## Spatio-temporal Hotelling observer for signal detection from image sequences

Optics Express, Vol. 17, Issue 13, pp. 10946-10958 (2009)

http://dx.doi.org/10.1364/OE.17.010946

Acrobat PDF (179 KB)

### Abstract

Detection of signals in noisy images is necessary in many applications, including astronomy and medical imaging. The optimal linear observer for performing a detection task, called the Hotelling observer in the medical literature, can be regarded as a generalization of the familiar prewhitening matched filter. Performance on the detection task is limited by randomness in the image data, which stems from randomness in the object, randomness in the imaging system, and randomness in the detector outputs due to photon and readout noise, and the Hotelling observer accounts for all of these effects in an optimal way. If multiple temporal frames of images are acquired, the resulting data set is a spatio-temporal random process, and the Hotelling observer becomes a spatio-temporal linear operator. This paper discusses the theory of the spatio-temporal Hotelling observer and estimation of the required spatio-temporal covariance matrices. It also presents a parallel implementation of the observer on a cluster of Sony PLAYSTATION 3 gaming consoles. As an example, we consider the use of the spatio-temporal Hotelling observer for exoplanet detection.

© 2009 Optical Society of America

## 1. Introduction

18. H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective Assessment of Image Quality. III. ROC Metrics, Ideal Observers, and Likelihood-generating Functions,” J. Opt. Soc. Am. A **15**, 1520–1535 (1998).
[CrossRef]

20. S. H. Park, J. M. Goo, and C.-H. Jo, “Receiver Operating Characteristic (ROC) Curve: Practical Review for Radiologists,” Korean J. Radiol. **5**, 11–18 (2004).
[CrossRef] [PubMed]

19. J. A. Hanley and B. J. McNeil, “The Meaning and Use of the Area Under a Receiver Operating Characteristic (ROC) Curve,” Radiology **143**, 29–36 (1982).
[PubMed]

22. H. Hotelling, “The Generalization of Student’s Ratio,” Ann. Math. Stat. **2**, 360–378 (1931).
[CrossRef]

## 2. The spatio-temporal Hotelling observer

**g**^{(1)}, …,

*g*^{(J)}} be a collection of

*J*images of the same object

*f*(assumed to be independent of time) taken over time. Each

**g**^{(j)}is, in turn, a collection of

*M*pixel intensities {

*g*

^{(j)}

_{1}, …,

*g*

*}. The data set {*

^{(j)}_{M}

**g**^{(1)}, …,

**g**^{(J)}} represents the intensities of a total of

*MJ*pixels, which we raster-scan and represent in compact form as the

*MJ*×1 vector

*.*

**G***, an observer must decide whether the object*

**G***f*that produced

*belongs to either the “signal-absent” class Γ*

**G**_{0}or to the “signal-present” class Γ

_{1}. The observer can also be said to decide between hypothesis

*H*

_{0}where the signal is absent, or hypothesis

*H*

_{1}where the signal is present. In any case, the observer evaluates a real-valued non-random function

*t*on the random data

*and compares*

**G***t*(

**) to a threshold**

*G**τ*. If

*t*(

*)>*

**G***τ*, hypothesis

*H*

_{1}is assumed. Otherwise, hypothesis

*H*

_{0}is concluded. All of the possible outcomes and associated terminology are summarized in Table 1.

*is a random vector, so*

**G***t*(

*) is a random variable. For any fixed value of*

**G***τ*, the decision taken depends on

*t*(

*), so it is a random variable as well. We can consider its probability density functions pr(*

**G***t*|

*H*

_{0}) and pr(

*t*|

*H*

_{1}) given, respectively, hypothesis

*H*

_{0}or

*H*

_{1}. These two densities allow us to formally define the true positive fraction (TPF) and the false positive fraction (FPF) as

*τ*.

*τ*, different values of TPF(

*τ*) and FPF(

*τ*) are obtained; a plot of TPF(

*τ*) versus FPF(

*τ*) as

*τ*is varied over the real line is called a receiver operating characteristic (ROC) curve [18

18. H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective Assessment of Image Quality. III. ROC Metrics, Ideal Observers, and Likelihood-generating Functions,” J. Opt. Soc. Am. A **15**, 1520–1535 (1998).
[CrossRef]

20. S. H. Park, J. M. Goo, and C.-H. Jo, “Receiver Operating Characteristic (ROC) Curve: Practical Review for Radiologists,” Korean J. Radiol. **5**, 11–18 (2004).
[CrossRef] [PubMed]

19. J. A. Hanley and B. J. McNeil, “The Meaning and Use of the Area Under a Receiver Operating Characteristic (ROC) Curve,” Radiology **143**, 29–36 (1982).
[PubMed]

*t*(

*):*

**G***t*(

*)〉*

**G**_{G|Hi}denotes the statistical expectation of the random variable

*t*(

*) conditioned to the knowledge that hypothesis*

**G***H*is true. Similarly, Var{

_{i}*t*(

*)|*

**G***H*} is the variance of

_{i}*t*(

*) under the hypothesis*

**G***H*. In (2), hypotheses

_{i}*H*

_{0}and

*H*

_{1}are assumed equiprobable. The AUCt(G) is a known, monotonic function of SNR

_{t(G)}if

*t*(

*) is normally distributed [21].*

**G***λ*(

*)=lnΛ(*

**G***). However, we note that Λ(*

**G***) requires knowledge of the multivariate densities pr(*

**G****|**

*G**H*), which are usually unknown or difficult to estimate. A more viable alternative can be found by restricting attention to linear observers, i.e., observers of the form

_{i}*t*(

*)=*

**G**

*W*^{T}

*, for an appropriate template vector*

**G***of the same size of*

**W***. Here, the symbol T denotes the transpose of a vector or matrix, so*

**G**

**W**^{T}

*is a scalar product. The optimal template vector can be derived by substituting*

**G***t*(

*)=*

**G**

**W**^{T}

*G*in (2) and maximizing SNR

^{2}

*(*

_{t}*) with respect to*

**G***. The resulting template vector, which we call the Hotelling template vector [21, 22*

**W**22. H. Hotelling, “The Generalization of Student’s Ratio,” Ann. Math. Stat. **2**, 360–378 (1931).
[CrossRef]

**K**

*is the covariance matrix of the data vector*

**G***and*

**G**

**W**_{Hot}is the Hotelling observer, defined as

*t*

_{Hot}(

*)=*

**G**

**W**^{T}

_{Hot}

*. The Hotelling observer is also called a prewhitening matched filter [21], and the prewhitening operation is both correcting for the correlation in a single image and also undoing the frame-to-frame correlation. Note that the Hotelling observer*

**G***t*

_{Hot}(

*) defined above is applied to spatio-temporal data, as opposed to the classical use of the Hotelling observer in the case of purely spatial data.*

**G**## 3. Analysis of the data covariance matrix

23. H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, “Objective Assessment of Image Quality: IV. Application to Adaptive Optics,” J. Opt. Soc. Am. A **23**, 3080–3105 (2006).
[CrossRef]

*:*

**G***represents the sequence {*

**P**

**p**^{(1)}, …,

**p**^{(J)}} of point spread functions (PSFs). We will allow the PSFs to be random, as in the adaptive optics problem. The PSFs will be assumed known statistically (PSF-known-statistically or PKS [21]), and their contribution

**K**̄

^{PSF}

*̄ to the data covariance matrix will be estimated by means of simulated data. It is important to note that full knowledge of the statistical properties of the PSFs is not needed. Instead, the Hotelling observer requires the knowledge of only the mean signal*

_{G}*and the data covariance matrix*

**S****K**. As long as these two quantities can be estimated with sufficient accuracy, the Hotelling observer will deliver high performance. No moments higher than the second are needed.

_{G}*and*

**P***f*. The resulting quantity is averaged over

*given*

**P***f*, and, finally, we average over

*f*. From (6) and adding and subtracting terms appropriately [23

23. H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, “Objective Assessment of Image Quality: IV. Application to Adaptive Optics,” J. Opt. Soc. Am. A **23**, 3080–3105 (2006).
[CrossRef]

24. E. Clarkson, “Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks,” J. Opt. Soc. Am. A **24**, B91–B98 (2007).
[CrossRef]

25. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the Hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A **24**, B13–B24 (2007).
[CrossRef]

*σ*

^{2}

*is the readout noise variance for the*

_{m}*m*-th pixel of the detector, and

*m*-th pixel of the

*j*-th image. The appropriateness of the Poisson model for the photon noise is justified by invoking the so-called Poisson postulates [21]. The readout noise variance

*σ*at each detector pixel is usually known, as provided by the detector’s manufacturer, or it can be measured. Matrix

^{2}_{m}**K**̿

^{noise}

*above is diagonal with no zero terms on the diagonal, which guarantees the invertibility of*

**G****K**

*.*

_{G}## 4. Estimation of the Hotelling template vector

*L*

_{1}realization of the PSF sequence

*and*

**P***L*

_{2}realization of a random signal. Consider

*L*

_{1}

*L*

_{2}simulated noiseless data sets

*̄*

**G**^{(ℓ1,ℓ2)}1,

*ℓ*

_{1}=1, …,

*L*

_{1},

*ℓ*

_{2}=1, …,

*L*

_{2}for the hypothesis

*H*

_{1}and, similarly,

*L*

_{1}noiseless data sets

*̄*

**G**^{(ℓ1)}

_{0}, for

*ℓ*

_{1}=1, …,

*L*

_{1}for the hypothesis

*H*

_{0}.

**K**̄

^{PSF}

*̄ is estimated from simulated data as well:*

_{G}**K**in (6) from noisy simulated data, without using the decomposition (4). A necessary (but not sufficient) condition for such estimate

_{G}**K̂**to be nonsingular is that the number

_{G}*L*=

*L*

_{1}

*L*

_{2}of simulated noisy sequences must be greater than

*MJ*, the order of

**K**itself. If, for example, each image

_{G}

**g**^{(j)}is of size 64×64 and there are 25 of them in each sequence

*, then*

**G***MJ*=64

^{2}·25≈10

^{5}. Simulating such a huge number of image sequences is clearly prohibitive. Instead, the decomposition (4), along with (7), guarantees the invertibility of

**K̂**. This can be proved by noting that

_{G}**K̂**is symmetric and (strictly) positive definite. Indeed:

**G****x**≠

**0**.

**K̂**]

_{G}^{−1}exists, the matrix we need to invert is huge, so computing the inverse by means of standard algorithms (such as Gaussian elimination) is computationally prohibitive [27]. Even if we could invert

**K̂**, storing it will require an incredible amount of disk space. However, we recognize the particular structure of the PSF covariance matrix [see (8)] and consider an algorithm that takes advantage of it. Indeed, if we introduce the matrix

_{G}**R**whose elements are

*MJ*×

*L*matrix

**R**contains in the

*ℓ*-th column the

*MJ*×1 vector obtained by raster-scanning the pixels in Δ

*̄*

**G**^{(ℓ)}

_{0}. The Woodbury matrix-inversion lemma [28

28. D. J. Tylavsky and G. R. L. Sohie, “Generalization of the Matrix Inversion Lemma,” Proc. IEEE **74**, 1050–1052 (1986).
[CrossRef]

31. H. V. Henderson and S. R. Searle, “On Deriving the Inverse of a Sum of Matrices,” SIAM Rev. **23**, 53–60 (1981).
[CrossRef]

**I**

*is the identity matrix of order*

_{N}*N*. If

**Q**is of size

*L*×

*L*and invertible. Note that

*L*is usually much smaller than

*MJ*, which implies that

**Q**

^{−1}can be calculated in much shorter time than [

**K̂**]

_{G}^{−1}. Standard Gaussian elimination with pivoting is a fast and numerically stable way to invert

**Q**. Overall, using (12) is a more tractable and stable problem than computing (11) directly.

## 5. Implementation

32. J. A. Kahle, M. N. Day, H. P. Hofstee, C. R. Johns, T. R. Maeurer, and D. Shippy, “Introduction to the Cell Multiprocessor,” IMB J. Res. Dev. **49**, 589–604 (2005).
[CrossRef]

35. S. Williams, J. Shalf, L. Oliker, S. Kamil, P. Husbands, and K. Yelick, “The potential of the Cell Processor for scientific computing,” in *Proceedings of the 3rd conference on Computing Frontiers*, pp. 9–20 (ACM, New York, NY, 2006).
[CrossRef]

39. M. Kachelrieß, M. Knaup, and O. Bockenbach, “Hyperfast parallel-beam and cone-beam backprojection using the Cell general purpose hardware,” Med. Phys. **34**, 1474–1486 (2007).
[CrossRef] [PubMed]

*̂ WHot according to (10) and (12) was implemented and run on the PLAYSTATION 3 cluster. The algorithm is composed of two different programs: one to be run as a master process and one to be run as a slave process.*

**W***J*←number of images in each sequence

*L*←number of simulated sequences

*N*←number of slave processes

**for all**

*n*∈{1, …,

*N*}

**do**

*initialize*” to slave process

*n*

**end for**

**for all**

*ℓ*∈ {1, …,

*L*}

**do**

*̄(ℓ)*

**G**_{0}from the disk

**for all**

*n*∈{1, …,

*N*}

**do**

**̄**

*G*^{(ℓ)}

_{0}to slave process

*n*

**end for**

**end for**

**for all**

*ℓ*∈{1, …,

*L*}

**do**

*G*̄

^{(ℓ)}

_{1}from the disk

**for all**

*n*∈{1, …,

*N*}

**do**

*̄*

**G**^{(ℓ)}

_{1}to slave process

*n*

**end for**

**end for**

*j*←1

*m*←0 {number of tasks completed}

**while**

*m*<

*J*

**do**

**while**

*m*<

*J*and there is an idle slave process

**do**

*n*←index of an idle slave process

*compute*[

*̂*

**W**_{Hot}]

^{(j)}” to slave process n

*j*←

*j*+1

**end while**

**if**

*m*<

*J*

**then**

*n*←slave process that has just computed [

*̂*

**W**_{Hot}]

^{(j′)}

*̂*

**W**_{Hot}]

^{(j′)}from slave process n

*̂*

**W**_{Hot}]

^{(j′)}to disk

*m*←

*m*+1

**end if**

**end while**

**for all**

*n*∈{1, …,

*N*}

**do**

*end of computation*” to slave process

*n*

**end for**

*J*←number of images in each sequence

*L*←number of simulated sequences

*σ*

^{2}

*←readout noise variance for all*

_{m}*m*∈ {1, …,

*M*}

**for all**

*ℓ*∈{1, …,

*L*}

**do**

*̄*

**G**^{(ℓ)}

_{0}from master process

*̄*

**G**^{(ℓ)}0 to disk

**end for**

**for all**

*ℓ*∈ {1, …,

*L*}

**do**

*G*̄

^{(ℓ)}

_{1}from master process

*̄*

**G**^{(ℓ)}

_{1}to disk

**end for**

**for all**

*j*∈ {1, …,

*J*}

**do**

**end for**

**Q**

^{−1}←inverse(

**Q**)

**while**not message “

*end of computation*” received

**do**

*compute*[

*̂*

**W**_{Hot}]

^{(j)}” from master process

*̂*

**W**_{Hot}]

^{(j)}←0

_{M×M}

**for all**

*j*′ ∈{1, …,

*J*}

**do**

*̂*

**W**_{Hot}]

^{(j)}←[

*̂*

**W**_{Hot}]

^{(j)}+

^{T}

^{(j, j′)}[

*̂]*

**S**^{(j′)}

**end for**

*̂*

**W**_{Hot}]

^{(j)}to master process

**end while**

**K̂**]

_{G}^{−1}as they were needed. Splitting the work load among all of the processing cores available on each PLAYSTATION 3 allowed more than a 15-fold reduction in the computation time with respect to an implementation that uses only one core and ignores their SIMD capabilities. Had we used single-precision values, the speed-up would have been even larger.

## 6. Simulation results

14. H. W. Babcock, “The Possibility of Compensating Astronomical Seeing,” Publ. Astron. Soc. Pac. **65**, 229–236 (1953).
[CrossRef]

*C*profile, given by [40

^{2}_{n}40. R. R. Parenti and R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A **11**, 288–309 (1994).
[CrossRef]

*C*trascurable when

^{2}_{n}(h)*h*≥24km. The Fried parameter

*r*

_{0}[41

41. D. L. Fried, “Statistics of a Geometric Representation of Wavefront Distortion,” J. Opt. Soc. Am **55**, 1427–1435 (1965).
[CrossRef]

*λ*=500nm. Our simulation took into account effects such as scintillation and anisoplanetism [15] as well. The wind speed was 1.25m/s. The telescope we simulated had a circular aperture of diameter 5m, and the diameter of the central obscuration due to the secondary mirror was 0.50m. The secondary mirror was supported by three arms. For the estimation of the template vector

*̂*

**W**_{Hot}in (10), we simulated

*L*=512 sequences containing

*J*=25 images each of size 64×64 (pixel size 5.96µm). The wavefront sensor apparatus of many AO systems includes a lenslet array. In our simulation, we simulated a 32×32 array of side length 0.02m. The total power entering the telescope was equally split between the wavefront sensor and the science camera by a 50/50 beamsplitter. The system was assumed idea, with an efficiency of 1 (i.e., no losses) and the AO loop was running at a speed of 1kHz. The quality of the AO correction can be quantified with an average Strehl ratio of about 0.67.

*H*

_{0}and

*H*

_{1}. The apparent magnitude of the star was

*m*=6 and the difference in apparent magnitude between the planet and the star was Δ

*m*=16.74. More specifically, we simulated

*L*=512 sequences for each hypothesis and, for each sequence, we used

*J*=25 frames containing

*M*=64

^{2}pixels each. The simulation was set up in order to mimic an astronomical observation. For example, we used typical values for the readout noise as found in [42

42. M. P. Fitzgerald and J. R. Graham, “Speckle Statistics in Adaptively Corrected Images,” Astrophys. J. **637**, 541–547 (2006).
[CrossRef]

*̂*

**W**_{Hot}was noiseless and no noise was considered in the wavefront sensor. On the other hand, data on which the detection task was applied were noisy and were obtained with different phase screens.

*t*

_{Hot}(

*) and the purely spatial matched filter [43*

**g**43. G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory **6**, 311–329 (1960).
[CrossRef]

*t*

_{mat}(

*)=*

**g**

**s**^{T}

*, where*

**g***is the signal to be detected. In an effort to mimic current practice in astronomical detection, both*

**s***t*

_{Hot}(

*) and*

**g***t*

_{mat}(

*) were applied to long-exposure data*

**g***obtained by on-chip integration of many short-exposure frames. The observers were run on simulated noisy data. We generated test noise-free image sequences for both the planet-absent and planet-present hypotheses and degraded them with Poisson photon noise and Gaussian readout noise to generate*

**g***n*=10,000 noisy sequences of short-exposure images for the planet-absent hypothesis and

*n*noisy sequences short-exposure images for the planet-present hypothesis. The corresponding long-exposure images were generated as well. These test data were supplied to the observers considered in this comparison, and the corresponding values of the test statistics

*t*

^{(1)}

_{0}, …,

*t*

^{(n)}

_{0}and

*t*

^{(1)}

_{1}, …,

*t*

^{(n)}

_{1}were collected. Binning these values would provide approximated plots of the densities pr(

*t*|

*H*

_{0}) and pr(

*t*|

*H*

_{1}) for a particular observer

*t*. For each observer, we estimated the values of the TPF and FPF [see (1)] as:

*S*| stands for the number of elements of the set

*S*, and we varied the value of

*τ*to obtain ROC curves. The ROC curves are reported in Fig. 1, and the corresponding values of the AUC, standard deviation

*σ*on the AUC, and SNR are reported in Table 2. The SNR for the three observers was computed according to (2), in which conditional means 〈…〉 and variances Var{…} were replaced by the sample means and sample variances of the

*t*

^{(i)}

_{0}and

*t*

^{(i)}

_{1}. The values of

*σ*were computed as described in [45

45. B. D. Gallas, “One-shot estimate of MRMC Variance: AUC,” Acad. Radiol. **13**, 353–362 (2006).
[CrossRef] [PubMed]

25. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the Hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A **24**, B13–B24 (2007).
[CrossRef]

25. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the Hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A **24**, B13–B24 (2007).
[CrossRef]

46. E. Bertin and S. Arnouts, “SEXTRACTOR: Software for source extraction,” Astron. Astrophys. Suppl. Ser. **117**, 393–404 (1996).
[CrossRef]

## 7. Conclusions

## Acknowledgments

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31. | H. V. Henderson and S. R. Searle, “On Deriving the Inverse of a Sum of Matrices,” SIAM Rev. |

32. | J. A. Kahle, M. N. Day, H. P. Hofstee, C. R. Johns, T. R. Maeurer, and D. Shippy, “Introduction to the Cell Multiprocessor,” IMB J. Res. Dev. |

33. | D. Pham, T. Aipperspach, D. Boerstler, M. Bolliger, R. Chaudhry, D. Cox, P. Harvey, P. Harvey, H. Hofstee, C. Johns, J. Kahle, A. Kameyama, J. Keaty, Y. Masubuchi, M. Pham, J. Pille, S. Posluszny, M. Riley, D. Stasiak, O. Suzuoki, M. Takahashi, J. Warnock, S. Weitzel, D. Wendel, and K. Yazawa, “Overview of the architecture, circuit design, and physical implementation of a first-generation Cell Processor,” IEEE J. Solid-State Circuits |

34. | P. Hofstee, “Introduction to the Cell Broadband Engine,” Tech. rep., IBM Corporation, Riverton, NJ (2005). |

35. | S. Williams, J. Shalf, L. Oliker, S. Kamil, P. Husbands, and K. Yelick, “The potential of the Cell Processor for scientific computing,” in |

36. | D. A. Bader, V. Agarwal, K. Madduri, and S. Kang, “High performance combinatorial algorithm design on the Cell Broadband Engine processor,” Parallel Comput. |

37. | I. W. C. Benthin, M. Scherbaum, and H. Friedrich, “Ray tracing on the Cell Processor,” in |

38. | M. Sakamoto and M. Murase, “Parallel implementation for 3-D CT image reconstruction on Cell Broadband Engine |

39. | M. Kachelrieß, M. Knaup, and O. Bockenbach, “Hyperfast parallel-beam and cone-beam backprojection using the Cell general purpose hardware,” Med. Phys. |

40. | R. R. Parenti and R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A |

41. | D. L. Fried, “Statistics of a Geometric Representation of Wavefront Distortion,” J. Opt. Soc. Am |

42. | M. P. Fitzgerald and J. R. Graham, “Speckle Statistics in Adaptively Corrected Images,” Astrophys. J. |

43. | G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory |

44. | D. Middleton, |

45. | B. D. Gallas, “One-shot estimate of MRMC Variance: AUC,” Acad. Radiol. |

46. | E. Bertin and S. Arnouts, “SEXTRACTOR: Software for source extraction,” Astron. Astrophys. Suppl. Ser. |

47. | L. Caucci, “Point Detection and Hotelling Discriminant: An Application in Adaptive Optics,” Master’s thesis, University of Arizona (2007). |

48. | C. K. Abbey and H. H. Barrett, “Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability,” J. Opt. Soc. Am. A |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(030.4280) Coherence and statistical optics : Noise in imaging systems

(110.2970) Imaging systems : Image detection systems

(110.3000) Imaging systems : Image quality assessment

(110.4155) Imaging systems : Multiframe image processing

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 20, 2009

Revised Manuscript: May 22, 2009

Manuscript Accepted: June 9, 2009

Published: June 16, 2009

**Virtual Issues**

Vol. 4, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Luca Caucci, Harrison H. Barrett, and Jeffrey J. Rodriguez, "Spatio-temporal Hotelling observer
for signal detection from image
sequences," Opt. Express **17**, 10946-10958 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10946

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