A bound on null functions for digital imaging systems with positivity constraints

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.

Bounds on null functions of linear digital imaging systems

Any linear digital imaging system produces a finite amount of data from a continuous object. This means that there are always null functions, so a reconstruction of the object, even without noise in the system, will differ from the actual object. With positivity constraints, the size of a null function is limited, provided that size is measured by the integral of the absolute value of the null function. When smoothing is used in reconstruction, then smoothed null functions become relevant. There are bounds on various measures of the size of smoothed null functions, and these bounds can be quite small. Smoothing will decrease the effects of null functions in object reconstructions, and this effect is greater if the smoothing operator is well matched to the system operator.