Bounds on null functions of linear digital imaging systems

Effect on null spaces of list-mode imaging systems due to increasing the number of attributes

There are two types of uncertainty in image reconstructions from list-mode data: statistical and deterministic. One source of statistical uncertainty is the finite number of attributes of the detected particles, which are sampled from a probability distribution on the attribute space. A deterministic source of uncertainty is the effect that null functions of the imaging operator have on reconstructed pixel or voxel values. Quantifying the reduction in this deterministic source of uncertainty when more attributes are measured for each detected particle is the subject of this work. Specifically, upper bounds on an error metric are derived to quantify the error introduced in the reconstruction by the presence of null functions, and these upper bounds are shown to be reduced when the number of attributes is increased. These bounds are illustrated with an example of a two-dimensional single photon emission computed tomography (SPECT) system where the depth of interaction in the scintillation crystal is added to the attribute vector.

Effect on null spaces of list-mode imaging systems due to increasing the size of attribute space

An upper bound is derived for a figure of merit that quantifies the error in reconstructed pixel or voxel values induced by the presence of null functions for any list-mode system. It is shown that this upper bound decreases as the region in attribute space occupied by the allowable attribute vectors expands. This upper bound allows quantification of the reduction in this error when this type of expansion is implemented. Of course, reconstruction error is also caused by system noise in the data, which has to be treated statistically, but we will not be addressing that problem here. This method is not restricted to pixelized or voxelized reconstructions and can in fact be applied to any region of interest. The upper bound for pixelized reconstructions is demonstrated on a list-mode 2D Radon transform example. The expansion in the attribute space is implemented by doubling the number of views. The results show how the pixel size and number of views both affect the upper bound on reconstruction error from null functions. This reconstruction error can be averaged over all pixels to give a single number or can be plotted as a function on the pixel grid. Both approaches are demonstrated for the example system. In conclusion, this method can be applied to any list-mode system for which the system operator is known and could be used in the design of the systems and reconstruction algorithms.

Comparison of the scanning linear estimator (SLE) and ROI methods for quantitative SPECT imaging

In quantitative emission tomography, tumor activity is typically estimated from calculations on a region of interest (ROI) identified in the reconstructed slices. In these calculations, unpredictable bias arising from the null functions of the imaging system affects ROI estimates. The magnitude of this bias depends upon the tumor size and location. In prior work it has been shown that the scanning linear estimator (SLE), which operates on the raw projection data, is an unbiased estimator of activity when the size and location of the tumor are known. In this work, we performed analytic simulation of SPECT imaging with a parallel-hole medium-energy collimator. Distance-dependent system spatial resolution and non-uniform attenuation were included in the imaging simulation. We compared the task of activity estimation by the ROI and SLE methods for a range of tumor sizes (diameter: 1-3 cm) and activities (contrast ratio: 1-10) added to uniform and non-uniform liver backgrounds. Using the correct value for the tumor shape and location is an idealized approximation to how task estimation would occur clinically. Thus we determined how perturbing this idealized prior knowledge impacted the performance of both techniques. To implement the SLE for the non-uniform background, we used a novel iterative algorithm for pre-whitening stationary noise within a compact region. Estimation task performance was compared using the ensemble mean-squared error (EMSE) as the criterion. The SLE method performed substantially better than the ROI method (i.e. EMSE(SLE) was 23-174 times lower) when the background is uniform and tumor location and size are known accurately. The variance of the SLE increased when a non-uniform liver texture was introduced but the EMSE(SLE) continued to be 5-20 times lower than the ROI method. In summary, SLE outperformed ROI under almost all conditions that we tested.

Scanning linear estimation: improvements over region of interest (ROI) methods

In tomographic medical imaging, a signal activity is typically estimated by summing voxels from a reconstructed image. We introduce an alternative estimation scheme that operates on the raw projection data and offers a substantial improvement, as measured by the ensemble mean-square error (EMSE), when compared to using voxel values from a maximum-likelihood expectation-maximization (MLEM) reconstruction. The scanning-linear (SL) estimator operates on the raw projection data and is derived as a special case of maximum-likelihood estimation with a series of approximations to make the calculation tractable. The approximated likelihood accounts for background randomness, measurement noise and variability in the parameters to be estimated. When signal size and location are known, the SL estimate of signal activity is unbiased, i.e. the average estimate equals the true value. By contrast, unpredictable bias arising from the null functions of the imaging system affect standard algorithms that operate on reconstructed data. The SL method is demonstrated for two different tasks: (1) simultaneously estimating a signal's size, location and activity; (2) for a fixed signal size and location, estimating activity. Noisy projection data are realistically simulated using measured calibration data from the multi-module multi-resolution small-animal SPECT imaging system. For both tasks, the same set of images is reconstructed using the MLEM algorithm (80 iterations), and the average and maximum values within the region of interest (ROI) are calculated for comparison. This comparison shows dramatic improvements in EMSE for the SL estimates. To show that the bias in ROI estimates affects not only absolute values but also relative differences, such as those used to monitor the response to therapy, the activity estimation task is repeated for three different signal sizes.