2.2.1 Linear Systems

The spatial resolution of an imaging modality or instrument denotes its ability to reproduce fine detail. Ideally, a tomographic image would exactly reproduce the object or, in the case of nuclear medicine, the activity distribution within the patient. If the properties of scaling, superposition, and space invariance are assumed to be preserved to allow the assumption of a linear system [Web88][BS81a][Mak83], then the physical response or capability of an imaging system to reproduce the limit of fine detail, a point source or function, may be characterized by its point spread response function (PSRF). The stimulus object is said to have been convoluted or distorted by the system to produce a blurry output image. The ``black box'' which represents the entirety of the imaging system consists of other cascaded linear systems representing the presence of scatter, attenuation, imperfect electronics, the apodizing functions used in the tomographic reconstruction and other factors which all contribute in an imaging system's capability to reproduce an object exactly [RK82b][RK82a][BS81b][BS81a] (see figure 2.2).

The degree of blurring introduced as described by the PSRF is most generally represented by a Gaussian plus a constant pedestal of activity where the information density follows a Poissonian distribution according to nuclear decay statistics (see figure 2.3). Better spatial resolution, or a narrower Gaussian distribution, may be obtained in practice by compensating for scatter and the depth dependent detector response in SPECT systems with the use of iterative filtered back projections and maximum a posteriori probability procedures [MCC+93][KZR92][Lia93]. The use of elliptical scans which more closely approximates the transaxial outline of the head also improves the spatial resolution by minimizing the source to detector distance. Attenuation correction studies [KPD+92] using simple first order techniques like the Sorenson hyperbolic sine method [Sor74][Kem89] have shown that the effect of attenuation on the spatial resolution is insignificant; it is nevertheless an important correction for quantitative studies. Methods which correct for the response of the different components in the cascaded linear systems are non-trivial because of the computational intensity required for their implementation (of the order of hours for typical 128128128 image arrays) [Lia93] and, as such, are often not clinically utilised.

The extension of the PSRF to a volume data set is naturally a 3-D Gaussian envelope. The width of the Gaussian on top of the background of noise in each dimension may be described by standard deviation. A good imaging system is said to have a narrow PSRF so that the spatial position of the original point source may be more accurately determined from information from the image alone. The dimensional components of the 3-D PSRF in image space are generally not equal. In figure 2.4 depicting point sources on and off the center of rotation in a tomographic plane perpendicular to the axis of rotation, the shape of the circular cross section of a point source tends to be stretched out in the radial direction to a more elliptical shape. Note that the x and y dimensions depicted in the figure are usually oriented from left to right and from the bottom to the top, respectively. In terms of the anatomical orientation of the brain, this corresponds to the neurological convention [KW92] of viewing transaxial cuts where top is the anterior, bottom is the posterior, left is the patient's left, etc. (please see the appendix). The radial component of the PSRF off axis tends to become greater than the tangential component to ultimately give this distorted oblong shape [SP87][OFBM88][UKMM81][Gre83][KPD+92]. This is due to increased rejection of large angle scatter by the collimator septa as the source moves farther away from the center of rotation and closer to the detector face (see figure 2.4). This is the same cause of resolution improvement which is achieved by using elliptical scans, as mentioned above, because the activity distribution within a patient is brought closer to the detector face. For an elliptical scan then, points radially further away from the axis of rotation along the major axis and in the plane perpendicular to the axis of rotation (usually the transaxial slice) will generally have diminished resolution. For scans which revolve about the inferior-superior axis, the orientation of diminished resolution therefore correspond to the anterior-posterior direction because the brain's dimension is largest in this direction.