Fast and efficient fully 3D PET image reconstruction using sparse system matrix factorization with GPU acceleration

Virtual cylindrical PET for efficient DOI image reconstruction with sub-millimetre resolution

Objective.Image reconstruction in high resolution, narrow bore PET scanners with depth of interaction (DOI) capability presents a substantial computational challenge due to the very high sampling in detector and image space. The aim of this study is to evaluate the use of a virtual cylinder in reducing the number of lines of response (LOR) for DOI-based reconstruction in high resolution PET systems while maintaining uniform sub-millimetre spatial resolution.Approach.Virtual geometry was investigated using the awake animal mousePET as a high resolution test case. Using GEANT4 Application for Tomographic Emission (GATE), we simulated the physical scanner and three virtual cylinder implementations with detector size 0.74 mm, 0.47 mm and 0.36 mm (vPET1, vPET2 and vPET3, respectively). The virtual cylinder condenses physical LORs stemming from various crystal pairs and DOI combinations, and which intersect a single virtual detector pair, into a single virtual LOR. Quantitative comparisons of the point spread function (PSF) at various positions within the field of view (FOV) were compared for reconstructions based on the vPET implementations and the physical scanner. We also assessed the impact of the anisotropic PSFs by reconstructing images of a micro Derenzo phantom.Main results.All virtual cylinder implementations achieved LOR data compression of at least 50% for DOI PET reconstruction. PSF anisotropy in radial and tangential profiles was chiefly influenced by DOI resolution and only marginally by virtual detector size. Spatial degradation introduced by virtual cylinders was most prominent in the axial profile. All virtual cylinders achieved sub-millimetre volumetric resolution across the FOV when 6-bin DOI reconstructions (3.3 mm DOI resolution) were performed. Using vPET2 with 6 DOI bins yielded nearly identical reconstructions to the non-virtual case in the transaxial plane, with an LOR compression factor of 86%. Resolution modelling significantly reduced the effects of the asymmetric PSF arising from the non-cylindrical geometry of mousePET.Significance.Narrow bore and high resolution PET scanners require detectors with DOI capability, leading to computationally demanding reconstructions due to the large number of LORs. In this study, we show that DOI PET reconstruction with 50%-86% LOR compression is possible using virtual cylinders while maintaining sub-millimetre spatial resolution throughout the FOV. The methodology and analysis can be extended to other scanners with DOI capability intended for high resolution PET imaging.

Efficient image reconstruction for a small animal PET system with dual-layer-offset detector design

BACKGROUND

A compact PET/SPECT/CT system Inliview-3000B has been developed to provide multi-modality information on small animals for biomedical research. Its PET subsystem employed a dual-layer-offset detector design for depth-of-interaction capability and higher detection efficiency, but the irregular design caused some difficulties in calculating the normalization factors and the sensitivity map. Besides, the relatively larger (2 mm) crystal cross-section size also posed a challenge to high-resolution image reconstruction.

PURPOSE

We present an efficient image reconstruction method to achieve high imaging performance for the PET subsystem of Inliview-3000B.

METHODS

List mode reconstruction with efficient system modeling was used for the PET imaging. We adopt an on-the-fly multi-ray tracing method with random crystal sampling to model the solid angle, crystal penetration and object attenuation effect, and modify the system response model during each iteration to improve the reconstruction performance and computational efficiency. We estimate crystal efficiency with a novel iterative approach that combines measured cylinder phantom data with simulated line-of-response (LOR)-based factors for normalization correction before reconstruction. Since it is necessary to calculate normalization factors and the sensitivity map, we stack the two crystal layers together and extend the conventional data organization method here to index all useful LORs. Simulations and experiments were performed to demonstrate the feasibility and advantage of the proposed method.

RESULTS

Simulation results showed that the iterative algorithm for crystal efficiency estimation could achieve good accuracy. NEMA image quality phantom studies have demonstrated the superiority of random sampling, which is able to achieve good imaging performance with much less computation than traditional uniform sampling. In the spatial resolution evaluation based on the mini-Derenzo phantom, 1.1 mm hot rods could be identified with the proposed reconstruction method. Reconstruction of double mice and a rat showed good spatial resolution and a high signal-to-noise ratio, and organs with higher uptake could be recognized well.

CONCLUSION

The results validated the superiority of introducing randomness into reconstruction, and demonstrated its reliability for high-performance imaging. The Inliview-3000B PET subsystem with the proposed image reconstruction can provide rich and detailed information on small animals for preclinical research.

Evaluation of applying space-variant resolution modeling to attenuation correction in PET

Attenuation correction aims to recover the underestimated tracer uptake and improve the image contrast recovery in positron emission tomography (PET). However, traditional ray-tracing-based projection of attenuation maps is inaccurate as some physical effects are not considered, such as finite crystal size, inter-crystal penetration and inter-crystal scatter. In this study, we evaluated the effects of applying resolution modeling (RM) to attenuation correction by implementing space-variant RM to complement physical effects which are usually omitted in the traditional projection model. We verified this method on a brain PET scanner developed by our group, in both Monte Carlo simulation and real-world data, in comparison with space-invariant Gaussian RM, average-depth-of-interaction, and multi-ray tracing methods. The results indicate that the space-variant RM is superior in terms of artifacts reduction and contrast recovery.

Direct Reconstruction of Linear Parametric Images From Dynamic PET Using Nonlocal Deep Image Prior

Direct reconstruction methods have been developed to estimate parametric images directly from the measured PET sinograms by combining the PET imaging model and tracer kinetics in an integrated framework. Due to limited counts received, signal-to-noise-ratio (SNR) and resolution of parametric images produced by direct reconstruction frameworks are still limited. Recently supervised deep learning methods have been successfully applied to medical imaging denoising/reconstruction when large number of high-quality training labels are available. For static PET imaging, high-quality training labels can be acquired by extending the scanning time. However, this is not feasible for dynamic PET imaging, where the scanning time is already long enough. In this work, we proposed an unsupervised deep learning framework for direct parametric reconstruction from dynamic PET, which was tested on the Patlak model and the relative equilibrium Logan model. The training objective function was based on the PET statistical model. The patient's anatomical prior image, which is readily available from PET/CT or PET/MR scans, was supplied as the network input to provide a manifold constraint, and also utilized to construct a kernel layer to perform non-local feature denoising. The linear kinetic model was embedded in the network structure as a 1 ×1 ×1 convolution layer. Evaluations based on dynamic datasets of ¹⁸F-FDG and ¹¹C-PiB tracers show that the proposed framework can outperform the traditional and the kernel method-based direct reconstruction methods.

Simulation Study of a Frame-Based Motion Correction Algorithm for Positron Emission Imaging

Positron emission tomography (PET) is a functional non-invasive imaging modality that uses radioactive substances (radiotracers) to measure changes in metabolic processes. Advances in scanner technology and data acquisition in the last decade have led to the development of more sophisticated PET devices with good spatial resolution (1-3 mm of full width at half maximum (FWHM)). However, there are involuntary motions produced by the patient inside the scanner that lead to image degradation and potentially to a misdiagnosis. The adverse effect of the motion in the reconstructed image increases as the spatial resolution of the current scanners continues improving. In order to correct this effect, motion correction techniques are becoming increasingly popular and further studied. This work presents a simulation study of an image motion correction using a frame-based algorithm. The method is able to cut the acquired data from the scanner in frames, taking into account the size of the object of study. This approach allows working with low statistical information without losing image quality. The frames are later registered using spatio-temporal registration developed in a multi-level way. To validate these results, several performance tests are applied to a set of simulated moving phantoms. The results obtained show that the method minimizes the intra-frame motion, improves the signal intensity over the background in comparison with other literature methods, produces excellent values of similarity with the ground-truth (static) image and is able to find a limit in the patient-injected dose when some prior knowledge of the lesion is present.

3D/4D Reconstruction and Quantitative Total Body Imaging

Total-body PET image reconstruction follows a similar procedure to the image reconstruction process for standard whole-body PET scanners. One unique aspect of total-body imaging is simultaneous coverage of the entire human body, which makes it convenient to perform total-body dynamic PET scans. Therefore, four-dimensional dynamic PET reconstruction and parametric imaging are of great interest in total-body imaging. This article covers some basics of PET image reconstruction and then focuses on three- and four-dimensional PET reconstruction for total-body imaging. Methods for image formation from raw measurements in total-body PET are described. Challenges and opportunities in total-body PET image reconstruction are discussed.

Deep-Learning Based Positron Range Correction of PET Images

Positron emission tomography (PET) is a molecular imaging technique that provides a 3D image of functional processes in the body in vivo. Some of the radionuclides proposed for PET imaging emit high-energy positrons, which travel some distance before they annihilate (positron range), creating significant blurring in the reconstructed images. Their large positron range compromises the achievable spatial resolution of the system, which is more significant when using high-resolution scanners designed for the imaging of small animals. In this work, we trained a deep neural network named Deep-PRC to correct PET images for positron range effects. Deep-PRC was trained with modeled cases using a realistic Monte Carlo simulation tool that considers the positron energy distribution and the materials and tissues it propagates into. Quantification of the reconstructed PET images corrected with Deep-PRC showed that it was able to restore the images by up to 95% without any significant noise increase. The proposed method, which is accessible via Github, can provide an accurate positron range correction in a few seconds for a typical PET acquisition.

Deep-Learning Based Positron Range Correction of PET Images

Positron emission tomography (PET) is a molecular imaging technique that provides a 3D image of functional processes in the body in vivo. Some of the radionuclides proposed for PET imaging emit high-energy positrons, which travel some distance before they annihilate (positron range), creating significant blurring in the reconstructed images. Their large positron range compromises the achievable spatial resolution of the system, which is more significant when using high-resolution scanners designed for the imaging of small animals. In this work, we trained a deep neural network named Deep-PRC to correct PET images for positron range effects. Deep-PRC was trained with modeled cases using a realistic Monte Carlo simulation tool that considers the positron energy distribution and the materials and tissues it propagates into. Quantification of the reconstructed PET images corrected with Deep-PRC showed that it was able to restore the images by up to 95% without any significant noise increase. The proposed method, which is accessible via Github, can provide an accurate positron range correction in a few seconds for a typical PET acquisition.

Generative adversarial network based regularized image reconstruction for PET

Positron emission tomography (PET) is an ill-posed inverse problem and suffers high noise due to limited number of detected events. Prior information can be used to improve the quality of reconstructed PET images. Deep neural networks have also been applied to regularized image reconstruction. One method is to use a pretrained denoising neural network to represent the PET image and to perform a constrained maximum likelihood estimation. In this work, we propose to use a generative adversarial network (GAN) to further improve the network performance. We also modify the objective function to include a data-matching term on the network input. Experimental studies using computer-based Monte Carlo simulations and real patient datasets demonstrate that the proposed method leads to noticeable improvements over the kernel-based and U-net-based regularization methods in terms of lesion contrast recovery versus background noise trade-offs.

Real-Time 3D PET Image with Pseudoinverse Reconstruction

Real-time positron emission tomography (PET) may provide information from first-shot images, enable PET-guided biopsies, and allow awake animal studies. Fully-3D iterative reconstructions yield the best images in PET, but they are too slow for real-time imaging. Analytical methods such as Fourier back projection (FBP) are very fast, but yield images of poor quality with artifacts due to noise or data incompleteness. In this work, an image reconstruction based on the pseudoinverse of the system response matrix (SRM) is presented. w. To implement the pseudoinverse method, the reconstruction problem is separated into two stages. First, the axial part of the SRM is pseudo-inverted (PINV) to rebin the 3D data into 2D datasets. Then, the resulting 2D slices can be reconstructed with analytical methods or by applying the pseudoinverse algorithm again. The proposed two-step PINV reconstruction yielded good-quality images at a rate of several frames per second, compatible with real time applications. Furthermore, extremely fast direct PINV reconstruction of projections of the 3D image collapsed along specific directions can be implemented.

Real-Time 3D PET Image with Pseudoinverse Reconstruction

Real-time positron emission tomography (PET) may provide information from first-shot images, enable PET-guided biopsies, and allow awake animal studies. Fully-3D iterative reconstructions yield the best images in PET, but they are too slow for real-time imaging. Analytical methods such as Fourier back projection (FBP) are very fast, but yield images of poor quality with artifacts due to noise or data incompleteness. In this work, an image reconstruction based on the pseudoinverse of the system response matrix (SRM) is presented. w. To implement the pseudoinverse method, the reconstruction problem is separated into two stages. First, the axial part of the SRM is pseudo-inverted (PINV) to rebin the 3D data into 2D datasets. Then, the resulting 2D slices can be reconstructed with analytical methods or by applying the pseudoinverse algorithm again. The proposed two-step PINV reconstruction yielded good-quality images at a rate of several frames per second, compatible with real time applications. Furthermore, extremely fast direct PINV reconstruction of projections of the 3D image collapsed along specific directions can be implemented.

Evaluation of quantitative, efficient image reconstruction for VersaPET, a compact PET system

PURPOSE

Previously we developed a high-resolution positron emission tomography (PET) system-VersaPET-characterized by a block geometry with relatively large axial and transaxial interblock gaps and a compact geometry susceptible to parallax blurring effects. In this work, we report the qualitative and quantitative evaluation of a graphic processing unit (GPU)-accelerated maximum-likelihood by expectation-maximization (MLEM) image reconstruction framework for VersaPET which features accurate system geometry and projection space point-spread-function (PSF) modeling.

METHODS

We combined the ray-tracing module from software for tomographic image reconstruction (STIR), an open-source PET image reconstruction package, with VersaPET's exact block geometry for the geometric system matrix. Point-spread-function modeling of crystal penetration and scattering was achieved by a custom Monte-Carlo simulation for projection space blurring in all dimensions. We also parallelized the reconstruction in GPU taking advantage of the system's symmetry for PSF computation. To investigate the effects of PSF width, we generated and studied multiple kernels between one that reflects the true LYSO density in the MC simulation and another that reflects geometry only (no PSF). GATE simulations of hot and cold-sphere phantoms with spheres of different sizes, real microDerenzo phantom, and human blood vessel data were used to characterize the quantitative and qualitative performances of the reconstruction.

RESULTS

Reconstruction with an accurate system geometry effectively improved image quality compared to STIR (version 3.0) which assumes an idealized system geometry. Reconstructions of GATE-simulated hot-sphere phantom data showed that all PSF kernels achieved superior performance in contrast recovery and bias reduction compared to using no PSF, but may introduce edge artifact and lumped background noise pattern depending on the width of PSF kernels. Cold-sphere phantom simulation results also indicated improvement in contrast recovery and quantification with PSF modeling (compared to no PSF) for 5 and 10 mm cold spheres. Real microDerenzo phantom images with the PSF kernel that reflects the true LYSO density showed degraded resolving power of small sectors that could be resolved more clearly by underestimated PSF kernels, which is consistent with recent literature despite differences in scanner geometries and in approaches to system model estimation. The human vessel results resemble those of the hot-sphere phantom simulation with the PSF kernel that reflects the true LYSO density achieving the highest peak in the time activity curve (TAC) and similar lumped noise pattern.

CONCLUSIONS

We fully evaluated a practical MLEM reconstruction framework that we developed for VersaPET in terms of qualitative and quantitative performance. Different PSF kernels may be adopted for improving the results of specific imaging tasks but the underlying reasons for the variation in optimal kernel for the real and simulation studies requires further study.

PET Image Reconstruction Using Deep Image Prior

Recently, deep neural networks have been widely and successfully applied in computer vision tasks and have attracted growing interest in medical imaging. One barrier for the application of deep neural networks to medical imaging is the need for large amounts of prior training pairs, which is not always feasible in clinical practice. This is especially true for medical image reconstruction problems, where raw data are needed. Inspired by the deep image prior framework, in this paper, we proposed a personalized network training method where no prior training pairs are needed, but only the patient's own prior information. The network is updated during the iterative reconstruction process using the patient-specific prior information and measured data. We formulated the maximum-likelihood estimation as a constrained optimization problem and solved it using the alternating direction method of multipliers algorithm. Magnetic resonance imaging guided positron emission tomography reconstruction was employed as an example to demonstrate the effectiveness of the proposed framework. Quantification results based on simulation and real data show that the proposed reconstruction framework can outperform Gaussian post-smoothing and anatomically guided reconstructions using the kernel method or the neural-network penalty.

4D digital tomosynthesis image reconstruction using brute force-based adaptive total variation (BF-ATV) in a prototype LINAC system

Respiratory-correlated cone-beam CT (CBCT) not only inhibits rapid scanning due to the slow speed of the LINAC head gantry rotation, but its implementation for routine patient imaging is impractical because of the high radiation dose delivered during the process. Digital tomosynthesis (DTS) is a potentially faster technique that delivers a much lower radiation dose by reducing the number of projections in a limited angular range. Unfortunately, 4D-DTS introduces strong aliasing artifacts in the reconstructed images due to the sparsely sampled projections in each respiratory phase bin. The authors hereby suggest a novel low-dose 4D-DTS image reconstruction method that achieves a compromise between the occurrence of aliasing artifacts and image smoothing using a brute force-based adaptive weighting parameter searching technique. We used a prototype LINAC system mounted with a flat-panel detector to acquire tomosynthesis projections of respiratory motion in a phantom in the anterior-posterior (AP) and lateral views. Three different 4D-DTS image reconstruction schemes that included conventional filtered back-projection (FBP), adaptive steepest descent projection onto convex sets (ASD-POCS), and the proposed brute force-based adaptive total variation (BF-ATV) were implemented in four different respiratory phase bins for both AP and lateral views. All reconstructions were accelerated using a single GPU card to reduce the computation time. To study the performance of the algorithm under various sparse conditions, we operated the prototype system in three different gantry sweep modes. The results indicate that the proposed BF-ATV method yields the largest structural similarities in the differenced image between the ground-truth dataset acquired using the slow gantry sweep mode and the sparse dataset from both moderate and fast sweep modes. In addition, the proposed method maintained the object sharpness with less streaking lines and small loss of sharpness compared to the conventional FBP and ASD-POCS methods. In conclusion, the proposed low-dose 4D-DTS reconstruction scheme may provide better performance due in part to its rapid scanning. Therefore, it is potentially applicable to practical 4D imaging for radiotherapy.

Iterative PET Image Reconstruction Using Convolutional Neural Network Representation

PET image reconstruction is challenging due to the ill-poseness of the inverse problem and limited number of detected photons. Recently, the deep neural networks have been widely and successfully used in computer vision tasks and attracted growing interests in medical imaging. In this paper, we trained a deep residual convolutional neural network to improve PET image quality by using the existing inter-patient information. An innovative feature of the proposed method is that we embed the neural network in the iterative reconstruction framework for image representation, rather than using it as a post-processing tool. We formulate the objective function as a constrained optimization problem and solve it using the alternating direction method of multipliers algorithm. Both simulation data and hybrid real data are used to evaluate the proposed method. Quantification results show that our proposed iterative neural network method can outperform the neural network denoising and conventional penalized maximum likelihood methods.

PET Image Denoising Using a Deep Neural Network Through Fine Tuning

Positron emission tomography (PET) is a functional imaging modality widely used in clinical diagnosis. In this work, we trained a deep convolutional neural network (CNN) to improve PET image quality. Perceptual loss based on features derived from a pre-trained VGG network, instead of the conventional mean squared error, was employed as the training loss function to preserve image details. As the number of real patient data set for training is limited, we propose to pre-train the network using simulation data and fine-tune the last few layers of the network using real data sets. Results from simulation, real brain and lung data sets show that the proposed method is more effective in removing noise than the traditional Gaussian filtering method.

Evaluation of Single-Chip, Real-Time Tomographic Data Processing on FPGA SoC Devices

A novel approach to tomographic data processing has been developed and evaluated using the Jagiellonian positron emission tomography scanner as an example. We propose a system in which there is no need for powerful, local to the scanner processing facility, capable to reconstruct images on the fly. Instead, we introduce a field programmable gate array system-on-chip platform connected directly to data streams coming from the scanner, which can perform event building, filtering, coincidence search, and region-of-response reconstruction by the programmable logic and visualization by the integrated processors. The platform significantly reduces data volume converting raw data to a list-mode representation, while generating visualization on the fly.

Enhancing the Image Quality via Transferred Deep Residual Learning of Coarse PET Sinograms

Increasing the image quality of positron emission tomography (PET) is an essential topic in the PET community. For instance, thin-pixelated crystals have been used to provide high spatial resolution images but at the cost of sensitivity and manufacture expense. In this paper, we proposed an approach to enhance the PET image resolution and noise property for PET scanners with large pixelated crystals. To address the problem of coarse blurred sinograms with large parallax errors associated with large crystals, we developed a data-driven, single-image super-resolution (SISR) method for sinograms, based on the novel deep residual convolutional neural network (CNN). Unlike the CNN-based SISR on natural images, periodically padded sinogram data and dedicated network architecture were used to make it more efficient for PET imaging. Moreover, we included the transfer learning scheme in the approach to process cases with poor labeling and small training data set. The approach was validated via analytically simulated data (with and without noise), Monte Carlo simulated data, and pre-clinical data. Using the proposed method, we could achieve comparable image resolution and better noise property with large crystals of bin sizes of thin crystals with a bin size from to . Our approach uses external PET data as the prior knowledge for training and does not require additional information during inference. Meanwhile, the method can be added into the normal PET imaging framework seamlessly, thus potentially finds its application in designing low-cost high-performance PET systems.

Theoretical study of the benefit of long axial field-of-view PET on region of interest quantification

The aim of this study is to evaluate the benefit of long axial field-of-view (AFOV) PET scanners on region of interest (ROI) quantification. We simulated a series of PET scanners with an AFOV ranging from 22 cm to 220 cm. A theoretical framework was used to predict the contrast recovery coefficient (CRC) and the variance of ROI quantification in penalized maximum likelihood (ML) image reconstruction, in which the resolution and noise tradeoff was controlled by a regularization parameter with a quadratic penalty function. The characterization was based on the converged penalized ML reconstruction with an accurate system model. We examined quantification of a 2 mm ROI and 10 mm ROI in a clinically relevant scan range of 110 cm. Multiple bed positions with 50% overlap were used for scanners with shorter AFOV to provide a relatively uniform sensitivity across the 110 cm axial range. A uniform water cylinder of 20 cm in diameter and 230 cm in length was chosen to model the attenuation and background activity. We computed the variance reduction factor at fixed resolution. Effects of different detector capabilities, including TOF (time-of-flight) resolution (320 ps, 500 ps, and non-TOF) and DOI (depth-of-interaction) resolution (4 mm, 10 mm, and no DOI), were evaluated. The results show that at a normal activity level (370 MBq), the 220 cm AFOV scanner offers a  ∼17-fold variance reduction for the 2 mm ROI and  ∼26-fold variance reduction for the 10 mm ROI (both measured at CRC  =  0.5) over the 22 cm AFOV scanner when both using detectors with 500 ps TOF resolution no DOI capability. The variance reduction factors of trues-only are higher than those of including scatters and randoms. Combining 320 ps TOF and 4 mm DOI, the 220 cm long scanner offers a  ∼45-fold variance reduction over the 22 cm long reference scanner (500 ps TOF, no DOI) for imaging 2 mm and 10 mm ROIs. The variance reduction factors are higher at a lower activity level due to lower random fraction. In conclusion, our study demonstrates that a long AFOV scanner can greatly improve the quantitative accuracy of PET imaging compared to current state-of-the-art clinical PET scanners.

Direct Patlak Reconstruction From Dynamic PET Data Using the Kernel Method With MRI Information Based on Structural Similarity

Positron emission tomography (PET) is a functional imaging modality widely used in oncology, cardiology, and neuroscience. It is highly sensitive, but suffers from relatively poor spatial resolution, as compared with anatomical imaging modalities, such as magnetic resonance imaging (MRI). With the recent development of combined PET/MR systems, we can improve the PET image quality by incorporating MR information into image reconstruction. Previously, kernel learning has been successfully embedded into static and dynamic PET image reconstruction using either PET temporal or MRI information. Here, we combine both PET temporal and MRI information adaptively to improve the quality of direct Patlak reconstruction. We examined different approaches to combine the PET and MRI information in kernel learning to address the issue of potential mismatches between MRI and PET signals. Computer simulations and hybrid real-patient data acquired on a simultaneous PET/MR scanner were used to evaluate the proposed methods. Results show that the method that combines PET temporal information and MRI spatial information adaptively based on the structure similarity index has the best performance in terms of noise reduction and resolution improvement.

Lesion detection and quantification performance of the Tachyon-I time-of-flight PET scanner: phantom and human studies

The first generation Tachyon PET (Tachyon-I) is a demonstration single-ring PET scanner that reaches a coincidence timing resolution of 314 ps using LSO scintillator crystals coupled to conventional photomultiplier tubes. The objective of this study was to quantify the improvement in both lesion detection and quantification performance resulting from the improved time-of-flight (TOF) capability of the Tachyon-I scanner. We developed a quantitative TOF image reconstruction method for the Tachyon-I and evaluated its TOF gain for lesion detection and quantification. Scans of either a standard NEMA torso phantom or healthy volunteers were used as the normal background data. Separately scanned point source and sphere data were superimposed onto the phantom or human data after accounting for the object attenuation. We used the bootstrap method to generate multiple independent noisy datasets with and without a lesion present. The signal-to-noise ratio (SNR) of a channelized hotelling observer (CHO) was calculated for each lesion size and location combination to evaluate the lesion detection performance. The bias versus standard deviation trade-off of each lesion uptake was also calculated to evaluate the quantification performance. The resulting CHO-SNR measurements showed improved performance in lesion detection with better timing resolution. The detection performance was also dependent on the lesion size and location, in addition to the background object size and shape. The results of bias versus noise trade-off showed that the noise (standard deviation) reduction ratio was about 1.1-1.3 over the TOF 500 ps and 1.5-1.9 over the non-TOF modes, similar to the SNR gains for lesion detection. In conclusion, this Tachyon-I PET study demonstrated the benefit of improved time-of-flight capability on lesion detection and ROI quantification for both phantom and human subjects.

Sparse domain approaches in dynamic SPECT imaging with high-performance computing

Iterative reconstruction algorithms often have relatively large computation time affecting their clinical deployment. This is especially true for 4D reconstruction in dynamic imaging (DI). In this work, we have shown how sparse domain approaches and parallelization for static 3D image reconstruction and 4D dynamic image reconstruction (directly from sinogram) in Single Photon Emission Computed Tomography (SPECT), without any intermediate 3D reconstructions, can improve computational efficiency. DI in SPECT is one of the hardest inverse problems in medical image reconstruction area and slow reconstruction is a challenge for this promising protocol. Our work hopefully, paves a new direction toward making DI in SPECT clinically viable. Our 4D reconstruction also is a novel application of non-negative matrix factorization (NNMF) in an inverse problem.

Sinogram Blurring Matrix Estimation From Point Sources Measurements With Rank-One Approximation for Fully 3-D PET

An accurate system matrix is essential in positron emission tomography (PET) for reconstructing high quality images. To reduce storage size and image reconstruction time, we factor the system matrix into a product of a geometry projection matrix and a sinogram blurring matrix. The geometric projection matrix is computed analytically and the sinogram blurring matrix is estimated from point source measurements. Previously, we have estimated a 2-D blurring matrix for a preclinical PET scanner. The 2-D blurring matrix only considers blurring effects within a transaxial sinogram and does not compensate for inter-sinogram blurring effects. For PET scanners with a long axial field of view, inter-sinogram blurring can be a major problem influencing the image quality in the axial direction. Hence, the estimation of a 4-D blurring matrix is desirable to further improve the image quality. The 4-D blurring matrix estimation is an ill-conditioned problem due to the large number of unknowns. Here, we propose a rank-one approximation for each blurring kernel image formed by a row vector of the sinogram blurring matrix to improve the stability of the 4-D blurring matrix estimation. The proposed method is applied to the simulated data as well as the real data obtained from an Inveon microPET scanner. The results show that the newly estimated 4-D blurring matrix can improve the image quality over those obtained with a 2-D blurring matrix and requires less point source scans to achieve similar image quality compared with an unconstrained 4-D blurring matrix estimation.

Quantitative image reconstruction for total-body PET imaging using the 2-meter long EXPLORER scanner

The EXPLORER project aims to build a 2 meter long total-body PET scanner, which will provide extremely high sensitivity for imaging the entire human body. It will possess a range of capabilities currently unavailable to state-of-the-art clinical PET scanners with a limited axial field-of-view. The huge number of lines-of-response (LORs) of the EXPLORER poses a challenge to the data handling and image reconstruction. The objective of this study is to develop a quantitative image reconstruction method for the EXPLORER and compare its performance with current whole-body scanners. Fully 3D image reconstruction was performed using time-of-flight list-mode data with parallel computation. To recover the resolution loss caused by the parallax error between crystal pairs at a large axial ring difference or transaxial radial offset, we applied an image domain resolution model estimated from point source data. To evaluate the image quality, we conducted computer simulations using the SimSET Monte-Carlo toolkit and XCAT 2.0 anthropomorphic phantom to mimic a 20 min whole-body PET scan with an injection of 25 MBq ¹⁸F-FDG. We compare the performance of the EXPLORER with a current clinical scanner that has an axial FOV of 22 cm. The comparison results demonstrated superior image quality from the EXPLORER with a 6.9-fold reduction in noise standard deviation comparing with multi-bed imaging using the clinical scanner.

Fast and Parallel Summed Area Table for Fabric Defect Detection

Automating fabric defect detection has a significant role in fabric industries. However, the existing fabric defect detection algorithms lack the real-time performance that is required in real applications due to their high demanding computation. To ensure real time, high accuracy and reliable fabric defect detection this paper developed a fast and parallel normalized cross-correlation algorithm based on summed-area table technique called PFDD-SAT. To meet real-time requirements, extensive use of the NVIDIA CUDA framework for Graphical Processing Unit (GPU) computing is made. The detailed implementation steps of the PFDD-SAT are illustrated in this paper. Several experiments have been carried out to evaluate the detection time and accuracy and then the robustness to illumination and Gaussian noises. The results show that the PFDD-SAT has robustness to noise and speeds the defect detection process more than 200 times than normal required time and that greatly met the needs for real-time automatic fabric defect detection.

A generalized reconstruction framework for unconventional PET systems

PURPOSE

Quantitative estimation of the radionuclide activity concentration in positron emission tomography (PET) requires precise modeling of PET physics. The authors are focused on designing unconventional PET geometries for specific applications. This work reports the creation of a generalized reconstruction framework, capable of reconstructing tomographic PET data for systems that use right cuboidal detector elements positioned at arbitrary geometry using a regular Cartesian grid of image voxels.

METHODS

The authors report on a variety of design choices and optimization for the creation of the generalized framework. The image reconstruction algorithm is maximum likelihood-expectation-maximization. System geometry can be specified using a simple script. Given the geometry, a symmetry seeking algorithm finds existing symmetry in the geometry with respect to the image grid to improve the memory usage/speed. Normalization is approached from a geometry independent perspective. The system matrix is computed using the Siddon's algorithm and subcrystal approach. The program is parallelized through open multiprocessing and message passing interface libraries. A wide variety of systems can be modeled using the framework. This is made possible by modeling the underlying physics and data correction, while generalizing the geometry dependent features.

RESULTS

Application of the framework for three novel PET systems, each designed for a specific application, is presented to demonstrate the robustness of the framework in modeling PET systems of unconventional geometry. Three PET systems of unconventional geometry are studied. (1) Virtual-pinhole half-ring insert integrated into Biograph-40: although the insert device improves image quality over conventional whole-body scanner, the image quality varies depending on the position of the insert and the object. (2) Virtual-pinhole flat-panel insert integrated into Biograph-40: preliminary results from an investigation into a modular flat-panel insert are presented. (3) Plant PET system: a reconfigurable PET system for imaging plants, with resolution of greater than 3.3 mm, is shown. Using the automated symmetry seeking algorithm, the authors achieved a compression ratio of the storage and memory requirement by a factor of approximately 50 for the half-ring and flat-panel systems. For plant PET system, the compression ratio is approximately five. The ratio depends on the level of symmetry that exists in different geometries.

CONCLUSIONS

This work brings the field closer to arbitrary geometry reconstruction. A generalized reconstruction framework can be used to validate multiple hypotheses and the effort required to investigate each system is reduced. Memory usage/speed can be improved with certain optimizations.

Designing a compact high performance brain PET scanner-simulation study

The desire to understand normal and disordered human brain function of upright, moving persons in natural environments motivates the development of the ambulatory micro-dose brain PET imager (AMPET). An ideal system would be light weight but with high sensitivity and spatial resolution, although these requirements are often in conflict with each other. One potential approach to meet the design goals is a compact brain-only imaging device with a head-sized aperture. However, a compact geometry increases parallax error in peripheral lines of response, which increases bias and variance in region of interest (ROI) quantification. Therefore, we performed simulation studies to search for the optimal system configuration and to evaluate the potential improvement in quantification performance over existing scanners. We used the Cramér-Rao variance bound to compare the performance for ROI quantification using different scanner geometries. The results show that while a smaller ring diameter can increase photon detection sensitivity and hence reduce the variance at the center of the field of view, it can also result in higher variance in peripheral regions when the length of detector crystal is 15 mm or more. This variance can be substantially reduced by adding depth-of-interaction (DOI) measurement capability to the detector modules. Our simulation study also shows that the relative performance depends on the size of the ROI, and a large ROI favors a compact geometry even without DOI information. Based on these results, we propose a compact 'helmet' design using detectors with DOI capability. Monte Carlo simulations show the helmet design can achieve four-fold higher sensitivity and resolve smaller features than existing cylindrical brain PET scanners. The simulations also suggest that improving TOF timing resolution from 400 ps to 200 ps also results in noticeable improvement in image quality, indicating better timing resolution is desirable for brain imaging.

Efficient system modeling for a small animal PET scanner with tapered DOI detectors

A prototype small animal positron emission tomography (PET) scanner for mouse brain imaging has been developed at UC Davis. The new scanner uses tapered detector arrays with depth of interaction (DOI) measurement. In this paper, we present an efficient system model for the tapered PET scanner using matrix factorization and a virtual scanner geometry. The factored system matrix mainly consists of two components: a sinogram blurring matrix and a geometrical matrix. The geometric matrix is based on a virtual scanner geometry. The sinogram blurring matrix is estimated by matrix factorization. We investigate the performance of different virtual scanner geometries. Both simulation study and real data experiments are performed in the fully 3D mode to study the image quality under different system models. The results indicate that the proposed matrix factorization can maintain image quality while substantially reduce the image reconstruction time and system matrix storage cost. The proposed method can be also applied to other PET scanners with DOI measurement.

Reconstruction for 3D PET Based on Total Variation Constrained Direct Fourier Method

This paper presents a total variation (TV) regularized reconstruction algorithm for 3D positron emission tomography (PET). The proposed method first employs the Fourier rebinning algorithm (FORE), rebinning the 3D data into a stack of ordinary 2D data sets as sinogram data. Then, the resulted 2D sinogram are ready to be reconstructed by conventional 2D reconstruction algorithms. Given the locally piece-wise constant nature of PET images, we introduce the total variation (TV) based reconstruction schemes. More specifically, we formulate the 2D PET reconstruction problem as an optimization problem, whose objective function consists of TV norm of the reconstructed image and the data fidelity term measuring the consistency between the reconstructed image and sinogram. To solve the resulting minimization problem, we apply an efficient methods called the Bregman operator splitting algorithm with variable step size (BOSVS). Experiments based on Monte Carlo simulated data and real data are conducted as validations. The experiment results show that the proposed method produces higher accuracy than conventional direct Fourier (DF) (bias in BOSVS is 70% of ones in DF, variance of BOSVS is 80% of ones in DF).

LOR-interleaving image reconstruction for PET imaging with fractional-crystal collimation

Positron emission tomography (PET) has become an important modality in medical and molecular imaging. However, in most PET applications, the resolution is still mainly limited by the physical crystal sizes or the detector's intrinsic spatial resolution. To achieve images with better spatial resolution in a central region of interest (ROI), we have previously proposed using collimation in PET scanners. The collimator is designed to partially mask detector crystals to detect lines of response (LORs) within fractional crystals. A sequence of collimator-encoded LORs is measured with different collimation configurations. This novel collimated scanner geometry makes the reconstruction problem challenging, as both detector and collimator effects need to be modeled to reconstruct high-resolution images from collimated LORs. In this paper, we present a LOR-interleaving (LORI) algorithm, which incorporates these effects and has the advantage of reusing existing reconstruction software, to reconstruct high-resolution images for PET with fractional-crystal collimation. We also develop a 3D ray-tracing model incorporating both the collimator and crystal penetration for simulations and reconstructions of the collimated PET. By registering the collimator-encoded LORs with the collimator configurations, high-resolution LORs are restored based on the modeled transfer matrices using the non-negative least-squares method and EM algorithm. The resolution-enhanced images are then reconstructed from the high-resolution LORs using the MLEM or OSEM algorithm. For validation, we applied the LORI method to a small-animal PET scanner, A-PET, with a specially designed collimator. We demonstrate through simulated reconstructions with a hot-rod phantom and MOBY phantom that the LORI reconstructions can substantially improve spatial resolution and quantification compared to the uncollimated reconstructions. The LORI algorithm is crucial to improve overall image quality of collimated PET, which can have significant implications in preclinical and clinical ROI imaging applications.

Isotope specific resolution recovery image reconstruction in high resolution PET imaging

PURPOSE

Measuring and incorporating a scanner-specific point spread function (PSF) within image reconstruction has been shown to improve spatial resolution in PET. However, due to the short half-life of clinically used isotopes, other long-lived isotopes not used in clinical practice are used to perform the PSF measurements. As such, non-optimal PSF models that do not correspond to those needed for the data to be reconstructed are used within resolution modeling (RM) image reconstruction, usually underestimating the true PSF owing to the difference in positron range. In high resolution brain and preclinical imaging, this effect is of particular importance since the PSFs become more positron range limited and isotope-specific PSFs can help maximize the performance benefit from using resolution recovery image reconstruction algorithms.

METHODS

In this work, the authors used a printing technique to simultaneously measure multiple point sources on the High Resolution Research Tomograph (HRRT), and the authors demonstrated the feasibility of deriving isotope-dependent system matrices from fluorine-18 and carbon-11 point sources. Furthermore, the authors evaluated the impact of incorporating them within RM image reconstruction, using carbon-11 phantom and clinical datasets on the HRRT.

RESULTS

The results obtained using these two isotopes illustrate that even small differences in positron range can result in different PSF maps, leading to further improvements in contrast recovery when used in image reconstruction. The difference is more pronounced in the centre of the field-of-view where the full width at half maximum (FWHM) from the positron range has a larger contribution to the overall FWHM compared to the edge where the parallax error dominates the overall FWHM.

CONCLUSIONS

Based on the proposed methodology, measured isotope-specific and spatially variant PSFs can be reliably derived and used for improved spatial resolution and variance performance in resolution recovery image reconstruction. The benefits are expected to be more substantial for more energetic positron emitting isotopes such as Oxygen-15 and Rubidium-82.

MAP reconstruction for Fourier rebinned TOF-PET data

Time-of-flight (TOF) information improves the signal-to-noise ratio in positron emission tomography (PET). The computation cost in processing TOF-PET sinograms is substantially higher than for nonTOF data because the data in each line of response is divided among multiple TOF bins. This additional cost has motivated research into methods for rebinning TOF data into lower dimensional representations that exploit redundancies inherent in TOF data. We have previously developed approximate Fourier methods that rebin TOF data into either three-dimensional (3D) nonTOF or 2D nonTOF formats. We refer to these methods respectively as FORET-3D and FORET-2D. Here we describe maximum a posteriori (MAP) estimators for use with FORET rebinned data. We first derive approximate expressions for the variance of the rebinned data. We then use these results to rescale the data so that the variance and mean are approximately equal allowing us to use the Poisson likelihood model for MAP reconstruction. MAP reconstruction from these rebinned data uses a system matrix in which the detector response model accounts for the effects of rebinning. Using these methods we compare the performance of FORET-2D and 3D with TOF and nonTOF reconstructions using phantom and clinical data. Our phantom results show a small loss in contrast recovery at matched noise levels using FORET compared to reconstruction from the original TOF data. Clinical examples show FORET images that are qualitatively similar to those obtained from the original TOF-PET data but with a small increase in variance at matched resolution. Reconstruction time is reduced by a factor of 5 and 30 using FORET3D+MAP and FORET2D+MAP respectively compared to 3D TOF MAP, which makes these methods attractive for clinical applications.

Efficient fully 3D list-mode TOF PET image reconstruction using a factorized system matrix with an image domain resolution model

A factorized system matrix utilizing an image domain resolution model is attractive in fully 3D time-of-flight PET image reconstruction using list-mode data. In this paper, we study a factored model based on sparse matrix factorization that is comprised primarily of a simplified geometrical projection matrix and an image blurring matrix. Beside the commonly-used Siddon's ray-tracer, we propose another more simplified geometrical projector based on the Bresenham's ray-tracer which further reduces the computational cost. We discuss in general how to obtain an image blurring matrix associated with a geometrical projector, and provide theoretical analysis that can be used to inspect the efficiency in model factorization. In simulation studies, we investigate the performance of the proposed sparse factorization model in terms of spatial resolution, noise properties and computational cost. The quantitative results reveal that the factorization model can be as efficient as a non-factored model, while its computational cost can be much lower. In addition we conduct Monte Carlo simulations to identify the conditions under which the image resolution model can become more efficient in terms of image contrast recovery. We verify our observations using the provided theoretical analysis. The result offers a general guide to achieve the optimal reconstruction performance based on a sparse factorization model with an image domain resolution model.

Development and evaluation of a LOR-based image reconstruction with 3D system response modeling for a PET insert with dual-layer offset crystal design

In this study we present a method of 3D system response calculation for analytical computer simulation and statistical image reconstruction for a magnetic resonance imaging (MRI) compatible positron emission tomography (PET) insert system that uses a dual-layer offset (DLO) crystal design. The general analytical system response functions (SRFs) for detector geometric and inter-crystal penetration of coincident crystal pairs are derived first. We implemented a 3D ray-tracing algorithm with 4π sampling for calculating the SRFs of coincident pairs of individual DLO crystals. The determination of which detector blocks are intersected by a gamma ray is made by calculating the intersection of the ray with virtual cylinders with radii just inside the inner surface and just outside the outer-edge of each crystal layer of the detector ring. For efficient ray-tracing computation, the detector block and ray to be traced are then rotated so that the crystals are aligned along the X-axis, facilitating calculation of ray/crystal boundary intersection points. This algorithm can be applied to any system geometry using either single-layer (SL) or multi-layer array design with or without offset crystals. For effective data organization, a direct lines of response (LOR)-based indexed histogram-mode method is also presented in this work. SRF calculation is performed on-the-fly in both forward and back projection procedures during each iteration of image reconstruction, with acceleration through use of eight-fold geometric symmetry and multi-threaded parallel computation. To validate the proposed methods, we performed a series of analytical and Monte Carlo computer simulations for different system geometry and detector designs. The full-width-at-half-maximum of the numerical SRFs in both radial and tangential directions are calculated and compared for various system designs. By inspecting the sinograms obtained for different detector geometries, it can be seen that the DLO crystal design can provide better sampling density than SL or dual-layer no-offset system designs with the same total crystal length. The results of the image reconstruction with SRFs modeling for phantom studies exhibit promising image recovery capability for crystal widths of 1.27-1.43 mm and top/bottom layer lengths of 4/6 mm. In conclusion, we have developed efficient algorithms for system response modeling of our proposed PET insert with DLO crystal arrays. This provides an effective method for both 3D computer simulation and quantitative image reconstruction, and will aid in the optimization of our PET insert system with various crystal designs.

Accurate and efficient modeling of the detector response in small animal multi-head PET systems

In fully three-dimensional PET imaging, iterative image reconstruction techniques usually outperform analytical algorithms in terms of image quality provided that an appropriate system model is used. In this study we concentrate on the calculation of an accurate system model for the YAP-(S)PET II small animal scanner, with the aim to obtain fully resolution- and contrast-recovered images at low levels of image roughness. For this purpose we calculate the system model by decomposing it into a product of five matrices: (1) a detector response component obtained via Monte Carlo simulations, (2) a geometric component which describes the scanner geometry and which is calculated via a multi-ray method, (3) a detector normalization component derived from the acquisition of a planar source, (4) a photon attenuation component calculated from x-ray computed tomography data, and finally, (5) a positron range component is formally included. This system model factorization allows the optimization of each component in terms of computation time, storage requirements and accuracy. The main contribution of this work is a new, efficient way to calculate the detector response component for rotating, planar detectors, that consists of a GEANT4 based simulation of a subset of lines of flight (LOFs) for a single detector head whereas the missing LOFs are obtained by using intrinsic detector symmetries. Additionally, we introduce and analyze a probability threshold for matrix elements of the detector component to optimize the trade-off between the matrix size in terms of non-zero elements and the resulting quality of the reconstructed images. In order to evaluate our proposed system model we reconstructed various images of objects, acquired according to the NEMA NU 4-2008 standard, and we compared them to the images reconstructed with two other system models: a model that does not include any detector response component and a model that approximates analytically the depth of interaction as detector response component. The comparisons confirm previous research results, showing that the usage of an accurate system model with a realistic detector response leads to reconstructed images with better resolution and contrast recovery at low levels of image roughness.

Improving PET imaging for breast cancer using virtual pinhole PET half-ring insert

A PET insert with detector having smaller crystals and placed near a region of interest in a conventional PET scanner can improve image resolution locally due to the virtual-pinhole PET (VP-PET) effect. This improvement is from the higher spatial sampling of the imaging area near the detector. We have built a prototype half-ring PET insert for head-and-neck cancer imaging applications. In this paper, we extend the use of the insert to breast imaging and show that such a system provides high resolution images of breast and axillary lymph nodes while maintaining the full imaging field of view capability of a clinical PET scanner. We characterize the resolution and contrast recovery for tumors across the imaging field of view. First, we model the system using Monte Carlo methods to determine its theoretical limit of improvement. Simulations were conducted with hot spherical tumors embedded in background activity at tumor-to-background contrast ranging from 3:1 to 12:1. Tumors are arranged in a Derenzo-like pattern with their diameters ranging from 2 to 12 mm. Experimental studies were performed using a chest phantom with cylindrical breast attachment. Tumors of different sizes arranged in a Derenzo-like pattern with tumor-to-background ratio of 6:1 are inserted into the breast phantom. Imaging capability of mediastinum and axillary lymph nodes is explored. Both Monte Carlo simulations and experiment show clear improvement in image resolution and contrast recovery with VP-PET half-ring insert. The degree of improvement in resolution and contrast recovery depends on location of the tumor. The full field of view imaging capability is shown to be maintained. Minor artifacts are introduced in certain regions.

Modelling the physics in the iterative reconstruction for transmission computed tomography

There is an increasing interest in iterative reconstruction (IR) as a key tool to improve quality and increase applicability of x-ray CT imaging. IR has the ability to significantly reduce patient dose; it provides the flexibility to reconstruct images from arbitrary x-ray system geometries and allows one to include detailed models of photon transport and detection physics to accurately correct for a wide variety of image degrading effects. This paper reviews discretization issues and modelling of finite spatial resolution, Compton scatter in the scanned object, data noise and the energy spectrum. The widespread implementation of IR with a highly accurate model-based correction, however, still requires significant effort. In addition, new hardware will provide new opportunities and challenges to improve CT with new modelling.

Modelling the physics in the iterative reconstruction for transmission computed tomography

There is an increasing interest in iterative reconstruction (IR) as a key tool to improve quality and increase applicability of x-ray CT imaging. IR has the ability to significantly reduce patient dose; it provides the flexibility to reconstruct images from arbitrary x-ray system geometries and allows one to include detailed models of photon transport and detection physics to accurately correct for a wide variety of image degrading effects. This paper reviews discretization issues and modelling of finite spatial resolution, Compton scatter in the scanned object, data noise and the energy spectrum. The widespread implementation of IR with a highly accurate model-based correction, however, still requires significant effort. In addition, new hardware will provide new opportunities and challenges to improve CT with new modelling.

Medical image processing on the GPU - past, present and future

Graphics processing units (GPUs) are used today in a wide range of applications, mainly because they can dramatically accelerate parallel computing, are affordable and energy efficient. In the field of medical imaging, GPUs are in some cases crucial for enabling practical use of computationally demanding algorithms. This review presents the past and present work on GPU accelerated medical image processing, and is meant to serve as an overview and introduction to existing GPU implementations. The review covers GPU acceleration of basic image processing operations (filtering, interpolation, histogram estimation and distance transforms), the most commonly used algorithms in medical imaging (image registration, image segmentation and image denoising) and algorithms that are specific to individual modalities (CT, PET, SPECT, MRI, fMRI, DTI, ultrasound, optical imaging and microscopy). The review ends by highlighting some future possibilities and challenges.

Resolution modeling in PET imaging: theory, practice, benefits, and pitfalls

In this paper, the authors review the field of resolution modeling in positron emission tomography (PET) image reconstruction, also referred to as point-spread-function modeling. The review includes theoretical analysis of the resolution modeling framework as well as an overview of various approaches in the literature. It also discusses potential advantages gained via this approach, as discussed with reference to various metrics and tasks, including lesion detection observer studies. Furthermore, attention is paid to issues arising from this approach including the pervasive problem of edge artifacts, as well as explanation and potential remedies for this phenomenon. Furthermore, the authors emphasize limitations encountered in the context of quantitative PET imaging, wherein increased intervoxel correlations due to resolution modeling can lead to significant loss of precision (reproducibility) for small regions of interest, which can be a considerable pitfall depending on the task of interest.