The Flutter Shutter Code Calculator

Tendero, Yohann

doi:10.5201/ipol.2015.108

Cited by 3 publications

(4 citation statements)

References 11 publications

(34 reference statements)

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“…Page 2 of 39 14,22,23,39] or in [15] that aims at adapting the sequence to the velocity. Diverse implementations of the method are presented in [17,18,24,32].…”

Section: Tendero and Osher Res Math Sci (2016) 3:4mentioning

confidence: 99%

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On a mathematical theory of coded exposure

Tendero

Osher

2016

Res Math Sci

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This paper proposes a mathematical model and formalism to study coded exposure (flutter shutter) cameras. The model includes the Poisson photon (shot) noise as well as any additive (readout) noise of finite variance. This is an improvement compared to our previous work that only considered the Poisson noise. Closed formulae for the mean square error and signal to noise ratio of the coded exposure method are given. These formulae take into account for the whole imaging chain, i.e., the Poisson photon (shot) noise, any additive (readout) noise of finite variance as well as the deconvolution and are valid for any exposure code. Our formalism allows us to provide a curve that gives an absolute upper bound for the gain of any coded exposure camera in function of the temporal sampling of the code. The gain is to be understood in terms of mean square error (or equivalently in terms of signal to noise ratio), with respect to a snapshot (a standard camera). Keywords: Coded exposure, Computational photography, Flutter shutter, Motion blur, Mean square error (MSE), Signal to noise ratio (SNR) Mathematics Subject Classification: Primary 68U10; Secondary 42A38, 60G99 BackgroundSince the seminal papers [1][2][3][4][5][6] of Agrawal and Raskar coded exposure (flutter shutter) method has received a lot of follow-ups . In a nutshell, the authors proposed to open and close the camera shutter, according to a sequence called "code," during the exposure time. By this clever exposure technique, the coded exposure method permits one to arbitrarily increase the exposure time when photographing (flat) scenes moving at a constant velocity. Note that with a coded exposure method only one picture is stored/transmitted. A rich body of empirical results suggest that the coded exposure method allows for a gain in terms of Mean Square Error (MSE) or Signal to Noise Ratio (SNR) compared to a classic camera, i.e., a snapshot. Therefore, the coded exposure method seems to be a magic tool that should equip all cameras.We now briefly expose the different applications, variants, and studies that surround the coded exposure method. An application of the coded exposure method to bar codes is given in [16,35], to fluorescent cell image in [27], to periodic events in [25,34,36], to multispectral imaging in [10], and to iris in [21]. Application to motion estimation/deblurring are presented in [9,19,20,26,31,33,37,38]. An extension for space-dependent blur is investigated in [28]. Methods to find better or optimal sequences as investigated in [12-© 2016 Tendero and Osher. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,-: vol Tendero and Osher Res Math Sci (2016) 3:4Page 2 of 39 14,22,23,39] o...

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“…Page 2 of 39 14,22,23,39] or in [15] that aims at adapting the sequence to the velocity. Diverse implementations of the method are presented in [17,18,24,32].…”

Section: Tendero and Osher Res Math Sci (2016) 3:4mentioning

confidence: 99%

“…An extension for space-dependent blur is investigated in [28]. Methods to find better or optimal sequences as investigated in [12][13][14]22,23,39] or in [15] that aims at adapting the sequence to the velocity. Diverse implementations of the method are presented in [17,18,24,32].…”

Section: Introductionmentioning

confidence: 99%

On a mathematical theory of coded exposure

Tendero

Osher

2016

Res Math Sci

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“…In addition, α is small on this set. Thus, we can apply Algorithm 1 by modifying α (or w) only slightly by replacing (4.2) by Algorithm 1, for which [44] also provides a peer-reviewed implementation and online demo, will be applied in section 6.2 to several classic (patented or not) codes to uncover their underlying velocity distribution ρÔvÕ.…”

Section: The Reverse Pathmentioning

confidence: 99%

“…Section 6.2 gives the reverse engineering of classic flutter shutter codes in the literature. We refer to [44] for a detailed pseudocode and an implementation.…”

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A Theory of Optimal Flutter Shutter for Probabilistic Velocity Models

Tendero¹,

Morel²

2016

SIAM J. Imaging Sci.

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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

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