Unbalanced Optimal Transport Dictionary Learning for Unsupervised Hyperspectral Image Clustering
Authors: Joshua Lentz, Nicholas Karris, Alex Cloninger, James M. Murphy
Organizations: Tufts University, Department of Mathematics, Medford, MA 02155, USA · University of California San Diego, Department of Mathematics, La Jolla, CA 92093, USA · University of California San Diego, Halıcıo˘glu Data Science Institute, La Jolla, CA 92093, USA
Hyperspectral images capture vast amounts of high-dimensional spectral information about a scene, making labeling an intensive task that is resistant to out-of-the-box statistical methods. Unsupervised learning of clusters allows for automated segmentation of the scene, enabling a more rapid understanding of the image. Partitioning the spectral information contained within the data via dictionary learning in Wasserstein space has proven an effective method for unsupervised clustering. However, this approach requires balancing the spectral profiles of the data, blurring the classes, and sacrificing robustness to outliers and noise. In this paper, we suggest improving this approach by utilizing unbalanced Wasserstein barycenters to learn a lower-dimensional representation of the underlying data. The deployment of spectral clustering on the learned representation results in an effective approach for the unsupervised learning of labels.
Figures & tables
Fig. 1 : Pictured here is the balanced (left) and unbalanced (right) barycentric interpolation between two Gaussian distributions with the same variance using τ=0.5 and ϵ=0.001 . Notice that the unbalanced barycenters do not obey strict mass conservation, but still take the general shape of the reference distributions.
Data set
Accuracy
τ
ϵ
Atoms
NN
Salinas A
0.89
1000
0.1
24
25
Salinas A
0.86
10000
0.05
30
20
Pavia Centre
0.84
1000
0.07
27
15
Pavia Centre
0.82
100
0.07
27
5
Pavia U
0.63
10000
0.07
18
5
Pavia U
0.58
100000
0.1
36
5
Table 1: Accuracy test results for various data sets.
Data set
Purity
τ
ϵ
Atoms
NN
c
Salinas A
0.92
1000
0.1
60
45
7
Pavia Centre
0.91
10000
0.09
24
10
10
Pavia U
0.72
10000
0.11
36
20
10
Indian Pines
0.47
1000
0.1
64
40
17
Salinas A
0.93
1000
0.1
30
45
35
Pavia Centre
0.91
10000
0.09
24
10
18
Table 2: Purity test results for various data sets. The first four rows highlight purity results in which the number of clusters exceeds the number of ground truth labels by just one. The following four rows allow c to further exceed the number of ground truth labels.
Fig. 2 : This image shows the in-painting process of a trial run of the Salinas A data set with 24 atoms, τ=1000 , and ϵ=0.1 achieving an accuracy before in-painting of 89% . We note that the bottom right corner of the image accounts for the majority of the mislabeling, and that this issue is common under unsupervised labeling schemes.
Data set
UBCSC
BCSC
τ
ϵ
Atoms
Salinas A
0.89
0.68
1000
0.1
24
Pavia Centre
0.84
0.73
1000
0.07
27
Pavia U
0.63
0.40
10000
0.07
18
Indian Pines
0.34
0.30
1000
0.06
32
Table 3: Comparison of UBCSC vs BCSC accuracy under identical hyperparameters.
Data set
UBCSC
BCSC
Salinas A
0.89
0.86
Pavia Centre
0.84
0.77
Pavia U
0.63
0.47
Indian Pines
0.34
0.32
Table 4: Comparison of UBCSC vs BCSC accuracy under best case performance.
Beijing Institute of Technology, Beijing 100081, China · Beijing Institute of Technology Chongqing Innovation Center, Chongqing 401135, China · Key Laboratory of Photoelectronic Imaging Technology and System, Ministry of Education of China, Beijing 100081, China