Uniform Motion Blur in Poissonian Noise: Blur/Noise Tradeoff

Boracchi, Giacomo; Foi, Alessandro

doi:10.1109/tip.2010.2062196

Cited by 22 publications

(34 citation statements)

References 15 publications

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“…If the exposure time is too long and the blur support exceeds two pixels, then the blur is no more invertible. In that case, the restoration process is an ill-posed problem [4]. The flutter shutter [1,2,3,12,10] (coded exposure) permits to ensure an invertible motion kernel for arbitrarily severe uniform motion blur.…”

Section: Analog and Numerical Flutter Shutter Methodsmentioning

confidence: 99%

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The Flutter Shutter Code Calculator

Tendero¹

2015

Image Processing on Line

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The goal of the flutter shutter is to make uniform motion blur invertible, by a "fluttering" shutter that opens and closes on a sequence of well chosen sub-intervals of the exposure time interval. In other words, the photon flux is modulated according to a well chosen sequence called flutter shutter code. This article provides a numerical method that computes optimal flutter shutter codes in terms of mean square error (MSE). We assume that the observed objects follow a known (or learned) random velocity distribution. In this paper, Gaussian and uniform velocity distributions are considered. Snapshots are also optimized taking the velocity distribution into account. For each velocity distribution, the gain of the optimal flutter shutter code with respect to the optimal snapshot in terms of MSE is computed. This symmetric optimization of the flutter shutter and of the snapshot allows to compare on an equal footing both solutions, i.e. camera designs. Optimal flutter shutter codes are demonstrated to improve substantially the MSE compared to classic (patented or not) codes. A numerical method that permits to perform a reverse engineering of any existing (patented or not) flutter shutter codes is also described and an implementation is given. In this case we give the underlying velocity distribution from which a given optimal flutter shutter code comes from. The combination of these two numerical methods furnishes a comprehensive study of the optimization of a flutter shutter that includes a forward and a backward numerical solution. Source CodeThe C++ source code, version 2.0, is available from the article web page 1 . The documentation is included in the archive. Basic compilation and usage instructions are included in the README.txt file. The demo permits to compute optimal flutter shutter codes for (truncated) Gaussian and uniform probability velocity distribution. It computes the optimal snapshot, as well. In this case the demo provides the ideal exposure time, taking the velocity model into account. It also computes the gain in terms of MSE of the flutter shutter compared to the optimal snapshot. This comparison permits to decide the viability of the flutter shutter apparatus for any application.1 http://dx

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Section: Analog and Numerical Flutter Shutter Methodsmentioning

confidence: 99%

“…compute the function w defined in (7) in the proposed implementation) using wpξq " |αpξq| 4 . This is implemented in routines.cpp.…”

Section: Compute and Write The Code Coefficientsmentioning

confidence: 99%

The Flutter Shutter Code Calculator

Tendero¹

2015

Image Processing on Line

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“…The photon emission follows a Poisson distribution see, e.g., [49] (tf X is a random variable that follows a Poisson distribution then all the possible realization of X are in N. In addition, the probability of the event …”

Section: A Mathematical Model Of Photons Acquisition By a Light Sensormentioning

confidence: 99%

“…In this case, the output of a pixel sensor centered at x is, in the noiseless case, We now extend our model to cope with the Poisson photon (shot) noise and then will add the readout noise. The photon emission follows a Poisson distribution see, e.g., [49] (tf X is a random variable that follows a Poisson distribution then all the possible realization of X are in N. In addition, the probability of the event X = k is P (X = k) = λ k e −λ k! , where λ > 0 is the intensity of the Poisson random variable).…”

Section: A Mathematical Model Of Photons Acquisition By a Light Sensormentioning

confidence: 99%

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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“…The support of this kernel increases linearly with the exposure time Δt 0 and the velocity v È Ê of the motion. If the exposure time is too long and the blur support exceeds two pixels, its deconvolution is an ill-posed problem [8]. Instead the flutter shutter, or coded exposure, ensures an invertible motion kernel for arbitrarily severe motion blur.…”

Section: 2mentioning

confidence: 99%

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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Uniform Motion Blur in Poissonian Noise: Blur/Noise Tradeoff

Cited by 22 publications

References 15 publications

The Flutter Shutter Code Calculator

The Flutter Shutter Code Calculator

On a mathematical theory of coded exposure

A Theory of Optimal Flutter Shutter for Probabilistic Velocity Models

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