Analysis of the Cherenkov light emitted from a single ATA beam pulse provides quantitative information regarding the distribution of beam particles in a four dimensional transverse phase space. Using the techniques described in this report we conclude that the experimentally observed spatial distribution of Cherenkov light implies a parabolic distribution of beam particles. Using the transfer matrix of the magnetic solenoid transport, quantitative values of the emittance, radius, tilt and energy at a position upstream of the Cherenkov foil may be obtained, and the beam brightness as a function of radius within the beam may be calculated. The brightness is defined as the ratio of the beam current within a given radius to the normalized four dimensional volume occupied by particles within that radius. The normalized four volume is the four dimensional volume multiplied by the relativistic parameters β2γ2. We find that the parabolic phase space distribution gives a core brightness three times the edge brightness and agreement with the core brightness measured by the two aperture emittance diagnostic.