Dual matrix ordered subsets reconstruction for accelerated 3D scatter compensation in single-photon emission tomography

Three-dimensional (3D) iterative maximum likelihood expectation maximization (ML-EM) algorithms for single-photon emission tomography (SPET) are capable of correcting image-degrading effects of non-uniform attenuation, distance-dependent camera response and patient shape-dependent scatter. However, the resulting improvements in quantitation, resolution and signal-to-noise ratio (SNR) are obtained at the cost of a huge computational burden. This paper presents a new acceleration method for ML-EM: dual matrix ordered subsets (DM-OS). DM-OS combines two acceleration methods: (a) different matrices for projection and back-projection and (b) ordered subsets of projections. DM-OS was compared with ML-EM on simulated data and on physical thorax phantom data, for both 180° and 360° orbits. Contrast, normalized standard deviation and mean squared error were calculated for the digital phantom experiment. DM-OS resulted in similar image quality to ML-EM, even for speed-up factors of 200 compared to ML-EM in the case of 120 projections. The thorax phantom data could be reconstructed 50 times faster (60 projections) using DM-OS with preservation of image quality. ML-EM and DM-OS with scatter compensation showed significant improvement of SNR compared to ML-EM without scatter compensation. Furthermore, inclusion of complex image formation models in the computer code is simplified in the case of DM-OS. It is thus shown that DM-OS is a fast and relatively simple algorithm for 3D iterative scatter compensation, with similar results to conventional ML-EM, for both 180° and 360° acquired data. Received 7 July and in revised form 27 September 1997     

Assessment of geometrical distortion and activity distribution after attenuation correction: A SPECT phantom study

BACKGROUND: If single photon emission computed tomography (SPECT) images are reconstructed with filtered backprojection (FBP), not accounting for photon attenuation, artifacts can occur related to geometrical distortion and inaccurate estimation of regional distribution of radioactivity. By reconstructing the images with an iterative algorithm such as the maximum likelihood-expectation maximization (ML-EM) that incorporates the attenuation distribution information, it is possible to compensate for nonuniform attenuation. The aim of this study was to assess whether correction for nonuniform attenuation in SPECT can reduce the geometrical distortion and improve the activity quantitation. METHODS AND RESULTS: Three capillary sources containing the same amount of technetium 99m were imaged by a dual-headed SPECT system provided with two gadolinium 153 scanning transmission line sources, in nonuniform attenuation conditions. The images were reconstructed (1) with the use of FBP, (2) with the iterative ML-EM algorithm, and (3) with the iterative ML-EM algorithm incorporating attenuation maps. The geometrical distortion was estimated by comparing the spread that occurred in 2 orthogonal directions in the reconstructed transverse slices, expressed by full width at half maximum related to the x-axis and y-axis line spread functions. The accuracy of activity quantitation was analyzed by comparing the counts in regions of interest placed over the transverse slices of the 3 sources, located in different attenuating areas. The FBP-reconstructed slices showed a spread of image intensity toward the direction of minor attenuation; the source shape improved in the iterative ML-EM images, as well as in the iterative attenuation-corrected ML-EM images. The sources located deep in the phantom showed an apparent decrease in image intensity in both FBP and ML-EM images, which became less evident in the iterative attenuation-corrected ML-EM images. CONCLUSIONS: Image reconstruction with the iterative ML-EM algorithm, without the use of attenuation maps, can reduce geometrical distortion and eliminate streak artifacts, leading to an improvement in the object's shape and size, but does not reduce activity underestimation and inaccurate quantitation. In the iterative attenuation-corrected ML-EM images, there was a significant improvement in the accurate quantitation of activity distribution and a further reduction in geometrical distortion. In conclusion, nonisotropic attenuation correction with iterative ML-EM reduced the geometrical distortion of images and improved the accuracy of activity quantitation.     

Evaluation of median filtering after reconstruction with maximum likelihood expectation maximization (ML-EM) by real space and frequency space

Maximum likelihood expectation maximization (ML-EM) image quality is sensitive to the number of iterations, because a large number of iterations leads to images with checkerboard noise. The use of median filtering in the reconstruction process allows both noise reduction and edge preservation. We examined the value of median filtering after reconstruction with ML-EM by comparing filtered back projection (FBP) with a ramp filter or ML-EM without filtering. SPECT images were obtained with a dual-head gamma camera. The acquisition time was changed from 10 to 200 (seconds/frame) to examine the effect of the count statistics on the quality of the reconstructed images. First, images were reconstructed with ML-EM by changing the number of iterations from 1 to 150 in each study. Additionally, median filtering was applied following reconstruction with ML-EM. The quality of the reconstructed images was evaluated in terms of normalized mean square error (NMSE) values and two-dimensional power spectrum analysis. Median filtering after reconstruction by the ML-EM method provided stable NMSE values even when the number of iterations was increased. The signal element of the image was close to the reference image for any repetition number of iterations. Median filtering after reconstruction with ML-EM was useful in reducing noise, with a similar resolution achieved by reconstruction with FBP and a ramp filter. Especially in images with poor count statistics, median filtering after reconstruction with ML-EM is effective as a simple, widely available method.     

Bayesian image reconstruction for emission tomography based on median root prior

The aim of the present study was to investigate a new type of Bayesian one-step late reconstruction method which utilizes a median root prior (MRP). The method favours images which have locally monotonous radioactivity concentrations. The new reconstruction algorithm was applied to ideal simulated data, phantom data and some patient examinations with PET. The same projection data were reconstructed with filtered back-projection (FBP) and maximum likelihood-expectation maximization (ML-EM) methods for comparison. The MRP method provided good-quality images with a similar resolution to the FBP method with a ramp filter, and at the same time the noise properties were as good as with Hann-filtered FBP images. The typical artefacts seen in FBP reconstructed images outside of the object were completely removed, as was the grainy noise inside the object. Quantitatively, the resulting average regional radioactivity concentrations in a large region of interest in images produced by the MRP method corresponded to the FBP and ML-EM results but at the pixel by pixel level the MRP method proved to be the most accurate of the tested methods. In contrast to other iterative reconstruction methods, e.g. ML-EM, the MRP method was not sensitive to the number of iterations nor to the adjustment of reconstruction parameters. Only the Bayesian parameter had to be set. The proposed MRP method is much more simple to calculate than the methods described previously, both with regard to the parameter settings and in terms of general use. The new MRP reconstruction method was shown to produce high-quality quantitative emission images with only one parameter setting in addition to the number of iterations.     

Evaluation of the number of SPECT projections in the ordered subsets-expectation maximization image reconstruction method

Filtered back projection (FBP) method, maximum likelihood-expectation maximization (ML-EM) method, and ordered subsets-expectation maximization (OS-EM) method are currently used for reconstruction of SPECT images in clinical studies. In the ML-EM method, images of good quality can be reconstructed even with a small sampling number of projection data, when compared with FBP. Shorter acquisition time and less radionuclide dose are preferable in the clinical setting if image quality is the same. In this study, we attempted to find optimal conditions for reconstruction of OS-EM images with commonly used sampling numbers of 30, 60 and 120 (step angles: 12°, 6°, and 3°, respectively), with acquisition counts/projection of 30, 60, 120 and 240 each. We adjusted the pixel counts of reconstructed images to be constant, by setting combination of sampling number and counts/projection (120 sampling number for 30 counts/projection, 60 for 60, and 30 for 120). Among the 3 acquisition conditions, the small sampling number of 30 had large acquisition counts per direction, resulting in low signal to noise ratio. Under this condition, the resolution was slightly low, but the uniformity of images was high. The combination of OS-EM and smaller sampling projection number may be clinically useful with reduction of the examination time, which is also beneficial to reduce dead time for gamma-camera rotation.     

Evaluation of the number of SPECT projections in the ordered subsets-expectation maximization image reconstruction method

Filtered back projection (FBP) method, maximum likelihood-expectation maximization(ML-EM) method, and ordered subsets-expectation maximization (OS-EM) method are currently used for reconstruction of SPECT images in clinical studies. In the ML-EM method, images of good quality can be reconstructed even with a small sampling number of projection data, when compared with FBP. Shorter acquisition time and less radionuclide dose are preferable in the clinical setting if image quality is the same. In this study, we attempted to find optimal conditions for reconstruction of OS-EM images with commonly used sampling numbers of 30, 60 and 120 (step angles: 12 degrees, 6 degrees, and 3 degrees, respectively), with acquisition counts/projection of 30, 60, 120 and 240 each. We adjusted the pixel counts of reconstructed images to be constant, by setting combination of sampling number and counts/projection (120 sampling number for 30 counts/projection, 60 for 60, and 30 for 120). Among the 3 acquisition conditions, the small sampling number of 30 had large acquisition counts per direction, resulting in low signal to noise ratio. Under this condition, the resolution was slightly low, but the uniformity of images was high. The combination of OS-EM and smaller sampling projection number may be clinically useful with reduction of the examination time, which is also beneficial to reduce dead time for gamma-camera rotation.     

A study of lesion contrast recovery for iterative PET image reconstructions versus filtered backprojection using an anthropomorphic thoracic phantom.

Iterative methods for the reconstruction of positron emission tomography images can produce results superior to filtered backprojection (FBP) due to their ability to explicitly model the Poisson statistics of photon pair coincidence detection. Many conventional implementations of these methods use simple forward and backward projection schemes based on computing the area of intersection of detection tubes with each voxel. Other important physical system factors, such as depth-dependent geometric sensitivity and spatially variant detector pair resolution are often ignored. One goal of this work is to examine the effect of a more accurate system model on iterative algorithm performance. A second factor that limits the performance of an iterative algorithm is the chosen objective function and the manner in which it is optimized. In this paper, performance of the following image reconstruction methods is evaluated: FBP, ordered subsets expectation maximization (OSEM) algorithm, and maximum a posteriori (MAP) estimation using Gibbs prior with convex potential functions. Using the contrast recovery coefficient (CRC) as a performance measure, this paper presents a lesion detection experiment based on an anthropomorphic thoracic phantom and demonstrates how the choices of reconstruction algorithm and projection matrix affect reconstruction accuracy. Plots of CRC versus background variance were generated by varying cut-off frequency in FBP, post-smoothing Gaussian kernel in OSEM, and smoothing hyper-parameter in MAP. The results of these studies show that all of the iterative methods evaluated produce superior CRCs than FBP at matched background variation. In addition, there is also considerable variation in performance within the class of statistical iterative methods depending on the choice of projection matrix and reconstruction algorithm.     

Improving spatial resolution in SPECT with the combination of PSPMT based detector and iterative reconstruction algorithms.

This paper investigates the possibility of developing a SPECT system that combines the high spatial resolution of position sensitive photomultiplier tubes (PSPMTs) with the excellent performance of iterative reconstruction algorithms. A small field of view (FOV) camera based on a PSPMT and a pixelized scintillation crystal made of CsI(Tl) have been used for the acquisition of the projections. With the use of maximum likelihood expectation maximization (ML-EM) and ordered subsets expectation maximization (OSEM) slices of the object are obtained while three-dimensional (3D) reconstruction of the object is carried out using a modified marching cubes (MMC) algorithm. The spatial resolution of tomographic images obtained with the system was 2-3mm. The spatial resolution of a conventional system that uses filtered backprojection (FBP) for slices reconstruction was more than 9 mm.     

Reconstruction of Compressed-sensing MR Imaging Using Deep Residual Learning in the Image Domain

Purpose:A deep residual learning convolutional neural network (DRL-CNN) was applied to improve image quality and speed up the reconstruction of compressed sensing magnetic resonance imaging. The reconstruction performances of the proposed method was compared with iterative reconstruction methods.Methods:The proposed method adopted a DRL-CNN to learn the residual component between the input and output images (i.e., aliasing artifacts) for image reconstruction. The CNN-based reconstruction was compared with iterative reconstruction methods. To clarify the reconstruction performance of the proposed method, reconstruction experiments using 1D-, 2D-random under-sampling and sampling patterns that mix random and non-random under-sampling were executed. The peak-signal-to-noise ratio (PSNR) and the structural similarity index (SSIM) were examined for various numbers of training images, sampling rates, and numbers of training epochs.Results:The experimental results demonstrated that reconstruction time is drastically reduced to 0.022 s per image compared with that for conventional iterative reconstruction. The PSNR and SSIM were improved as the coherence of the sampling pattern increases. These results indicate that a deep CNN can learn coherent artifacts and is effective especially for cases where the randomness of k-space sampling is rather low. Simulation studies showed that variable density non-random under-sampling was a promising sampling pattern in 1D-random under-sampling of 2D image acquisition.Conclusion:A DRL-CNN can recognize and predict aliasing artifacts with low incoherence. It was demonstrated that reconstruction time is significantly reduced and the improvement in the PSNR and SSIM is higher in 1D-random under-sampling than in 2D. The requirement of incoherence for aliasing artifacts is different from that for iterative reconstruction.     

Space-domain non-iterative approach for SPECT/CT systems considering attenuation and space-variant detector response.

A quantitative analysis of emission planar image reconstruction from projections by an object dependent, exact, direct approach in the space-domain considering both object attenuation and space-variant impulse response of SPECT/CT systems is proposed. That approach is compared with iterative methods and non-object-dependent exact methods in both the space domain and the frequency one. Since the mean-projection precorrection method is the concern of some actual 3D methods of compensation for distance-dependent spatial resolution and is thought right for competing with different methods able to quantify the tracer density in the object of interest, it is also examined in the course of the analysis. The direct approach may also augment the simulation power of the Matlab Image Processing Toolbox concerning the direct and inverse Radon transform from parallel projection data, the Toolbox being actually restricted to the ideal transform in the frequency domain.     

Evaluation of the X-ray digital linear tomosynthesis reconstruction processing method for metal artifact reduction

We evaluated the X-ray digital linear tomosynthesis reconstruction processing method for metal artifact reduction. A volumetric X-ray digital linear tomosynthesis instrument was used to image a hip prosthesis. Artifacts caused by high-attenuation features in hip prostheses were observed in digital linear tomosynthesis reconstruction due to the few projections and narrow angular range typically employed in tomosynthesis imaging. We developed artifact reduction methods based on a modified Shepp and Logan reconstruction filter kernel realized by taking into account additional weighing by direct current (DC) components in frequency domain space. Processing leads to an increase in the ratio of low-frequency components in an image. The effectiveness of the method in enhancing the visibility of a prosthetic case was quantified in terms of removal of ghosting artifacts. The potential of artifact reduction processing for digital linear tomosynthesis in the evaluation of hip prostheses was demonstrated. A modified Shepp and Logan reconstruction filter kernel causes artifact reduction and improved the quality of images affected by metal artifacts. Future investigations will study the ability of digital linear tomosynthesis to quantify the spatial relationship between the metallic components of these devices as well as the ability of the technique to identify bony changes of diagnostic significance.     

Reconstruction of undersampled radial PatLoc imaging using total generalized variation

In the case of radial imaging with nonlinear spatial encoding fields, a prominent star‐shaped artifact has been observed if a spin distribution is encoded with an undersampled trajectory. This work presents a new iterative reconstruction method based on the total generalized variation, which reduces this artifact. For this approach, a sampling operator (as well as its adjoint) is needed that maps data from PatLoc k‐space to the final image space. It is shown that this can be realized as a type‐3 nonuniform fast Fourier transform, which is implemented by a combination of a type‐1 and type‐2 nonuniform fast Fourier transform. Using this operator, it is also possible to implement an iterative conjugate gradient SENSE based method for PatLoc reconstruction, which leads to a significant reduction of computation time in comparison to conventional PatLoc image reconstruction methods. Results from numerical simulations and in vivo PatLoc measurements with as few as 16 radial projections are presented, which demonstrate significant improvements in image quality with the total generalized variation‐based approach. Magn Reson Med, 2013. © 2012 Wiley Periodicals, Inc.     

The relation of low frequency restoration methods to the Gerchberg-Papoulis algorithm

In magnetic resonance imaging, low frequency components can be allowed to saturate the analog to digital converter to reduce the quantization noise. These components can be estimated using least squares error estimation based low frequency restoration methods or the iterative Gerchberg-Papoulis algorithm. In this paper, we show the relationship between the closed form estimation methods and the iterative algorithm, propose a method for improving the speed of iteration, and discuss the advantages and disadvantages of two types of methods.     

Pinhole single-photon emission tomography reconstruction based on median root prior

The maximum likelihood expectation maximisation (ML-EM) algorithm can be used to reduce reconstruction artefacts produced by filtered backprojection (FBP) methods in pinhole single-photon emission tomography (SPET). However, ML-EM suffers from noise propagation along iterations, which leads to quantitatively unpleasant reconstruction results. To avoid this increase in noise, the median root prior (MRP) algorithm for pinhole SPET was implemented. Projection data of a line source and Picker's thyroid phantom were collected using a single-head gamma camera with a pinhole collimator. MRP was added to existing pinhole ML-EM reconstruction algorithm and the phantom studies were reconstructed using MRP, ML-EM and FBP for comparison. Coefficients of variation, contrasts and full-widths at half-maximum were calculated and showed a clear reduction in noise without significant loss of resolution or decrease in contrast when MRP was applied. MRP also produced visually pleasing images even with high iteration numbers, free of the checkerboard-type noise patterns which are typical of ML-EM images.     

Gibbs artifact reduction by nonnegativity constraint

This paper proposes a 2-step image reconstruction method in which the nonnegativity constraint in the iterative maximum-likelihood expectation maximization (MLEM) algorithm is used to effectively reduce Gibbs ringing artifacts. Methods: Gibbs artifacts are difficult to control during imaging reconstruction. The proposed method uses the postprocessing strategy to suppress Gibbs artifacts. In the first step, a raw image is reconstructed from projections without correction for point spread function (PSF). The attenuation correction can be performed in the first step by using, for example, the iterative MLEM or ordered-subsets expectation maximization (OS-EM) algorithm. The second step is a postprocessing procedure that corrects for the PSF blurring effect. If the target features (e.g., hot lesions) have a positive background, removing the background before application of the postprocessing filter significantly helps with target deblurring and Gibbs artifact suppression. This postprocessing filter is the image-domain MLEM algorithm. The background activity is attached back to the foreground after lesion sharpening. Results: Computer simulations and PET phantom studies were performed using the proposed 2-step method. The background removal strategy significantly reduced Gibbs artifacts. Conclusion: Gibbs ringing artifacts generated during image reconstruction are difficult to avoid if compensation for the PSF of the system is needed. The strategy of separating image reconstruction from PSF compensation has been shown effective in removal of Gibbs ringing artifacts.     

SPECT in the year 2000: basic principles

Objective: SPECT has become a routine procedure in most nuclear medicine departments. SPECT provides significant technical challenges for the nuclear medicine technologist, as compared with planar imaging, in the areas of SPECT acquisition, image reconstruction, and data processing. Many new advances in SPECT methodology are becoming available, such as iterative reconstruction, multimodality fusion, and advanced gated cardiac SPECT. SPECT imaging is demanding and requires careful attention to proper acquisition protocols, whether circular or noncircular orbits, and postprocessing is becoming more complex with the addition of iterative reconstruction and attenuation correction algorithms, among others. Understanding the principles of SPECT is essential not only to produce the highest quality scans but also to identify image artifacts. After reading this article, the nuclear medicine technologist should be able to: (a) describe the historical development and benefits of SPECT imaging; (b) state the impact of image matrix size, number of projections, and arc of rotation on final SPECT image quality; (c) discuss the trade-offs between image noise content and spatial and contrast resolution in SPECT reconstruction; (d) discuss SPECT filters and their impact on image quality; (e) explain the differences between filtered backprojection and iterative reconstruction; and (f) describe the impact of attenuation and scatter in SPECT imaging and the advantages and pitfalls of attenuation correction methods.     

Enhancement of SPECT images by Fourier filtering the projection image set

Tomographic images from rotating gamma camera systems are often difficult to interpret because of poor contrast and high noise levels. A method is presented for improving the quality of these images by Fourier filtering the projection image set prior to reconstruction. A two-dimensional circularly symmetric Gaussian function is used as the spatial frequency filter. This filter can be optimized to enhance contrast and suppress noise in the projection image set in a straightforward and simple manner from the power spectra of representative projections. Preprocessing of the projections makes it possible to use a ramp reconstruction filter. The resulting tomographic sections show a dramatic improvement in image quality.     

不完全投影医学图像的多目标向量优化重建算法

目的解决少数据投影图像重建中分辨率差、伪影严重等问题。方法提出了一种基于多目标决策的交叉熵向量优化图像重建算法。算法中折衷考虑了图像重建中最小交叉熵、范数极小化以及最大熵3个目标,并建立了一种新的动态权系数迭代法。应用该算法对模拟的有噪声投影数据和SIEMENS SOMATOM DR3的头部实际扫描数据分别进行了重建。结果文中算法较传统的重建算法．如卷积反投影法、代数重建法以及单目标优化算法,在误差、平滑性以及分辨率方面均有显著改善。结论本算法将对采用迭代法求解少数据投影重建问题产生一定的影响。     

Maximum likelihood estimation of cerebral blood flow in dynamic susceptibility contrast MRI.

For quantification of cerebral blood flow (CBF) using dynamic susceptibility contrast magnetic resonance imaging (DSC-MRI), knowledge of the tissue response function is necessary. To obtain this, the tissue contrast passage measurement must be corrected for the arterial input. This study proposes an iterative maximum likelihood expectation maximization (ML-EM) algorithm for this correction, which takes into account the noise in T2- or T2*-weighted image sequences. The ML-EM algorithm does not assume a priori knowledge of the shape of the response function; it automatically corrects for arrival time offsets and inherently yields positive response values. The results on synthetic image sequences are presented, for which the recovered flow values and the response functions are in good agreement with their expectation values. The method is illustrated by calculating the gray and white matter flow in a clinical example.     

Use of scanner characteristics in iterative image reconstruction for high-resolution positron emission tomography studies of small animals

The purpose of this work was to improve of the spatial resolution of a whole-body positron emission tomography (PET) system for experimental studies of small animals by incorporation of scanner characteristics into the process of iterative image reconstruction. The image-forming characteristics of the PET camera were characterized by a spatially variant line-spread function (LSF), which was determined from 49 activated copper-64 line sources positioned over a field of view (FOV) of 21.0 cm. This information was used to model the image degradation process. During the course of iterative image reconstruction, the forward projection of the estimated image was blurred with the LSF at each iteration step before the estimated projections were compared with the measured projections. The imaging characteristics of the high-resolution algorithm were investigated in phantom experiments. Moreover, imaging studies of a rat and two nude mice were performed to evaluate the imaging properties of our approach in vivo. The spatial resolution of the scanner perpendicular to the direction of projection could be approximated by a one-dimensional Gaussian-shaped LSF with a full-width at half-maximum increasing from 6.5 mm at the centre to 6.7 mm at a radial distance of 10.5 cm. The incorporation of this blurring kernel into the iteration formula resulted in a significantly improved spatial resolution of about 3.9 mm over the examined FOV As demonstrated by the phantom and the animal experiments, the high-resolution algorithm not only led to a better contrast resolution in the reconstructed emission scans but also improved the accuracy for quantitating activity concentrations in small tissue structures without leading to an amplification of image noise or image mottle. The presented data-handling strategy incorporates the image restoration step directly into the process of algebraic image reconstruction and obviates the need for ill-conditioned deconvolution procedures to be performed on the projections or on the reconstructed image. In our experience, the proposed algorithm is of special interest in experimental studies of small animals.This paper is dedicated to Professor Dr. Walter J. Lorenz, on the occasion of his 65th birthday