Unsupervised Diffusion and Volume MaximizationBased Clustering of Hyperspectral Images
Abstract
:1. Introduction
2. Background
2.1. Background on Unsupervised HSI Clustering
2.2. Background on Spectral Graph Theory
Background on Diffusion Geometry
2.3. Background on Spectral Unmixing
2.3.1. Background on the HySime Algorithm
2.3.2. Background on the AVMAX Algorithm
3. Diffusion and Volume MaximizationBased Image Clustering
Algorithm 1: Diffusion and Volume maximizationbased Image Clustering 
Input: X (HSI), N (# nearest neighbors), ${\sigma}_{0}$ (KDE scale), t (diffusion time), K (# clusters) Output: $\mathcal{C}$ (clustering)

3.1. Computational Complexity
3.2. Comparison with Learning by Unsupervised Nonlinear Diffusion
4. Experiments and Discussion
4.1. Analysis of Benchmark HSI Datasets
 1.
 Salinas A (Figure 5a) was recorded by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor over farmland in Salinas Valley, California, USA, in 1998 at a spatial resolution of 1.3 m. Spectral signatures, ranging in recorded wavelength from 380 nm to 2500 nm across 224 spectral bands, were recorded across $83\times 86$ pixels ($n=7138$). Gaussian noise (with mean 0 and standard deviation $={10}^{7}$) was added to each pixel to differentiate two pixels with identical spectral signatures. The Salinas A scene contains $K=6$ ground truth classes corresponding to crop types.
 2.
 Jasper Ridge (Figure 5b) was recorded by the AVIRIS sensor over the Jasper Ridge Biological Preserve, California, USA, in 1989 at a spatial resolution of 5 m. Spectral signatures, ranging in recorded wavelength from 380 nm to 2500 nm across 224 spectral bands, were recorded across spatial dimensions of $100\times 100$ pixels ($n=$ 10,000). The Jasper Ridge scene contains $K=4$ ground truth endmembers: road, soil, water, and trees. Ground truth labels were recovered by selecting the material of the highest ground truth abundance for each pixel.
 3.
 Indian Pines (Figure 5c) was recorded by the AVIRIS sensor over farmland in northwest Indiana, USA, in 1992 at a low spatial resolution of 20 m. Spectral signatures, ranging in recorded wavelength from 400 nm to 2500 nm across 224 spectral bands, were recorded across spatial dimensions of $145\times 145$ pixels ($n=$ 21,025). The Indian Pines scene contains $K=16$ ground truth classes (e.g., crop types and manufactured structures) and many unlabeled pixels.
4.1.1. Discussion of Benchmark HSI Experiments
4.1.2. Runtime Analysis
4.1.3. Robustness to Hyperparameter Selection
4.2. Analysis of the Madingley HSI Dataset
Discussion of Madingley Experiments
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Hyperparameter Optimization
Parameter 1 Grid  Parameter 2 Grid  Parameter 3 Grid  

KMeans  —  —  — 
KMeans + PCA  —  —  — 
GMM + PCA  —  —  — 
DPC [135]  $N\in \mathcal{N}$  ${\sigma}_{0}\in \mathcal{D}$  — 
SC [41]  $N\in \mathcal{N}$  —  — 
SymNMF [21]  $N\in \mathcal{N}$  —  — 
KNNSSC [19,20]  $N\in \mathcal{N}$  $\lambda =10$  — 
FSSC [46]  $N\in \mathcal{N}$  ${\alpha}_{u}\in \mathcal{A}$  $\ell ={2}^{11}$ 
LUND [42]  $N\in \mathcal{N}$  ${\sigma}_{0}\in \mathcal{D}$  $t\in \mathcal{T}$ 
DVIC  $N\in \mathcal{N}$  ${\sigma}_{0}\in \mathcal{D}$  $t\in \mathcal{T}$ 
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Dataset  Spatial Resolution  Spectral Range  Spatial Dimensions  Num. Pixels  Num. Spectral Bands  Num. Clusters 

Salinas A  1.3 m  380–2500 nm  $83\times 86$  $n=7138$  $D=224$  $K=6$ 
Jasper Ridge  5.0 m  380–2500 nm  $100\times 100$  $n=$ 10,000  $D=224$  $K=4$ 
Indian Pines  20 m  400–2500 nm  $145\times 145$  $n=$ 21,025  $D=224$  $K=16$ 
Salinas A  Jasper Ridge  Indian Pines  

OA  $\mathit{\kappa}$  OA  $\mathit{\kappa}$  OA  $\mathit{\kappa}$  
KMeans  0.764  0.703  0.784  0.703  0.383  0.315 
KMeans + PCA  0.764  0.703  0.785  0.703  0.382  0.316 
GMM + PCA  0.611  0.512  0.789  0.701  0.364  0.292 
DPC  0.629  0.529  0.809  0.727  0.410  0.271 
SC  0.834  0.797  0.760  0.670  0.382  0.314 
SymNMF  0.828  0.791  0.662  0.542  0.365  0.304 
KNNSSC  0.844  0.809  0.726  0.629  0.371  0.308 
FSSC  0.830  0.793  0.780  0.691  0.396  0.281 
LUND  0.887  0.860  0.815  0.737  0.404  0.312 
DVIC  0.976  0.970  0.865  0.805  0.445  0.350 
Salinas A  Jasper Ridge  Indian Pines  

KMeans  0.04  0.10  1.04 
KMeans + PCA  0.10  0.14  0.58 
GMM + PCA  0.13  0.23  2.19 
DPC  3.20  6.41  25.77 
SC  1.82  3.15  14.54 
SymNMF  3.50  4.42  48.29 
KNNSSC  4.11  7.91  103.05 
FSSC  13.53  30.40  130.72 
LUND  2.35  4.14  14.74 
DVIC  4.95  7.64  23.70 
KM  KM + PCA  GMM + PCA  DPC  SC  SymNMF  KNNSSC  FSSC  LUND  DVIC  

OA  0.570  0.570  0.477  0.555  0.595  0.630  0.651  0.608  0.648  0.645 
$\kappa $  0.245  0.245  0.099  0.000  0.300  0.243  0.328  0.262  0.296  0.287 
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Polk, S.L.; Cui, K.; Chan, A.H.Y.; Coomes, D.A.; Plemmons, R.J.; Murphy, J.M. Unsupervised Diffusion and Volume MaximizationBased Clustering of Hyperspectral Images. Remote Sens. 2023, 15, 1053. https://doi.org/10.3390/rs15041053
Polk SL, Cui K, Chan AHY, Coomes DA, Plemmons RJ, Murphy JM. Unsupervised Diffusion and Volume MaximizationBased Clustering of Hyperspectral Images. Remote Sensing. 2023; 15(4):1053. https://doi.org/10.3390/rs15041053
Chicago/Turabian StylePolk, Sam L., Kangning Cui, Aland H. Y. Chan, David A. Coomes, Robert J. Plemmons, and James M. Murphy. 2023. "Unsupervised Diffusion and Volume MaximizationBased Clustering of Hyperspectral Images" Remote Sensing 15, no. 4: 1053. https://doi.org/10.3390/rs15041053