## Algorithm validation using multicolor phantoms |

Biomedical Optics Express, Vol. 3, Issue 6, pp. 1300-1311 (2012)

http://dx.doi.org/10.1364/BOE.3.001300

Acrobat PDF (1593 KB)

### Abstract

We present a framework for hyperspectral image (HSI) analysis validation, specifically abundance fraction estimation based on HSI measurements of water soluble dye mixtures printed on microarray chips. In our work we focus on the performance of two algorithms, the Least Absolute Shrinkage and Selection Operator (LASSO) and the Spatial LASSO (SPLASSO). The LASSO is a well known statistical method for simultaneously performing model estimation and variable selection. In the context of estimating abundance fractions in a HSI scene, the “sparse” representations provided by the LASSO are appropriate as not every pixel will be expected to contain every endmember. The SPLASSO is a novel approach we introduce here for HSI analysis which takes the framework of the LASSO algorithm a step further and incorporates the rich spatial information which is available in HSI to further improve the estimates of abundance. In our work here we introduce the dye mixture platform as a new benchmark data set for hyperspectral biomedical image processing and show our algorithm’s improvement over the standard LASSO.

© 2012 OSA

## 1. Introduction

5. M. Clarke, D. Allen, D. Samarov, and J. Hwang, “Characterization of hyperspectral imaging and analysis via microarray printing of dyes,” Proc. SPIE. **7891**, 78910W (2011). [CrossRef]

5. M. Clarke, D. Allen, D. Samarov, and J. Hwang, “Characterization of hyperspectral imaging and analysis via microarray printing of dyes,” Proc. SPIE. **7891**, 78910W (2011). [CrossRef]

*μ*m in diameter) are spaced 250

*μ*m apart to ensure no signal crosstalk between spots.

7. L. Nieman, M. Sinclair, J. Timlin, H. Jones, and D. Haaland, “Hyperspectral imaging system for quantitative identification and discrimination of fluorescent labels in the presence of autofluorescence,” in 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006 (2006), pp. 1288–1291. [CrossRef]

5. M. Clarke, D. Allen, D. Samarov, and J. Hwang, “Characterization of hyperspectral imaging and analysis via microarray printing of dyes,” Proc. SPIE. **7891**, 78910W (2011). [CrossRef]

*abundance fraction*(i.e. dye concentration) estimation. This benchmarking sample allows us to see how well an algorithm is capable of performing with various issues present. It also provides insight into some of the instrument and acquisition limitations such as the imager’s limit of detection for various endmember mixture proportions and combinations. Similar to complex tissue phantoms, this sample also exhibits effects of scattering and edge diffraction despite the intent of creating a simple system of variable absorbance spectra.

### 1.1. Sample Description

11. D. Samarov, M. Clarke, J. Lee, D. Allen, M. Litorja, and J. Hwang, “Validating the lasso algorithm by unmixing spectral signatures in multicolor phantoms,” Proc. SPIE **8229**, 82290Z (2012). [CrossRef]

**7891**, 78910W (2011). [CrossRef]

12. C.-I. Chang and Q. Du, “Estimation of the number of spectrally distinct signal sources in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. **42**, 608–619 (2004). [CrossRef]

13. J. Bioucas-Dias and J. Nascimento, “Hyperspectral subspace identification,” IEEE Trans. Geosci. Remote Sens. **46**, 2435–2445 (2008). [CrossRef]

14. J. Nascimento and J. Dias, “Vertex component analysis: A fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. **43**, 898–910 (2005). [CrossRef]

15. J. Bioucas-Dias, “A variable splitting augmented lagrangian approach to linear spectral unmixing,” in First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, 2009. WHISPERS ’09 (2009), pp. 1–4. [CrossRef]

### 1.2. Results and discussion

_{100}, and NC

_{100}. Similar 7 × 7 regions are then selected and averages calculated at each spot; we call these values AR

*, and NC*

_{i}*, where*

_{i}*i*∈ {1,..., 45} denotes the spot location shown in Fig. 4. The final estimated calibrated abundance fraction for each dye at each location are then calculated as

*not*present, the SPLASSO tends to produce estimates closer to 0% CAF. In biomedical imaging applications this is particularly important as avoiding false positives can be as important as identifying true positives.

## 2. Methods

**y**

*= (*

_{i}*y*

_{i}_{1},...,

*y*)

_{ip}*,*

^{T}*i*= 1,...,

*n*to be the set of spectral response vectors,

*n*corresonding to the total number of pixels in the image. Let

**x**

*= (*

_{j}*x*

_{1}

*,...,*

_{j}*x*)

_{pj}*,*

^{T}*j*= 1,...,

*m*be the set endmembers (where each of the

*p*entries maps to a specific wavelength), which are collected in the matrix

**X**= [

**x**

_{1},...,

**x**

*]. Finally let*

_{m}*β*= (

_{i}*β*

_{i}_{1},...,

*β*),

_{im}*i*= 1,...,

*n*be the set of abundance vectors whose entries tell us the proportion and concentration of an endmember at a pixel. In order to ensure that these abundances have a physical meaning it is typically required that each element of

*β*be nonnegative and that the sum of the elements of

_{i}*β*are less than or equal to one. More generally Note, it is more commonly assumed in the HSI literature that ∑

_{i}

_{i}*β*= 1. This constraint reflects the assumption that we have captured all or most of the relevant endmembers; however in the majority of cases this will not be true. One of the major issues with forcing the

_{il}*β*’s to sum to one is that it has the potential to introduce noise artifacts into the estimates; for example, if an endmember actually present in the image has not been included, a sum-to-one constraint may artificially inflate the estimate abundance of another endmember in order to compensate. Allowing for an inequality as in (1) will help avoid such situations and will generally be more robust to noise.

_{il}*β*(i.e. possibly many

_{i}*β*’s being equal to 0) arises naturally in hyperspectral imaging as most pixels are typically composed of only a subset of the

_{ij}*m*endmembers. For example, in the the dye mixture data we know that many of the spots are made up of one or a mixture of two dyes. In some applications large dictionaries of endmembers specific to the types of objects being analyzed are available, with only a subset of the endmembers in the dictionary being present in the image at all. By explicitly taking into account the sparse nature of the endmember abundance vectors we are able to reduce the number of false positives (saying an endmember is present in a pixel when it is not) and therefore the accuracy of the estimation.

### 2.1. LASSO

**X**is orthonormal, i.e.

**X**

^{T}**X**=

**I**and

**I**is the identity matrix. Then it can be shown that the solution of the LASSO problem in (2) has the closed form solution where

*u*)

_{+}= max(0,

*u*). Thus for

*λ*/2 ≥ |

*β̂*(OLS)|,

_{il}*β̂*(LASSO) = 0.

_{il}20. B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Stat. **32**, 407–499 (2004). [CrossRef]

21. J. Friedman, T. Hastie, H. Hofling, and R. Tibshirani, “Pathwise coordinate optimization,” Ann. Appl. Stat. **1**, 302–332 (2007). [CrossRef]

21. J. Friedman, T. Hastie, H. Hofling, and R. Tibshirani, “Pathwise coordinate optimization,” Ann. Appl. Stat. **1**, 302–332 (2007). [CrossRef]

### 2.2. SPLASSO

22. A. Zymnis, S.-J. Kim, J. Skaf, M. Parente, and S. Boyd, “Hyperspectral image unmixing via alternating projected subgradients,” in Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers, 2007. ACSSC 2007 (2007), pp. 1164–1168. [CrossRef]

18. M.-D. Iordache, J. Bioucas-Dias, and A. Plaza, “Sparse unmixing of hyperspectral data,” IEEE Trans. Geosci. Remote Sens. **49**, 2014–2039 (2011). [CrossRef]

23. A. Zare, “Spatial-spectral unmixing using fuzzy local information,” in *2011 IEEE International Geoscience and Remote Sensing Symposium (IGARSS)* (IEEE, 2011), pp. 1139–1142. [CrossRef]

_{j∈N(yi)}||

*β*−

_{i}*β*||

_{j}^{2}

*w*into the LASSO objective (2) giving us the SPLASSO loss function Here

_{ij}*λ*

_{1}and

*λ*

_{2}are nonnegative regularization parameters,

*N*(

**y**

*) is the set of neighboring pixels about*

_{i}**y**

*and*

_{i}*w*∈ [0, 1] is a spatial weight function capturing the similarity between observation

_{ij}*i*and its neighbors

*j*∈

*N*(

**y**

*). The neighborhood defined by*

_{i}*N*(·) can take on a number of different forms; for our purposes we take

*N*=

*N*, the symmetric

_{k}*k*-neighborhood on a regular 2D grid. To illustrate the form of

*N*, suppose we are at grid point

_{k}*g*in a

_{rs}*M*

_{1}×

*M*

_{2}image, 1 ≤

*r*≤

*M*

_{1}, 1 ≤

*s*≤

*M*

_{2}. For

*k*= 1 our neighborhood would be defined as the set of points

*N*

_{1}= {

*g*

_{r}_{−1,}

*,*

_{s}*g*

_{r}_{+1,}

*,*

_{s}*g*

_{r}_{,}

_{s}_{−1},

*g*

_{r}_{,}

_{s}_{+1},

*g*

_{r}_{−1,}

_{s}_{+1},

*g*

_{r}_{+1,}

_{s}_{+1},

*g*

_{r}_{−1,}

_{s}_{−1},

*g*

_{r}_{+1,}

_{s}_{−1}}.

_{j∈Nk (yi)}||

*β*−

_{i}*β*||

_{j}^{2}

*w*in (4) has the effect of “encouraging” the

_{ij}*β*’s to be similar to their

_{i}*k*-neighbors, introducing a smoothness to the coefficient vectors. In hyperspectral unmixing this has several appealing aspects: in particular it allows our estimates to be more robust to instrument and sample variability. Intuitively this makes sense as the variability introduced from these different sources will tend to be smoothed out. Of course, as in any smoothing method, care needs to be taken to avoid removing actual features by oversmoothing.

*w*and regularization parameters

_{ij}*λ*

_{1}and

*λ*

_{2}are extremely important. In the application of the SPLASSO to hyperspectral imaging it is desirable to have a weight function which uses both spatial and spectral information. Let us suppose that the spectral signature,

**y**

*whose abundances we are estimating corresponds to the*

_{i}*rs*pixel in the image (for illustrative purpose we refer to this point as

^{th}**y**

*). The spatial component of the weight function can then be captured by Our decision to use (5) is because it provides a decrease in the effect a neighboring pixel has the further we move out from the current observation being estimated. However, the decrease is not so rapid as to make the contribution of the surrounding observations negligible. Next, to leverage spectral information we use the weights which is the cosine of the angle between the spectra. This is a similarity measure commonly used in hyperspectral image analysis applications. For our purposes it is appealing because it allows our spatial weight function to be adaptive to local features in the image, e.g. if we are at the edge of an object. We have also found it useful in practice to include a threshold on the angle between spectra, so that if acos(*

_{rs}*c*(

_{rs}*lm*)) >

*t*,

*t*∈ [0,

*π*] then

*c*(

_{rs}*lm*) = 0. Putting the spatial (5) and spectral (6) weights together the weight function is defined as

*λ*

_{2}we once again considering the case where

**X**is taken to be orthonormal. Let

*γ*= 1/(1 + λ

_{2}), ∑

_{j∈Nk (yi)}

*w*= 1, (note, the latter does not need to hold in general, we do so here for illustrative purposes),

_{ij}*α*= ∑

_{i,l}_{j∈Nk (yi)}

*β*

_{j,l}*w*and then it can be shown that

_{ij}*γ*controls the tradeoff between the OLS estimate and a smoothly weighted average of its neighboring pixels.

## 3. Conclusions

## Acknowledgment

## Footnotes

Certain commercial equipment, instruments, or materials are identified in this manuscript are to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.DisclaimerCertain commercial equipment, instruments, or materials are identified in this manuscript are to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. |

## References and links

1. | B. Sorg, B. Moeller, O. Donovan, Y. Cao, and M. Dewhirst, “Hyperspectral imaging of hemoglobin saturation in tumor microvasculature and tumor hypoxia development,” J. Biomed. Opt. |

2. | M. Martin, M. Wabuyele, P. Chen, M. Panjehpour, M. Phan, B. Overholt, G. Cunningham, D. Wilson, R. DeNovo, and T. Vo-Dinh, “Development of an advanced hyperspectral imaging (hsi) system with applications for cancer detection,” Ann. Biomed. Eng. |

3. | K. Zuzak, R. Francis, E. Wehner, M. Litorja, J. Cadeddu, and E. Livingston, “Active dlp hyperspectral illumination: a noninvasive, in vivo, system characterization visualizing tissue oxygenation at near video rates,” Anal. Chem. |

4. | B. Pogue and M. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. |

5. | M. Clarke, D. Allen, D. Samarov, and J. Hwang, “Characterization of hyperspectral imaging and analysis via microarray printing of dyes,” Proc. SPIE. |

6. | M. Clarke, J. Lee, D. Samarov, D. Allen, M. Litorja, and J. Hwang, “Designing microarray phantoms for hyper-spectral imaging validation,” Biomed. Opt. Express (to be published). |

7. | L. Nieman, M. Sinclair, J. Timlin, H. Jones, and D. Haaland, “Hyperspectral imaging system for quantitative identification and discrimination of fluorescent labels in the presence of autofluorescence,” in 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006 (2006), pp. 1288–1291. [CrossRef] |

8. | R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc. B |

9. | D. Samarov, J. Hwang, J. Lee, and M. Clarke, “The spatial lasso with applications to unmixing hyperspectral images,” |

10. | F. Green, |

11. | D. Samarov, M. Clarke, J. Lee, D. Allen, M. Litorja, and J. Hwang, “Validating the lasso algorithm by unmixing spectral signatures in multicolor phantoms,” Proc. SPIE |

12. | C.-I. Chang and Q. Du, “Estimation of the number of spectrally distinct signal sources in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. |

13. | J. Bioucas-Dias and J. Nascimento, “Hyperspectral subspace identification,” IEEE Trans. Geosci. Remote Sens. |

14. | J. Nascimento and J. Dias, “Vertex component analysis: A fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. |

15. | J. Bioucas-Dias, “A variable splitting augmented lagrangian approach to linear spectral unmixing,” in First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, 2009. WHISPERS ’09 (2009), pp. 1–4. [CrossRef] |

16. | L. Breiman, “Better subset rergression using the nonnegative garotte,” em Technometrics |

17. | J. Fan and R. Li, “Variable selection via nonconcave penalized likelihood and its oracle properties,” J. Am. Stat. Assoc. |

18. | M.-D. Iordache, J. Bioucas-Dias, and A. Plaza, “Sparse unmixing of hyperspectral data,” IEEE Trans. Geosci. Remote Sens. |

19. | J. Bioucas-Dias and A. Plaza, “Hyperspectral unmixing: geometrical, statistical, and sparse regression approaches,” Proc. SPIE |

20. | B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Stat. |

21. | J. Friedman, T. Hastie, H. Hofling, and R. Tibshirani, “Pathwise coordinate optimization,” Ann. Appl. Stat. |

22. | A. Zymnis, S.-J. Kim, J. Skaf, M. Parente, and S. Boyd, “Hyperspectral image unmixing via alternating projected subgradients,” in Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers, 2007. ACSSC 2007 (2007), pp. 1164–1168. [CrossRef] |

23. | A. Zare, “Spatial-spectral unmixing using fuzzy local information,” in |

**OCIS Codes**

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

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(180.0180) Microscopy : Microscopy

(350.4800) Other areas of optics : Optical standards and testing

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

**ToC Category:**

Calibration, Validation and Phantom Studies

**History**

Original Manuscript: March 12, 2012

Revised Manuscript: April 26, 2012

Manuscript Accepted: April 26, 2012

Published: May 9, 2012

**Virtual Issues**

Phantoms for the Performance Evaluation and Validation of Optical Medical Imaging Devices
(2012) *Biomedical Optics Express*

**Citation**

Daniel V. Samarov, Matthew L. Clarke, Ji Youn Lee, David W. Allen, Maritoni Litorja, and Jeeseong Hwang, "Algorithm validation using multicolor phantoms," Biomed. Opt. Express **3**, 1300-1311 (2012)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-6-1300

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### References

- B. Sorg, B. Moeller, O. Donovan, Y. Cao, and M. Dewhirst, “Hyperspectral imaging of hemoglobin saturation in tumor microvasculature and tumor hypoxia development,” J. Biomed. Opt.10, 044004 (2005). [CrossRef]
- M. Martin, M. Wabuyele, P. Chen, M. Panjehpour, M. Phan, B. Overholt, G. Cunningham, D. Wilson, R. DeNovo, and T. Vo-Dinh, “Development of an advanced hyperspectral imaging (hsi) system with applications for cancer detection,” Ann. Biomed. Eng.34, 1061–1068 (2006). [CrossRef] [PubMed]
- K. Zuzak, R. Francis, E. Wehner, M. Litorja, J. Cadeddu, and E. Livingston, “Active dlp hyperspectral illumination: a noninvasive, in vivo, system characterization visualizing tissue oxygenation at near video rates,” Anal. Chem.83, 7424–7430 (2011). [CrossRef] [PubMed]
- B. Pogue and M. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt.16, 16272–16283 (2006).
- M. Clarke, D. Allen, D. Samarov, and J. Hwang, “Characterization of hyperspectral imaging and analysis via microarray printing of dyes,” Proc. SPIE.7891, 78910W (2011). [CrossRef]
- M. Clarke, J. Lee, D. Samarov, D. Allen, M. Litorja, and J. Hwang, “Designing microarray phantoms for hyper-spectral imaging validation,” Biomed. Opt. Express (to be published).
- L. Nieman, M. Sinclair, J. Timlin, H. Jones, and D. Haaland, “Hyperspectral imaging system for quantitative identification and discrimination of fluorescent labels in the presence of autofluorescence,” in 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006 (2006), pp. 1288–1291. [CrossRef]
- R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc. B58, 267–288 (1996).
- D. Samarov, J. Hwang, J. Lee, and M. Clarke, “The spatial lasso with applications to unmixing hyperspectral images,” Tech. Rep., National Institute of Standards and Technology (2012).
- F. Green, The Sigma-Aldrich Handbook of Stains, Dyes and Indicators (Aldrich Chem Co Library, 1990).
- D. Samarov, M. Clarke, J. Lee, D. Allen, M. Litorja, and J. Hwang, “Validating the lasso algorithm by unmixing spectral signatures in multicolor phantoms,” Proc. SPIE8229, 82290Z (2012). [CrossRef]
- C.-I. Chang and Q. Du, “Estimation of the number of spectrally distinct signal sources in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens.42, 608–619 (2004). [CrossRef]
- J. Bioucas-Dias and J. Nascimento, “Hyperspectral subspace identification,” IEEE Trans. Geosci. Remote Sens.46, 2435–2445 (2008). [CrossRef]
- J. Nascimento and J. Dias, “Vertex component analysis: A fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens.43, 898–910 (2005). [CrossRef]
- J. Bioucas-Dias, “A variable splitting augmented lagrangian approach to linear spectral unmixing,” in First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, 2009. WHISPERS ’09 (2009), pp. 1–4. [CrossRef]
- L. Breiman, “Better subset rergression using the nonnegative garotte,” em Technometrics37, 373–384 (1995).
- J. Fan and R. Li, “Variable selection via nonconcave penalized likelihood and its oracle properties,” J. Am. Stat. Assoc.96, 1348–1360 (2001). [CrossRef]
- M.-D. Iordache, J. Bioucas-Dias, and A. Plaza, “Sparse unmixing of hyperspectral data,” IEEE Trans. Geosci. Remote Sens.49, 2014–2039 (2011). [CrossRef]
- J. Bioucas-Dias and A. Plaza, “Hyperspectral unmixing: geometrical, statistical, and sparse regression approaches,” Proc. SPIE783078300A (2010). [CrossRef]
- B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Stat.32, 407–499 (2004). [CrossRef]
- J. Friedman, T. Hastie, H. Hofling, and R. Tibshirani, “Pathwise coordinate optimization,” Ann. Appl. Stat.1, 302–332 (2007). [CrossRef]
- A. Zymnis, S.-J. Kim, J. Skaf, M. Parente, and S. Boyd, “Hyperspectral image unmixing via alternating projected subgradients,” in Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers, 2007. ACSSC 2007 (2007), pp. 1164–1168. [CrossRef]
- A. Zare, “Spatial-spectral unmixing using fuzzy local information,” in 2011 IEEE International Geoscience and Remote Sensing Symposium (IGARSS) (IEEE, 2011), pp. 1139–1142. [CrossRef]

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