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We recently developed the tomographic ultrafast retrieval of transverse light E-fields (TURTLE) as a self-referenced method for retrieving vector-shaped ultrashort laser pulses. TURTLE combines three or more linear polarization projections of the pulse, which are measured with existing single-polarization pulse characterization techniques, to determine the full polarization state. We present in detail the two approaches for retrieving the full vector-shaped field: an analytic inversion and a parameter-fitting technique. By examining the impact of measurement noise on retrieval fidelity, we find that the fitting approach, which makes use of the additional redundancy inherent in some single-polarization measurements, is greatly superior to the analytic.