  We present a continuous variable tomography scheme that reconstructs the
Husimi Q-function (Wigner function) by Lagrange interpolation, using
measurements of the Q-function (Wigner function) at the Padua points, the
optimal sampling points for two dimensional reconstruction. Our approach
drastically reduces the number of measurements required compared to using
equidistant points on a regular grid, although reanalysis of such experiments
is possible. The reconstruction algorithm produces a reconstructed function
with exponentially decreasing error and quasi-linear runtime in the number of
Padua points. Moreover, using the interpolating polynomial of the Q-function,
we present a technique to directly estimate the density matrix elements of the
continuous variable state, with only linear propagation of input measurement
error. Furthermore, we derive a state-independent analytical bound on this
error, such that our estimate of the density matrix is accompanied by a measure
of its uncertainty.
