## 2. FormalismIn this section we briefly review the classical Richardson-Lucy algorithm and its features in order to provide a basic understanding of the multiple-data Richardson-Lucy algorithm. ## 2.1. Richardson-Lucy algorithmThe projection of observables along the LOS can be formulated as an inverse problem, mathematically such problems can be cast into the form of a Fredholm integral equation of the first kind, where is the function of interest, is the function accessible through observation, and the integral kernel reflects the measurement process. In general, and represent probability density functions, which implies that they and the kernel obey normalization and non-negativity constraints. is the probability - presumed to be known - that will fall in the interval when it is known that . Richardson (1972) and Lucy (1974, 1994) recognized that the problem is statistical in nature and proposed an iterative inversion algorithm, consisting of the following two steps: where the denotes the observed
distribution, while is the measured
quantity having sampling errors. A derivation of the iterative
algorithm based on The iteratively constructed functions satisfy the nonnegativity constraint: From Eq. (2b) it follows that if . The normalization constraint is fulfilled, as one can prove by integrating Eq. (2b) with respect to and using the normalizations of the probabilities and . Ideally we would like an iterative algorithm to converge to the
exact solution, and from Eq. 2 it can be seen that the above
scheme converges if is sufficiently
close to for all points ## 2.2. Richardson-Lucy algorithm: the axisymmetric caseAfter deriving the basic RL-algorithm and discussing its features, we need to specify a cluster model in order to derive the kernel and formulate the RL-scheme for this specific model. Consider a cluster of galaxies covered by a system of cartesian
coordinates . We are interested in
recovering the distribution of some physical quantity
, which we assume to have axial
symmetry with respect to the Furthermore, we assume that the projection of
is observed as some quantity
, where the observer's coordinate
system is inclined by an angle
and the observed projection is given as the LOS integral over ,
If we want to apply the RL-algorithm to recover from , we first have to bring the fundamental LOS integral 5 into the form of Eq. 1, where has to be normalized to unity. The derivation follows closely the derivation for elliptical galaxies given in Binney et al. (1990) and is reproduced in Appendix A. We can identify the kernel and find that it is properly normalized: Having the explicit expression 7 for the kernel , we could now start to apply the RL-scheme for recovering from an observed by means of Eqs. (2.1 a,b). However, since the probability kernel contains a -function, which is notoriously difficult to deal with in the context of discretized grid-data, we have chosen to reformulate the main Eqs. (2.1 a,b) for this special axisymetric case. For the first integral of the iterative scheme the formulation A.5 of the integral using the probability kernel is not necessary. Instead, this integral can be evaluated as a simple integral along the LOS, where the coordinate transformation (4c) is used to evaluate the
integration along The second step 2b in the iterative RL-scheme reads in our case For this second integral the evaluation of the probability kernel
cannot be avoided. However, it is
possible to eliminate the -function
by again applying the same rule as reproduced in the Appendix to
and by subsequently integrating over
Please note that the integral (11) describes a full ellipse on the sky. The difference between Eq. (11) and the Appendix of Binney et al. (1990) is, that Eq. (11) uses the full information on the ellipse, while Binney et al. (1990), use only half of it. This formulation has the advantage that no assumptions on the data are made, i.e. the latter are not assumed to be symmetric along any of the projection axes. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |