruction method at present. It is an approximate algorithm which uses 3D cone-beam back projection and 1D row-wise filtering. It is approximate in the sense that reconstruction results deviate from the actual object regardless of the measurement resolution. Sufficiency condition given by Tuy states that a circular trajectory cannot deliver sufficient data for exact reconstruction. This non-exactness is because of planes parallel to the circular trajectory do not contain a conebeam source point. It results in artifacts that appear in FDK reconstructions at larger cone angles. Moderate cone angles, however, give results that are acceptable. Approximate nature of this algorithm motivates error estimation strategies in reconstruction of 3D objects. Artifacts in reconstruction cannot be predicted/detected easily without going into Sobolev space type approximations. The present work is an attempt toward understanding this approximate nature of the FDK algorithm from the point-of-view of the user dependent filter functions in convolution of projection data. We use First Kanpur Theorem (KT-1) that was initially proposed by Munshi, et al. for error estimation in 2D CBP algorithms and it was subsequently simplified for practical applications [5]. THEORY Engineering objects generally have finite support in spatial domain. It is well known that such functions cannot be simultaneously band-limited in the frequency domain. It is required to use band-limiting filter for computational feasibility with user selected Fourier cut-off frequency that is generally dependant on data collection configuration. This makes reconstruction approximation band limited thus incorporating inherent error, E1, in the resulting tomographic image. Band-limiting filter along with Ram-Lak window [3] has a tendency to boost inherent noise present in the projection data. It is always advisable to use additional smoothing window in frequency domain. Most commonly used smoothing windows are Hamming, Hanning and sinc function [3]. Window function plays a crucial role in any filtered backprojection algorithm. Approximate error formula gives E1 as [5], (1) We see E1 is dependent on three entities, Fourier cut-off frequency, A the partial derivative of object function (∇2 f ) and the second-derivative of the window function at the Fourier space origin (W"(0)). The values of A and ∇2 f will remain unchanged for a fixed object cross section and a particular data-collection geometry and in that case is E1 directly proportional to W"(0). It has been shown already that different window functions give same reconstructions if their W"(0) have the same value [6]. We know that maximum value (NMAX) of gray-level in a tomographic image is related to the inverse of inherent error associated with that image [7]. A plot of 1/NMAX and W"(0) gives the KT-1 signature for that image. Slope and intercept of the linear fit indicate the frequency content of the cross section. Here we extend this approach to 3D volume reconstruction using FDK algorithm. Object is reconstructed using five different Hamming classes of filters and KT-1 signatures obtained to further quantify the cross section. EXPERIMENTS Two phantoms are selected in this study. Shepp-Logan: 3D Shepp-Logan phantom is low contrast medical phantom consists of spherical and ellipsoidal objects with known constant densities [8]. KT-1 Signature for Circular Cone-Beam Tomography Nitin Jain, M. S. Kalra, and Prabhat Munshi Nuclear Engineering and Technology Programme Indian Institute of Technology Kanpur Tel. +91 512 259 7902; e-mail: nitin@iitk.ac.in, pmunshi@iitk.ac.in INTRODUCTION Cone-beam X-ray tomography is used widely to scan simultaneously several 2D slices of a 3D object. Feldkamp et al. [1] proposed a circular cone-beam reconstruction algorithm (abbreviated as FDK) or this purpose and it is the most popular reconstruction method at present. It is an approximate algorithm which uses 3D cone-beam back projection and 1D row-wise filtering. It is approximate in the sense that reconstruction results deviate from the actual object regardless of the measurement resolution. Sufficiency condition given by Tuy [2] states that a circular trajectory can not deliver sufficient data for exact reconstruction. This non-exactness is because of planes parallel to the circular trajectory do not contain a cone-beam source point. It results in artifacts that appear in FDK reconstructions at larger cone angles. Moderate cone angles, however, give results that are acceptable. Approximate nature of this algorithm motivates error estimation strategies in reconstruction of 3D objects. Artifacts in reconstruction cannot be predicted/detected easily without going into Sobolev space type approximations [3]. The present work is an attempt towards understanding this approximate nature of the FDK algorithm from the pointof- view of the user dependent filter functions in convolution of projection data. We use First Kanpur Theorem (KT- 1) that was initially proposed by Munshi et al. [4] for error estimation in 2D CBP algorithms and it was subsequently simplified for practical applications

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