Noise characterization of block-iterative reconstruction algorithms: I. Theory

Researchers have shown increasing interest in block-iterative image reconstruction algorithms due to the computational and modeling advantages they provide. Although their convergence properties have been well documented, little is known about how they behave in the presence of noise. In this work, we fully characterize the ensemble statistical properties of the rescaled block-iterative expectation-maximization (RBI-EM) reconstruction algorithm and the rescaled block-iterative simultaneous multiplicative algebraic reconstruction technique (RBI-SMART). Also included in the analysis are the special cases of RBI-EM, maximum-likelihood EM (ML-EM) and ordered-subset EM (OS-EM), and the special case of RBI-SMART, SMART. A theoretical formulation strategy similar to that previously outlined for ML-EM is followed for the RBI methods. The theoretical formulations in this paper rely on one approximation, namely, that the noise in the reconstructed image is small compared to the mean image. In a second paper, the approximation will be justified through Monte Carlo simulations covering a range of noise levels, iteration points, and subset orderings. The ensemble statistical parameters could then be used to evaluate objective measures of image quality.

Attenuation compensation in 99mTc SPECT brain imaging: a comparison of the use of attenuation maps derived from transmission versus emission data in normal scans

Brain SPECT imaging using 99mTc lipophilic tracers such as hexamethyl propyleneamine oxime (HMPAO) attempts to estimate cerebral, cerebellar and subcortical perfusion by assessing the relative amount of tracer uptake among these regions. Most commonly, comparison is made with cerebellar activity. Because the assessment of relative tracer uptake may be rendered inaccurate by photon attenuation by the nonuniform attenuation properties of the head, brain SPECT reconstructions have been compared using attenuation correction (AC) with various methods for estimating the attenuation map.

METHODS

Patients underwent 99mTc-HMPAO brain SPECT with transmission line source AC hardware. In addition to the emission dataset, emission downscatter and transmission datasets were acquired. Iterative reconstructions using three different attenuation maps were investigated. These included: (a) that obtained from transmission imaging, (b) that obtained from segmentation of a reconstruction from a lower energy Compton scatter window and (c) a slice-independent, uniform, elliptical attenuation map. No AC was also compared.

RESULTS

Count profiles in patients having brain perfusion SPECT scans showed a significant difference in region count estimates in the brain depending on whether AC is used as well as on the attenuation map used. Scatter-based AC is able to provide external contour detection and attenuation compensation based on that contour, whereas transmission-based AC provides external contour detection as well as internal, nonuniform attenuation estimation and AC. If one considers transmission AC to be the clinical "gold standard," non-attenuation-corrected as well as fixed-ellipsoid, uniform attenuation-corrected studies provided unreliable regional estimates of tracer activity.

CONCLUSION

This study shows the significant difference in clinical brain SPECT count profiles depending on how and whether there is compensation for attenuation. Based on prior studies validating the improved quantitative accuracy of SPECT using transmission-based AC, this study suggests that clinical 99mTc brain perfusion SPECT would benefit from and, in situations demanding rigorous quantitative assessment, requires transmission-based AC. Estimating attenuation maps with scatter-based methods was the next most accurate (clinical) method tested and can be used if and when transmission imaging cannot be used.

Comparison of frequency-distance relationship and Gaussian-diffusion-based methods of compensation for distance-dependent spatial resolution in SPECT imaging

The goal of this investigation was to compare resolution recovery versus noise level of two methods for compensation of distance-dependent resolution (DDR) in SPECT imaging. The two methods of compensation were restoration filtering based on the frequency-distance relationship (FDR) prior to iterative reconstruction, and modelling DDR in the projector/backprojector pair employed in iterative reconstruction. FDR restoration filtering was computationally faster than modelling the detector response in iterative reconstruction. Using Gaussian diffusion to model the detector response in iterative reconstruction sped up the process by a factor of 2.5 over frequency domain filtering in the projector/backprojector pair. Gaussian diffusion modelling resulted in a better resolution versus noise tradeoff than either FDR restoration filtering or solely modelling attenuation in the projector/backprojector pair of iterative reconstruction. For the pixel size investigated herein (0.317 cm), accounting for DDR in the projector/backprojector pair by Gaussian diffusion, or by applying a blurring function based on the distance from the face of the collimator at each distance, resulted in very similar resolution recovery and slice noise level.

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

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

Attenuation compensation for cardiac single-photon emission computed tomographic imaging: Part 2. Attenuation compensation algorithms

Attenuation is believed to be one of the major causes of false-positive cardiac single-photon emission computed tomographic perfusion images. This article provides an introduction to the approaches used to correct for nonuniform attenuation once a patient-specific attenuation map is available. Comparison is made of specific attenuation-correction algorithms from each of three major categories of compensation methods that are or will be available commercially. Examples of the use of the algorithms on simulated projections of a mathematic phantom modeling the anatomy of the upper torso are used to illustrate the ability of the methods to compensate for attenuation. The advantages and disadvantages of each approach are summarized, as well as areas that need further investigation.

An analytical approach for compensation of non-uniform attenuation in cardiac SPECT imaging

Photon attenuation can reduce the diagnostic accuracy of cardiac SPECT imaging. Bellini et al have previously derived a mathematically exact method to compensate for attenuation in a uniform attenuator. Since the human thorax contains structures with differing attenuation properties, non-uniform attenuation compensation is required in cardiac SPECT. Given an estimate of the patient attenuation map, we show that the Bellini attenuation compensation method can be used in cardiac SPECT to provide a quantitatively accurate reconstruction of a central region in the image which includes the heart and surrounding soft tissue. Simulations using a mathematical cardiac-torso phantom were conducted to evaluate the Bellini method and to compare its performance to the ML-EM iterative algorithm, and to 180 degrees and 360 degrees filtered backprojection (FBP) with no attenuation compensation. 'Bulls-eye' polar maps and circumferential profiles showed that both the Bellini method and the ML-EM algorithm provided quantitatively accurate reconstructions of the myocardium, with a substantial reduction in attenuation-induced artifacts that were observed in the FBP images. The computational load required to implement the Bellini method is approximately equivalent to that required for one iteration of the ML-EM algorithm, thus it is suitable for routine clinical use.

Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging

A filtering approach is described, which accurately compensates for the 2D distance-dependent detector response, as well as for photon attenuation in a uniform attenuating medium. The filtering method is based on the frequency distance principle (FDP) which states that points in the object at a specific source-to-detector distance provide the most significant contribution to specified frequency regions in the discrete Fourier transform (DFT) of the sinogram. By modeling the detector point spread function as a 2D Gaussian function whose width is dependent on the source-to-detector distance, a spatially variant inverse filter can be computed and applied to the 3D DFT of the set of all sinogram slices. To minimize noise amplification the inverse filter is rolled off at high frequencies by using a previously published Wiener filter strategy. Attenuation compensation is performed with Bellini's method. It was observed that the tomographic point response, after distance-dependent filtering with the FDP, was approximately isotropic and varied substantially less with position than that obtained with other correction methods. Furthermore, it was shown that processing with this filtering technique provides reconstructions with minimal degradation in image fidelity.

The effect of intrinsic attenuation correction methods on the stationarity of the 3-D modulation transfer function of SPECT

The application of stationary restoration techniques to SPECT images assumes that the modulation transfer function (MTF) of the imaging system is shift invariant. It was hypothesized that using intrinsic attenuation correction (i.e., methods which explicitly invert the exponential radon transform) would yield a three-dimensional (3-D) MTF which varies less with position within the transverse slices than the combined conjugate view two-dimensional (2-D) MTF varies with depth. Thus the assumption of shift invariance would become less of an approximation for 3-D post- than for 2-D pre-reconstruction restoration filtering. SPECT acquisitions were obtained from point sources located at various positions in three differently shaped, water-filled phantoms. The data were reconstructed with intrinsic attenuation correction, and 3-D MTFs were calculated. Four different intrinsic attenuation correction methods were compared: (1) exponentially weighted backprojection, (2) a modified exponentially weighted backprojection as described by Tanaka et al. [Phys. Med. Biol. 29, 1489-1500 (1984)], (3) a Fourier domain technique as described by Bellini et al. [IEEE Trans. ASSP 27, 213-218 (1979)], and (4) the circular harmonic transform (CHT) method as described by Hawkins et al. [IEEE Trans. Med. Imag. 7, 135-148 (1988)]. The dependence of the 3-D MTF obtained with these methods, on point source location within an attenuator, and on shape of the attenuator, was studied. These 3-D MTFs were compared to: (1) those MTFs obtained with no attenuation correction, and (2) the depth dependence of the arithmetic mean combined conjugate view 2-D MTFs.(ABSTRACT TRUNCATED AT 250 WORDS)

A dual-photopeak window method for scatter correction

The imaging of scattered photons degrades contrast and is a major source of error in the quantitation of activity. It was hypothesized that, if the photopeak was divided into two nonoverlapping energy windows, a regression relation could be obtained between the ratio of counts within these windows and the scatter fraction for counts within the total region. This idea was tested by acquiring dual photopeak window acquisitions of a 99mTc point source in an elliptical attenuator, and at the same locations in air. From these, a regression between the scatter fraction and window ratio was determined. When this regression was applied to estimate the scatter distribution for acquisitions in both uniform and nonuniform elliptical attenuators, the residual scatter fraction was reduced approximately ten-fold and the estimated scatter line spread functions matched very closely the tails of the total line spread functions. In SPECT acquisitions, dual-photopeak window scatter correction was observed to significantly increase the contrast of "cold" spheres, improve the accuracy of estimating activity at the center of "hot" spheres, and return the three-dimensional modulation transfer function for point sources in an elliptical attenuator to near their in-air shape.

Activity quantitation in SPECT: a study of prereconstruction Metz filtering and use of the scatter degradation factor

A study of activity quantitation with prereconstruction Metz filtering and use of the scatter degradation factor (SDF) to numerically correct for scatter was conducted. The ratio of the count rate per unit activity for source locations within a 30 x 23-cm water-filled tub phantom to the count rate per unit activity for Tc-99m point sources of known activity imaged in air was used to judge the accuracy of activity determination. The investigation was conducted for certain locations within the tub when it was uniformly filled with Tc-99m activity, and for the same locations at the center of 5, 4, 3, and 2-cm diam, hot spheres imaged in a cold background. The source locations were the center, and one-fourth, one-half, and three-fourths the major axis. Various methods of combining the conjugate views for use with prereconstruction attenuation correction (arithmetic and geometric mean), and extent to which the Metz filter followed the inverse filter before rolling off to suppress noise were investigated. Without Metz filtering, attenuation correction was performed using a transmission curve that included buildup. With Metz filtering, the good-geometry attenuation coefficient was used and the combined views were scaled by the SDF calculated for the average body thickness. Depending on the size of the sphere and the extent to which the inverse filter was followed, Metz filtering combined with use of the SDF improved the accuracy of activity quantitation.

Relative importance of the error sources in Wiener restoration of scintigrams

Through simulation studies, the relative importance of three error sources in Wiener filtering as applied to scintigrams is quantified. The importance of these error sources has been quantified using the percentage changed in squared error (compared to that of an image restored using an ideal Wiener filter) which is caused by estimating one of three factors in the Wiener filter. Estimating the noise power spectrum using the total image count produced to appreciable change in the squared error (less than 1%). Estimating the power spectrum of the true image from that of the degraded image produced small to moderate increases in the squared error (4-139%). In scintigraphic imaging, the modular transfer function (MTF) is dependent on source depth; hence, this study underscores the importance of using methods which reduce the depth dependence of the effective MTF prior to applying restoration filters. A novel method of estimating the power spectrum of the true image from that of the degraded images is also described and evaluated. Wiener restoration filters based on this spectral estimation method are found to be competitive with the image-dependent Metz restoration filter.

Investigation of causes of geometric distortion in 180 degrees and 360 degrees angular sampling in SPECT

To investigate geometric distortion when 180 degrees or 360 degrees angular sampling techniques are used in single photon emission computed tomography (SPECT), a study of point sources imaged at different positions in a water filled cylindrical phantom, and reconstructed using filtered back projection, was conducted. A simulation study, based upon a serial model of the system point spread function (PSF), was used to investigate the contributions of attenuation, spatial resolution and scatter on distortion of the reconstructed PSFs. To study the geometric distortion in transverse (x-y plane), coronal (x-z plane), and sagittal (y-z plane) sections, the ratios of the full widths at half maximum (FWHM) and full widths at tenth maximum (FWTM) in the x/y, x/z, and y/z directions were calculated for the real and simulated PSFs. These results showed that, in an attenuating medium, there is more distortion of point sources into ovals for 180 degrees than for 360 degrees sampling. The simulation study indicated that the primary cause of geometrical distortion in SPECT studies, is the inconsistency of projections due to variable attenuation and spatial resolution. The impact of scatter on geometric distortion was small as measured by the ratios of FWHMs and FWTMs for PSFs. Attenuation correction applied to acquired PSFs significantly reduced geometric distortion in both 180 degrees and 360 degrees studies. To investigate distortion in extended objects, an Iowa heart phantom was placed inside an Alderson body phantom and 201Tl heart SPECT studies acquired. The phantom images confirmed the conclusion that in transverse sections of 360 degrees studies with arithmetic averaging of opposite views, geometric distortion is reduced compared to 180 degrees. The coronal and sagittal sections were equally distorted in both, the 180 degrees and 360 degrees studies, and the 180 degrees studies yielded better contrast.

Unreliable visual estimation of the incidence and amount of turbidity, hemolysis, and icterus in serum from hospitalized patients

We examined the frequency of occurrence for turbidity, hemolysis, or icterus in 2599 serum samples submitted for chemistry testing in an acute-care general hospital. Each specimen was compared visually with full-color photographs of adulterated serum, and designated as either "0" (containing no interferent), or trace, 1+, 2+, 3+, 4+, or 5+. Visible interferents (1+ or greater) were thought to be present in 838 (31%) of the specimens (icterus, 525; hemolysis, 244; lipemia, 69). To assess the accuracy of such visual grading, we determined the concentration of triglycerides, hemoglobin, or bilirubin in the specimens considered to be contaminated. There was little agreement between the actual concentration of each interferent and the assigned grade of turbidity, hemolysis, or icterus, confirming the unreliability of human visual estimation of these potentially interfering substances.

Unreliable visual estimation of the incidence and amount of turbidity, hemolysis, and icterus in serum from hospitalized patients

We examined the frequency of occurrence for turbidity, hemolysis, or icterus in 2599 serum samples submitted for chemistry testing in an acute-care general hospital. Each specimen was compared visually with full-color photographs of adulterated serum, and designated as either "0" (containing no interferent), or trace, 1+, 2+, 3+, 4+, or 5+. Visible interferents (1+ or greater) were thought to be present in 838 (31%) of the specimens (icterus, 525; hemolysis, 244; lipemia, 69). To assess the accuracy of such visual grading, we determined the concentration of triglycerides, hemoglobin, or bilirubin in the specimens considered to be contaminated. There was little agreement between the actual concentration of each interferent and the assigned grade of turbidity, hemolysis, or icterus, confirming the unreliability of human visual estimation of these potentially interfering substances.

Characterization of the modulation transfer function of discrete filtered backprojection

A mathematical expression for the modulation transfer function (MTF) of image reconstruction by discrete filtered backprojection (DFBP) is derived. A simulation study is used to investigate the dependence of the MTF of DFBP on: (1) the number of projection views; (2) the type of ramp filter used; (3) the interpolation method used during backprojection; and (4) the position of the object. These results were compared to MTFs calculated from point-source single-photon-emission computed tomographic (SPECT) acquisitions in air. The experimentally obtained MTFs contained much of the same structure as the MTFs of DFBP obtained through simulation. It is shown that the discretization of the filtered backprojection process can cause the tomographic transfer function to be anisotropic and nonstationary. However, through proper selection of the methods used in reconstruction, a nearly isotropic and stationary MTF can be obtained.

An image-dependent Metz filter for nuclear medicine images

To provide optimal image quality, digital filters should account for both the count level and the object imaged. That is, they should be image-dependent. By using the constraint equation of constrained least-squares (CLS) restoration to determine one parameter of the Metz filter, a filter which adapts to the image has been developed. This filter has been named the Constrained Least-Squares Metz filter. The filter makes use of a regression relation to convert the Metz filter parameter determined using the CLS criterion to the value which would minimize the normalized mean square error (NMSE). The regression relation and the parameters which specify the general form of the Metz filter were determined using images of the Alderson liver and spleen phantoms. The designed filter was tested for its ability to adapt to other objects with images from each of three different test objects. When the values of the Metz filter parameters for these images determined by the CLS-Metz filter were compared by a regression analysis to those which minimized the NMSE for each image, a correlation coefficient of 0.98, a slope of 0.95, and a zero intercept of 0.1 were obtained. With clinical images, the CLS-Metz filter has been shown to provide consistently good image quality with images as diverse as heart perfusion images and bone studies.

Interactive visual optimization of SPECT prereconstruction filtering

A number of factors must be considered when forming a digital filter to two-dimensionally filter single photon emission computed tomographic (SPECT) acquisition images. In an effort to provide subjectively optimal filtering, a program has been developed which provides "real-time" visual feedback. This allows a user to select from among a family of Metz filters tailored for the imaging conditions (i.e., formed to deconvolve scatter, septal penetration, and combined collimator and intrinsic spatial resolution losses). Also, a guideline for assisting the user in selecting from among the possible Metz filters has been formulated. This guideline is based upon knowledge of the probability distribution of the noise power spectrum, and consists of choosing the filter which has a value of 1.0 when the one-dimensional compression of the image power spectrum equals the 90% confidence limit for noise fluctuations. The program starts by filtering a planar reference image with the Metz filter computed for the radionuclide, collimator, magnification, and count-level of the image. This filter is displayed beside the image where it is overlayed on a plot of the logarithm of the one-dimensional compression of the image power spectrum. The user is then allowed to vary the filter parameters through movement of a joystick. By doing the filtering using an array processor, a new filtered image is formed and displayed less than a second after movement of the joystick. Visual feedback from the series of filtered images thus produced as well as the plots of the filter overlayed on the estimated blurred object power spectrum are used to obtain a visually "optimal" filter. The filter can be adapted to the visual preferences of the individual reader, and serves as a useful teaching tool on the effects of filtering.