## Coded aperture spectroscopy with denoising through sparsity |

Optics Express, Vol. 20, Issue 3, pp. 2297-2309 (2012)

http://dx.doi.org/10.1364/OE.20.002297

Acrobat PDF (785 KB)

### Abstract

We compare noise and classification metrics for three aperture codes in dispersive spectroscopy. In contrast with previous theory, we show that multiplex codes may be advantageous even in systems dominated by Poisson noise. Furthermore, ill-conditioned codes with a regularized estimation strategy are shown to perform competitively with well-conditioned codes.

© 2011 OSA

## 1. Introduction

1. M. T. E. Golay, “Multi-slit
spectrometry,” J. Opt. Soc. Am. **39**(6),
437–437 (1949). URL
http://www.opticsinfobase.org/abstract.cfm?URI=josa-39-6-437. [CrossRef] [PubMed]

3. A. Barducci, D. Guzzi, C. Lastri, V. Nardino, P. Marcoionni, and I. Pippi, “Radiometric and signal-to-noise ratio
properties of multiplex dispersive spectrometry,”
Appl. Opt. **49**(28),
5366–5373 (2010). URL
http://ao.osa.org/abstract.cfm?URI=ao-49-28-5366. [CrossRef] [PubMed]

4. A. A. Wagadarikar, M. E. Gehm, and D. J. Brady, “Performance comparison of aperture
codes for multimodal, multiplex spectroscopy,”
Appl. Opt. **46**(22),
4932–4942 (2007). URL
http://ao.osa.org/abstract.cfm?URI=ao-46-22-4932. [CrossRef] [PubMed]

5. S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, “Hadamard spectroscopy with a
two-dimensional detecting array,” Appl.
Opt. **32**(34),
7095–7105 (1993). URL
http://ao.osa.org/abstract.cfm?URI=ao-32-34-7095. [CrossRef] [PubMed]

5. S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, “Hadamard spectroscopy with a
two-dimensional detecting array,” Appl.
Opt. **32**(34),
7095–7105 (1993). URL
http://ao.osa.org/abstract.cfm?URI=ao-32-34-7095. [CrossRef] [PubMed]

6. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding
for multimodal, multiplex spectroscopy,”
Appl. Opt. **45**(13),
2965–2974 (2006). URL
http://ao.osa.org/abstract.cfm?URI=ao-45-13-2965. [CrossRef] [PubMed]

7. M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral
imaging with a dual-disperser architecture,”
Opt. Express **15**(21),
14,013–14,027 (2007). URL
http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013. [CrossRef]

8. A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded
aperture snapshot spectral imaging,” Appl.
Opt. **47**(10),
B44–B51 (2008). URL
http://ao.osa.org/abstract.cfm?URI=ao-47-10-B44. [CrossRef] [PubMed]

9. R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Realization of the hadamard multiplex
advantage using a programmable optical mask in a dispersive flat-field
near-infrared spectrometer,” Appl.
Spectrosc. **54**(12),
1751–1758 (2000). URL
http://as.osa.org/abstract.cfm?URI=as-54-12-1751. [CrossRef]

*MSE*decreases by a factor of

*N*/4, where

*N*is the dimension of the code, for independent Gaussian noise limited systems, the

*MSE*increases by a factor of two, regardless of the code dimension, for Poisson noise limited systems. These results, however, apply only to the linear unbiased estimator, which was assumed to be the Moore-Penrose pseudo-inverse. While the computing power in the 1970’s limited the algorithms available to Harwit and Sloane, they noted “... there are also good arguments in favor of a biased estimate...” This paper studies Harwit and Sloane’s suggestion using modern non-linear estimators. In using regularization techniques based on a sparsity prior, certain solutions are favored over others. Regularization allows for measurement energy to be focused into a subspace. The rest of the space is filled in by the prior knowledge of the signal class. These biased estimators could reveal codes previously determined to be suboptimal to outperform optimal codes for the linear unbiased estimator.

## 2. Signal Estimation

### 2.1. Mathematical Framework

**g**is the measured data from object

**f**, sensing matrix

**H**and noise

**n**. There are two goals of linear measurement which are not mutually exclusive. The first goal is to optimally select a measurement system which defines

**H**and therefore the mathematical properties of the system. The second goal is to select an estimator given knowledge of

**H**and the class of signals from which

**f**is drawn. Selecting an estimator independent of knowledge of

**f**would be to select a linear unbiased estimator. The results for the linear unbiased estimator are the results presented in [2], and are uniquely determined by the condition number and throughput, the amount of energy transferred, of the matrix

**H**[10]. This paper is concerned with an estimator for a special class of signals from which

**f**may be drawn. This class, the class of sparse signals, has special properties which allow for efficient denoising [11

11. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis
pursuit,” SIAM J. Sci. Comput.
(USA) **20**, 33–61
(1998). [CrossRef]

12. J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative
shrinkage/thresholding algorithms for image
restoration,” IEEE Trans. Image
Process. **16**(12), 2992
–3004 (2007). [CrossRef] [PubMed]

**f̂**is the estimate,

*τ*is the weight of the sparsity inducing prior, classically a Laplacian distribution, and

**Ψ**is the inverse transform from the basis on which

**f**is sparse to the canonical basis. The

*l*

_{2}norm in the functional is from a Gaussian negative log-likelihood function. The noise models observed in optics are most often a Poisson noise model, or a mixed Poisson and Gaussian noise model. Clearly the more optimal negative log-likelihood function would be the Poisson negative log-likelihood function. An even better solution would be to use a minimum mean squared error estimator (MMSE) for the distributions of interest. Since there is no simple analytical form for the posterior distribution with the priors and penalties we have chosen, computing a MMSE would be impractical, so we do not consider it. For the Poisson only noise model the negative log-likelihood to be used in a more optimal MAP estimator is where

**e**

*is a canonical basis vector with the*

_{i}*i*

^{th}component is non-zero, and

*g*is the

_{i}*i*

^{th}component of the measurement [13]. However, the results from [13] from solving for the MAP estimate with the optimal negative log-likelihood are not staggeringly better than solving for the Gaussian MAP estimate, at most 7 percent error improvement was demonstrated in [13], and the algorithms require non-negativity of the elements of the matrix

**H**. While

**H**contains only non-negative entries in any singly encoded coded aperture spectrometer, the structure of the principal components of the data, along with the structure of the measurement matrix, may make the system amenable to truncated-SVD, truncated singular value decomposition, reconstructions as described in chapter 8 of [14]. Truncated-SVD aids the reconstruction process, because the components of the data which correspond to weak singular vectors are removed, which makes the reconstruction more reliable. The system matrix of reduced dimensionality may have negative entries, and is therefore not usable in the framework laid by [13]. When using truncated-SVD the objective function becomes where

**S**

^{†}

*denotes the pseudo-inverse of the*

_{n}*n*largest components of the diagonal matrix

**S**, and

**H**. It is important to note that truncating the singular values of the matrix makes the inversion ill-posed, but not unsolvable. Because the signals we are trying to estimate are sparse, they can be estimated on their full dimension using the theory of compressive measurement [15

15. D. Donoho, “Compressed
sensing,” IEEE Trans. Inf. Theory **52**(4), 1289
–1306 (2006). [CrossRef]

### 2.2. Simulation Setup

*n*+ 1) where

*n*is the dimension of the spectrum to be measured. The pseudo-random matrix is a matrix with half the entries equal to one, and half the entries equal to zero uniformly distributed throughout the matrix. The sparse data set consists of 9,10, or 11 randomly placed spikes of random heights in a spectrum of 127 different wavelengths. The reflectance set is downloadable from http://www.planetary.brown.edu/relab/. It is taken from a Bidirection Visible Near Infrared Spectrometer and is downsampled to 5nm resolution on a 450–1080nm window, which leads to a signal of 127 dimensions. The data is denoised before simulated measurement by a total variation(TV) method [16

16. A. Chambolle, “An algorithm for total variation
minimization and applications,” J. Math.
Imaging Vision **20**, 89–97
(2004). URL http://dx.doi.org/10.1023/B:JMIV.0000011325.36760.1e. [CrossRef]

12. J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative
shrinkage/thresholding algorithms for image
restoration,” IEEE Trans. Image
Process. **16**(12), 2992
–3004 (2007). [CrossRef] [PubMed]

*σ*is

_{g}*σ*of the mean spike value for the synthetic set, and

_{p}*σ*is

_{g}*σ*of the mean signal value for the real data set. The reason for setting the levels of the Gaussian standard deviation as such is that this situation represents the transition point between a Poisson limited and Gaussian limited measurement for the slit spectrometer. We are not concerned with the zero-level signal locations in the canonically sparse signals, because they do not contribute to noise in the Poisson limited slit spectrometer measurements. The smoothly-varying signals on the other hand are assumed to be approximately constant, and therefore the transition between a Poisson limited and Gaussian limited measurement is assumed to happen when the Gaussian noise standard deviation is set to the square root of the mean value of the signal. For example, if the canonical data is being measured with 1000 photons on average, then each spike generates 100 photons on average. This generates a shot noise level of 10 photons for each peak, which is what the

_{p}*σ*is set to for the mixed noise measurements. On the other hand, if the reflectance data is being measured, it is assumed to have uniform intensity throughout the spectrum. Therefore, a signal level of 10000 photons generates about 80 photons per band so the

_{g}*σ*is set to 9 photons. The only difference in the estimation process is the denoising step is canonical

_{g}*l*

_{1}, i.e.

**Ψ**is the identity matrix, for the synthetic data, and is a total variation penalty, i.e.

**Ψ**

^{−1}transforms

**f**into a gradient pseudo-basis, for the reflectance data. In measuring the performance, each signal is first estimated using the optimal regularization parameter

*τ*, as determined by lowest

*MSE*, from (4) on a training set. This means that each signal is reconstructed for many

*τ*over a large range. (For the pseudo-random matrix the optimal truncation level is selected as well.) After the performance is evaluated on the training data sets with optimal parameters, a new random draw of one hundred signals from each data set is performed generating a disjoint test set, and the process is repeated again using only the most commonly used parameters for each measurement system, signal level, noise type combination as determined by the results from the training set data.

*MSE*, and is defined by The baseline for comparison is established by using the linear unbiased estimator on the same data. The results of these control experiments should match the predictions of [2].

### 2.3. Simulation Results

*MSE*was a factor of two lower for the slit spectrometer versus the cyclic-s spectrometer for nearly all input signal levels and for both data sets. The ill-conditioned pseudo-random coded spectrometer performs orders of magnitude worse than the well-conditioned systems. When a mixed noise model is employed, however, the

*MSE*deteriorates by a substantially greater amount for the slit spectrometer than the cyclic-s spectrometer. For the synthetic data, the cyclic-s spectrometer outperforms the slit spectrometer by a large margin, and for the real reflectance data the cyclic-s spectrometer performs comparably well to the slit spectrometer. Recall the Gaussian signal level was chosen to be on the same order as the Poisson noise in the

*native*signals. The multiplexed signals, however, have many, many more photons per measurement. So while the additive noise ramps quickly for the slit spectrometer, the multiplexed spectrometers effectively stay photon-limited. Clearly as more and more Gaussian noise is added to the point where the noise model is Gaussian dominated the multiplexing advantage will be recognized to its full extent.

*MSE*, and the test set, in which the most commonly used parameters from the test set are exclusively employed, are considered in the convex optimization analysis and represent two different measurement scenarios. The training parameter set represents having each spectrum reconstructed by a skilled operator. This is common practice and the results shown in [17

17. D. Kittle, K. Choi, A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded
aperture snapshot spectral imagers,” Appl.
Opt. **49**(36),
6824–6833 (2010). URL
http://ao.osa.org/abstract.cfm?URI=ao-49-36-6824. [CrossRef] [PubMed]

*l*

_{1}minimization impractical. Another main result is that in the presence of Poisson only noise the multiplexed systems perform better relative to the slit spectrometer than the predictions of [2] suggest. In some cases in this empirical study of two sparse signal classes the multiplexed systems even outperform the slit system in the presence of Poisson only noise.

5. S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, “Hadamard spectroscopy with a
two-dimensional detecting array,” Appl.
Opt. **32**(34),
7095–7105 (1993). URL
http://ao.osa.org/abstract.cfm?URI=ao-32-34-7095. [CrossRef] [PubMed]

*l*

_{1}denoising optimally removes the noise on the zero components. Once Gaussian noise is added the effect become even more pronounced. While the

*l*

_{1}denoising allows for the slit code to remove noise from the zero components similarly to the multiplexed codes, the less accurate matching of the peak intensities demonstrated in the spectroscopy discussions in chapter 9 of [14] becomes more apparent. The one exception to this is on the cross-validation data the pseudo-random spectrometer performance is unpredictable relative to the slit spectrometer. Since the pseudo-random spectrometer reconstructions require two parameters, the effects of non-optimal parameter selection are more strongly felt.

17. D. Kittle, K. Choi, A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded
aperture snapshot spectral imagers,” Appl.
Opt. **49**(36),
6824–6833 (2010). URL
http://ao.osa.org/abstract.cfm?URI=ao-49-36-6824. [CrossRef] [PubMed]

## 3. Signal Classification

### 3.1. Mathematical Framework

*μ*_{1},

*μ*_{2}} and covariances {

**Σ**

_{1},

**Σ**

_{2}}. Unfortunately, the difference in means may be small relative to the covariances of the samples and power of the measured noise. This would make it difficult for an observer to tell from which distribution each sample was drawn from trying to distinguish reconstructed spectra with the naked eye. For this reason a computational method of distinguishing the spectra, or classifying them, is employed. Any classifier needs a set of labeled, or pre-classified data, off of which it will base future classifications. However, acquiring the labels of the pre-classified data is expensive, else the researcher would exclusively classify the data through those outside means. We sketch one such process of selecting data to be labeled, and how to use that labeled data to build an accurate classifier. This is not intended to be a full discussion of the inner workings of classification, detailed discussions of the methods of this work can be found in [19

19. S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897. [CrossRef]

20. Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active
selection of labeled data: application to sensing subsurface
UXO,” IEEE Trans. Geosci. Remote
Sens. **42**(11),
2535–2543 (2004). [CrossRef]

19. S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897. [CrossRef]

20. Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active
selection of labeled data: application to sensing subsurface
UXO,” IEEE Trans. Geosci. Remote
Sens. **42**(11),
2535–2543 (2004). [CrossRef]

**x**

*, drawn from data set*

_{i}**X**= {

**x**

_{1}...

**x**

*}, the corresponding label*

_{N}*t*is assigned by The weights,

_{i}**w**, weigh the importance of matching each basis function,

**b**

*. How closely a vector*

_{j}**x**

*matches a basis function*

_{i}**b**

*is determined by the radial basis function*

_{j}*K*(

**x**

*,*

_{i}**b**

*).*

_{j}**t**, for the remaining unlabeled data in

**X**. A reliable classifier would minimize the entropy of the posterior distribution of the weights

*p*(

**w**|

**t**), where the labels

**t**correspond to a small subset of the data which is labeled. Assuming a Gaussian distribution on the prior and posterior distribution of

**w**with constant diagonal covariance, minimizing the entropy of the posterior is equivalent to maximizing the determinant precision matrix

**A**Luckily, this precision matrix depends only on the set

**X**being considered, and the basis functions with which they are compared. This means we can first select the basis functions assuming the precision matrix covers all

**x**

*ɛ*

**X**. Selecting a next basis function can be done by a greedy process for reasons explained in [19

19. S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897. [CrossRef]

20. Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active
selection of labeled data: application to sensing subsurface
UXO,” IEEE Trans. Geosci. Remote
Sens. **42**(11),
2535–2543 (2004). [CrossRef]

**Φ**that best conditions the inverse of

**ΦΦ**

*.*

^{T}**B**. This yields a precision matrix from the labeled set of data where the diagonal loading is used to insure invertibility and

**Φ**

*corresponds to the columns of*

_{L}**Φ**representative of the labeled set

**X**

*. Now the goal is to add more labels once again with the goal of maximizing the determinant of the precision matrix, this time defined by*

_{L}**A**. Since

**Φ**

*, by the matrix determinant lemma the vector*

_{L}21. K. B. Petersen and M. S. Pedersen, “The matrix cookbook,” (2008). URL http://matrixcookbook.com/.

**Φ**

*to find the weights. With the weights and basis set we can accurately classify any of the remaining vectors in the class. The main advantage of this method, is that none of the vectors need to labeled until the last step. In the second step we obtain knowledge of each next best vector to be labeled, but we can label the entire labeled set in one step right before we determine the weights. This means that a researcher could obtain all the desired spectral data, and only need to make one shipment to outside labeling resources to generate the labels necessary to build the classifier.*

^{T}### 3.2. Simulation Setup

**X**. The true spectra set,

**F**, is generated by drawing 300 signals from each of the multivariate Gaussian distributions defined in table 3. The table shows that each of the signals has 9 peaks, and the two distributions similar means and covariances. The 9 peaks occur at the same wavelengths in all spectra. Each spectrum is then measured at a signal level of 100 photons, and is reconstructed using the framework of Section 2 to generate a data set

**X**of samples for classification. However, opposed to the spectra in Section 2, the exact sparsity of the signals to be input into the estimator is known. For this reason a different optimization algorithm is employed to solve (2), which removes the need for selection of an accurate regularization parameter. Orthogonal matching pursuit (OMP) as described in [22

22. J. A. Tropp, “Greed is good: algorithmic results for
sparse approximation,” IEEE Trans. Inf.
Theory **50**, 2231–2242
(2004). [CrossRef]

*n*characteristic peaks, the algorithm will reconstruct an input signal such that the reconstruction has only the

*n*peaks that best match the data. The only regularization parameter required is the truncation level of the ill-conditioned pseudo-random measurement matrix.

*λ*= .001 and

*γ*= 25. The basis function selection procedure was run to exhaustion (until more basis functions decreased the determinant of

**A**), and the label selection procedure was run until the change in determinant was less than .1. The data set of reconstructed spectra was normalized and centered, as is standard procedure in classification. The classifier was built using signals reconstructed two times: once to the full support of the data, and once without the weakest peak included. The reduction in reconstruction size for classification was because OMP is not guaranteed to select weak peaks correctly in the presence of noise as is demonstrated in Figure 3 [18]. Finally, it is important to note that in many cases it is more optimal to try to classify on the projection data than the reconstructed data. That indeed may even be true in this case. We present the results of classification on the reconstructed data, because we are trying to show which projection matrices provide the best projections for

*l*

_{1}denoising.

### 3.3. Simulation Results

*PD*) versus false positive rate (

*PF*). If the researcher in the problem described cared equally about distinguishing between the two species, then the intersection of the curve with the

*PD*= 1 –

*PF*line is most informative. The second metric is the number of basis functions and labels required to meet the stopping thresholds. If a data set is classified to high accuracy but takes many labels to do so its utility is limited.

*PD*= 1 –

*PF*line with a 96% detection rate, the slit spectrometer with a 93% detection rate, and the pseudo-random spectrometer at various lower rates depending on the truncation level. These results, however, could be misleading. The cyclic-s spectrometer requires nearly twice the number of labels to meet the stopping criteria in building the classifier. This is overcome when the weakest peak of the spectrum is removed from the reconstruction. As aforementioned, the weakest peak cannot be reconstructed reliably by the multiplexing spectrometers. Upon its removal, the cyclic-s and pseudo-random spectrometer ROCs all improve slightly, but what is more important is that the number of labels required for classification with the multiplex spectrometers falls in line with the number of labels required for classification for the slit spectrometer.

*MSE*analysis is presented. It would be very easy to conceive of a spectrum with many strong peaks and one weak peak on which lies the most important information to identify the substance, and it would be just as easy to conceive of a spectrum with the converse properties.

## 4. Conclusion

*MSE*is the evaluation criterion, an optimal spectrometer will be a multiplex spectrometer with optimal conditioning if the sparsity is exact. If the sparsity is approximate, the results become dependent on signal level. As the signal level increases, the ideal measurement system becomes the slit spectrometer, as the slit spectrometer best measures weak features. The weak feature hypothesis is solidified through classification results. When the weak feature is removed from the data set, multiplex systems outperform the slit spectrometer. However, when the classfication performance depends on the weak feature the data does not cluster well for the multiplex systems, and many more labeled spectra are required for accurate classification. There is no overarching rule for what spectrometer coding a scientist should use based on noise model. To make optimal spectral estimates or classifications the scientist must use knowledge of the data of interest.

## References and links

1. | M. T. E. Golay, “Multi-slit
spectrometry,” J. Opt. Soc. Am. |

2. | M. Harwit and N. J. A. Sloane, |

3. | A. Barducci, D. Guzzi, C. Lastri, V. Nardino, P. Marcoionni, and I. Pippi, “Radiometric and signal-to-noise ratio
properties of multiplex dispersive spectrometry,”
Appl. Opt. |

4. | A. A. Wagadarikar, M. E. Gehm, and D. J. Brady, “Performance comparison of aperture
codes for multimodal, multiplex spectroscopy,”
Appl. Opt. |

5. | S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, “Hadamard spectroscopy with a
two-dimensional detecting array,” Appl.
Opt. |

6. | M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding
for multimodal, multiplex spectroscopy,”
Appl. Opt. |

7. | M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral
imaging with a dual-disperser architecture,”
Opt. Express |

8. | A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded
aperture snapshot spectral imaging,” Appl.
Opt. |

9. | R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Realization of the hadamard multiplex
advantage using a programmable optical mask in a dispersive flat-field
near-infrared spectrometer,” Appl.
Spectrosc. |

10. | B. Noble, |

11. | S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis
pursuit,” SIAM J. Sci. Comput.
(USA) |

12. | J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative
shrinkage/thresholding algorithms for image
restoration,” IEEE Trans. Image
Process. |

13. | Z. Harmany, R. Marcia, and R. Willett, “This is SPIRAL-TAP: sparse poisson intensity reconstruction algorithms - theory and practice,” ArXiv e-prints (2010). 1005.4274. |

14. | D. J. Brady, |

15. | D. Donoho, “Compressed
sensing,” IEEE Trans. Inf. Theory |

16. | A. Chambolle, “An algorithm for total variation
minimization and applications,” J. Math.
Imaging Vision |

17. | D. Kittle, K. Choi, A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded
aperture snapshot spectral imagers,” Appl.
Opt. |

18. | T. T. Cai and L. Wang, “Orthogonal matching pursuit for sparse signal recovery,” Technical Report (2010). |

19. | S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897. [CrossRef] |

20. | Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active
selection of labeled data: application to sensing subsurface
UXO,” IEEE Trans. Geosci. Remote
Sens. |

21. | K. B. Petersen and M. S. Pedersen, “The matrix cookbook,” (2008). URL http://matrixcookbook.com/. |

22. | J. A. Tropp, “Greed is good: algorithmic results for
sparse approximation,” IEEE Trans. Inf.
Theory |

**OCIS Codes**

(300.6190) Spectroscopy : Spectrometers

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: August 10, 2011

Revised Manuscript: October 26, 2011

Manuscript Accepted: October 28, 2011

Published: January 18, 2012

**Citation**

Alex Mrozack, Daniel L. Marks, and David J. Brady, "Coded aperture spectroscopy with denoising through sparsity," Opt. Express **20**, 2297-2309 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2297

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### References

- M. T. E. Golay, “Multi-slit spectrometry,” J. Opt. Soc. Am.39(6), 437–437 (1949). URL http://www.opticsinfobase.org/abstract.cfm?URI=josa-39-6-437 . [CrossRef] [PubMed]
- M. Harwit and N. J. A. Sloane, Hadamard transform optics (Academic Press, 1979).
- A. Barducci, D. Guzzi, C. Lastri, V. Nardino, P. Marcoionni, and I. Pippi, “Radiometric and signal-to-noise ratio properties of multiplex dispersive spectrometry,” Appl. Opt.49(28), 5366–5373 (2010). URL http://ao.osa.org/abstract.cfm?URI=ao-49-28-5366 . [CrossRef] [PubMed]
- A. A. Wagadarikar, M. E. Gehm, and D. J. Brady, “Performance comparison of aperture codes for multimodal, multiplex spectroscopy,” Appl. Opt.46(22), 4932–4942 (2007). URL http://ao.osa.org/abstract.cfm?URI=ao-46-22-4932 . [CrossRef] [PubMed]
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