Information content of a photon and how to extract it

Medical imaging techniques can be used for two types of tasks. Firstly, detection tasks decide whether a signal of interest is present in the data, perhaps indicating a tumor, arterial blockage, skeletal abnormality, or some other interesting phenomenon that deserves a closer look. Second, estimation tasks determine one (or more) numerical value about a patient. In both task types, the performances of an imaging system and the data collected with it have to be mathematically defined in terms of how well the specific task is performed. This goal can be accomplished by introducing appropriate figures of merit (FOMs) for task performance.^1–4

We have taken the mathematical formalism of objective assessment of medical imaging task performance, and applied it to gamma-ray detectors and quantities estimated about every photon. The formalism and FOMs together provide a method to quantify the contribution, or information content, of each photon to task performance. By providing a mathematical expression for the information content of a photon, we can compare two or more imaging systems and decide which is more suited for a particular task. Alternatively, an imaging system can be designed and configured in a way that maximizes what we can learn about the patient with the minimum dose of gamma radiation.

A typical position-sensitive imaging device for gamma-ray photons consists of a scintillation crystal paired to an array of photomultiplier tubes.^5–8 From the photomultiplier tube outputs, different photon parameters—such as position, energy, and time of arrival—can be estimated.^9–11 The data collected can be organized in two possible ways while imaging the patient, usually referred to as ‘binned’ and ‘list-mode’. In a binned-data representation, the detector space is divided into a large number of bins. During data acquisition, bin counts are accumulated. Bin counts are then read from the detector circuitry and stored in the memory of a computer. This is the way standard photon-counting imaging devices work. An example of binned data is provided in Figure 1. The alternative—list-mode data—consists of creating a list in which each entry corresponds to parameters estimated for each photon. In Figure 2, we have represented each parameter (consisting, in this case, of just the 2D photon location) as a point on a plane.

Figure 1. Example of binned data representation.

Figure 2. Example of list-mode data representation.

A quick comparison of Figure 1 with Figure 2 emphasizes the loss of information incurred with binned data. For this reason, we have developed a formalism for assessment of task performance when list-mode data (as opposed to binned data) is used. Our formalism also enables us to quantify the contribution of each photon to task performance, naturally leading to a way to measure the information that each photon conveys.

In our recent work, we used a meaningful FOM to assess the performance of signal detection with list-mode data.¹² We considered two different methods for signal detection: the Hotelling observer^13,14 and the ideal observer.^{3,4,7,12–15} For any signal detection problem and given any data (in any representation), the ideal observer is the mathematical tool that maximizes the probability of a correct detection for any given value of the probability of a false positive. In a sense, the ideal observer is the ‘best’ observer we can possibly use. Hence, by introducing an appropriate FOM for the performance of the ideal observer, we are indirectly defining a FOM for the data the ideal observer uses. We can, in turn, compare list-mode data with binned data by considering the performance of the ideal observer when applied to these two data representations. In a practical case, however, the ideal observer's mathematical expression can be too complex to use. Linear observers tend to have a simpler mathematical expression, which makes them more practical. Among all the linear observers, the Hotelling observer is the one that maximizes task performance.⁷

The Hotelling observer produces a scalar by calculating the inner product between the data vector and the Hotelling template vector. The Hotelling template vector is obtained by first considering a generic linear observer, and then by finding the template that maximizes the observer's signal-to-noise ratio.

Our research shed further light on the benefits of list-mode data, and some interesting results regarding task performance with list-mode data were obtained.¹²By considering one photon at a time, we could quantify how much each photon contributed to the value of the FOM and, ultimately, to task performance.

Building on the idea of information content of a photon, we introduced the concept of the ideal dose utilizer (IDU). The IDU uses maximum-likelihood methods^7,16 to estimate list-mode photon parameters, which are then fed to the ideal observer³ to perform the task of interest. In other words, we not only defined optimal methods that fully use all the information in the data, we also established what type of data we have to use (and how we calculate it) so that task performance is optimized and what we learn by imaging the patient is used fully. The IDU paradigm provides an optimal way to use the dose delivered to the patient by careful extraction of all the information that can be learned about the object.

We now plan to apply the same theory to estimation problems. For example, we could use list-mode data to localize a tumor or calculate its volume. Our next project will use an ultra-fast camera and computing equipment to extract list-mode information from x-ray photons.

The authors would like to acknowledge National Institutes of Health grants R37 EB000803 and P41 EB002035 for financial support.