  We propose a method for characterizing a photodetector by directly
reconstructing the Wigner functions of the detector's
Positive-Operator-Value-Measure (POVM) elements. This method extends the works
of S. Wallentowitz and Vogel [Phys. Rev. A 53, 4528 (1996)] and Banaszek and
W\'odkiewicz [Phys. Rev. Lett. 76, 4344 (1996)] for quantum state tomography
via weak-field homodyne technique to characterize quantum detectors. The scheme
uses displaced thermal mixtures as probes to the detector and reconstructs the
Wigner function of the photodetector POVM elements from its outcome statistics.
In order to make the reconstruction robust to the experimental noise, we use
techniques from quadratic convex optimization.
