International Journal of Biomedical Imaging

Volume 2017, Article ID 1867025, 14 pages

https://doi.org/10.1155/2017/1867025

## An Improved Extrapolation Scheme for Truncated CT Data Using 2D Fourier-Based Helgason-Ludwig Consistency Conditions

^{1}Pattern Recognition Lab, Friedrich-Alexander-University Erlangen-Nuremberg, Erlangen, Germany^{2}Erlangen Graduate School in Advanced Optical Technologies (SAOT), Erlangen, Germany^{3}Siemens Healthcare GmbH, Forchheim, Germany

Correspondence should be addressed to Andreas Maier; ed.uaf@reiam.saerdna

Received 15 January 2017; Revised 17 May 2017; Accepted 4 June 2017; Published 20 July 2017

Academic Editor: Marc Kachelriess

Copyright © 2017 Yan Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We improve data extrapolation for truncated computed tomography (CT) projections by using Helgason-Ludwig (HL) consistency conditions that mathematically describe the overlap of information between projections. First, we theoretically derive a 2D Fourier representation of the HL consistency conditions from their original formulation (projection moment theorem), for both parallel-beam and fan-beam imaging geometry. The derivation result indicates that there is a zero energy region forming a double-wedge shape in 2D Fourier domain. This observation is also referred to as the Fourier property of a sinogram in the previous literature. The major benefit of this representation is that the consistency conditions can be efficiently evaluated via 2D fast Fourier transform (FFT). Then, we suggest a method that extrapolates the truncated projections with data from a uniform ellipse of which the parameters are determined by optimizing these consistency conditions. The forward projection of the optimized ellipse can be used to complete the truncation data. The proposed algorithm is evaluated using simulated data and reprojections of clinical data. Results show that the root mean square error (RMSE) is reduced substantially, compared to a state-of-the-art extrapolation method.

#### 1. Introduction

It is known that traditional computed tomography (CT) reconstruction algorithms, for example, filtered backprojection methods, are not compatible to laterally truncated projection data, which appears often in the case of either when the object extends outside of the field of view (FOV) or X-ray beam collimation for the purpose of dose reduction. If data truncation is not effectively compensated for, it will result in cupping-like artifacts and incorrect gray-value levels in the reconstruction. A typical approach to reduce truncation artifacts is to perform extrapolation, for example, with the symmetric mirroring method [1], water cylinder extrapolation method [2], optimization-based extrapolation scheme [3], or implicit extrapolation method performed in the second-order derivative domain [4]. However, these heuristic extrapolation methods typically rely on techniques that complete the truncated data by means of a continuity assumption and thus appear to be ad hoc.

It has been demonstrated that any physically consistent sinogram has a strong restriction in its functional form [5]. This restriction is expressed by Helgason-Ludwig (HL) consistency conditions [6, 7], which are a mathematical expression to precisely describe the overlap of information between different projections. The HL consistency conditions play an important role in image reconstruction from imperfect projection data (e.g., due to noise, motion, and truncation) since these projections no longer satisfy the HL conditions. Related work uses HL conditions to estimate motion parameters directly from sinograms [8–10] or to solve the problem of limited angle tomography using a variational formulation that incorporates HL conditions [11]. In PET/SPECT, the HL consistency conditions were also used for attenuation correction if no transmission data is available [12].

This work addresses consistency-based sinogram completion. The methods proposed in [2, 13] implicitly used the zeroth-order HL consistency condition; that is, the DC term is the same for all projections, as a constraint for data extrapolation. The first-order condition, which corresponds to the first moment of the projections and describes the so-called “center of mass,” was also used to guide the extrapolation procedure [14]. Later, the elliptical extrapolation suggested in [15] explicitly used a small subset of the consistency conditions in the original HL formulation (projection moment theorem) so that large numerical instability can be avoided when computing the moment terms. The approach in [16] modified this original formulation by expanding the Radon transform in terms of its basis functions and incorporated not only one or two HL consistency conditions, but theoretically an infinite number of such constraints. However, the HL consistency conditions proposed in [16] were represented in the Chebyshev-Fourier domain, which increased computational complexity for practical applications. To simplify the computation, the method in [17] refined the Chebyshev-Fourier representation of HL conditions using an FFT with additional cosine transform along the detector channel. Furthermore, fan-beam to parallel-beam rebinning is required since the consistency conditions were only derived for parallel-beam geometry.

In this paper, we first derive the HL consistency conditions in the 2D Fourier domain from their original formulation. The Fourier representation shows that there is a zero energy region appearing in the Fourier transform of a sinogram (symmetric for parallel-beam and asymmetric for fan-beam geometry). This property was also demonstrated in [18, 19], which is referred to as the Fourier property of a sinogram and which was approximately arrived at using the parallel-/fan-beam sinogram of a delta point object. If the projection data is imperfect or incomplete, the zero energy double-wedge region contains nonnegligible values that indicate the corresponding inconsistent components. Several applications using this Fourier property of a sinogram can be found in [8, 20–23]. In this work, we theoretically prove the equivalence between the HL consistency conditions and the Fourier property of a sinogram and show the advantages of applying these Fourier-based consistency conditions: first, an infinite number of conditions are considered; and second, 2D Fourier transform via FFT is computationally more efficient than the Chebyshev-Fourier transform [16] or Lagrange-Fourier transform [11]. These features allow us to develop an efficient data extrapolation method by optimization of a cost function based on the Fourier-based HL conditions. First investigation on the method was also reported in [24].

The organization of the paper is as follows. In Section 2, we review the HL consistency conditions in its original formulation and the modified Chebyshev-Fourier representation. Then, we derive 2D Fourier-based HL consistency conditions and extend the conditions from parallel-beam to fan-beam geometry for centered objects. In Section 3, we design a cost function based on HL consistency conditions, which we use in a constrained optimization over a uniform ellipse that describes the object outline. In Section 4, we present experimental results from both a simulated phantom and reprojections of clinical data. In Sections 5 and 6, we discuss the relevant issues and draw conclusion.

#### 2. Consistency Conditions

##### 2.1. Helgason-Ludwig (HL) Consistency Conditions

In this section, we review the original formulation of HL consistency conditions, which is also referred to as the projection moment theorem in the literature [16]. Suppose the object is supported on the unit disk centered at the origin. Let be the th moment of the sinogram with respect to the detector bin , which is defined as

Then, the function does not change arbitrarily when the rotation angle varies. The Fourier series expansion of can be written as follows: with Fourier coefficients given by Then, it is readily proven [25] that all necessarily satisfy

##### 2.2. Chebyshev-Fourier Representation of HL Conditions

The derivation of the Chebyshev-Fourier version of HL conditions is similar to the one in [16]. But here we use the Chebyshev polynomial of the* first kind* to replace the monomial term , instead of the Chebyshev polynomial of the* second kind* as shown in [16].

Note that the functions do not form a set of orthogonal basis functions on (the Radon transform maps the Hilbert space consisting of finite norm objects to the Hilbert space consisting of finite norm sinograms ), where . In the following we show that the monomial can be replaced by the th-order orthogonal polynomial, such that the HL consistency conditions are tractable to use in reconstruction.

The sinogram can be expanded in a series as follows:where denote the expansion coefficients and denotes the th-order Chebyshev polynomial of the* first kind*, which is defined bywith

Let ; we define the inner product of and as follows: From Appendix A we will prove that form an orthogonal basis of .

Then, we can obtain an expression of the expansion coefficients as a scalar product

By comparing (5) and (10), it is noted that the coefficients are related to by the following combination: Then, we will have the HL conditions as

In sum,

##### 2.3. 2D Fourier-Based HL Consistency Conditions

We perform the 2D Fourier transform to both sides of (6)

Because the term is uniformly convergent, the order of the integral operator and the sum operator can be changed:

Because of the orthogonality of complex exponentials, we know that Then, we have where is the* first kind* Bessel function of order .

According to Debye’s relation [25], we know that decays exponentially when . From (13) we also know that when . Thus, we will have 2D Fourier representation of the HL condition as follows:

So far, we only assume that the object is supported on the unit disk. If the object is supported by a disk with a radius , then we replace by (where ) in (14) and (18) such that we can obtain the following conditions:where is the largest object support. Note that this property was also found in [18] by investigating the parallel-beam sinogram of a point object.

Figure 1 illustrates a double-wedge region of zero coefficients in the 2D Fourier transform of a sinogram, when .