# Geometrical Approximated Principal Component Analysis for Hyperspectral Image Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Materials and Methods

#### 3.1. The gaPCA Algorithm

- the first step consists of seeking the projection vector defined by two points separated by the maximum distance and
- the second step consists of reducing the dimension of the data by projecting it onto the subspace orthogonal to the previous projection.

Algorithm 1: gaPCA. |

Algorithm 2: computeMaximumDistance. |

Algorithm 3: computeProjectionsHyperplane. |

#### 3.2. Datasets

^{-6}m. The Indian Pines scene is a subset of a larger one and contains approximately 60 percent agriculture, and 30 percent forest or other natural perennial vegetation. There are two major dual-lane highways, a rail line, and some low density housing, other built structures, and smaller roads. The scene was taken in June and some of the crops present, corn, soybeans, are in early stages of growth with less than 5% coverage [30].

#### 3.3. Performance Evaluation

#### 3.3.1. Textural Analysis Metrics

#### 3.3.2. Quality of the Reconstruction Metrics

#### 3.3.3. Redundancy of the Principal Components Metric

#### 3.3.4. Classification Accuracy Assesment Metrics

## 4. Results and Discussion

#### 4.1. GLCM Textural Analysis Metrics

#### 4.2. Quality of the Reconstruction Metrics

#### 4.3. Redundancy of the Principal Components Metric

#### 4.4. Land Classification Accuracy

#### 4.4.1. Indian Pines Dataset

#### 4.4.2. Pavia University Dataset

#### 4.4.3. DC Mall Dataset

#### 4.4.4. AHS Dataset

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chi, M.; Plaza, A.; Benediktsson, J.A.; Sun, Z.; Shen, J.; Zhu, Y. Big data for remote sensing: Challenges and opportunities. Proc. IEEE
**2016**, 104, 2207–2219. [Google Scholar] [CrossRef] - Rodarmel, C.; Shan, J. Principal component analysis for hyperspectral image classification. Surv. Land Inf. Sci.
**2002**, 62, 115–122. [Google Scholar] - Cheng, G.; Han, J.; Lu, X. Remote Sensing Image Scene Classification: Benchmark and State of the Art. Proc. IEEE
**2017**, 105, 1865–1883. [Google Scholar] [CrossRef] [Green Version] - Dixon, B.; Uddameri, V. GIS and Geocomputation for Water Resource Science and Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2016. [Google Scholar]
- Norko, A. Simple Image Classification Using Principal Component Analysis (PCA); GMU Volgenau School of Engineering: Fairfax, VA, USA, 2015; Available online: https://ece.gmu.edu/hayes/courses/MachineLearning/Projects/Presentations/Norko.pdf (accessed on 10 April 2020).
- Bajwa, I.S.; Naweed, M.; Asif, M.N.; Hyder, S.I. Feature based image classification by using principal component analysis. ICGST Int. J. Graph. Vis. Image Process. GVIP
**2009**, 9, 11–17. [Google Scholar] - Qahtan, A.A.; Alharbi, B.; Wang, S.; Zhang, X. A pca-based change detection framework for multidimensional data streams: Change detection in multidimensional data streams. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Sydney, Australia, 10–15 August 2015; pp. 935–944. [Google Scholar]
- Lin, H.; Zhang, A. Summarization of hyperspectral image visualization methods. In Proceedings of the 2014 IEEE International Conference on Progress in Informatics and Computing, Shanghai, China, 16–18 May 2014; pp. 355–358. [Google Scholar] [CrossRef]
- Báscones, D.; González, C.; Mozos, D. Hyperspectral Image Compression Using Vector Quantization, PCA and JPEG2000. Remote Sens.
**2018**, 10, 907. [Google Scholar] [CrossRef] [Green Version] - Loizzo, R.; Daraio, M.; Guarini, R.; Longo, F.; Lorusso, R.; Dini, L.; Lopinto, E. Prisma Mission Status and Perspective. In Proceedings of the IGARSS 2019—2019 IEEE International Geoscience and Remote Sensing Symposium, Yokohama, Japan, 28 July–2 August 2019. [Google Scholar] [CrossRef]
- Naik, G.R. Advances in Principal Component Analysis: Research and Development; Springer: Basel, Switzerland, 2017. [Google Scholar]
- Khatami, R.; Mountrakis, G.; Stehman, S.V. A meta-analysis of remote sensing research on supervised pixel-based land-cover image classification processes: General guidelines for practitioners and future research. Remote Sens. Environ.
**2016**, 177, 89–100. [Google Scholar] [CrossRef] [Green Version] - Raadt, A.D.; Warrens, M.J.; Bosker, R.J.; Kiers, H.A.L. Kappa Coefficients for Missing Data. Educ. Psychol. Meas.
**2019**, 79, 558–576. [Google Scholar] [CrossRef] [Green Version] - Jolliffe, I.T.; Cadima, J. Principal component analysis: A review and recent developments. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2016**, 374, 20150202. [Google Scholar] [CrossRef] - Ramsay, J. Functional data analysis. Encycl. Stat. Behav. Sci.
**2005**, 4. [Google Scholar] [CrossRef] - Hall, P.; Müller, H.G.; Wang, J.L. Properties of principal component methods for functional and longitudinal data analysis. Ann. Stat.
**2006**, 34, 1493–1517. [Google Scholar] - Ke, Q.; Kanade, T. Robust L/sub 1/norm factorization in the presence of outliers and missing data by alternative convex programming. In Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, USA, 20–25 June 2005; pp. 739–746. [Google Scholar]
- Candès, E.J.; Li, X.; Ma, Y.; Wright, J. Robust principal component analysis? J. ACM (JACM)
**2011**, 58, 11. [Google Scholar] [CrossRef] - Lee, T.W. Independent Component Analysis; Springer: Boston, MA, USA, 1998; pp. 27–66. [Google Scholar]
- Scholz, M.; Kaplan, F.; Guy, C.L.; Kopka, J.; Selbig, J. Non-linear PCA: A missing data approach. Bioinformatics
**2005**, 21, 3887–3895. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mori, Y.; Kuroda, M.; Makino, N. Nonlinear Principal Component Analysis and Its Applications; Springer: New York, NY, USA, 2016. [Google Scholar]
- Zhang, R.; Wang, W.; Ma, Y. Approximations of the standard principal components analysis and kernel PCA. Expert Syst. Appl.
**2010**, 37, 6531–6537. [Google Scholar] [CrossRef] - Kumar, D.; Singh, R.; Kumar, A.; Sharma, N. An adaptive method of PCA for minimization of classification error using Naïve Bayes classifier. Procedia Comput. Sci.
**2015**, 70, 9–15. [Google Scholar] [CrossRef] [Green Version] - Gupta, A.; Barbu, A. Parameterized principal component analysis. Pattern Recognit.
**2018**, 78, 215–227. [Google Scholar] [CrossRef] [Green Version] - Bigot, J.; Gouet, R.; Lopez, A. Geometric PCA of images. SIAM J. Imaging Sci.
**2013**, 6, 1851–1879. [Google Scholar] [CrossRef] [Green Version] - Ifarraguerri, A.; Chang, C.I. Unsupervised hyperspectral image analysis with projection pursuit. IEEE Trans. Geosci. Remote Sens.
**2000**, 38, 2529–2538. [Google Scholar] - Machidon, A.L.; Ciobanu, C.B.; Machidon, O.M.; Ogrutan, P.L. On Parallelizing Geometrical PCA Approximation. In Proceedings of the 2019 18th RoEduNet Conference: Networking in Education and Research (RoEduNet), Galati, Romania, 10–12 October 2019; pp. 1–6. [Google Scholar] [CrossRef]
- Härdle, W.; Klinke, S.; Turlach, B.A. XploRe: An Interactive Statistical Computing Environment; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Dayal, M. A New Algorithm for Exploratory Projection Pursuit. arXiv
**2018**, arXiv:1112.4321. [Google Scholar] - Baumgardner, M.F.; Biehl, L.L.; Landgrebe, D.A. 220 Band AVIRIS Hyperspectral Image Data Set: June 12, 1992 Indian Pine Test Site 3. 2015. Available online: https://purr.purdue.edu/publications/1947/1 (accessed on 20 November 2019). [CrossRef]
- Huang, X.; Zhang, L. A comparative study of spatial approaches for urban mapping using hyperspectral ROSIS images over Pavia City, northern Italy. Int. J. Remote Sens.
**2009**, 30, 3205–3221. [Google Scholar] [CrossRef] - Hajnsek, I.; Bianchi, R.; Davidson, M.; D’Urso, G.; Gomez-Sanches, A.; Hausold, A.; Horn, R.; Howse, J.; Löw, A.; Lopez-Sanchez, J.M.; et al. AgriSAR 2006—Airborne SAR and optics campaigns for an improved monitoring of agricultural processes and practices. In the Proceedings of the AGRISAR and EAGLE campaigns, Noordwijk, The Netherlands, 15–16 October 2007; Volume 9, p. 04085. [Google Scholar]
- Haralick, R.M.; Shanmugam, K.; Dinstein, I. Textural Features for Image Classification. IEEE Trans. Syst. Man Cybern.
**1973**, 6, 610–621. [Google Scholar] [CrossRef] [Green Version] - Gong, P.; Marceau, D.J.; Howarth, P.J. A comparison of spatial feature extraction algorithms for land-use classification with SPOT HRV data. Remote Sens. Environ.
**1992**, 40, 137–151. [Google Scholar] [CrossRef] - Barber, D.; Ledrew, E. SAR sea ice discrimination using texture statistics—A multivariate approach. Photogramm. Eng. Remote Sens.
**1991**, 57, 385–395. [Google Scholar] - Baraldi, A.; Panniggiani, F. An investigation of the textural characteristics associated with gray level cooccurrence matrix statistical parameters. IEEE Trans. Geosci. Remote Sens.
**1995**, 33, 293–304. [Google Scholar] [CrossRef] - Gadkari, D. Image Quality Analysis Using GLCM. Master’s Thesis, University of Central Florida, Orlando, FL, USA, December 2004. [Google Scholar]
- Sulochana, S.; Vidhya, R. Texture based image retrieval using framelet transform-gray level co-occurrence matrix (GLCM). Int. J. Adv. Res. Artif. Intell.
**2013**, 2, 68–73. [Google Scholar] [CrossRef] [Green Version] - Shi, Y.Q.; Sun, H. Image and Video Compression for Multimedia Engineering: Fundamentals, Algorithms, and Standards; CRC Press: Boca Raton, FL, USA, 1999. [Google Scholar]
- Paul, S.; Kumar, D.N. Spectral-spatial classification of hyperspectral data with mutual information based segmented stacked autoencoder approach. ISPRS J. Photogramm. Remote Sens.
**2018**, 138, 265–280. [Google Scholar] [CrossRef] - Johnson, K.; Cole-Rhodes, A.; Zavorin, I.; Moigne, J.L. Mutual information as a similarity measure for remote sensing image registration. In Proceedings of the Geo-Spatial Image and Data Exploitation II, Orlando, FL, USA, 16–20 April 2001; pp. 51–61. [Google Scholar] [CrossRef]
- Guo, B.; Gunn, S.R.; Damper, R.I.; Nelson, J.D.B. Band Selection for Hyperspectral Image Classification Using Mutual Information. IEEE Geosci. Remote Sens. Lett.
**2006**, 3, 522–526. [Google Scholar] [CrossRef] [Green Version] - Liang, J.; Liu, X.; Huang, K.; Li, X.; Wang, D.; Wang, X. Automatic Registration of Multisensor Images Using an Integrated Spatial and Mutual Information (SMI) Metric. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 603–615. [Google Scholar] - Fauvel, M.; Chanussot, J.; Benediktsson, J.A. Kernel principal component analysis for the classification of hyperspectral remote sensing data over urban areas. EURASIP J. Adv. Signal Process.
**2009**, 2009, 783194. [Google Scholar] [CrossRef] [Green Version] - Aktar, M.; Mamun, M.; Hossain, M.; Shuvo, M. Weighted normalized mutual information based change detection in remote sensing images. In Proceedings of the 2016 19th International Conference on Computer and Information Technology (ICCIT), Dhaka, Bangladesh, 18–20 December 2016; pp. 257–260. [Google Scholar]
- Lu, D.; Weng, Q. A survey of image classification methods and techniques for improving classification performance. Int. J. Remote Sens.
**2007**, 28, 823–870. [Google Scholar] - ENVI Image Analysis Software. Available online: https://www.harrisgeospatial.com/Software-Technology/ENVI (accessed on 24 April 2020).
- McNemar, Q. Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika
**1947**, 12, 153–157. [Google Scholar] [CrossRef]

**Figure 3.**2D clouds of points with gaPCA axes vs. PCA axes for various correlation coefficients: (

**a**) $\rho $ = 0.5, (

**b**) $\rho $ = 0.7, (

**c**) $\rho $ = 0.9.

**Figure 9.**Mutual Information for all the principal components for PCA (

**a**) and gaPCA (

**b**) images of the Indian Pines dataset.

**Figure 10.**Standard PCA (

**a**) and gaPCA (

**b**) images classified (Maximum Likelihood) vs. the ground-truth image (

**c**) of the Indian Pines dataset.

**Figure 11.**Standard PCA (

**a**) and gaPCA (

**b**) images classified (Maximum Likelihood) vs. the groundtruth (

**c**) of the Pavia University dataset.

**Figure 12.**Standard PCA (

**a**) and gaPCA (

**b**) images classified (Maximum Likelihood) vs. the groundtruth (

**c**) of the DC Mall dataset.

**Figure 13.**Standard PCA (

**a**) and gaPCA (

**b**) images classified (Maximum Likelihood) vs. the groundtruth (

**c**) of the AHS dataset.

Indian Pines | Pavia University | DC Mall | AHS | |
---|---|---|---|---|

PCA | 0.96 | 0.14 | 0.25 | 0.17 |

gaPCA | 0.34 | 0.12 | 0.32 | 0.18 |

Indian Pines | Pavia University | DC Mall | AHS | |
---|---|---|---|---|

PCA | 0.14 | 0.58 | 0.29 | 0.23 |

gaPCA | 0.21 | 0.53 | 0.20 | 0.21 |

Indian Pines | Pavia University | DC Mall | AHS | |
---|---|---|---|---|

PCA | 6.95 | 5.28 | 6.07 | 6.72 |

gaPCA | 6.61 | 5.17 | 6.38 | 6.45 |

Method | 1PC | 2PC | 10PC | 100PC | 200PC |
---|---|---|---|---|---|

gaPCA | 13.47 | 15.39 | 24.84 | 42.19 | 275.66 |

PCA | 10.97 | 24.33 | 26.46 | 35.86 | 303.67 |

Method | 1PC | 2PC | 10PC | 100PC | 200PC |
---|---|---|---|---|---|

gaPCA | 24.87 | 26.79 | 36.24 | 53.60 | 287.06 |

PCA | 22.37 | 35.73 | 37.86 | 47.26 | 315.03 |

Class | Training Pixels | PCA ML | gaPCA ML | PCA SVM | gaPCA SVM |
---|---|---|---|---|---|

Alfalfa | 32 | 98.7 | 80.5 | 18.2 | 18.2 |

Corn notill | 1145 | 30.6 | 47.6 | 65.2 | 69.3 |

Corn mintill | 595 | 51.6 | 69.2 | 34.9 | 46.1 |

Corn | 167 | 84.9 | 100 | 31.4 | 37.7 |

Grass pasture | 328 | 55.7 | 80.5 | 64.6 | 71.9 |

Grass trees | 463 | 96.1 | 90.6 | 91.2 | 92.5 |

Grass pasture mowed | 19 | 68.3 | 71.7 | 60 | 60 |

Hay windrowed | 528 | 88.5 | 96.7 | 99.5 | 99.6 |

Oats | 20 | 100 | 96.9 | 15.6 | 6.3 |

Soybean notill | 681 | 83.7 | 77.1 | 40.9 | 56.1 |

Soybean mintill | 1831 | 46.6 | 47.7 | 79.4 | 78.3 |

Soybean clean | 457 | 36.9 | 77.7 | 11.8 | 36.1 |

Wheat | 150 | 97.2 | 97 | 91.1 | 93.1 |

Woods | 884 | 98.7 | 96.9 | 97.3 | 97.3 |

Buildings Drives | 263 | 33.7 | 61.4 | 45.5 | 52.1 |

Stone Steel Towers | 103 | 100 | 100 | 95.5 | 97.2 |

z_{ML} = 25.1 (signif = yes) | OA(%) | 62.1 | 70.2 | 67.2 | 72.1 |

z_{SVM} = 24.8 (signif = yes) | Kappa | 0.57 | 0.67 | 0.62 | 0.68 |

Class | Training Pixels | PCA ML | gaPCA ML | PCA SVM | gaPCA SVM |
---|---|---|---|---|---|

Asphalt (grey) | 1766 | 60.5 | 61.5 | 67.2 | 78.3 |

Meadows (light green) | 2535 | 68.3 | 80 | 65 | 86.9 |

Gravel (cyan) | 923 | 100 | 100 | 33.3 | 40 |

Trees (dark green) | 599 | 88.2 | 89.7 | 100 | 67.7 |

Metal sheets (magenta) | 872 | 100 | 100 | 100 | 100 |

Bare soil (brown) | 1579 | 77.8 | 79.4 | 53.2 | 68.3 |

Bitumen (purple) | 565 | 89.7 | 89.7 | 89.7 | 55.2 |

Bricks (red) | 1474 | 68.3 | 72 | 81.7 | 86.6 |

Shadows (yellow) | 876 | 100 | 100 | 100 | 100 |

z_{ML} = 4.87 (signif = yes) | OA(%) | 72.2 | 78 | 69 | 78 |

z_{SVM} = 5.97 (signif = yes) | Kappa | 0.65 | 0.72 | 0.61 | 0.72 |

Class | True | False |
---|---|---|

Asphalt (PCA) | 60.5 Asphalt | 29.5 Bitumen |

Asphalt (gaPCA) | 61.5 Asphalt | 21.8 Bitumen |

Meadows (PCA) | 68.3 Meadows | 25.8 Bare soil |

Meadows (gaPCA) | 80 Meadows | 17.6 Bare soil |

Bricks (PCA) | 68.3 Bricks | 25.6 Gravel |

Bricks (gaPCA) | 72 Bricks | 24.3 Gravel |

Class | Training Pixels | PCA ML | gaPCA ML | PCA SVM | gaPCA SVM |
---|---|---|---|---|---|

Road (dark brown) | 862 | 90 | 100 | 100 | 100 |

Trees (dark green) | 413 | 75.9 | 82.7 | 75.9 | 75.9 |

Water (blue) | 466 | 86.7 | 83.3 | 86.7 | 86.7 |

Grass (light green) | 992 | 86.9 | 91.3 | 67.4 | 71.7 |

Shadows (black) | 121 | 87.5 | 75 | 37.5 | 50 |

Roofs and paths(brown) | 358 | 64.7 | 94.1 | 52.9 | 52.9 |

z_{ML} = 2 (signif = yes) | OA(%) | 82 | 88 | 72 | 74 |

z_{SVM} = 1.13 (signif = no) | Kappa | 0.77 | 0.85 | 0.65 | 0.67 |

Class | Training Pixels | PCA ML | gaPCA ML | PCA SVM | gaPCA SVM |
---|---|---|---|---|---|

Oilseed rape (dark yellow) | 2786 | 93.3 | 93.3 | 93.2 | 97.7 |

Oilseed rape (light yellow) | 1013 | 80 | 80 | 90 | 95 |

Maize (pink) | 969 | 100 | 100 | 100 | 100 |

Winter wheat (orange) | 4429 | 100 | 98.1 | 97.3 | 97.3 |

Pasture (light green) | 1788 | 66.7 | 66.7 | 84.6 | 92.3 |

Grassland (dark green) | 1242 | 60 | 80 | 52.4 | 95.2 |

Urban (grey) | 1079 | 60 | 90 | 64 | 92 |

z_{ML} = 1.97 (signif = yes) | OA(%) | 90.6 | 93.8 | 90 | 96.6 |

z_{SVM} = 3.92 (signif = yes) | Kappa | 0.86 | 0.91 | 0.86 | 0.95 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Machidon, A.L.; Del Frate, F.; Picchiani, M.; Machidon, O.M.; Ogrutan, P.L.
Geometrical Approximated Principal Component Analysis for Hyperspectral Image Analysis. *Remote Sens.* **2020**, *12*, 1698.
https://doi.org/10.3390/rs12111698

**AMA Style**

Machidon AL, Del Frate F, Picchiani M, Machidon OM, Ogrutan PL.
Geometrical Approximated Principal Component Analysis for Hyperspectral Image Analysis. *Remote Sensing*. 2020; 12(11):1698.
https://doi.org/10.3390/rs12111698

**Chicago/Turabian Style**

Machidon, Alina L., Fabio Del Frate, Matteo Picchiani, Octavian M. Machidon, and Petre L. Ogrutan.
2020. "Geometrical Approximated Principal Component Analysis for Hyperspectral Image Analysis" *Remote Sensing* 12, no. 11: 1698.
https://doi.org/10.3390/rs12111698