  We propose a new approach that allows for the separate numerical calculation
of the real and imaginary parts of finite loop integrals. We find that at
one-loop the real part is given by the Loop-Tree Duality integral supplemented
with suitable counterterms and the imaginary part is a sum of two-body phase
space integrals, constituting a locally finite representation of the
generalised optical theorem. These expressions are integrals in momentum space,
whose integrands were specially designed to feature local cancellations of
threshold singularities. Such a representation is well suited for Monte Carlo
integration and avoids the drawbacks of a numerical contour deformation around
remaining singularities. Our method is directly applicable to a range integrals
with certain geometric properties but not yet fully generalised for arbitrary
one-loop integrals. We demonstrate the computational performance with examples
of one-loop integrals with various kinematic configurations, which gives
promising prospects for an extension to multi-loop integrals.
