We present a new method for obtaining ring optical depth profiles and equivalent depth measurements from stellar occultation data possessing a large statistical noise. Our procedure is to model the ring by the smoothest possible profile that is consistent with the data and its noise properties. This method has the advantage of a vastly increased signal-to-noise ratio, but without as severe a loss in radial resolution as is caused by simple averaging. We show the resulting optical depth profiles and calculated equivalent depths for the Uranian ɛ Ring in several Voyager 2 Photopolarimeter (PPS) occultation cuts. By combining our profiles with the Voyager radio occultation data, we are able to calculate streamlines and to measure the local gradient in eccentricity within the ring. Our measurements indicate that the streamlines are well modeled by Keplerian ellipses in the inner portion of Ring ɛ but not near the outer edge. The eccentricity gradients that we compute appear to be incompatible with the self-gravity model for the maintenance of ring eccentricity against the differential precession induced by planetary oblateness.