Fourier correction for spatially variant collimator blurring in SPECT

Space-variant deconvolution of Cerenkov light images acquired from a curved surface

PURPOSE

Cerenkov photons are generated by high-energy radiation used in external beam radiation therapy (EBRT). This study expands upon the Cerenkov light dosimetry formula previously developed to relate an image of Cerenkov photons to the primary beam fluence. Extension of this formulation allows for deconvolution to be performed on images acquired from curved geometries.

METHODS

The integral equation, which represented the formation of Cerenkov photon image from an incident high-energy photon beam, was expanded to allow for space-variance of the convolution kernel called as the Cerenkov scatter function (CSF). The GAMOS (Geant4-based Architecture for Medicine-Oriented Simulations) Monte Carlo (MC) particle simulation software was used to obtain the CSF for different incident beam angles. The image of a curved surface was first projected to a flat plane by using a perspective correction method. Then, the planar image was partitioned into small segments (or blocks), where a CSF corresponding to a specific beam incident angle was applied for deconvolution. The block size and the margin around the block were optimized by studying the effects of those parameters on the deconvolution accuracy for a test image. We evaluated three deconvolution techniques: Richardson-Lucy, Blind, and Total Variation minimization (TV/L2) algorithms, to select the most accurate method for the current applications.

RESULTS

Analysis of deconvolution algorithms showed that the TV/L2 method provided the most accurate solution to the deconvolution problem for Cerenkov imaging. Optimization of space-variant deconvolution parameters showed that including a margin that is at least 42.9% of the image width provided the most accurate product image. There was no optimal size for the deconvolution area and should be chosen based on the presence of unique CSF kernels within an image. Space-variant deconvolution improved measured field size in Cerenkov photon images by 7.37%, as compared with 1.74% by space-invariant deconvolution. Space-variant deconvolution improved measured penumbra by 99.3%, as compared with 76.7% by space-invariant deconvolution. Space-variant deconvolution introduced artifacts in flat regions of the beam. Artifacts were avoided through selective space-variant deconvolution in only the penumbra region.

CONCLUSIONS

Primary photon fluence distributions of a curved surface can be obtained by using space-variant deconvolution methods in Cerenkov light dosimetry. The TV/L2 algorithm is the best method for deconvolution of Cerenkov photon images from an open-field beam derived from either a flat or curved surface. The partition size chosen for space-variant deconvolution should be at least six times the full width at half maximum (FWHM) of the corresponding scatter kernel used in deconvolution. Space-variant deconvolution is necessary if the incident beam angle difference is larger than 6 ∘ between regions of an image.

Comparison of image reconstruction techniques for optical projection tomography

We present a comparison of image reconstruction techniques for optical projection tomography. We compare conventional filtered back projection, sinogram filtering using the frequency-distance relationship (FDR), image deconvolution, and 2D point-spread-function-based iterative reconstruction. The latter three methods aim to remove the spatial blurring in the reconstructed image originating from the limited depth of field caused by the point spread function of the imaging system. The methods are compared based on simulated data, experimental optical projection tomography data of single fluorescent beads, and high-resolution optical projection tomography imaging of an entire zebrafish larva. We demonstrate that the FDR method performs poorly on data acquired with high numerical aperture optical imaging systems. We show that the deconvolution technique performs best on highly sparse data with low signal-to-noise ratio. The point-spread-function-based reconstruction method is superior for nonsparse objects and data of high signal-to-noise ratio.

Point spread function based image reconstruction in optical projection tomography

As a result of the shallow depth of focus of the optical imaging system, the use of standard filtered back projection in optical projection tomography causes space-variant tangential blurring that increases with the distance to the rotation axis. We present a novel optical tomographic image reconstruction technique that incorporates the point spread function of the imaging lens in an iterative reconstruction. The technique is demonstrated using numerical simulations, tested on experimental optical projection tomography data of single fluorescent beads, and applied to high-resolution emission optical projection tomography imaging of an entire zebrafish larva. Compared to filtered back projection our results show greatly reduced radial and tangential blurring over the entire [Formula: see text] mm² field of view, and a significantly improved signal to noise ratio.

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

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.

Image resolution and deconvolution in optical tomography

We present a frequency domain analysis of the image resolution of optical tomography systems. The result of our analysis is a description of the spatially-variant resolution in optical tomographic image after reconstruction as a function of the properties of the imaging system geometry. We validate our model using optical projection tomography (OPT) measurements of fluorescent beads embedded in agarose gel. Our model correctly describes both the radial and tangential resolution of the measured images. In addition, we present a correction of the tomographic images for the spatially-varying resolution using a deconvolution algorithm. The resulting corrected tomographic reconstruction shows a homogeneous and isotropic pixel-limited resolution across the entire image. Our method is applied to OPT measurements of a zebrafish, showing improved resolution. Aside from allowing image correction and providing a resolution measure for OPT systems, our model provides a powerful tool for the design of optical tomographic systems.

Quantification and reduction of the collimator-detector response effect in SPECT by applying a system model during iterative image reconstruction: a simulation study

BACKGROUND

Detector blurring and non-ideal collimation decrease the spatial resolution of the single-photon emission computed tomography (SPECT) images. Iterative reconstruction algorithms such as ordered subsets expectation maximization (OSEM) can incorporate degrading factors during reconstruction. We investigated the quantitative errors associated with poor SPECT resolution and evaluated the importance of two-dimensional (2D) and three-dimensional (3D) resolution recovery by modelling system response during iterative image reconstruction.

METHODS

Different phantoms consisted of the NURBS-based cardiac-torso (NCAT) liver phantom with small tumors, the Zubal brain phantom and the NCAT heart phantom were used in this study. Monte Carlo simulation was used to create SPECT projections. Gaussian functions were used to model collimator detector response (CDR). Modeled CDRs were applied during OSEM. Both noise-free and noisy projections were created.

RESULTS

Even with noise-free projections, conventional OSEM algorithm provided limited quantitative accuracy compared to both 2D and 3D resolution recovery. The 3D implementation of resolution recovery, however, yielded superior results compared to its 2D implementation. For the liver phantom, the ability to distinguish small tumors in both transverse and axial planes was improved. For the brain phantom, gray to white matter activity ratio was increased from 3.14 ± 0.04 in simple OSEM to 3.84 ± 0.06 in 3D OSEM. For the NCAT heart phantom, 3D resolution recovery, results in images with thinner wall and higher contrast for different noise levels.

CONCLUSION

There are considerable quantitative errors associated with CDR, especially when the size of the target is comparable with the spatial resolution of the system. Between different reconstruction algorithms, 3D OSEM that consider the 3D nature of CDR, improve both the visual quality and the quantitative accuracy of any SPECT studies.

Incorporation of an experimentally determined MTF for spatial frequency filtering and deconvolution during optical projection tomography reconstruction

We demonstrate two techniques to improve the quality of reconstructed optical projection tomography (OPT) images using the modulation transfer function (MTF) as a function of defocus experimentally determined from tilted knife-edge measurements. The first employs a 2-D binary filter based on the MTF frequency cut-off as an additional filter during back-projection reconstruction that restricts the high frequency information to the region around the focal plane and progressively decreases the spatial frequency bandwidth with defocus. This helps to suppress "streak" artifacts in OPT data acquired at reduced angular sampling, thereby facilitating faster OPT acquisitions. This method is shown to reduce the average background by approximately 72% for an NA of 0.09 and by approximately 38% for an NA of 0.07 compared to standard filtered back-projection. As a biological illustration, a Fli:GFP transgenic zebrafish embryo (3 days post-fertilisation) was imaged to demonstrate the improved imaging speed (a quarter of the acquisition time). The second method uses the MTF to produce an appropriate deconvolution filter that can be used to correct for the spatial frequency modulation applied by the imaging system.

Fourier properties of the fan-beam sinogram

PURPOSE

P. R. Edholm, R. M. Lewitt, and B. Lindholm, "Novel properties of the Fourier decomposition of the sinogram," in Proceedings of the International Workshop on Physics and Engineering of Computerized Multidimensional Imaging and Processing [Proc. SPIE 671, 8-18 (1986)] described properties of a parallel beam projection sinogram with respect to its radial and angular frequencies. The purpose is to perform a similar derivation to arrive at corresponding properties of a fan-beam projection sinogram for both the equal-angle and equal-spaced detector sampling scenarios.

METHODS

One of the derived properties is an approximately zero-energy region in the two-dimensional Fourier transform of the full fan-beam sinogram. This region is in the form of a double-wedge, similar to the parallel beam case, but different in that it is asymmetric with respect to the frequency axes. The authors characterize this region for a point object and validate the derived properties in both a simulation and a head CT data set. The authors apply these results in an application using algebraic reconstruction.

RESULTS

In the equal-angle case, the domain of the zero region is (q,k) for which / k/(k-q) / > R/L, where q and k are the frequency variables associated with the detector and view angular positions, respectively, R is the radial support of the object, and L is the source-to-isocenter distance. A filter was designed to retain only sinogram frequencies corresponding to a specified radial support. The filtered sinogram was used to reconstruct the same radial support of the head CT data. As an example application of this concept, the double-wedge filter was used to computationally improve region of interest iterative reconstruction.

CONCLUSIONS

Interesting properties of the fan-beam sinogram exist and may be exploited in some applications.

Fully three-dimensional defocus-gradient corrected backprojection in cryoelectron microscopy

Recognizing that the microscope depth of field is a significant resolution-limiting factor in 3D cryoelectron microscopy, Jensen and Kornberg proposed a concept they called defocus-gradient corrected backprojection (DGCBP) and illustrated by computer simulations that DGCBP can effectively eliminate the depth of field limitation. They did not provide a mathematical justification for their concept. Our paper provides this, by showing (in the idealized case of noiseless data being available for all projection directions) that the reconstructions obtained based on DGCBP from data produced with distance-dependent blurring are essentially the same as what is obtained by a classical method of reconstruction of a 3D object from its line integrals. The approach is general enough to be applicable for correcting for any distance-dependent blurring during projection data collection. We present a new implementation of the DGCBP concept, one that closely follows the mathematics of its justifications, and illustrate it using mathematically described phantoms and their reconstructions from finitely many distance-dependently blurred projections.

Sinogram bow-tie filtering in FBP PET reconstruction

Low-pass filtering of sinograms in the radial direction is the most common practice to limit noise amplification in filtered back projection (FBP) reconstruction of positron emission tomography studies. Other filtering strategies have been proposed to prevent the loss in resolution due to low-pass radial filters, although results have been diverse. Using the well-known properties of the Fourier transform of a sinogram, the authors defined a binary mask that matches the expected shape of the support region in the Fourier domain of the sinogram ("bow tie"). This mask was smoothed by a convolution with a ten-point Gaussian kernel which not only avoids ringing but also introduces a pre-emphasis at low frequencies. A new filtering scheme for FBP is proposed, comprising this smoothed bow-tie filter combined with a standard radial filter and an axial filter. The authors compared the performance of the bow-tie filtering scheme with that of other previously reported methods: Standard radial filtering, angular filtering, and stackgram-domain filtering. All the quantitative data in the comparisons refer to a baseline reconstruction using a ramp filter only. When using the smallest size of the Gaussian kernel in the stackgram domain, the authors achieved a noise reduction of 33% at the cost of degrading radial and tangential resolutions (14.5% and 16%, respectively, for cubic interpolation). To reduce the noise by 30%, the angular filter produced a larger degradation of contrast (3%) and tangential resolution (46% at 10 mm from the center of the field of view) and showed noticeable artifacts in the form of circular blurring dependent on the distance to the center of the field of view. For a similar noise reduction (33%), the proposed bow-tie filtering scheme yielded optimum results in resolution (gain in radial resolution of 10%) and contrast (1% increase) when compared with any of the other filters alone. Experiments with rodent images showed noticeable image quality enhancement when using the proposed bow-tie filtering scheme.

The geometric response function for convergent slit-slat collimators

We have derived an analytic geometric transfer function (GTF) for a convergent slit-slat collimator that treats the parallel slit-slat collimator as a special case. The effective point spread function (EPSF) is then derived from the GTF through the Fourier transform. The results of these derivations give an accurate description of the complete geometric response for a slit-slat collimator that includes the effects of the shape and orientation of the slit and slats. We have also derived exact and approximate sensitivity formulae and spatial resolution formulae using the EPSF.

Scatter and blurring compensation in inhomogeneous media using a postprocessing method

An efficient postprocessing method to compensate for the scattering and blurring effects in inhomogeneous medium in SPECT is proposed. A two-dimensional point spread function (2D-PSF) was estimated in the image domain to model the combination of these two physical effects. This 2D-PSF in the inhomogeneous medium is fitted with an asymmetric Gaussian function based on Monte Carlo simulation results. An efficient further blurring and deconvolution method was used to restore images from the spatially variant 2D-PSF kernel. The compensation is performed using a computer-simulated NCAT phantom and a flanged Jaszczak experimental phantom. The preliminary results demonstrate an improvement in image quality and quantity accuracy with increased image contrast (25% increase compared to uncompensated image) and decreased error (40% decrease compared to uncompensated image). This method also offers an alternative to compensate for scatter and blurring in a more time efficient manner compared to the popular iterative methods. The execution time for this efficient postprocessing method is only a few minutes, which is within the clinically acceptable range.

Three-dimensional analysis of vascular development in the mouse embryo

Key vasculogenic (de-novo vessel forming) and angiogenic (vessel remodelling) events occur in the mouse embryo between embryonic days (E) 8.0 and 10.0 of gestation, during which time the vasculature develops from a simple circulatory loop into a complex, fine structured, three-dimensional organ. Interpretation of vascular phenotypes exhibited by signalling pathway mutants has historically been hindered by an inability to comprehensively image the normal sequence of events that shape the basic architecture of the early mouse vascular system. We have employed Optical Projection Tomography (OPT) using frequency distance relationship (FDR)-based deconvolution to image embryos immunostained with the endothelial specific marker PECAM-1 to create a high resolution, three-dimensional atlas of mouse vascular development between E8.0 and E10.0 (5 to 30 somites). Analysis of the atlas has provided significant new information regarding normal development of intersomitic vessels, the perineural vascular plexus, the cephalic plexus and vessels connecting the embryonic and extraembryonic circulation. We describe examples of vascular remodelling that provide new insight into the mechanisms of sprouting angiogenesis, vascular guidance cues and artery/vein identity that directly relate to phenotypes observed in mouse mutants affecting vascular development between E8.0 and E10.0. This atlas is freely available at http://www.mouseimaging.ca/research/mouse_atlas.html and will serve as a platform to provide insight into normal and abnormal vascular development.

SPECT/CT

In view of the commercial success of integrated PET/CT scanners, there is an increasing interest in comparable SPECT/CT systems. SPECT in combination with CT enables a direct correlation of anatomic information and functional information, resulting in better localization and definition of scintigraphic findings. Besides anatomic referencing, the added value of CT coregistration is based on the attenuation correction capabilities of CT. The number of clinical studies is limited, but pilot studies have indicated a higher specificity and a significant reduction in indeterminate findings. The superiority of SPECT/CT over planar imaging or SPECT has been demonstrated in bone scintigraphy, somatostatin receptor scintigraphy, parathyroid scintigraphy, and adrenal gland scintigraphy. Also, rates of detection of sentinel nodes by biopsy can be increased with SPECT/CT. This review highlights recent technical developments in integrated SPECT/CT systems and summarizes the current literature on potential clinical uses and future directions for SPECT/CT in cardiac, neurologic, and oncologic applications.

Resolution improvement in emission optical projection tomography

A new imaging technique called emission optical projection tomography (eOPT), essentially an optical version of single-photon emission computed tomography (SPECT), provides molecular specificity, resolution on the order of microns to tens of microns, and large specimen coverage ( approximately 1 cubic centimetre). It is ideally suited to gene expression studies in embryos. Reconstructed eOPT images suffer from blurring that worsens as the distance from the axis of rotation increases. This blur is caused in part by the defocusing of the lens' point-spread function, which increases with object distance from the focal plane. In this paper, we describe a frequency space filter based on the frequency-distance relationship of sinograms to deconvolve the distance-dependent point-spread function and exclude highly defocused data from the eOPT sinograms prior to reconstruction. The method is shown to reduce the volume at half-maximum of the reconstructed point-spread function to approximately 20% the original, and the volume at 10% maximum to approximately 6% the original. As an illustration, the visibility of fine details in the vasculature of a 9.5 day old mouse embryo is dramatically improved.

Quantification in simultaneous (99m)Tc/(123)I brain SPECT using generalized spectral factor analysis: a Monte Carlo study

In SPECT, simultaneous (99m)Tc/(123)I acquisitions allow comparison of the distribution of two radiotracers in the same physiological state, without any image misregistration, but images can be severely distorted due to cross-talk between the two isotopes. We propose a generalized spectral factor analysis (GSFA) method for solving the cross-talk issue in simultaneous (99m)Tc/(123)I SPECT. In GSFA, the energy spectrum of the photons in any pixel is expressed as a linear combination of five common spectra: (99m)Tc and (123)I photopeaks and three scatter spectra. These basis spectra are estimated from a factor analysis of all spectra using physical priors (e.g. Klein-Nishina distributions). GSFA was evaluated on (99m)Tc/(123)I Monte Carlo simulated data and compared to images obtained using recommended spectral windows (WIN) and to the gold standard (GS) images (scatter-free, cross-talk-free and noise-free). Using GSFA, activity concentration differed by less than 9% compared to GS values against differences from -23% to 110% with WIN in the (123)I and (99m)Tc images respectively. Using GSFA, simultaneous (99m)Tc/(123)I imaging can yield images of similar quantitative accuracy as when using sequential and scatter-free (99m)Tc/(123)I imaging in brain SPECT.

Iterative reconstruction techniques in emission computed tomography

In emission tomography statistically based iterative methods can improve image quality relative to analytic image reconstruction through more accurate physical and statistical modelling of high-energy photon production and detection processes. Continued exponential improvements in computing power, coupled with the development of fast algorithms, have made routine use of iterative techniques practical, resulting in their increasing popularity in both clinical and research environments. Here we review recent progress in developing statistically based iterative techniques for emission computed tomography. We describe the different formulations of the emission image reconstruction problem and their properties. We then describe the numerical algorithms that are used for optimizing these functions and illustrate their behaviour using small scale simulations.

Iterative three-dimensional expectation maximization restoration of single photon emission computed tomography images: Application in striatal imaging

Single photon emission computed tomography imaging suffers from poor spatial resolution and high statistical noise. Consequently, the contrast of small structures is reduced, the visual detection of defects is limited and precise quantification is difficult. To improve the contrast, it is possible to include the spatially variant point spread function of the detection system into the iterative reconstruction algorithm. This kind of method is well known to be effective, but time consuming. We have developed a faster method to account for the spatial resolution loss in three dimensions, based on a postreconstruction restoration method. The method uses two steps. First, a noncorrected iterative ordered subsets expectation maximization (OSEM) reconstruction is performed and, in the second step, a three-dimensional (3D) iterative maximum likelihood expectation maximization (ML-EM) a posteriori spatial restoration of the reconstructed volume is done. In this paper, we compare to the standard OSEM-3D method, in three studies (two in simulation and one from experimental data). In the two first studies, contrast, noise, and visual detection of defects are studied. In the third study, a quantitative analysis is performed from data obtained with an anthropomorphic striatal phantom filled with 123-I. From the simulations, we demonstrate that contrast as a function of noise and lesion detectability are very similar for both OSEM-3D and OSEM-R methods. In the experimental study, we obtained very similar values of activity-quantification ratios for different regions in the brain. The advantage of OSEM-R compared to OSEM-3D is a substantial gain of processing time. This gain depends on several factors. In a typical situation, for a 128 x 128 acquisition of 120 projections, OSEM-R is 13 or 25 times faster than OSEM-3D, depending on the calculation method used in the iterative restoration. In this paper, the OSEM-R method is tested with the approximation of depth independent resolution. For the striatum this approximation is appropriate, but for other clinical situations we will need to include a spatially varying response. Such a response is already included in OSEM-3D.

Image distortion and correction in single photon emission CT

The task of single photon emission CT (SPECT) is to visualize the physiological function of various organs with the help of radiopharmaceuticals. But the projection data used for image reconstruction are distorted by several factors, making the reconstruction of a quantitative SPECT image very difficult in most cases. These factors include the attenuation and scattering of gamma rays, collimator aperture, data acquisition method, movement of organs, and washout of radiopharmaceuticals. This review article classifies the causes of the distortion in SPECT images and describes correction methods.

Compensation for nonuniform resolution using penalized-likelihood reconstruction in space-variant imaging systems

Imaging systems that form estimates using a statistical approach generally yield images with nonuniform resolution properties. That is, the reconstructed images possess resolution properties marked by space-variant and/or anisotropic responses. We have previously developed a space-variant penalty for penalized-likelihood (PL) reconstruction that yields nearly uniform resolution properties. We demonstrated how to calculate this penalty efficiently and apply it to an idealized positron emission tomography (PET) system whose geometric response is space-invariant. In this paper, we demonstrate the efficient calculation and application of this penalty to space-variant systems. (The method is most appropriate when the system matrix has been precalculated.) We apply the penalty to a large field of view PET system where crystal penetration effects make the geometric response space-variant, and to a two-dimensional single photon emission computed tomography system whose detector responses are modeled by a depth-dependent Gaussian with linearly varying full-width at half-maximum. We perform a simulation study comparing reconstructions using our proposed PL approach with other reconstruction methods and demonstrate the relative resolution uniformity, and discuss tradeoffs among estimators that yield nearly uniform resolution. We observe similar noise performance for the PL and post-smoothed maximum-likelihood (ML) approaches with carefully matched resolution, so choosing one estimator over another should be made on other factors like computational complexity and convergence rates of the iterative reconstruction. Additionally, because the postsmoothed ML and the proposed PL approach can outperform one another in terms of resolution uniformity depending on the desired reconstruction resolution, we present and discuss a hybrid approach adopting both a penalty and post-smoothing.

Emission tomography with a large-hole collimator (CACAO): a possible new way to improve radionuclide imaging

This work aims to improve the quality of scintigraphy. It evaluates the use of a large-hole collimator, the Computer Aided Collimation Gamma Camera Project (CACAO), in SPECT. Acquisition data from the same object were simulated for CACAO and for a conventional collimator. Better signal-to-noise ratios were found for CACAO images, whatever the number of emitted photons. This work demonstrates that high-resolution images may be obtained with large-hole collimators. The combination of CACAO and pixilated detectors may further improve radionuclide imaging.

Performance evaluation of OSEM reconstruction algorithm incorporating three-dimensional distance-dependent resolution compensation for brain SPECT: a simulation study

Iterative reconstruction techniques such as an ordered subsets-expectation maximization (OSEM) algorithm can easily incorporated various physical models of attenuation or scatter. We implemented OSEM reconstruction algorithm incorporating compensation for distance-dependent blurring due to the collimator in SPECT. The algorithm was examined by computer simulation to estimate the accuracy for brain perfusion study.

METHODS

The detector response was assumed to be a two-dimensional Gauss function and the width of the function varied linearly with the source-to-detector distance. The attenuation compensation (AC) was also included. To investigate the properties of the algorithm, we performed computer simulations with the point source and digital brain phantoms. In the point source phantom, the uniformity of FWHM for the radial, tangential and longitudinal directions was evaluated on the reconstruction image. As for the brain phantom, quantitative accuracy was estimated by comparing the reconstructed images with the true image by the mean square error (MSE) and the ratio of gray and white matter counts (G/W). Both noise free and noisy simulations were examined.

RESULTS

In the point source simulation, FWHM in radial, tangential and longitudinal directions were 14.7, 14.7 and 15.0 mm at the image center and were 15.9, 9.83 and 10.6 mm at a distance of 15 cm from the center by using FBP, respectively. On the other hand, they were 8.12, 8.12 and 7.83 mm at the image center, and were 7.45, 7.44 and 7.01 mm at 15 cm from the center by OSEM with distance-dependent resolution compensation (DRC). An isotropic and stationary resolution was obtained at any location by OSEM with DRC. The spatial resolution was also improved about 6.5 mm by OSEM with DRC at the image center. In the brain phantom simulation, the blurring at the edge of the brain structure was eliminated by using OSEM with both DRC and AC. The G/W was 2.95 and 2.68 for noise free and noisy cases, respectively, when no compensation was performed. But the values for G/W without and with noise became 3.45 and 3.21 with AC only and were improved to 3.75 and 3.71 with both AC and DRC. The G/W approached the true value (4.00) by using OSEM with both AC and DRC even when there was statistical noise.

CONCLUSION

In conclusion, OSEM reconstruction including the distance-dependent resolution compensation algorithm was reasonably successful in achieving isotropic and stationary resolution and improving the quantitative accuracy for brain perfusion SPECT.

Image-correction techniques in SPECT

This overview takes a look at different correction techniques for Single Photon Emission Computed Tomography (SPECT). We discuss the influence of the detection system followed by the scatter and attenuation caused by the object of investigation. When possible we describe how the correction methods for the different physical effects can be incorporated in the reconstruction method, being either filtered backprojection or iterative reconstruction.

A three-dimensional ray-driven attenuation, scatter and geometric response correction technique for SPECT in inhomogeneous media

The qualitative and quantitative accuracy of SPECT images is degraded by physical factors of attenuation, Compton scatter and spatially varying collimator geometric response. This paper presents a 3D ray-tracing technique for modelling attenuation, scatter and geometric response for SPECT imaging in an inhomogeneous attenuating medium. The model is incorporated into a three-dimensional projector-backprojector and used with the maximum-likelihood expectation-maximization algorithm for reconstruction of parallel-beam data. A transmission map is used to define the inhomogeneous attenuating and scattering object being imaged. The attenuation map defines the probability of photon attenuation between the source and the scattering site, the scattering angle at the scattering site and the probability of attenuation of the scattered photon between the scattering site and the detector. The probability of a photon being scattered through a given angle and being detected in the emission energy window is approximated using a Gaussian function. The parameters of this Gaussian function are determined using physical measurements of parallel-beam scatter line spread functions from a non-uniformly attenuating phantom. The 3D ray-tracing scatter projector-backprojector produces the scatter and primary components. Then, a 3D ray-tracing projector-backprojector is used to model the geometric response of the collimator. From Monte Carlo and physical phantom experiments, it is shown that the best results are obtained by simultaneously correcting attenuation, scatter and geometric response, compared with results obtained with only one or two of the three corrections. It is also shown that a 3D scatter model is more accurate than a 2D model. A transmission map is useful for obtaining measurements of attenuation and scatter in SPECT data, which can be used together with a model of the geometric response of the collimator to obtain corrected images with quantitative and diagnostically accurate information.

Non-iterative methods incorporating a priori source distribution and data information for suppression of image noise and artefacts in 3D SPECT

Non-iterative methods have been developed for image reconstruction in 3D SPECT with uniform attenuation and distance-dependent spatial resolution. It was observed that these methods can, in general, be susceptible to data noise and other errors, yielding conspicuous image artefacts. In this work, we developed and evaluated a regularized inverse-filtering approach for effective suppression of noise and artefacts in 3D SPECT images without significantly compromising image resolution. The proposed approach allows the incorporation of a priori random image field and data information and can thus robustly control the degree of suppression of noise and artefacts in 3D SPECT images. Using computer simulations, we evaluated and compared quantitatively images reconstructed from data sets of various noise levels by the use of the proposed methods and the existing non-iterative methods. These numerical results clearly demonstrated that the proposed regularized inverse-filtering approach can effectively suppress image noise and artefacts that plague the existing non-iterative methods, thus yielding quantitatively more accurate 3D SPECT images. The proposed regularized inverse-filtering approach can also be generalized to other imaging modalities.

Combined corrections for attenuation, depth-dependent blur, and motion in cardiac SPECT: a multicenter trial

BACKGROUND

The diagnostic accuracy of cardiac single photon emission computed tomography (SPECT) is limited by image-degrading factors, such as heart or subject motion, depth-dependent blurring caused by the collimator, and photon scatter and attenuation. We developed correction approaches for motion, depth-dependent blur, and attenuation and performed a multicenter validation.

METHODS AND RESULTS

Motion was corrected both transversely and axially with a cross-correlation technique. Depth-dependent blurring was corrected by first back-projecting each projection and then applying a depth-dependent Wiener filter row by row. Attenuation was corrected with an iterative, nonuniform Chang algorithm, based on a transmission scan-generated attenuation map. We validated these approaches in 112 subjects, including 36 women (20 healthy volunteers, 8 angiographically normal patients, and 8 patients with coronary artery disease [CAD] found by means of angiography) and 76 men (23 healthy volunteers, 10 angiographically normal patients, and 43 patients with CAD found by means of angiography). Either technetium 99m or thallium 201 was used for emission; either gadolinium 153 or Tc-99m was used for transmission. Images were reconstructed and blindly interpreted with a 5-point scale for receiver operating characteristic analysis in 2 ways: motion correction plus a Butterworth filter, and combined motion and blur and attenuation corrections. The interpretation by means of consensus was for the overall presence of CAD and vascular territory. The receiver operating characteristic curves for overall presence and each of the 3 main coronary arteries were all shifted upward and to the left and had larger areas under the curve, for combined corrections compared with motion correction and Butterworth. Sensitivity/specificity for motion correction and Butterworth were 84/69, 64/71, 32/94, and 71/81 overall for the left anterior descending, the right coronary artery, and circumflex territories, respectively, compared with 88/92, 77/93, 50/97, and 74/95, respectively, for the combined corrections.

CONCLUSIONS

The proposed combined corrections for motion, depth-dependent blur, and attenuation significantly improve diagnostic accuracy, when compared with motion correction alone.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Application of distance-dependent resolution compensation and post-reconstruction filtering for myocardial SPECT

Compensation for distance-dependent resolution can be directly incorporated in maximum likelihood reconstruction. Our objective was to examine the effectiveness of this compensation using either the standard expectation maximization (EM) algorithm or an accelerated algorithm based on use of ordered subsets (OSEM). We also investigated the application of post-reconstruction filtering in combination with resolution compensation. Using the MCAT phantom, projections were simulated for 360 degrees data, including attenuation and distance-dependent resolution. Projection data were reconstructed using conventional EM and OSEM with subset size 2 and 4, with/without 3D compensation for detector response (CDR). Also post-reconstruction filtering (PRF) was performed using a 3D Butterworth filter of order 5 with various cutoff frequencies (0.2-1.2 cycles cm(-1)). Image quality and reconstruction accuracy were improved when CDR was included. Image noise was lower with CDR for a given iteration number. PRF with cutoff frequency greater than 0.6 cycles cm(-1) improved noise with no reduction in recovery coefficient for myocardium but the effect was less when CDR was incorporated in the reconstruction. CDR alone provided better results than use of PRF without CDR. Results suggest that using CDR without PRF, and stopping at a small number of iterations, may provide sufficiently good results for myocardial SPECT. Similar behaviour was demonstrated for OSEM.

Quasi-bandlimited properties of radon transforms and their implications for increasing angular sampling densities

The n-dimensional (n-D) radon transform, which forms the mathematical basis for a broad variety of tomographic imaging applications, can be viewed as an n-D function in n-D sinogram space. Accurate reconstruction of continuous or discrete tomographic images requires full knowledge of the radon transform in the corresponding n-D sinogram space. In practice, however, one can have only a finite set of discrete samples of the radon transform in the sinogram space. One often derives the desired full knowledge of the radon transform from its discrete samples by invoking various interpolation algorithms. According to the Wittaker-Shannon sampling theorem, a necessary condition for a full and unique recovery of the radon transform from its discrete samples is that the radon transform itself be bandlimited. Therefore, it is necessary to analyze the bandlimited properties of the radon transform. In this work, we analyze explicitly the bandlimited properties of the radon transform and show that the radon transform is mathematically quasi-bandlimited [or essentially bandlimited] in two quantitative senses and can essentially be treated as bandlimited in practice. The quasi-bandlimited properties can be used for increasing the angular sampling density of the radon transform.

Performance of the Fourier rebinning algorithm for PET with large acceptance angles

The recently proposed Fourier rebinning (FORE) technique of 3D PET reconstruction is investigated over a wide range of axial acceptance angles. In this study we evaluate the performance of the FORE technique using spatial resolution, contrast and noise figures of merit and compare reconstruction performance of the FORE (followed by multislice 2D reconstruction) to the 3D-RP technique for large-acceptance-angle data (+/-26.25 degrees). Our results show that the FORE technique does not affect the transverse resolution. On the other hand the axial resolution using FORE deteriorates faster, compared with the 3D-RP, at large radii as the acceptance angle increases. Concerning the noise behaviour, we have found that filtering has better ability to suppress the noise in the FORE reconstruction, compared with the 3D-RP reconstruction, especially in the slices near the edge of the axial field of view. Overall, the combination of good performance and fast reconstruction time makes the FORE technique a practical choice for 3D PET applications.

Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modelling of spatial resolution in iterative reconstruction

Quantitative parameters such as the maximum and total counts in a volume are influenced by the partial volume effect. The magnitude of this effect varies with the non-stationary and anisotropic spatial resolution in SPECT slices. The objective of this investigation was to determine whether iterative reconstruction which includes modelling of the three-dimensional (3D) spatial resolution of SPECT imaging can reduce the impact of the partial volume effect on the quantitation of activity compared with filtered backprojection (FBP) techniques which include low-pass, and linear restoration filtering using the frequency distance relationship (FDR). The iterative reconstruction algorithms investigated were maximum-likelihood expectation-maximization (MLEM), MLEM with ordered subset acceleration (ML-OS), and MLEM with acceleration by the rescaled-block-iterative technique (ML-RBI). The SIMIND Monte Carlo code was used to simulate small hot spherical objects in an elliptical cylinder with and without uniform background activity as imaged by a low-energy ultra-high-resolution (LEUHR) collimator. Centre count ratios (CCRs) and total count ratios (TCRs) were determined as the observed counts over true counts. CCRs were unstable while TCRs had a bias of approximately 10% for all iterative techniques. The variance in the TCRs for ML-OS and ML-RBI was clearly elevated over that of MLEM, with ML-RBI having the smaller elevation. TCRs obtained with FDR-Wiener filtering had a larger bias (approximately 30%) than any of the iterative reconstruction methods but near stationarity is also reached. Butterworth filtered results varied by 9.7% from the centre to the edge. The addition of background has an influence on the convergence rate and noise properties of iterative techniques.

Projection space image reconstruction using strip functions to calculate pixels more "natural" for modeling the geometric response of the SPECT collimator

The spatially varying geometric response of the collimator-detector system in single photon emission computed tomography (SPECT) causes loss in resolution, shape distortions, reconstructed density nonuniformity, and quantitative inaccuracies. A projection space image reconstruction algorithm is used to correct these reconstruction artifacts. The projectors F use strip functions to calculate pixels more "natural" for modeling the two-dimensional (2-D) geometric response of the SPECT collimator transaxially to the axis of rotation. These projectors are defined by summing the intersection of an array of multiple strips rotated at equal angles to approximate the ideal system geometric response of the collimator. Two projection models were evaluated for modeling the system geometric response function. For one projector each strip is of equal weight, for the other projector a Gaussian weighting is used. Parallel beam and fan beam projections of a physical three-dimensional (3-D) Hoffman brain phantom and a Jaszczak cold rod phantom were used to evaluate the geometric response correction. Reconstructions were obtained by using the singular value decomposition (SVD) method and the iterative conjugate gradient algorithm to solve for q in the imaging equation FGq = p, where p is the projection measurement. The projector F included the new models for the geometric response, whereas, the backprojector G did not always model the geometric response in order to increase the computational speed. The final reconstruction was obtained by sampling the backprojection Gq at a discrete array of points. Reconstructions produced by the two proposed projectors showed improved resolution when compared against a unit-strip "natural" pixel model, the conventional image pixelized model with ray tracing to calculate the geometric response, and the filtered backprojection algorithm. When the reconstruction is displayed on fine grid points, the continuity and resolution of the image is preserved without the ring artifacts seen in the unit-strip "natural" pixel model. With present computing power, the geometric response correction using the proposed projection space reconstruction approach is not yet feasible for routine clinical use.

Exact and approximate rebinning algorithms for 3-D PET data

This paper presents two new rebinning algorithms for the reconstruction of three-dimensional (3-D) positron emission tomography (PET) data. A rebinning algorithm is one that first sorts the 3-D data into an ordinary two-dimensional (2-D) data set containing one sinogram for each transaxial slice to be reconstructed; the 3-D image is then recovered by applying to each slice a 2-D reconstruction method such as filtered-backprojection. This approach allows a significant speedup of 3-D reconstruction, which is particularly useful for applications involving dynamic acquisitions or whole-body imaging. The first new algorithm is obtained by discretizing an exact analytical inversion formula. The second algorithm, called the Fourier rebinning algorithm (FORE), is approximate but allows an efficient implementation based on taking 2-D Fourier transforms of the data. This second algorithm was implemented and applied to data acquired with the new generation of PET systems and also to simulated data for a scanner with an 18 degrees axial aperture. The reconstructed images were compared to those obtained with the 3-D reprojection algorithm (3DRP) which is the standard "exact" 3-D filtered-backprojection method. Results demonstrate that FORE provides a reliable alternative to 3DRP, while at the same time achieving an order of magnitude reduction in processing time.

Fast direct Fourier methods, based on one- and two-pass coordinate transformations, yield accurate reconstructions of x-ray CT clinical images

The conversion from polar to Cartesian coordinates can be carried out with two-pass algorithms. The paper describes two different methods based on concentric square frames and octagonal frames and their results, obtained with accurate interpolations based on the "moving window Shannon reconstruction' (MWSR). The embedding of these algorithms in direct Fourier methods (DFMs) of tomographic reconstruction is discussed. With respect to one-pass methods and to the use of octagonal frames, the square frame method makes it possible to carry out the first pass, a radial resampling, in the direct space, before computing 1D Fourier transforms (FTs) of projections. Reconstructions of clinical images from the raw data of a third-generation x-ray tomograph are presented and compared with those obtained with one-pass DFMs and with the convolution back-projection method (CBPM) performed by the instrument. The simple algorithm using square frames yields results in complete agreement with other DFM protocols and the CBPM. On a general-purpose computer, the execution of DFM protocols based on one-pass and two-pass coordinate transformations is 35 to 55 times faster than the CBPM and make the algorithms attractive for modern instrumentation.