Abstract

The spectra in spectral reflectance datasets tend to be quite correlated and therefore they can be represented more compactly using standard techniques such as principal components analysis (PCA) as part of a lossy compression strategy. However, the presence of outlier spectra can often increase the overall error of the reconstructed spectra. This paper introduces a new outlier modeling (OM) method that detects, clusters, and separately models outliers with their own set of basis vectors. Outliers are defined in terms of the robust Mahalanobis distance using the fast minimum covariance determinant algorithm as a robust estimator of the multivariate mean and covariance from which it is computed. After removing the outliers from the main dataset, the performance of PCA on the remaining data improves significantly; however, since outlier spectra are a part of the image, they cannot simply be ignored. The solution is to cluster the outliers into a small number of clusters and then model each cluster separately using its own cluster-specific PCA-derived bases. Tests show that OM leads to lower spectral reconstruction errors of reflectance spectra in terms of both normalized RMS and goodness of fit.

© 2014 Optical Society of America

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2011 (1)

R. Vidal, “A tutorial on subspace clustering,” IEEE Signal Process. Mag. 28(2), 52–68 (2011).
[CrossRef]

2008 (1)

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

2007 (2)

Q. Du and J. E. Fowler, “Hyperspectral image compression using JPEG2000 and principal component analysis,” IEEE Geosci. Remote Sens. Lett. 4, 201–205 (2007).

B. Penna, T. Tillo, E. Magli, and G. Olmo, “Hyperspectral image compression employing a model of anomalous pixels,” IEEE Geosci. Remote Sens. Lett. 4, 664–668 (2007).
[CrossRef]

2006 (1)

F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvector,” J. Opt. Soc. Am. A. 23, 2020–2026 (2006).
[CrossRef]

2005 (1)

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

2003 (1)

M. Okuyama, N. Tsumura, and Y. Miyake, “Evaluating a multispectral imaging system for mapping pigments in human skin,” Opt. Rev. 10, 580–584 (2003).
[CrossRef]

2001 (2)

R. S. Berns, “The science of digitizing paintings for color-accurate image archives: a review,” J. Imaging Sci. Technol. 4, 305–325 (2001).

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

1999 (1)

P. J. Rousseeuw and K. Van Driessen, “A Fast algorithm for the minimum covariance determinant estimator,” Technometrics. 41, 212–223 (1999).

1998 (2)

J. Hernandez-Andres, J. Romero, A. García-Beltrán, and J. L. Nieves, “Testing linear models on spectral daylight measurements,” Appl. Opt. 37, 971–977 (1998).
[CrossRef]

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

1984 (1)

P. J. Rousseeuw, “Least median of squares regression,” J. Am. Stat. Assoc. 79, 871–880 (1984).

1933 (1)

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educ. Psychol. 24, 417–444 (1933).

1901 (1)

K. Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2(11), 559–572 (1901).
[CrossRef]

Agahian, F.

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Amirshahi, S. A.

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Amirshahi, S. H.

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Ayala, F.

F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvector,” J. Opt. Soc. Am. A. 23, 2020–2026 (2006).
[CrossRef]

Berns, R. S.

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

R. S. Berns, “The science of digitizing paintings for color-accurate image archives: a review,” J. Imaging Sci. Technol. 4, 305–325 (2001).

F. H. Imai, M. R. Rosen, and R. S. Berns, “Multispectral imaging of van Gogh’s self portrait at the National Gallery of Art,” in Proc. PICS: Image Processing, Image Quality, Image Capture Systems Conference (IS&T) (2001), pp. 185–189.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using matrix R method in the multispectral image archives,” in Proceedings of the 10th Congress of the International Colour Association (AIC) (2005), pp. 469–472.

Crettez, J.

H. Maitre, F. Schmitt, J. Crettez, Y. Wu, and J. Y. Hardeberg, “Spectrophotometric image analysis of fine art paintings,” in Proceedings of the IS&T/SID Color Imaging Conference (1996), pp. 50–53.

Du, Q.

Q. Du and J. E. Fowler, “Hyperspectral image compression using JPEG2000 and principal component analysis,” IEEE Geosci. Remote Sens. Lett. 4, 201–205 (2007).

Echavarri, J. F.

F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvector,” J. Opt. Soc. Am. A. 23, 2020–2026 (2006).
[CrossRef]

Filzmoser, P.

P. Filzmoser, “A multivariate outlier detection method,” in Proceedings of International Conference on Computer Data Analysis and Modeling (2004), pp. 18–22.

Finlayson, G.

S. Hordley, G. Finlayson, and P. Morovic, “A multispectral image database and an application to image rendering across illumination,” in Proceedings of Third International Conference on Image and Graphics (2004), pp. 394–397.

Fowler, J. E.

Q. Du and J. E. Fowler, “Hyperspectral image compression using JPEG2000 and principal component analysis,” IEEE Geosci. Remote Sens. Lett. 4, 201–205 (2007).

Garcia-Beltran, A.

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

García-Beltrán, A.

Hardeberg, J. Y.

H. Maitre, F. Schmitt, J. Crettez, Y. Wu, and J. Y. Hardeberg, “Spectrophotometric image analysis of fine art paintings,” in Proceedings of the IS&T/SID Color Imaging Conference (1996), pp. 50–53.

Hernandez-Andres, J.

J. Hernandez-Andres, J. Romero, A. García-Beltrán, and J. L. Nieves, “Testing linear models on spectral daylight measurements,” Appl. Opt. 37, 971–977 (1998).
[CrossRef]

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

Hordley, S.

S. Hordley, G. Finlayson, and P. Morovic, “A multispectral image database and an application to image rendering across illumination,” in Proceedings of Third International Conference on Image and Graphics (2004), pp. 394–397.

Hotelling, H.

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educ. Psychol. 24, 417–444 (1933).

Imai, F. H.

F. H. Imai, M. R. Rosen, and R. S. Berns, “Multispectral imaging of van Gogh’s self portrait at the National Gallery of Art,” in Proc. PICS: Image Processing, Image Quality, Image Capture Systems Conference (IS&T) (2001), pp. 185–189.

Iso, D.

F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: post-capture control of resolution, dynamic range and spectrum,” (Department of Computer Science, Columbia University, 2008).

Johnson, R. A.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice-Hall, 1992).

Jolliffe, I. T.

I. T. Jolliffe, Principal Component Analysis, 2nd ed., Springer Series in Statistics (Springer-Verlag, 2002).

Kaarna, A.

A. Kaarna, P. Zemcik, H. Kalviainen, and J. Parkkinen, “Multispectral image compression,” in Proceedings of the 14th International Conference on Pattern Recognition (1998), pp. 1264–1267.

Kalviainen, H.

A. Kaarna, P. Zemcik, H. Kalviainen, and J. Parkkinen, “Multispectral image compression,” in Proceedings of the 14th International Conference on Pattern Recognition (1998), pp. 1264–1267.

Magli, E.

B. Penna, T. Tillo, E. Magli, and G. Olmo, “Hyperspectral image compression employing a model of anomalous pixels,” IEEE Geosci. Remote Sens. Lett. 4, 664–668 (2007).
[CrossRef]

Maitre, H.

H. Maitre, F. Schmitt, J. Crettez, Y. Wu, and J. Y. Hardeberg, “Spectrophotometric image analysis of fine art paintings,” in Proceedings of the IS&T/SID Color Imaging Conference (1996), pp. 50–53.

Mitsunaga, T.

F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: post-capture control of resolution, dynamic range and spectrum,” (Department of Computer Science, Columbia University, 2008).

Miyake, Y.

M. Okuyama, N. Tsumura, and Y. Miyake, “Evaluating a multispectral imaging system for mapping pigments in human skin,” Opt. Rev. 10, 580–584 (2003).
[CrossRef]

Morovic, P.

S. Hordley, G. Finlayson, and P. Morovic, “A multispectral image database and an application to image rendering across illumination,” in Proceedings of Third International Conference on Image and Graphics (2004), pp. 394–397.

Motomura, H.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Murakami, Y.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Nayar, S. K.

F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: post-capture control of resolution, dynamic range and spectrum,” (Department of Computer Science, Columbia University, 2008).

Nezamabadi, M.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using matrix R method in the multispectral image archives,” in Proceedings of the 10th Congress of the International Colour Association (AIC) (2005), pp. 469–472.

Nieves, J. L.

J. Hernandez-Andres, J. Romero, A. García-Beltrán, and J. L. Nieves, “Testing linear models on spectral daylight measurements,” Appl. Opt. 37, 971–977 (1998).
[CrossRef]

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

Ohsawa, K.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Ohyama, N.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Okuyama, M.

M. Okuyama, N. Tsumura, and Y. Miyake, “Evaluating a multispectral imaging system for mapping pigments in human skin,” Opt. Rev. 10, 580–584 (2003).
[CrossRef]

Olmo, G.

B. Penna, T. Tillo, E. Magli, and G. Olmo, “Hyperspectral image compression employing a model of anomalous pixels,” IEEE Geosci. Remote Sens. Lett. 4, 664–668 (2007).
[CrossRef]

Parkkinen, J.

A. Kaarna, P. Zemcik, H. Kalviainen, and J. Parkkinen, “Multispectral image compression,” in Proceedings of the 14th International Conference on Pattern Recognition (1998), pp. 1264–1267.

Pearson, K.

K. Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2(11), 559–572 (1901).
[CrossRef]

Penna, B.

B. Penna, T. Tillo, E. Magli, and G. Olmo, “Hyperspectral image compression employing a model of anomalous pixels,” IEEE Geosci. Remote Sens. Lett. 4, 664–668 (2007).
[CrossRef]

Renet, P.

F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvector,” J. Opt. Soc. Am. A. 23, 2020–2026 (2006).
[CrossRef]

Romero, J.

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

J. Hernandez-Andres, J. Romero, A. García-Beltrán, and J. L. Nieves, “Testing linear models on spectral daylight measurements,” Appl. Opt. 37, 971–977 (1998).
[CrossRef]

Rosen, M. R.

F. H. Imai, M. R. Rosen, and R. S. Berns, “Multispectral imaging of van Gogh’s self portrait at the National Gallery of Art,” in Proc. PICS: Image Processing, Image Quality, Image Capture Systems Conference (IS&T) (2001), pp. 185–189.

Rousseeuw, P. J.

P. J. Rousseeuw and K. Van Driessen, “A Fast algorithm for the minimum covariance determinant estimator,” Technometrics. 41, 212–223 (1999).

P. J. Rousseeuw, “Least median of squares regression,” J. Am. Stat. Assoc. 79, 871–880 (1984).

Schmitt, F.

H. Maitre, F. Schmitt, J. Crettez, Y. Wu, and J. Y. Hardeberg, “Spectrophotometric image analysis of fine art paintings,” in Proceedings of the IS&T/SID Color Imaging Conference (1996), pp. 50–53.

Taplin, L. A.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using matrix R method in the multispectral image archives,” in Proceedings of the 10th Congress of the International Colour Association (AIC) (2005), pp. 469–472.

Teraji, T.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Tillo, T.

B. Penna, T. Tillo, E. Magli, and G. Olmo, “Hyperspectral image compression employing a model of anomalous pixels,” IEEE Geosci. Remote Sens. Lett. 4, 664–668 (2007).
[CrossRef]

Tsumura, N.

M. Okuyama, N. Tsumura, and Y. Miyake, “Evaluating a multispectral imaging system for mapping pigments in human skin,” Opt. Rev. 10, 580–584 (2003).
[CrossRef]

Tzeng, D. Y.

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Uchiyama, T.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Van Driessen, K.

P. J. Rousseeuw and K. Van Driessen, “A Fast algorithm for the minimum covariance determinant estimator,” Technometrics. 41, 212–223 (1999).

Vidal, R.

R. Vidal, “A tutorial on subspace clustering,” IEEE Signal Process. Mag. 28(2), 52–68 (2011).
[CrossRef]

Wichern, D. W.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice-Hall, 1992).

Wu, Y.

H. Maitre, F. Schmitt, J. Crettez, Y. Wu, and J. Y. Hardeberg, “Spectrophotometric image analysis of fine art paintings,” in Proceedings of the IS&T/SID Color Imaging Conference (1996), pp. 50–53.

Yamaguchi, M.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Yasuma, F.

F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: post-capture control of resolution, dynamic range and spectrum,” (Department of Computer Science, Columbia University, 2008).

Zemcik, P.

A. Kaarna, P. Zemcik, H. Kalviainen, and J. Parkkinen, “Multispectral image compression,” in Proceedings of the 14th International Conference on Pattern Recognition (1998), pp. 1264–1267.

Zhao, Y.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using matrix R method in the multispectral image archives,” in Proceedings of the 10th Congress of the International Colour Association (AIC) (2005), pp. 469–472.

Appl. Opt. (1)

Color Res. Appl. (3)

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

IEEE Geosci. Remote Sens. Lett. (2)

Q. Du and J. E. Fowler, “Hyperspectral image compression using JPEG2000 and principal component analysis,” IEEE Geosci. Remote Sens. Lett. 4, 201–205 (2007).

B. Penna, T. Tillo, E. Magli, and G. Olmo, “Hyperspectral image compression employing a model of anomalous pixels,” IEEE Geosci. Remote Sens. Lett. 4, 664–668 (2007).
[CrossRef]

IEEE Signal Process. Mag. (1)

R. Vidal, “A tutorial on subspace clustering,” IEEE Signal Process. Mag. 28(2), 52–68 (2011).
[CrossRef]

J. Am. Stat. Assoc. (1)

P. J. Rousseeuw, “Least median of squares regression,” J. Am. Stat. Assoc. 79, 871–880 (1984).

J. Educ. Psychol. (1)

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educ. Psychol. 24, 417–444 (1933).

J. Imaging Sci. Technol. (1)

R. S. Berns, “The science of digitizing paintings for color-accurate image archives: a review,” J. Imaging Sci. Technol. 4, 305–325 (2001).

J. Opt. Soc. Am. A. (1)

F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvector,” J. Opt. Soc. Am. A. 23, 2020–2026 (2006).
[CrossRef]

Opt. Rev. (1)

M. Okuyama, N. Tsumura, and Y. Miyake, “Evaluating a multispectral imaging system for mapping pigments in human skin,” Opt. Rev. 10, 580–584 (2003).
[CrossRef]

Philos. Mag. (1)

K. Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2(11), 559–572 (1901).
[CrossRef]

Proc. SPIE (1)

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2001).
[CrossRef]

Technometrics. (1)

P. J. Rousseeuw and K. Van Driessen, “A Fast algorithm for the minimum covariance determinant estimator,” Technometrics. 41, 212–223 (1999).

Other (12)

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice-Hall, 1992).

P. Filzmoser, “A multivariate outlier detection method,” in Proceedings of International Conference on Computer Data Analysis and Modeling (2004), pp. 18–22.

I. T. Jolliffe, Principal Component Analysis, 2nd ed., Springer Series in Statistics (Springer-Verlag, 2002).

F. H. Imai, M. R. Rosen, and R. S. Berns, “Multispectral imaging of van Gogh’s self portrait at the National Gallery of Art,” in Proc. PICS: Image Processing, Image Quality, Image Capture Systems Conference (IS&T) (2001), pp. 185–189.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using matrix R method in the multispectral image archives,” in Proceedings of the 10th Congress of the International Colour Association (AIC) (2005), pp. 469–472.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of the proposed method for spectral data reduction: (a) separate the outliers (yellow squares) from the nonoutlier spectra (red circles), (b) apply K-means clustering to the outliers, and (c) apply PCA data reduction to all clusters (inliers and outliers).

Fig. 2.
Fig. 2.

Multispectral images from the database of Hordley et al. [22].

Fig. 3.
Fig. 3.

Multispectral images from the Columbia University database [23].

Fig. 4.
Fig. 4.

“Fruits and Flowers” image from the Eastern Finland University spectral image database [24].

Fig. 5.
Fig. 5.

Comparison of the classic Mahalanobis distance (MDclassic) and robust distance (MDmcd) for the 19,200 Fruits and Flowers spectra: (a) MDclassic distance versus sample number, (b) MDmcd distance versus sample number, and (c) MDmcd versus MDclassic. The horizontal red lines represent the quantile cutoffs defining the inlier/outlier boundary.

Fig. 6.
Fig. 6.

Mean reconstruction error for the outlier spectra (excluding the inliers) as a function of the number of clusters used.

Fig. 7.
Fig. 7.

Mean of reconstruction error for the outlier spectra versus the number of iterations.

Fig. 8.
Fig. 8.

Comparison between the accuracy of the encoding and reconstruction of the 10 multispectral images from the Hordley database using classic PCA versus OM.

Fig. 9.
Fig. 9.

Comparison of the accuracy of the reconstruction of reflectance spectra from seven multispectral images from the Columbia University database using classic PCA and OM.

Tables (5)

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Table 1. Spectral Accuracy of Reflectance Reconstruction for the Fruits and Flowers Image Using Standard PCA Versus Outlier Modeling and Associated CRa

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Table 2. Accuracy of Reflectance Reconstruction for 10 Multispectral Images from the Hordley Database Using Classic PCA with a 4D Basis Versus the OM Approach Using Three Basis Vectorsa

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Table 3. Accuracy of Reflectance Reconstruction for Seven Multispectral Images from the Columbia University Database Using Classic PCA with Five Eigenvectors as well as the OM Approach Using Three Basis Vectorsa

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Table 4. Colorimetric Accuracy of Reflectance Reconstruction of 18 Spectral Images Taken from 3 Multispectral Databases Using Classic PCA and the Proposed OM Methoda

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Table 5. OM and Classic PCA Running Time (in Seconds) for the Smallest and Largest Images in the Datasets

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

e=λ(Rm(λ)Re(λ))2,
MD(x)=(xμ)TS1(xμ).
NRMS=1λ(Rm(λ)Re(λ))2λ(Rm(λ)mean(Re))2,
GFC=|λRm(λ)Re(λ)||λ[Rm(λ)]2||λ[Re(λ)]2|.

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