Back-Projection Corrections

Gamma Ray Imaging with a Rotating Modulator Astronomy And Astrophysics 120, 150-155 (1983) P. Durouchoux H. Hudson G. Hurford K. Hurley J. Matteson E. Orsal [ Excerpt from Section 2] Any modulator/detector configuration may be evaluated by considering the three following properties: sensitivity, angular resolution, and selectivity. Each of these may be defined in terms of its vector interpretation or its physical interpretation (See Table 1). That the sensitivity in a given pixel is proportional to the square root of the variance of the sky vector for that pixel may be shown as follows. Suppose that a single point is observed in pixel j which would cause each detector to have a constant counting rate s counts/s if no other modulator was present, while the background counting rate, assumed constant on the average from detector to detector, is b counts/s. If the observation lasts a time T seconds, Oi = (b + s Si)T, In practice, the source strength s may be extracted from the observation vector O by first multiplying through by a function whose average is zero, namely Sji-<Sj>, where <Sj> = Σi(Sji)/Nc Then, summing over the i components of the vector, the background contribution drops out: Σi[Oi(Sji-<Sj>)] = Σi[bT(Sji-<Sj>)] + Σi[sTSji(Sji-<Sj>)]      = sTNc(<Sj2> - <Sj>2) The standard deviation associated with the number of counts in the ith component is. for Gaussian statistics, (Oi)½ = ({b+sSji}T)½ Since the standard deviations add as the square root of the sum of the squares, the standard deviation associated with the above is (Σi[Oi(Sji-<Sj>)2] )½      = Nc½(Σi[bT(Sji-<Sj>)2]+ Σi[sTSji(Sji-<Sj>)2])½      =(bTNc)½ ((<Sj2> - <Sj>2))½ for b >> s. Thus the statistical significance of the observation is s(TNc/b)½ ((<Sj2> - <Sj>2))½ or just proportional to the square root of the variance of the sky vector.

Notes transcribed from GH.

Notes by GH, April 1999

O_i =(B + P_im S_m) AT (1)

  O_i=observed counts in time bin i
  B=background (assumed constant)
  P_im=probability of photon passage at pixel m, time i
  S_m=true source strength at pixel m (presumed a point source?)
  A=effective area of collimator
  T=time in time bin i (assumed equal for all time bins)
  i=1,...,N

from which GH derived:

S_m= Σ_i O_i ( P_im - <P_m >)/ {AT ( < P_m² > - <P_m >²)} (2)

where < P_m ² > is the mean square of P_im, averaged over time, and <P_m >² is the squared mean, averaged over time.

The expressions for the point source strength in back projection and clean.pro (also MEM?) must be replaced with the time-averaged values of equation (2) for S_m.