A discrete-time attitude state estimation scheme that uses a global representation of the configuration space for rigid body attitude motion, is presented. This estimation scheme uses discrete-time state measurements of inertially known vectors along with rate gyro measurements of the angular velocity, to obtain state estimates in the filtering stage. Additionally, a set of sigma points is obtained from the unscented transform based on exponential coordinates, with re-sampling centered at the current state estimate at each measurement instant. The state estimates along with sampled sigma points are propagated between measurement instants, using a discrete-time attitude state observer that is almost globally finite-time stable. The propagated sigma points and state estimate are enclosed by a minimum volume ellipsoid at the measurement instant. It is assumed that all states are measured at a constant measurement sample rate and that state measurement errors are bounded by an ellipsoidal bound. The update stage of the filter consists of finding the minimum volume enclosing ellipsoid that contains the propagated sigma points and the measurement uncertainty bound. This updated ellipsoid provides the filtered uncertainty bound and its center provides the updated state estimate. A new set of sigma points is selected from this ellipsoid and the propagation and update steps are repeated between measurement instants. Numerical simulation results confirm the analytically obtained stability properties of the attitude state observer. Numerical results also show that state estimate errors are bounded in the presence of bounded measurement noise and bounded disturbance torque.