Edward J. Soares

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.

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, the authors 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.

Application of the Karhunen-Loeve transform to 4D reconstruction of cardiac gated SPECT images

Reconstruction of gated SPECT images is intrinsically a four-dimensional problem. Gated SPECT studies are normally reconstructed frame by frame; thus, the time frames are treated independently. This approach fails to exploit the strong signal correlations among the time frames which are critical for noise reduction and resolution recovery. The authors investigated two reconstruction approaches, developed for dynamic PET by Wernick et al. (1997, 1999), which utilize the compression and decorrelation properties of the Karhunen-Loeve (KL) transform. In Method I, the authors temporally filter the data by using only the first few KL components, then perform frame-by-frame reconstruction. In Method II, the authors KL transform the projection data, reconstruct only the significant KL component images, then perform an inverse KL transformation to obtain the full 4D image sequence. Results indicate that Methods I and II provide better noise performance than ordinary frame-by-frame reconstruction. Additionally, Method II requires substantially fewer computations than conventional reconstruction methods.

Implementation and evaluation of an analytical solution to the photon attenuation and nonstationary resolution reconstruction problem in SPECT

The implementation and evaluation of an analytical solution to the photon attenuation and nonstationary resolution reconstruction problem in single photon emission computed tomography (SPECT) are presented. The method is based on a combination of the Bellini attenuation compensation and the Appledorn nonstationary resolution correction algorithms. It is termed the simultaneous correction algorithm (SCA). The results of testing the method with point source acquisitions showed a significant improvement in reducing the nonstationary behavior of the collimator response. The ratios of the FWHMs showed better agreement with the ideal value of 1.0, and the modulation transfer functions (MTFs) of the point spread functions exhibited less positional variation when using the SCA over other methods. In addition, the initial quantitative values were restored to acceptable levels.<<ETX>>

Reconstruction of 2D PET data with Monte Carlo generated system matrix for generalized natural pixels

In discrete detector PET, natural pixels are image basis functions calculated from the response of detector pairs. By using reconstruction with natural pixel basis functions, the discretization of the object into a predefined grid can be avoided. Here, we propose to use generalized natural pixel reconstruction. Using this approach, the basis functions are not the detector sensitivity functions as in the natural pixel case but uniform parallel strips. The backprojection of the strip coefficients results in the reconstructed image. This paper proposes an easy and efficient way to generate the matrix M directly by Monte Carlo simulation. Elements of the generalized natural pixel system matrix are formed by calculating the intersection of a parallel strip with the detector sensitivity function. These generalized natural pixels are easier to use than conventional natural pixels because the final step from solution to a square pixel representation is done by simple backprojection. Due to rotational symmetry in the PET scanner, the matrix M is block circulant and only the first blockrow needs to be stored. Data were generated using a fast Monte Carlo simulator using ray tracing. The proposed method was compared to a listmode MLEM algorithm, which used ray tracing for doing forward and backprojection. Comparison of the algorithms with different phantoms showed that an improved resolution can be obtained using generalized natural pixel reconstruction with accurate system modelling. In addition, it was noted that for the same resolution a lower noise level is present in this reconstruction. A numerical observer study showed the proposed method exhibited increased performance as compared to a standard listmode EM algorithm. In another study, more realistic data were generated using the GATE Monte Carlo simulator. For these data, a more uniform contrast recovery and a better contrast-to-noise performance were observed. It was observed that major improvements in contrast recovery were obtained with MLEM when the correct system matrix was used instead of simple ray tracing. The correct modelling was the major cause of improved contrast for the same background noise. Less important factors were the choice of the algorithm (MLEM performed better than ART) and the basis functions (generalized natural pixels gave better results than pixels).

Noise characterization of combined Bellini-type attenuation correction and frequency-distance principle restoration filtering [SPECT]

We investigate how successive processing of the projection data with Bellini-type (BT) attenuation correction, frequency distance principle (FDP) restoration filtering, and filtered-backprojection (FBP) reconstruction propagates noise into the reconstructed image. The BT methods, which are Fourier-domain correction techniques, are all exact solutions to the attenuated Radon transform under the constraints of uniform attenuation and a convex object contour. The FDP states that points in the object at a specified distance from the center-of-rotation will contribute predominantly to particular regions of the Fourier transform-series expansion of the sinogram. Using this frequency-distance information, the nonstationary response of the collimator can ideally be deconvolved using an inverse filter. In the presence of noise, however, the FDP filter needs to be regularized in order to control noise amplification caused by the deconvolution. The noise is characterized by calculating the population covariance matrix of the projection data after successive processing with BT attenuation correction and FDP filtering and of the reconstructed image after FBP reconstruction. This is done for a simulated point-source located near the edge of an elliptical, uniformly attenuating medium. In our study, we consider the two implementational combinations of BT correction and FDP filtering that allow for sequential processing of the projection data, which we call mirroring and no mirroring. The results show that the mirroring strategy introduces local positive angular noise correlations in the processed sinogram that are caused by smoothing of the data in the projection angular direction. Furthermore, mirroring highly correlates the noise in opposing projection views. The impact on the image covariance is the presence of positive angular (arc-like) noise correlations. In contrast, the no mirroring strategy yields local positive and negative noise correlations with negative side-lobes in the processed sinogram. Also, no mirroring does not correlate noise in opposing projection views. The result on the image covariance structure is the presence of positive and negative angular correlations with negative side-lobes. It was also noted that the no mirroring scenario produces a much higher noise variance relative to the mirroring implementation of BT attenuation correction and FDP filtering.

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)

Noise characterization of block-iterative reconstruction algorithms: II. Monte Carlo simulations

In Soares et al. (2000), the ensemble statistical properties of the rescaled block-iterative expectation-maximization (RBI-EM) reconstruction algorithm and rescaled block-iterative simultaneous multiplicative algebraic reconstruction technique (RBI-SMART) were derived. Included in this analysis were 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. Explicit expressions were found for the ensemble mean, covariance matrix, and probability density function of RBI reconstructed images, as a function of iteration number. The theoretical formulations relied on one approximation, namely that the noise in the reconstructed image was small compared to the mean image. We evaluate the predictions of the theory by using Monte Carlo methods to calculate the sample statistical properties of each algorithm and then compare the results with the theoretical formulations. In addition, the validity of the approximation will be justified.

Noise properties of attenuation correction methods for SPECT

The authors investigate the noise properties of SPECT images reconstructed with the attenuation correction methods of Bellini, Chang, and Tretiak and Metz. The general model for the image covariance matrix can be described by two terms, the first representing object variability, the second representing the object-dependent quantum noise. The model assumes the reconstruction operation is non-iterative and linear, and the noise in the projection data is nonstationary. All aspects of digital reconstruction are included in the model. The three attenuation correction methods are tested to demonstrate the noise character of SPECT images for a uniformly emitting and attenuating disk. In addition, image quality assessment is performed for the task of detecting a cold signal on a uniformly emitting and attenuating circular background. Comparing the local noise power spectrums for various pixel locations, it is shown that image noise is both globally and locally nonstationary for all three methods, except for a small, uniform region near the center of the disk. It is also seen that the noise properties of all three methods are similar in this region. The method of Tretiak-Metz increases the noise variance at the edge of the disk, whereas it is reduced with the Bellini and Chang methods. Finally, the ideal, nonprewhitening, and region-of-interest observers are shown to be invariant across reconstruction method, however the human observer performs better with the Bellini attenuation correction method.<<ETX>>

Noise characteristics of SPECT iterative reconstruction with a mis-matched projector-backprojector pair

SPECT reconstructed images are degraded by photon attenuation, the limited spatial resolution of the gamma camera, and scatter. Iterative reconstruction with the expectation-maximization (EM) algorithm provides an attractive solution for SPECT in that the physics of photon transport can be modeled into the projector and backprojector. Unfortunately, the implementation of this algorithm (including modelling of the physics) requires a large computational load which makes it unsuitable for routine clinical use. One acceleration method that has been suggested is to use a mis-matched projector and backprojector pair. By doing this, the computational time can be reduced. Barrett et al. (1994) have recently presented a theory which describes how noise in the projection data is propagated into the reconstructed image when using the EM algorithm. The authors have used this theory to evaluate and compare the differences in pixel mean and variance, as well as performance for the task of estimating activity within a region-of-interest, when reconstructing using the EM algorithm with a matched and mismatched projector-backprojector pair. Results indicate that EM reconstruction can be accelerated by not modeling attenuation into the backprojection process, with little penalty in terms of estimating activity in regions-of-interest.