Frédéric Noo

A two-step Hilbert transform method for 2D image reconstruction

The paper describes a new accurate two-dimensional (2D) image reconstruction method consisting of two steps. In the first step, the backprojected image is formed after taking the derivative of the parallel projection data. In the second step, a Hilbert filtering is applied along certain lines in the differentiated backprojection (DBP) image. Formulae for performing the DBP step in fanbeam geometry are also presented. The advantage of this two-step Hilbert transform approach is that in certain situations, regions of interest (ROIs) can be reconstructed from truncated projection data. Simulation results are presented that illustrate very similar reconstructed image quality using the new method compared to standard filtered backprojection, and that show the capability to correctly handle truncated projections. In particular, a simulation is presented of a wide patient whose projections are truncated laterally yet for which highly accurate ROI reconstruction is obtained.

Cone-beam reconstruction using the backprojection of locally filtered projections

This paper describes a flexible new methodology for accurate cone beam reconstruction with source positions on a curve (or set of curves). The inversion formulas employed by this methodology are based on first backprojecting a simple derivative in the projection space and then applying a Hilbert transform inversion in the image space. The local nature of the projection space filtering distinguishes this approach from conventional filtered-backprojection methods. This characteristic together with a degree of flexibility in choosing the direction of the Hilbert transform used for inversion offers two important features for the design of data acquisition geometries and reconstruction algorithms. First, the size of the detector necessary to acquire sufficient data for accurate reconstruction of a given region is often smaller than that required by previously documented approaches. In other words, more data truncation is allowed. Second, redundant data can be incorporated for the purpose of noise reduction. The validity of the inversion formulas along with the application of these two properties are illustrated with reconstructions from computer simulated data. In particular, in the helical cone beam geometry, it is shown that 1) intermittent transaxial truncation has no effect on the reconstruction in a central region which means that wider patients can be accommodated on existing scanners, and more importantly that radiation exposure can be reduced for region of interest imaging and 2) at maximum pitch the data outside the Tam-Danielsson window can be used to reduce image noise and thereby improve dose utilization. Furthermore, the degree of axial truncation tolerated by our approach for saddle trajectories is shown to be larger than that of previous methods.

Truncated Hilbert transform and image reconstruction from limited tomographic data

A data sufficiency condition for 2D or 3D region-of-interest (ROI) reconstruction from a limited family of line integrals has recently been introduced using the relation between the backprojection of a derivative of the data and the Hilbert transform of the image along certain segments of lines covering the ROI. This paper generalizes this sufficiency condition by showing that unique and stable reconstruction can be achieved from an even more restricted family of data sets, or, conversely, that even larger ROIs can be reconstructed from a given data set. The condition is derived by analysing the inversion of the truncated Hilbert transform, here defined as the problem of recovering a function of one real variable from the knowledge of its Hilbert transform along a segment which only partially covers the support of the function but has at least one end point outside that support. A proof of uniqueness and a stability estimate are given for this problem. Numerical simulations of a 2D thorax phantom are presented to illustrate the new data sufficiency condition and the good stability of the ROI reconstruction in the presence of noise.

Tiny a priori knowledge solves the interior problem in computed tomography

Based on the differentiated backprojection (DBP) framework [1-3], this paper shows that the solution to the interior problem in computed tomography is unique if a tiny a priori knowledge on the object f(x,y) is available in the form that f(x,y) is known on a small region located inside the region of interest. Furthermore, we advance the uniqueness result to obtain a more general uniqueness result which can be applied to a wider class of imaging configurations. The experimental results show evidence that the inversion corresponding to each obtained uniqueness result is stable.

A solution to the long-object problem in helical cone-beam tomography

This paper presents a new algorithm for the long-object problem in helical cone-beam (CB) computerized tomography (CT). This problem consists in reconstructing a region-of-interest (ROI) bounded by two given transaxial slices, using axially truncated CB projections corresponding to a helix segment long enough to cover the ROI, but not long enough to cover the whole axial extent of the object. The new algorithm is based on a previously published method, referred to as CB-FBP (Kudo et al 1998 Phys. Med. Biol. 43 2885-909), which is suitable for quasi-exact reconstruction when the helix extends well beyond the support of the object. We first show that the CB-FBP algorithm simplifies dramatically, and furthermore constitutes a solution to the long-object problem, when the object under study has line integrals which vanish along all PI-lines. (A PI line is a line which connects two points of the helix separated by less than one pitch.) Exploiting a geometric property of the helix, we then show how the image can be expressed as the sum of two images, where the first image can be reconstructed from the measured CB projections by a simple backprojection procedure, and the second image has zero PI-line integrals and hence can be reconstructed using the simplified CB-FBP algorithm. The resulting method is a quasi-exact solution to the long-object problem, called the ZB method. We present its implementation and illustrate its performance using simulated CB data of the 3D Shepp phantom and of a more challenging head-like phantom.

Exact helical reconstruction using native cone-beam geometries.

This paper is about helical cone-beam reconstruction using the exact filtered backprojection formula recently suggested by Katsevich (2002a Phys. Med. Biol. 47 2583-97). We investigate how to efficiently and accurately implement Katsevichs formula for direct reconstruction from helical cone-beam data measured in two native geometries. The first geometry is the curved detector geometry of third-generation multi-slice CT scanners, and the second geometry is the flat detector geometry of C-arms systems and of most industrial cone-beam CT scanners. For each of these two geometries, we determine processing steps to be applied to the measured data such that the final outcome is an implementation of the Katsevich formula. These steps are first described using continuous-form equations, disregarding the finite detector resolution and the source position sampling. Next, techniques are presented for implementation of these steps with finite data sampling. The performance of these techniques is illustrated for the curved detector geometry of third-generation CT scanners, with 32, 64 and 128 detector rows. In each case, resolution and noise measurements are given along with reconstructions of the FORBILD thorax phantom.

Cone-beam filtered-backprojection algorithm for truncated helical data

This paper investigates 3D image reconstruction from truncated cone-beam (CB) projections acquired with a helical vertex path. First, we show that a rigorous derivation of Grangeats formula for truncated projections leads to a small additional term compared with previously published similar formulations. This correction term is called the boundary term. Next, this result is used to develop a CB filtered-backprojection (FBP) algorithm for truncated helical projections. This new algorithm only requires the CB projections to be measured within the region that is bounded, in the detector, by the projections of the upper and lower turns of the helix. Finally, simulations with mathematical phantoms demonstrate that: (i) the boundary term is necessary to obtain high-quality imaging of low-contrast structures and (ii) good image quality is obtained even with large values of the pitch of the helix.

Cone-beam reconstruction using 1D filtering along the projection of M-lines

In this paper, three exact formulae are derived for cone-beam reconstruction with source positions on a curve or set of curves. For reconstruction at a single point, these formulae all operate by applying a filtration step followed by a backprojection step to cone-beam data. The filtering is performed along a 1D curve which is defined as the intersection of the detector surface with a filtering plane. Two of these formulae allow a flexibility in the choice of the filtering direction. In some cases, this flexibility allows the efficiency of volume reconstruction to be improved. Alternatively, the flexibility can be used to reduce the detector size necessary to avoid truncation artefacts in the reconstruction or to change the noise properties of the reconstruction.

Exact Radon rebinning algorithm for the long object problem in helical cone-beam CT

This paper addresses the long object problem in helical cone-beam computed tomography. The authors present the PHI-method, a new algorithm for the exact reconstruction of a region-of-interest (ROI) of a long object from axially truncated data extending only slightly beyond the ROI. The PHI-method is an extension of the Radon-method, published by Kudo et al. in Phys. in Med. and Biol., vol. 43, p. 2885-909 (1998). The key novelty of the PHI-method is the introduction of a virtual object f/sub /spl phi//(x) for each value of the azimuthal angle /spl phi/ in the image space, with each virtual object having the property of being equal to the true object f(x) in some ROI /spl Omega//sub m/. The authors show that, for each /spl phi/, one can calculate exact Radon data corresponding to the two-dimensional (2-D) parallel-beam projection of f/sub /spl phi//(x) onto the meridian plane of angle /spl phi/. Given an angular range of length /spl pi/ of such parallel-beam projections, the ROI /spl Omega//sub m/ can be exactly reconstructed because f(x) is identical to f/sub /spl phi//(x) in /spl Omega//sub m/. Simulation results are given for both the Radon-method and the PHI-method indicating that (1) for the case of short objects, the Radon- and PHI-methods produce comparable image quality, (2) for the case of long objects, the PHI-method delivers the same image quality as in the short object case, while the Radon-method fails, and (3) the image quality produced by the PHI-method is similar for a large range of pitch values.

Image covariance and lesion detectability in direct fan-beam x-ray computed tomography

We consider noise in computed tomography images that are reconstructed using the classical direct fan-beam filtered backprojection algorithm, from both full- and short-scan data. A new, accurate method for computing image covariance is presented. The utility of the new covariance method is demonstrated by its application to the implementation of a channelized Hotelling observer for a lesion detection task. Results from the new covariance method and its application to the channelized Hotelling observer are compared with results from Monte Carlo simulations. In addition, the impact of a bowtie filter and x-ray tube current modulation on reconstruction noise and lesion detectability are explored for full-scan reconstruction.