The Flutter Shutter Paradox

Acquiring good quality images of moving objects by a digital camera remains a valid question, particularly if the velocity of the photographed object is only partially known, it is virtually impossible to tune an optimal exposure time. The same question arises when a solid object is being photographed by a moving camera. This is not only frequent, but even necessary when the camera is embarked on terrestrial vehicle, a plane, or an Earth observation satellite. In the latter cases, the camera velocity is constrained by physical laws. For this reason the recent Raskar and al. flutter shutter and the Durand and al. motion invariant photography methods have generated much interest. This paper proposes a mathematical analysis of moving photography which permits to treat in the same formalism the flutter shutter and the motion invariant photography. A general flutter shutter method is defined and explored, which allows for nonbinary shutter sequences and permits to formalize the question of an optimal flutter shutter code. The first result of this analysis is actually negative, and raises what we call the flutter shutter paradox. We prove that both the flutter shutter and the motion-invariant photography methods cannot increase indefinitely the signal to noise ratio compared to an exposure short enough to avoid motion blur despite the use of a possibly infinite exposure time. Bounds are shown. We also resolve the flutter shutter paradox under the assumption that the relative velocity of object and camera is not a priori known. Then a significant increase of the average SNR can be expected provided there is an a priori knowledge of the probability density for the observed velocities. In this set up we show how to compute analytically an optimal flutter shutter strategy.