The Mathematics of HYPR

Introduction A promising new method for accelerated imaging, HighlY constrained backPRojection (HYPR) [1] uses the spatial information of a time averaged image (composite) to improve images at individual time frames. The original HYPR algorithm [1] was derived heuristically based on the idea that the location of the object was known using the composite image and that the role of the data from each time frame was to extract the intensity profile at that time. That perspective led to a practical understanding of the algorithm that has been generalized and modified, as in the HW-HYPR method [2] which modified HYPR so that the method converges to the composite when the time frame uses all projections. The HYPR approach has also been extended to an iterative method (I-HYPR) [3] in which that version of the HYPR algorithm was identified by the authors as the expectation maximization (EM) algorithm. The objective of this work is to clarify the mathematical understanding of the original HYPR algorithm and HW-HYPR as the first step of iterative methods used in statistical image reconstruction in other imaging modalities which enforce a positivity constraint (if the initial guess is non-negative, the resulting image will be nonnegative also) with an initial condition that encodes the spatial location of the object being reconstructed (i.e. the composite). Mathematical Background The two main mathematical ideas needed to explain HYPR are maximum likelihood estimation (MLE) and fixed point iteration (FPI) [4]. In MLE the parameter being estimated is chosen such that the probability that the data was generated by that parameter is maximized. Let H = + g f n , where g is the