Impact of Artificial Intelligence in the Brave New Medical World
The Journal of craniofacial surgery (Print)
Introduction Mathematical tomography centers around integral transforms in which both the object and the image are treated as functions of continuous variables; we refer to an integral transform as a continuous-to-continuous (CC) operator. The ubiquitous practice of representing real-life objects (such as organs, bones, tissues, etc in the case of medical imaging) with a finite set of intensities over a 2D or 3D grid of pixels or voxels often leads to inaccuracies when the object itself
... ject itself presents many features that cannot be represented using a grid of pixels or voxels. Similarly, sets of discrete data (such as projection images in the case of emission tomography) are typically used as the starting point to perform reconstruction. These discretizations of the object and the data produced by the imaging system are almost always assumed, leading to matrix representations or discrete-to-discrete (DD) operators. In this paper, we discuss our approach to approximating CC operators for real single-photon emission computed tomography (SPECT) imaging systems in which we use measured calibration data from real detectors and real multi-pinhole imaging systems, thus potentially avoiding discrete representations of the object or the raw data. We use photon-processing detectors to collect data with no binning involved, and show how such data sets can be reconstructed using iterative maximum-likelihood (ML) estimation algorithms. Our approach does not introduce any error due to discretization of the measurement, and it allows reconstructions over an arbitrary sets of points, not necessarily arranged as a 3D uniform grid of voxels. Prior art in list-mode reconstruction can be found in the work by Levkovitz et al (2001) . In it, the authors derive an algorithm for direct reconstruction of list-mode data for positron emission tomography (PET) by starting from an expression of the maximum-likelihood expectation maximization algorithm for binned data, and then they consider the limit of no more than one count per bin. An alternative formulation, also for PET imaging, is provided in Parra and . Similar to Barrett et al (1997) and Parra and Barrett (1998) , our algorithm assumes that the list-mode data are positions of interaction as estimated (Moore et al 2007) from data collected with the gamma-ray cameras in the SPECT scanner. These estimates are allowed to vary over a continuous domain, and probability density functions are evaluated on-the-fly for each item of the list. This paper is organized as follows. Section 2 presents the mathematical notation used in this paper; section 3 discusses photon-processing detectors and introduces a point process we use to mathematically represent listmode data. In section 4 we discuss the CC imaging operator as a mapping from the object space to data space. This treatment allows us to derive an explicit expression for the kernel of this CC operator. In section 5 we present Abstract Imaging systems are often modeled as continuous-to-discrete mappings that map the object (i.e. a function of continuous variables such as space, time, energy, wavelength, etc) to a finite set of measurements. When it comes to reconstruction, some discretized version of the object is almost always assumed, leading to a discrete-to-discrete representation of the imaging system. In this paper, we discuss a method for single-photon emission computed tomography (SPECT) imaging that avoids discrete representations of the object or the imaging system, thus allowing reconstruction on an arbitrarily fine set of points.