  Tomography of a quantum state is usually based on positive operator-valued
measure (POVM) and on their experimental statistics. Among the available
reconstructions, the maximum-likelihood (MaxLike) technique is an efficient
one. We propose an extension of this technique when the measurement process
cannot be simply described by an instantaneous POVM. Instead, the tomography
relies on a set of quantum trajectories and their measurement records. This
model includes the fact that, in practice, each measurement could be corrupted
by imperfections and decoherence, and could also be associated with the record
of continuous-time signals over a finite amount of time. The goal is then to
retrieve the quantum state that was present at the start of this measurement
process. The proposed extension relies on an explicit expression of the
likelihood function via the effective matrices appearing in quantum smoothing
and solutions of the adjoint quantum filter. It allows to retrieve the initial
quantum state as in standard MaxLike tomography, but where the traditional POVM
operators are replaced by more general ones that depend on the measurement
record of each trajectory. It also provides, aside the MaxLike estimate of the
quantum state, confidence intervals for any observable. Such confidence
intervals are derived, as the MaxLike estimate, from an asymptotic expansion of
multi-dimensional Laplace integrals appearing in Bayesian Mean estimation. A
validation is performed on two sets of experimental data: photon(s) trapped in
a microwave cavity subject to quantum non-demolition measurements relying on
Rydberg atoms; heterodyne fluorescence measurements of a superconducting qubit.
