A Theory of Optimal Flutter Shutter for Probabilistic Velocity Models

Abstract

Abstract. Flutter shutter (coded exposure) is a new paradigm for cameras that allows for an arbitrary increase of the exposure time when the relative camera/scene motion is uniform. The photon flux is interrupted according to a flutter shutter code. For arbitrarily severe uniform motion blur a well chosen code guarantees an invertible blur kernel. Yet, when the relative camera/scene velocity is a known constant, a flutter shutter cannot gain more than a 1.17 factor in terms of root mean-squared error compared to the optimal snapshot. In this paper, we prove that this optimality bound can be relaxed under the realistic assumption that a random model for the velocities is available. We give analytical formulae for the optimal flutter shutter code and the optimal snapshots associated with a random velocity distribution. Conversely we also prove formulae that reveal the velocity distribution underlying a given flutter shutter code. 1. Introduction. Digital cameras count at each pixel sensor the number of photons emitted by the observed scene during a time span called the exposure time. The photon count follows a Poisson random variable. Its mean is the ideal (noiseless) pixel value. The difference between this ideal pixel value and the observed sensor photon count is called (shot) noise. The ratio of the mean of the photon count over its standard-deviation is called the signal-to-noise ratio (SNR). In passive imaging systems there is no control over the scene lighting. Thus, the only safe way to increase the SNR is to integrate more photons by increasing the exposure time. Yet, when the scene or the camera moves during the exposure process, the resulting images are degraded by a motion blur. If the support of a motion blur kernel exceeds two pixels, the blur is no longer invertible. This limits the exposure time and therefore the image quality of a snapshot. A setup solving this photography trade-off between noise and blur was proposed in [3,4,6,37,38,39] for uniform camera-scene motions. The authors of these papers attached a flutter shutter to a camera to get an invertible motion blur kernel. A flutter shutter