# Hyperspectral Image Classification with Localized Graph Convolutional Filtering

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The usual supervised setting regarding fitting the graph-based learning models is designed through collecting the patch-based feature cubes and localized graph adjacent matrices.
- (2)
- The graph convolution layer is used to learn the spatially local graph representation and to represent the localized topological patterns of the graph nodes.
- (3)
- The experiments demonstrate that the presented study could achieve promising classification performance based on the localized graph convolutional filter.

## 2. Related Work

## 3. Preliminaries

#### 3.1. Graph Structure

#### 3.2. Adjacency Matrix

- (1)
- the radial basis function ${a}_{i,j}=\mathrm{exp}\left(-\frac{{\Vert {x}_{i}-{x}_{j}\Vert}^{2}}{{\sigma}^{2}}\right)$ [8] or the Gaussian similarity function ${a}_{i,j}=\mathrm{exp}\left(-\frac{{\Vert {x}_{i}-{x}_{j}\Vert}^{2}}{2{\sigma}^{2}}\right)$ [2], where $\sigma $ is a parameter to control the width of the neighborhoods, the vectors ${x}_{i}$ and ${x}_{j}$ denote the spectral signatures associated to the vertexes ${v}_{i}$ and ${v}_{j}$, respectively;
- (2)
- the distance function ${a}_{ij}={\Vert x\Vert}_{p}={\left({\displaystyle \sum _{c=1}^{C}{\left|{x}_{ic}-{x}_{jc}\right|}^{p}}\right)}^{\frac{1}{p}},p\ge 1$ is defined between two samples ${x}_{i}$ and ${x}_{j}$, where $p$ is an optional parameter, and $C$ is the dimension of the feature vector. When $p$ is 1 or 2, it becomes Manhattan distance or Euclidean distance (i.e., used in this study), respectively. The distance metric of all sample pairs can form a symmetric distance matrix ${A}_{m}=\left[{a}_{ij}\right]\in {\mathrm{R}}^{n\times n}$. For example, ${a}_{ij}$ at row $i$ and column $j$ in the matrix ${A}_{m}$ denotes the distance between the ${i}^{th}$ pixel and the ${j}^{th}$ pixel [20].

#### 3.3. Graph Laplacian

#### 3.4. Graph Fourier Transform

## 4. Proposed Method

#### 4.1. Graph Construction

#### 4.2. Graph Convolution Filter

#### 4.3. Localized Graph Convolution

#### 4.4. Graph-Based CNN

## 5. Experiments and Analysis

#### 5.1. Datasets and Settings

#### 5.2. Classification Maps

^{st}run for all hyperspectral datasets were illustrated as well (see Figure 7).

#### 5.3. Classification Accuracies

#### 5.4. Probability Maps

#### 5.5. Time Consumption

^{5}; GCN: 2.50 × 10

^{5}).

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The overview of hyperspectral image (HSI) classification with localized graph convolutional network (GCN).

**Figure 2.**The presented graph representation learning framework for HSI classification with the localized graph convolutional filter. Here, the localized feature cubes were created by using a principal component analysis (PCA) transformation.

**Figure 3.**The architecture of the proposed convolutional neural network (CNN) and graph convolutional network (GCN). Here, S

_{i}{i = 1, 2} denotes the key sub-flows, and P

_{j}{j = 1, 2, 3} represents several sets of parameter settings. Note that the CNN and GCN are completely independent networks.

**Figure 4.**The pseudo-color images and ground truth data for four real hyperspectral datasets, i.e., (

**a**) the IA dataset, (

**b**) the SA dataset, (

**c**) the SV dataset, and (

**d**) the PU dataset.

**Figure 5.**The accuracy & loss curves of the CNN and GCN models in the 1st run for the SV and PU datasets, i.e., (

**a**) CNN-SV, (

**b**) CNN-PU, (

**c**) GCN-SV, (

**d**) GCN-PU, corresponding to the experiment with five random runs.

**Figure 6.**The classification maps of the presented GCN and its competitors, i.e., SVM and CNN, in five times random run for the IA dataset. The 1st, 2nd, and 3rd rows correspond to the (

**a**) SVM, (

**b**) CNN, and (

**c**) GCN algorithms, respectively. Meanwhile, the 1st, 2nd, 3rd, 4th, and 5th columns correspond to different random runs, respectively.

**Figure 7.**The classification maps of the presented GCN (the 4th column) and its competitors, i.e., SVM (the 2nd column) and CNN (the 3rd column), in the 1st run for four real hyperspectral datasets, i.e., (

**a**) the IA dataset, (

**b**) the SA dataset, (

**c**) the SV dataset, (

**d**) the PU dataset, corresponding to the experiment with five random runs.

**Figure 8.**The confusion matrices of the presented GCN for different datasets, corresponding to the experiment with five random runs. (

**a**) The IA dataset was at the 5th random run, (

**b**) the SA dataset was at the 3rd random run, (

**c**) the SV dataset was at the 3rd random run, and (

**d**) the PU dataset was at the 2nd random run.

**Figure 9.**The probability maps of the presented GCN and its competitors in the 1st run for four real hyperspectral datasets, i.e., (

**a**) the IA dataset, (

**b**) the SA dataset, (

**c**) the SV dataset, (

**d**) the PU dataset, corresponding to the experiment with five random runs. Note that the deeper the color, the weaker the prediction.

**Table 1.**The division of ground truth samples for the Indian Pines-A (IA), Salinas Valley-A (SA), Salinas Valley (SV), and Pavia University (PU) datasets.

Datasets | Codes | Classes | Total | Training | Test | Validation |
---|---|---|---|---|---|---|

IA | C0 | Not-ground truth | 1534 | 0 | 0 | 0 |

C1 | Corn-notill | 1005 | 60 | 885 | 60 | |

C2 | Grass-trees | 730 | 60 | 610 | 60 | |

C3 | Soybean-notill | 741 | 60 | 621 | 60 | |

C4 | Soybean-mintill | 1924 | 60 | 1804 | 60 | |

SA | C0 | Not-ground truth | 1790 | 0 | 0 | 0 |

C1 | Brocoli_green_weeds_1 | 391 | 60 | 271 | 60 | |

C2 | Corn_senesced_green_weeds | 1343 | 60 | 1223 | 60 | |

C3 | Lettuce_romaine_4wk | 616 | 60 | 496 | 60 | |

C4 | Lettuce_romaine_5wk | 1525 | 60 | 1405 | 60 | |

C5 | Lettuce_romaine_6wk | 674 | 60 | 554 | 60 | |

C6 | Lettuce_romaine_7wk | 799 | 60 | 679 | 60 | |

SV | C0 | Not-ground truth | 56,975 | 0 | 0 | 0 |

C1 | Brocoli_green_weeds_1 | 2009 | 60 | 1889 | 60 | |

C2 | Brocoli_green_weeds_2 | 3726 | 60 | 3606 | 60 | |

C3 | Fallow | 1976 | 60 | 1856 | 60 | |

C4 | Fallow_rough_plow | 1394 | 60 | 1274 | 60 | |

C5 | Fallow_smooth | 2678 | 60 | 2558 | 60 | |

C6 | Stubble | 3959 | 60 | 3839 | 60 | |

C7 | Celery | 3579 | 60 | 3459 | 60 | |

C8 | Grapes_untrained | 11,271 | 60 | 11,151 | 60 | |

C9 | Soil_vinyard_develop | 6203 | 60 | 6083 | 60 | |

C10 | Corn_senesced_green_weeds | 3278 | 60 | 3158 | 60 | |

C11 | Lettuce_romaine_4wk | 1068 | 60 | 948 | 60 | |

C12 | Lettuce_romaine_5wk | 1927 | 60 | 1807 | 60 | |

C13 | Lettuce_romaine_6wk | 916 | 60 | 796 | 60 | |

C14 | Lettuce_romaine_7wk | 1070 | 60 | 950 | 60 | |

C15 | Vinyard_untrained | 7268 | 60 | 7148 | 60 | |

C16 | Vinyard_vertical_trellis | 1807 | 60 | 1687 | 60 | |

PU | C0 | Not-ground truth | 164,624 | 0 | 0 | 0 |

C1 | Asphalt | 6631 | 60 | 6511 | 60 | |

C2 | Meadows | 18,649 | 60 | 18,529 | 60 | |

C3 | Gravel | 2099 | 60 | 1979 | 60 | |

C4 | Trees | 3064 | 60 | 2944 | 60 | |

C5 | Painted metal sheets | 1345 | 60 | 1225 | 60 | |

C6 | Bare Soil | 5029 | 60 | 4909 | 60 | |

C7 | Bitumen | 1330 | 60 | 1210 | 60 | |

C8 | Self-Blocking Bricks | 3682 | 60 | 3562 | 60 | |

C9 | Shadows | 947 | 60 | 827 | 60 |

**Table 2.**The statistical classification accuracies for four real hyperspectral datasets with 5 and 10 random runs. Note that the experiments regarding a different number of random runs were carried out independently.

Alg. | SVM | CNN | GCN | ||||||
---|---|---|---|---|---|---|---|---|---|

Dat./Acc. | K | OA | AA | K | OA | AA | K | OA | AA |

IA^{5} | 0.7646 ± 0.0174 | 0.8343 ± 0.0120 | 0.8574 ± 0.0136 | 0.8990 ± 0.0103 | 0.9294 ± 0.0076 | 0.9506 ± 0.0043 | 0.9550 ± 0.0064 | 0.9685 ± 0.0045 | 0.9692 ± 0.0062 |

SA^{5} | 0.9793 ± 0.0054 | 0.9836 ± 0.0043 | 0.9818 ± 0.0065 | 0.9477 ± 0.0153 | 0.9586 ± 0.0122 | 0.9701 ± 0.0079 | 0.9983 ± 0.0010 | 0.9987 ± 0.0008 | 0.9982 ± 0.0011 |

SV^{5} | 0.8370 ± 0.0076 | 0.8533 ± 0.0070 | 0.9229 ± 0.0027 | 0.7895 ± 0.0126 | 0.8101 ± 0.0115 | 0.8244 ± 0.0057 | 0.9360 ± 0.0099 | 0.9426 ± 0.0088 | 0.9614 ± 0.0090 |

PU^{5} | 0.7015 ± 0.0065 | 0.7651 ± 0.0051 | 0.8270 ± 0.0090 | 0.7238 ± 0.0235 | 0.7856 ± 0.0204 | 0.8423 ± 0.0121 | 0.9113 ± 0.0080 | 0.9336 ± 0.0059 | 0.8927 ± 0.0128 |

IA^{10} | 0.7671 ± 0.0156 | 0.8355 ± 0.0114 | 0.8613 ± 0.0106 | 0.8930 ± 0.0249 | 0.9250 ± 0.0182 | 0.9475 ± 0.0094 | 0.9579 ± 0.0104 | 0.9706 ± 0.0072 | 0.9702 ± 0.0075 |

SA^{10} | 0.9795 ± 0.0046 | 0.9837 ± 0.0036 | 0.9823 ± 0.0055 | 0.9637 ± 0.0074 | 0.9713 ± 0.0058 | 0.9783 ± 0.0038 | 0.9978 ± 0.0020 | 0.9982 ± 0.0016 | 0.9977 ± 0.0021 |

SV^{10} | 0.8389 ± 0.0080 | 0.8551 ± 0.0073 | 0.9224 ± 0.0037 | 0.7896 ± 0.0069 | 0.8105 ± 0.0064 | 0.8234 ± 0.0098 | 0.9405 ± 0.0136 | 0.9465 ± 0.0123 | 0.9663 ± 0.0063 |

PU^{10} | 0.7038 ± 0.0168 | 0.7674 ± 0.0148 | 0.8274 ± 0.0088 | 0.7218 ± 0.0314 | 0.7836 ± 0.0267 | 0.8424 ± 0.0152 | 0.9079 ± 0.0153 | 0.9309 ± 0.0117 | 0.8916 ± 0.0185 |

**Table 3.**Total network parameters and time consumption (i.e., the average time of 5 and 10 random runs). Note that the numbers in parentheses regarding datasets and models indicate the number of samples and the number of network parameters, respectively.

Alg. (Para.)/Dat. (Num.)/Time (s) | IA (86 × 69 × 200) | SA (83 × 86 × 204) | SV (512 × 217 × 204) | PU (610 × 340 × 103) | ||||
---|---|---|---|---|---|---|---|---|

Training (240) | Test (3920) | Training (360) | Test (4628) | Training (960) | Test (52209) | Training (540) | Test (41696) | |

SVM^{5} | 1.00 ± 0.04 | 0.01 ± 0.01 | 2.22 ± 0.01 | 0.02 ± 0.01 | 17.35 ± 0.25 | 1.29 ± 0.05 | 5.77 ± 0.01 | 0.55 ± 0.04 |

CNN^{5}(1.22 × 10 ^{5}) | 14.65 ± 4.47 | 0.31 ± 0.01 | 17.07 ± 3.33 | 0.39 ± 0.02 | 18.49 ± 0.27 | 5.00 ± 0.14 | 15.03 ± 0.94 | 3.21 ± 0.05 |

GCN^{5}(2.50 × 10 ^{5}) | 13.22 ± 1.22 | 0.24 ± 0.00 | 20.41 ± 0.72 | 0.31 ± 0.00 | 43.39± 0.48 | 3.61 ± 0.07 | 28.10 ± 0.59 | 3.03 ± 0.06 |

SVM^{10} | 1.05 ± 0.06 | 0.01 ± 0.00 | 2.24 ± 0.03 | 0.02 ± 0.01 | 17.47 ± 0.20 | 1.27 ± 0.03 | 5.80 ± 0.10 | 0.56 ± 0.04 |

CNN^{10}(1.22 × 10 ^{5}) | 12.69 ± 2.98 | 0.33 ± 0.03 | 14.79 ± 2.34 | 0.37 ± 0.01 | 18.06 ± 1.16 | 4.79 ± 0.31 | 13.72 ± 0.85 | 3.04 ± 0.04 |

GCN^{10}(2.50 × 10 ^{5}) | 13.50 ± 4.12 | 0.24 ± 0.01 | 21.52 ± 1.31 | 0.32 ± 0.01 | 50.04 ± 2.85 | 4.08 ± 0.13 | 35.08 ± 2.69 | 3.62 ± 0.14 |

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**MDPI and ACS Style**

Pu, S.; Wu, Y.; Sun, X.; Sun, X.
Hyperspectral Image Classification with Localized Graph Convolutional Filtering. *Remote Sens.* **2021**, *13*, 526.
https://doi.org/10.3390/rs13030526

**AMA Style**

Pu S, Wu Y, Sun X, Sun X.
Hyperspectral Image Classification with Localized Graph Convolutional Filtering. *Remote Sensing*. 2021; 13(3):526.
https://doi.org/10.3390/rs13030526

**Chicago/Turabian Style**

Pu, Shengliang, Yuanfeng Wu, Xu Sun, and Xiaotong Sun.
2021. "Hyperspectral Image Classification with Localized Graph Convolutional Filtering" *Remote Sensing* 13, no. 3: 526.
https://doi.org/10.3390/rs13030526