  In this paper, we proposed a new Monte Carlo radiative transport (MCRT)
scheme, which is based completely on the Neumann series solution of Fredholm
integral equation. This scheme indicates that the essence of MCRT is the
calculation of infinite terms of multiple integrals in Neumann solution
simultaneously. Under this perspective we redescribed MCRT procedure
systematically, in which the main work amounts to choose an associated
probability distribution function (PDF) for a set of random variables and the
corresponding unbiased estimation functions. We can select a relatively optimal
estimation procedure that has a lower variance from an infinite possible
choices, such as the term by term estimation. In this scheme, MCRT can be
regarded as a pure problem of integral evaluation, rather than as the tracing
of random walking photons. Keeping this in mind, one can avert some subtle
intuitive mistakes. In addition the $\delta$-functions in these integrals can
be eliminated in advance by integrating them out directly. This fact together
with the optimal chosen random variables can remarkably improve the Monte Carlo
(MC) computational efficiency and accuracy, especially in systems with axial or
spherical symmetry. An MCRT code, Lemon (Linear Integral Equations' Monte Carlo
Solver Based on the Neumann solution), has been developed completely based on
this scheme. Finally, we intend to verify the validation of Lemon, a suite of
test problems mainly restricted to flat spacetime have been reproduced and the
corresponding results are illustrated in detail.
