WO2002067201A1 - Reconstruction par statistique d'une image tomographique calculee par rayons x avec correcteur de durcissement de faisceau - Google Patents

Reconstruction par statistique d'une image tomographique calculee par rayons x avec correcteur de durcissement de faisceau Download PDF

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WO2002067201A1
WO2002067201A1 PCT/US2001/004894 US0104894W WO02067201A1 WO 2002067201 A1 WO2002067201 A1 WO 2002067201A1 US 0104894 W US0104894 W US 0104894W WO 02067201 A1 WO02067201 A1 WO 02067201A1
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calculating
image
ray
gradient
algorithm
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PCT/US2001/004894
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Idris A. Elbakri
Jeffrey A. Fessler
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The Regents Of The University Of Michigan
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T11/00—2D [Two Dimensional] image generation
    • G06T11/003—Reconstruction from projections, e.g. tomography
    • G06T11/006—Inverse problem, transformation from projection-space into object-space, e.g. transform methods, back-projection, algebraic methods
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2211/00—Image generation
    • G06T2211/40—Computed tomography
    • G06T2211/408—Dual energy
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2211/00—Image generation
    • G06T2211/40—Computed tomography
    • G06T2211/424—Iterative

Definitions

  • the present invention relates to statistical methods for reconstructing a polyenergetic X-ray computed tomography image and image reconstructor apparatus and, in particular, to methods and reconstructor apparatus which reconstruct such images from a single X-ray CT scan having a polyenergetic source
  • X-ray computed or computerized tomography provides structural information about tissue anatomy. Its strength lies in the fact that it can provide "slice" images, taken through a three-dimensional volume with enhanced 15 contrast and reduced structure noise relative to projection radiography.
  • Figure 1 illustrates a simple CT system.
  • An X-ray source is collimated and its rays are scanned through the plane of interest.
  • the intensity of the X-ray photons is diminished by tissue attenuation.
  • a detector measures the photon flux that emerges from the object. This procedure is repeated at sufficiently
  • Figures 2a-2c illustrate the evolution of CT geometries.
  • Figure 2a is a parallel-beam (single ray) arrangement, much like what was found in a first- generation CT scanner.
  • the major drawback of this arrangement is long scan time, since the source detector arrangement has to be translated and rotated.
  • the fan-beam geometry of Figure 2b reduces the scan time to a fraction of a second by eliminating the need for translation.
  • Figure 2c It further reduces scan time by providing three-dimensional information in one rotation. It is most efficient in its usage of the X-ray tube, but it suffers from high scatter ( ⁇ 40%). It is also the most challenging in terms of reconstruction algorithm implementation.
  • the linear attenuation coefficient ⁇ (x,y,z,E) characterizes the overall attenuation property of tissue. It depends on the spatial coordinates and the beam energy, and has units of inverse distance. For a ray of infinitesimal width, the mean photon flux detected along a particular projection line L, is given by:
  • FBP Filtered back projection
  • the attenuation coefficient ⁇ is energy dependent and the X-ray beam is polyenergetic. Lower energy X-rays are preferentially attenuated.
  • Figure 7 shows the energy dependence of the attenuation coefficients of water (density 1.0 gm/cm 3 ) and bone (at density 1.92 gm/cm 3 ).
  • a hard X-ray beam is one with higher average energy. Beam hardening is a process whereby the average energy of the X-ray beam increases as the beam propagates through a material. This increase in average energy is a direct consequence of the energy dependence of the attenuation coefficient.
  • the expected detected photon flux along path L is given by (1). If one were to ignore the energy dependence of the measurements and simply apply FBP to the log processed data, some attenuation map ⁇ would be reconstructed that is indirectly related to the source spectrum and object attenuation properties.
  • Figures 8 and 9 show the effect of beam hardening on the line integral in bone and water.
  • the line integral increases linearly with thickness.
  • the soft tissue line integral departs slightly from the linear behavior. The effect is more pronounced for high Z (atomic number) tissue such as bone.
  • Dual-energy imaging has been described as the most theoretically elegant approach to eliminate beam hardening artifacts.
  • the approach is based on expressing the spectral dependence of the attenuation coefficient as a linear combination of two basis function, scaled by constants independent of energy.
  • the two basis functions are intended to model the photo-electric effect and Compton scattering.
  • This technique provides complete energy dependence information for CT imaging.
  • An attenuation coefficient image can, in principle, be presented at any energy, free from beam hardening artifacts.
  • the method's major drawback is the requirement for two independent energy measurements. This has inhibited its use in clinical applications, despite the potential diagnostic benefit of energy information.
  • Recently, some work has been presented on the use of multi-energy X-ray CT for imaging small animals. For that particular application, the CT scanner was custom built with an energy-selective detector arrangement.
  • Pre-processing works well when the object consists of homogeneous soft tissue. Artifacts caused by high Z materials such as bone mandate the use of post-processing techniques to produce acceptable images.
  • the attenuation coefficient of some material k is modeled as the product of the energy -dependent mass attenuation coefficient m k (E) (cm 2 /g) and the energy-independent density p(x,y) (g/cm 3 ) of the tissue.
  • ⁇ (x,y,E) ⁇ m k (E)p k (x,y)r k (x,y) (2)
  • ⁇ (x,y, E) m l (E)p (x,y) (3) * m w (E)p sof ' (x,y) (4)
  • m w (E) is the mass attenuation coefficient of water
  • f f o ⁇ is the effective soft tissue density.
  • T is the line integral of the density along path L
  • the goal of the pre-processing method is to estimate ⁇ t ⁇ and from that reconstruct (using FBP) an estimate p(x,y) of the energy-independent density f f o ⁇ .
  • This pre-processing approach is inaccurate when bone is present, but is often the first step in a post-processing bone correction algorithm.
  • Post-processing techniques first pre-process and reconstruct the data for soft tissue correction, as explained above.
  • the resulting effective density image is then segmented into bone and soft tissue. This classification enables one to estimate the contributions of soft tissue and bone to the line integrals. These estimates are used to correct for non-linear effects in the line integrals.
  • the final artifact-free image is produced using FBP and displays density values independent of energy according to the following relationship:
  • ⁇ 0 is some constant independent of energy that maintains image contrast.
  • post-processing accomplishes its goal of eliminating energy dependence, it suffers from quantitative inaccuracy.
  • the parameter ⁇ 0 is somewhat heuristically estimated.
  • Another beam hardening correction of interest is known. This algorithm is iterative (but not statistical). At each pixel, it assumes that the attenuation coefficient is a linear combination of the known attenuation coefficients of two base materials, and it iteratively determines the volume fractions of the base materials.
  • the algorithm depends on an empirically -determined estimate of effective X-ray spectrum of the scanner. The main limitations of this approach is that the spectrum estimate captures the imaging characteristics for a small FOV only and that prior knowledge of the base materials at each pixel is necessary.
  • Statistical methods are a subclass of iterative techniques, although the two terms are often used interchangeably in the literature.
  • the broader class of iterative reconstruction techniques includes non-statistical methods such as the Algebraic Reconstruction Technique (ART) which casts the problem as an algebraic system of equations.
  • Successive substitution methods such as Joseph and Spital's beam-hardening correction algorithm, are also iterative but not statistical. Hence, statistical methods are iterative, but the opposite is not necessarily true.
  • Dual-energy systems operate based on the principle that the attenuation coefficient can be expressed as a linear combination of two energy basis functions and are capable of providing density images independent of energy.
  • An object of the present invention is to provide a method for reconstructing a polyenergetic X-ray computed tomography image and an image reconstructor apparatus, both of which utilize a statistical algorithm which explicitly accounts for a polyenergetic source spectrum and resulting beam hardening effects.
  • Another object of the present invention is to provide a method for reconstructing a polyenergetic X-ray computed tomography image and an image reconstructor apparatus, both of which utilize a statistical algorithm which is portable to different scanner geometries.
  • a method for statistically reconstructing a polyenergetic X-ray computed tomography image to obtain a corrected image includes providing a computed tomography initial image produced by a single X-ray CT scan having a polyenergetic source spectrum.
  • the initial image has components of materials which cause beam hardening artifacts.
  • the method also includes separating the initial image into different sections to obtain a segmented image and calculating a series of intermediate corrected images based on the segmented image utilizing a statistical algorithm which accounts for the polyenergetic source spectrum and which converges to obtain a final corrected image which has significantly reduced beam hardening artifacts.
  • the step of calculating may include the steps of calculating a gradient of a cost function having an argument and utilizing the gradient to minimize the cost function with respect to its argument.
  • the step of calculating the gradient may include the step of back projecting.
  • the cost function preferably has a regularizing penalty term.
  • the step of calculating the gradient may include the step of calculating thicknesses of the components.
  • the step of calculating thicknesses may include the step of reprojecting the segmented image.
  • the step of calculating the gradient may include the step of calculating means of data along paths and gradients based on the thicknesses of the components.
  • the argument may be density of the materials at each image voxel.
  • the method may further include calibrating the spectrum of the X-ray
  • the method may further include displaying the final corrected image.
  • the step of calculating the gradient may include the step of utilizing ordered subsets to accelerate convergence of the algorithm.
  • an image reconstructor apparatus for statistically reconstructing a polyenergetic X-ray computed tomography image to obtain a corrected image.
  • the apparatus includes means for providing a computed tomography initial image produced by a single X-ray CT scan having a polyenergetic source spectrum wherein the initial image has components of materials which cause beam hardening artifacts.
  • the apparatus further includes means for separating the initial image into different sections to obtain a segmented image and means for calculating a series of intermediate corrected images based on the segmented image utilizing a statistical algorithm which accounts for the polyenergetic source spectrum and which converges to obtain a final corrected image which has significantly reduced beam hardening artifacts.
  • the means for calculating may include means for calculating a gradient of a cost function having an argument and means for utilizing the gradient to minimize the cost function with respect to its argument.
  • the cost function preferably has a regularizing penalty term.
  • the means for calculating the gradient may include means for back projecting.
  • the means for calculating the gradient may include means for calculating thicknesses of the components.
  • the means for calculating thicknesses may include means for reprojecting the segmented image.
  • the means for calculating the gradient may include means for calculating means of data along paths and gradients based on the thicknesses of the components.
  • the argument may be density of the materials at each image voxel.
  • the means for calculating the gradient may include means for utilizing ordered subsets to accelerate convergence of the algorithm.
  • FIGURE 1 is a schematic view of a basic CT system
  • FIGURES 2a-2c are schematic views of various CT geometries wherein Figure 2a shows a parallel-beam (single ray) arrangement, Figure 2b shows a fan-beam geometry and Figure 2c shows a cone-beam arrangement;
  • FIGURE 3 is a schematic view which illustrates system matrix computation for the fan-beam geometry
  • FIGURE 4 shows graphs which illustrate convex penalty functions
  • FIGURE 5 shows graphs which illustrate the optimization transfer principle
  • FIGURE 6 shows graphs which are quadratic approximations to the Poisson log likelihood
  • FIGURE 7 shows graphs which illustrate attenuation coefficient energy dependence
  • FIGURE 8 shows graphs which illustrate beam hardening induced deviation of line integral from linearity in water
  • FIGURE 9 shows graphs which illustrate beam hardening induced deviation of line integral from linearity in bone.
  • FIGURE 10 is a block diagram flow chart of the method of the present invention.
  • the method and system of the present invention utilize a statistical approach to CT reconstruction and, in particular, iterative algorithms for transmission X-ray CT.
  • the method and system of the invention are described herein for a single-slice fan-beam geometry reconstruction, the method and system may also be used with cone-beam geometries and helical scanning. The method and system may also be used with flat-panel detectors.
  • Statistical Reconstruction for X-ray CT is described herein for a single-slice fan-beam geometry reconstruction, the method and system may also be used with cone-beam geometries and helical scanning.
  • the method and system may also be used with flat-panel detectors.
  • the image in object space (attenuation coefficient) is parameterized using square pixels.
  • the goal of the algorithm becomes to estimate the value of the discretized attenuation coefficient at those pixels.
  • ⁇ [ ⁇ ! , ... , ⁇ p ] ' be the vector of unknown attenuation coefficients having units of inverse length.
  • the measurements in a photon-limited counting process are reasonably modeled as independently distributed Poisson random variables.
  • the mean number of detected photons is related exponentially to the projections (line integrals) of the attenuation map.
  • the measurements are also contaminated by extra background counts, caused primarily by scatter in X-ray CT.
  • the following model is assumed for measurements:
  • N the number of measurements (or, equivalently, the number of detector bins).
  • Figure 3 illustrates one method for computing the elements of A in the fan-beam case.
  • a l ⁇ is the normalized area of overlap between the ray beam and the pixel.
  • the term r is the mean number of background events, b, is the blank scan factor and Y, represents the photon flux measured by the ith detector.
  • the 7,'s are assumed independent and that b l ,r l and ⁇ y ⁇ are known non-negative constants, ⁇ is also assumed to be independent of energy.
  • ML Maximum Likelihood
  • Regularization penalized-likelihood
  • the penalty function improves the conditioning of the problem.
  • the penalty functions with certain desirable properties such as edge preservation.
  • a general form for the regularizing penalty is the following:
  • ⁇ 's are potential functions acting on the soft constraints C ⁇ * 0 and K is the number of such constraints.
  • the potential functions are symmetric, convex, non-negative and differentiable.
  • Non-convex penalties can be useful for preserving edges, but are more difficult to analyze.
  • One can think of the penalty as imposing a degree of smoothness or as a Bayesian prior. Both views are practically equivalent.
  • penalty function penalize differences in the neighborhood of any particular pixel. They can be expressed as:
  • Equation (14) forces the estimator to match the measured data.
  • the second term imposes some degree of smoothness leading to visually appealing images.
  • the scalar parameter ⁇ controls the tradeoff between the two terms (or, alternatively, between resolution and noise).
  • the goal of the reconstruction technique becomes to maximize (14) subject to certain object constraints such as non-negativity:
  • OSTR Ordered Subsets Transmission Reconstruction
  • OS-PWLS Ordered Subsets Penalized Weighted Least Squares
  • the optimization transfer principle is a very useful and intuitive principle that underlies many iterative methods, including the ones described herein.
  • De Pierro introduced it in the context of inverse problems (emission tomography, to be specific).
  • the process is repeated iteratively, using a new surrogate function at each iteration. If the surrogate is chose appropriately, then the maximizer of ⁇ ( ⁇ ) can be found. Sufficient conditions that ensure that the surrogate leads to a monotonic algorithm are known.
  • Paraboloidal surrogates are used because they are analytically simple, and can be easily maximized. One can also take advantage of the convexity of these surrogates to parallelize the algorithm. Separable Paraboloidal Surrogates
  • the parameter ⁇ controls the tradeoff between the data-fit and penalty terms, and R( ⁇ ) imposes a degree of smoothness on the solution.
  • curvature c must be such that the surrogate satisfies the monotonicity conditions (19).
  • This formulation decouples the pixels.
  • Each pixel effectively has its own cost function q ⁇ .
  • the q s can be minimized for all pixels simultaneously, resulting in a parallelizable algorithm.
  • the pre-computed curvature may violate the conditions of monotonicity. It does, however, give an almost-monotonic algorithm, where the surrogate becomes a quadratic approximation of the log likelihood.
  • the pre- computed curvature seems to work well in practice, and the computational savings seem well worth the sacrifice.
  • Major computational savings come from the use of ordered subsets, discussed hereinbelow.
  • Ordered subsets are useful when an algorithm involves a summation over sinogram indices (i.e. , a back projection).
  • the basic idea is to break the set of sinogram angles into subsets, each of which subsamples the sinogram in the angular domain.
  • the back projection process over the complete sinogram is replaced with successive back projections over the subsets of the sinogram.
  • One iteration is complete after going through all of the subsets.
  • Ordered subsets have been applied to emission tomography with a good degree of success. Improvements in convergence rate by a factor approximately equal to the number of subsets have been reported. Ordered subsets have also been used with transmission data for attenuation map estimation in SPECT. Ordered subsets where applied to the convex algorithm, and an increase in noise level with number of subsets have been reported. Ordered subsets have been used with the transmission EM algorithm and a cone-beam geometry. The OSTR algorithm was originally developed for attenuation correction in PET scans with considerable success.
  • OSTR combines the accuracy of statistical reconstruction with the accelerated rate of convergence that one gets from ordered subsets.
  • the separability of the surrogates makes the algorithm easily parallelizable.
  • the algorithm also naturally enforces the non-negativity constraint. The monotonicity property has been satisfied, but that seems to hardly make a difference in practice if a reasonable starting image is used.
  • OSTR uses Poisson statistics to model the detection process.
  • the Gaussian model is a reasonable approximation to the Poisson distribution.
  • the Gaussian model leads to a simpler quadratic objection function and weighted-least-squares minimization. With high counts, PWLS leads to negligible bias and the simpler objective function reduces computation time.
  • Figure 6 illustrates how the quadratic approximation to the likelihood improves with count number.
  • the algorithm is reformulated by deriving a quadratic approximation to the Poisson likelihood, which leads to a simpler objective function.
  • the regularization term and the use of ordered subsets are retained.
  • This variation of the method of the invention is called Ordered Subset Penalized Weight Least Squares (OS-PWLS).
  • Taylor's expansion is applied to h,(l) around some value , and first and second order terms only are retained.
  • the first term in (37) is independent of /, and can be dropped.
  • the subscript q indicates that this objective function is based on a quadratic approximation to the log likelihood. Subsequently, the subscript is dropped. The penalty term is also added. Minimizing this objective function over ⁇ ⁇ 0 will lead to an estimator with negligible bias, since the number of counts is large.
  • the numerator (first derivative) in (41) involves no exponential terms and the denominator (second derivative) in (42) can be pre-computed and stored.
  • the sum over sinogram indices can also be broken into sums over ordered subsets, further accelerating the algorithm.
  • CT transmission model is generalized hereinbelow to account for the broad spectrum of the beam. From the model, a cost function is derived and an iterative algorithm is developed for finding the minimizer of the cost function.
  • a model for X-ray CT is described now that incorporates the energy dependence of the attenuation coefficient.
  • a prior art algorithm could be applied to an image reconstructed with OS-PWLS. Instead, beam hardening correction is developed as an integral element of the statistical algorithm.
  • An iterative algorithm that generalizes OS- PWLS naturally emerges from the model.
  • R k ⁇ set of pixels classified as tissue type k], ( 4 )
  • K vector quantities of length p each representing the density of one kind of tissue
  • the non-overlapping assumption of the tissue types enables one to keep the number of unknowns equal to p, as is the case in the monoenergetic model. This is possible when prior segmentation of the object is available. This can be obtained from a good FBP image, for example.
  • the Poisson log likelihood is set up in terms of the density p and the vector v,-.
  • a quadratic cost function one follows a similar procedure to that described hereinabove, using the second-order Taylor's expansion.
  • the function Y t represents the expected value of the measurement
  • the gradient VA is a row vector and the Laplacian operator V 2 gives a K x K matrix of partial derivatives.
  • Y t is assumed to be close enough to l ⁇ (v ( ) for one to drop the first term on the right of (57). This also ensures that the resulting Hessian approximation is non-negative definite.
  • V k denotes the km element of the gradient vector.
  • A ⁇ y ⁇ is the geometrical system matrix.
  • the matrix B ⁇ b y ⁇ is a weighted system matrix, with the weights expressed as the non-zero elements of a diagonal matrix D(-), to the left of A.
  • the term Z combines constants independent of p.
  • This algorithm globally converges to the minimizer of the cost function ⁇ q (p) when one subset is used, provided the penalty is chosen so that ⁇ q is strictly convex. When two or more subsets are used, it is expected to be monotone in the initial iterations.
  • the spectrum of the incident X-ray beam is calibrated.
  • the correction image is then subtracted and non-negativity is enforced.
  • the number of iterations is checked against a predetermined number or other criteria and if the iterative part of the method is complete, then the final corrected image is displayed. If not done, the iterative part of the method is re- entered at the reprojection step. At least some of the results obtained after subtraction may be used in the segmentation step as described herein as indicated by the dashed line from the "DONE" block.
  • beam hardening correction in the method of the invention depends on the availability of accurate classification of the different substances in the object.
  • the bone/tissue distribution was known exactly.
  • such a classification would be available from segmenting an initial image reconstructed with FBP. Using this segmentation map for all iterations may adversely affect the accuracy of the reconstruction.
  • joint likelihoods and penalties are used to estimate both pixel density values and tissue classes.
  • the tissue classes are treated as random variables with a Markov random field model and are estimated jointly with the attenuation map.
  • the joint likelihood will be a function of both the pixel density value and the pixel class.
  • the joint penalties involve two parameters that balance the tradeoff between data fit and smoothness.
  • joint penalties must account for the fact that pixels tend to have similar attenuation map values if the underlying tissue classes are the same, and vice versa. Such penalties would encourage smoothness in the same region but allow discontinuities between regions of different tissues.
  • Scatter is a major problem in cone-beam CT, where it can range from 40% up to 200% of the unscattered data. Collimation reduces scatter, but collimating flat-panel detectors is challenging. Among several factors that affect the performance of a cone-beam, flat-panel detector computed tomography system, scatter was shown to degrade the detector quantum efficiency (DQE) and to influence the optimal magnification factor. Larger air gaps were needed to cope with high scatter, especially if imaging a large FOV.
  • DQE detector quantum efficiency
  • Scatter can either be physically removed (or reduced) before detection or can be numerically estimated and its effect compensated for.
  • Ways to physically remove scatter include air gaps and grids, but are not very practical once flat-panel detectors are used, due to the small size of detector pixels.
  • idle detectors can be used to provide a scatter estimate.
  • Such measurements can be combined with analytic models for scatter that depend, among other factors, on the energy of the radiation used, the volume of scattering material and system geometry.
  • a scatter estimate may be incorporated in the model of the CT problem as well as in the reconstruction algorithm using the ri terms above.
  • One approach to scatter estimation and correction is a numerical one, i.e. , ways to physically eliminate scatter will not be considered. This makes the approach portable to different systems, and less costly.
  • the statistical measurement model has the potential to be extended to take into account the time dimension when imaging the heart, and thus free the designer from synchronization constraints. Moreover, statistical reconstruction eliminates the need for rebinning and interpolation. This may lead to higher helical scanning pitch and cardiac imaging with good temporal and axial resolutions.

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Abstract

L'invention concerne un procédé de reconstruction par statistique d'une image tomographique calculée par rayons X, générée par un seul tomodensitogramme CT à rayons X possédant un spectre de source polyénergétique et un dispositif de reconstruction d'image utilisant un algorithme statistique convergent représentant de manière explicite le spectre de la source polyénergétique. L'invention concerne également un premier et un second procédé itératifs statistiques associés de reconstruction CT basés sur un modèle statistique de Poisson. Les deux procédés sont accélérés par l'utilisation de sous-ensembles ordonnés, remplaçant des sommes sur l'indice angulaire d'un sinogramme par une série de sommes sur des sous-ensembles angulaires du sinogramme. Le premier procédé est généralisé pour modéliser le cas le plus réaliste de tomographie (CT) calculée polyénergétique. Le second procédé élimine les artéfacts de durcissement de faisceau discernés lorsqu'on utilise une rétroprojection filtrée (FBP) sans correction de post-traitement. Ces procédés sont supérieurs à la reconstruction FBP en ce qui concerne la réduction du bruit. Le procédé et le dispositif de reconstruction d'image de l'invention servent à produire efficacement des images corrigées non affectées par les effets de durcissement du faisceau.
PCT/US2001/004894 2001-02-15 2001-02-15 Reconstruction par statistique d'une image tomographique calculee par rayons x avec correcteur de durcissement de faisceau WO2002067201A1 (fr)

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