# Hyperspectral Image Classification Using Geodesic Spatial–Spectral Collaborative Representation

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Contributions

- Adoption of geodesic distance, instead of the traditional Euclidean distance, for selecting spectral nearest-neighbor information, which fully utilizes the nearest-neighbor information of hyperspectral images.
- Fusion of spatial information and spectral information based on geodesic distance, enabling the full utilization of spatial–spectral information in hyperspectral images.
- Establishment of a spatial–spectral joint representation model with the combination of spatial and spectral information, followed by the classification of hyperspectral images using the minimum residual method.

## 2. Related Works

#### 2.1. Data Classification

#### 2.2. Sparse Representation

#### 2.3. Collaborative Representation

## 3. The Proposed Method

#### 3.1. Extracting Spatial Neighborhood Information of HSIs Based on Euclidean Distance

- 1.
- Obtain the coordinates of the test sample.
- 2.
- Obtain the coordinates of the training samples.
- 3.
- Calculate the Euclidean distance between the coordinates of the test sample and the training samples.
- 4.
- Convert the calculated Euclidean distances into matrix form to obtain the spatial information matrix, D.

#### 3.2. Extracting Spectral Neighborhood Information of HSIs Based on Geodesic Distance

- 1.
- Determine the connected sample points in the sample set by calculating the Euclidean distance and constructing a weighted graph. Any similarity measurement model can be utilized to determine neighborhood relationships. Typically, Euclidean distance ${d}_{E}$ is employed to determine whether two sample points are considered neighbors. A pair of neighboring points must satisfy that one sample point is a k-nearest neighbor of the other sample point.
- 2.
- Utilize either the Floyd algorithm or the Dijkstra algorithm to identify the nearest neighbors for each test sample point based on geodesic distance.

- 1.
- Calculate Euclidean distance ${d}_{y}(y,{x}_{m,j})$ along the spectral dimension between test sample y and training sample ${x}_{m,j}$. If y is a k-nearest neighbor of ${x}_{m,j}$, then y is considered a neighbor of ${x}_{m,j}$. In this case, the weight of the edge is ${d}_{y}(y,{x}_{m,j})$.
- 2.
- Determine the shortest path between y and ${x}_{m,j}$. If an edge exists between them, consider the shortest path to be ${d}_{G}(y,{x}_{m,j})={d}_{y}(y,{x}_{m,j})$. If no edge exists, consider the shortest path to be ${d}_{G}(y,{x}_{m,j})=\infty $.
- 3.
- Calculate the spectral geodesic distance using Formula (2):$${d}_{G}(y,{x}_{m,j})=min({d}_{G}(y,{x}_{m,j}),{d}_{G}(y,{x}_{m,l})+{d}_{G}({x}_{m,l},{x}_{m,j}))$$
- 4.
- Obtain spectral information matrix $S=\left\{{d}_{G}(y,{x}_{m,j})\right\}$, with $S\in {\mathbb{R}}^{n\times n}$.

#### 3.3. Building a Spatial–Spectral Collaborative Representation Model Based on Geodesic Distance

## 4. Experiments

#### 4.1. Experimental Dataset

#### 4.1.1. Indian Pines Dataset

#### 4.1.2. Salinas Dataset

#### 4.1.3. PaviaU Dataset

#### 4.2. Experimental Results and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Indian Pines dataset: (

**a**) pseudo-color images, (

**b**) real terrain maps, (

**c**) category labels.

**Figure 3.**Salinas dataset: (

**a**) pseudo-color image, (

**b**) class diagram of real features, (

**c**) category labels.

**Figure 4.**PaviaU dataset: (

**a**) pseudo-color image, (

**b**) class diagram of real features, (

**c**) category labels.

**Figure 5.**Classification performance of GSSCRC algorithm and other algorithms on the Indian Pines dataset: (

**a**) ground truth, (

**b**) SVM, (

**c**) SRC, (

**d**) KSRC, (

**e**) JCR, (

**f**) GSSCRC.

**Figure 6.**Classification performance of GSSCRC algorithm and other algorithms on the Salinas dataset: (

**a**) ground truth, (

**b**) SVM, (

**c**) SRC, (

**d**) KSRC, (

**e**) JCR, (

**f**) GSSCRC.

**Figure 7.**Classification effects of GSSCRC algorithm and other algorithms on PaviaU dataset: (

**a**) ground truth, (

**b**) SVM, (

**c**) SRC, (

**d**) KSRC, (

**e**) JCR, (

**f**) GSSCRC.

**Figure 8.**Impact of different algorithms on overall classification accuracy with different training sample sizes: (

**a**) Indian Pines dataset, (

**b**) Salinas dataset, (

**c**) PaviaU dataset.

Index | Class Name | Sample Size |
---|---|---|

1 | Alfalfa | 46 |

2 | Corn-notill | 1428 |

3 | Corn-mintill | 830 |

4 | Corn | 237 |

5 | Grass-pasture | 483 |

6 | Grass-trees | 730 |

7 | Grass-pasture-mowed | 28 |

8 | Hay-windrowed | 478 |

9 | Oats | 20 |

10 | Soybean-nottill | 972 |

11 | Soybean-mintill | 2455 |

12 | Soybean-clean | 593 |

13 | Wheat | 205 |

14 | Woods | 1265 |

15 | Buildings-Grass-Trees-Drives | 386 |

16 | Stone-Steel-Towers | 93 |

Index | Class Name | Sample Size |
---|---|---|

1 | Brocoli_green_weeds_1 | 2009 |

2 | Brocoli_green_weeds_2 | 3726 |

3 | Fallow | 1976 |

4 | Fallow_rough_plow | 1394 |

5 | Fallow_smooth | 2678 |

6 | Stubble | 3959 |

7 | Celery | 3579 |

8 | Grapes_untrained | 11,271 |

9 | Soil_vinyard_develop | 6203 |

10 | Corn_senesced_green_weeds | 3278 |

11 | Lettuce_romaine_4wk | 1068 |

12 | Lettuce_romaine_5wk | 1927 |

13 | Lettuce_romaine_6wk | 916 |

14 | Lettuce_romaine_7wk | 1070 |

15 | Vinyard_ untrained | 7268 |

16 | Vinyard_vertical_trellis | 1807 |

Index | Class Name | Sample Size |
---|---|---|

1 | Asphalt | 6631 |

2 | Meadows | 18,649 |

3 | Gravel | 2099 |

4 | Trees | 3064 |

5 | Metal sheets | 1345 |

6 | Bare soil | 5029 |

7 | Bitumen | 1330 |

8 | Bricks | 3682 |

9 | Shadows | 947 |

**Table 4.**Detailed classification results of GSSCRC algorithm and other algorithms for various features in the Indian Pines dataset (%).

Category | Sample Size | Classification Methods | |||||
---|---|---|---|---|---|---|---|

Training Samples | Test Samples | SVM | SRC | KSRC | JCR | GSSCRC | |

1 | 5 | 41 | 87.96 | 100 | 100 | 100 | 100 |

2 | 143 | 1285 | 60.38 | 81.58 | 86.97 | 88.44 | 90.97 |

3 | 83 | 747 | 45.36 | 80.60 | 84.58 | 86.50 | 88.07 |

4 | 24 | 213 | 42.61 | 56.12 | 72.80 | 78.48 | 84.39 |

5 | 48 | 435 | 80.12 | 90.68 | 92.75 | 95.03 | 95.65 |

6 | 73 | 657 | 93.65 | 98.63 | 97.53 | 98.76 | 98.77 |

7 | 3 | 25 | 92.80 | 100 | 100 | 100 | 100 |

8 | 48 | 430 | 88.62 | 97.48 | 97.90 | 99.79 | 99.79 |

9 | 2 | 18 | 93.25 | 100 | 100 | 100 | 100 |

10 | 97 | 875 | 57.68 | 81.17 | 85.70 | 89.40 | 90.53 |

11 | 246 | 2209 | 74.06 | 82.20 | 86.27 | 86.29 | 86.76 |

12 | 59 | 534 | 60.85 | 85.16 | 88.70 | 87.18 | 94.94 |

13 | 21 | 184 | 93.67 | 99.02 | 99.02 | 99.51 | 99.51 |

14 | 127 | 1138 | 90.76 | 97.87 | 96.92 | 97.54 | 97.63 |

15 | 39 | 347 | 50.27 | 62.17 | 73.83 | 79.53 | 79.53 |

16 | 9 | 84 | 94.38 | 100 | 100 | 100 | 100 |

OA (%) | 75.40 | 84.82 | 88.33 | 89.92 | 91.33 | ||

AA (%) | 77.71 | 87.34 | 90.66 | 92.32 | 93.81 | ||

Kappa (%) | 73.86 | 82.68 | 86.70 | 88.53 | 90.13 |

**Table 5.**Detailed classification results of various features in the Salinas dataset using GSSCRC algorithm and other algorithms (%).

Category | Sample Size | Classification Methods | |||||
---|---|---|---|---|---|---|---|

Training Samples | Test Samples | SVM | SRC | KSRC | JCR | GSSCRC | |

1 | 101 | 1908 | 99.65 | 99.25 | 99.60 | 99.70 | 100 |

2 | 187 | 3539 | 97.69 | 98.87 | 99.97 | 100 | 100 |

3 | 99 | 1877 | 98.93 | 98.38 | 99.79 | 99.70 | 99.80 |

4 | 70 | 1324 | 99.56 | 99.64 | 100 | 99.71 | 99.92 |

5 | 144 | 2648 | 96.71 | 98.84 | 98.58 | 98.14 | 99.10 |

6 | 198 | 3761 | 99.04 | 99.62 | 99.62 | 99.72 | 99.72 |

7 | 179 | 3400 | 99.46 | 99.61 | 99.80 | 99.86 | 99.86 |

8 | 564 | 10,707 | 80.25 | 96.51 | 81.68 | 89.62 | 89.76 |

9 | 311 | 6173 | 99.45 | 97.34 | 99.84 | 99.98 | 100 |

10 | 164 | 3248 | 90.91 | 96.92 | 97.83 | 98.28 | 98.32 |

11 | 54 | 1012 | 94.75 | 99.53 | 99.91 | 98.60 | 98.65 |

12 | 97 | 1830 | 98.91 | 100 | 100 | 100 | 100 |

13 | 46 | 870 | 99.56 | 97.92 | 99.45 | 99.12 | 99.13 |

14 | 54 | 1016 | 80.56 | 98.59 | 99.34 | 95.70 | 95.70 |

15 | 364 | 7238 | 97.48 | 97.33 | 96.79 | 78.23 | 77.43 |

16 | 91 | 1716 | 96.84 | 99.33 | 99.61 | 99.43 | 99.45 |

OA (%) | 89.34 | 91.19 | 92.21 | 94.45 | 95.62 | ||

AA (%) | 93.21 | 95.81 | 96.76 | 97.28 | 97.30 | ||

Kappa(%) | 88.15 | 90.19 | 91.33 | 93.76 | 93.84 |

**Table 6.**Detailed classification results of GSSCRC algorithm and other algorithms for various features in the PaviaU dataset (%).

Category | Sample Size | Classification Methods | |||||
---|---|---|---|---|---|---|---|

Training Samples | Test Samples | SVM | SRC | KSRC | JCR | GSSCRC | |

1 | 663 | 5968 | 86.79 | 92.76 | 91.38 | 92.09 | 96.20 |

2 | 1864 | 16,785 | 88.78 | 96.65 | 96.75 | 95.93 | 98.44 |

3 | 210 | 1889 | 83.33 | 82.56 | 75.27 | 81.89 | 83.66 |

4 | 306 | 2758 | 97.81 | 96.41 | 96.41 | 96.96 | 96.21 |

5 | 135 | 1210 | 99.63 | 99.70 | 99.18 | 99.63 | 99.63 |

6 | 503 | 4526 | 93.39 | 91.91 | 90.97 | 94.65 | 93.82 |

7 | 133 | 1197 | 93.98 | 90.98 | 92.93 | 92.26 | 90.60 |

8 | 368 | 3314 | 85.85 | 88.29 | 91.01 | 92.50 | 91.91 |

9 | 95 | 852 | 100 | 99.89 | 100 | 99.89 | 99.89 |

OA (%) | 89.81 | 93.86 | 93.35 | 94.18 | 95.77 | ||

AA (%) | 91.99 | 92.98 | 92.14 | 93.73 | 94.13 | ||

Kappa (%) | 86.77 | 91.88 | 91.21 | 92.33 | 94.38 |

Method | Indian Pines Dataset | Salinas Dataset | PaviaU Dataset |
---|---|---|---|

SVM | 293.97 | 2168.53 | 1301.62 |

SRC | 334.91 | 2865.72 | 1425.13 |

KSRC | 338.34 | 4114.06 | 1476.75 |

JCR | 348.08 | 4800.50 | 1685.32 |

GSSCRC | 365.36 | 5176.28 | 2244.06 |

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## Share and Cite

**MDPI and ACS Style**

Zheng, G.; Xiong, X.; Li, Y.; Xi, J.; Li, T.; Tolba, A.
Hyperspectral Image Classification Using Geodesic Spatial–Spectral Collaborative Representation. *Electronics* **2023**, *12*, 3777.
https://doi.org/10.3390/electronics12183777

**AMA Style**

Zheng G, Xiong X, Li Y, Xi J, Li T, Tolba A.
Hyperspectral Image Classification Using Geodesic Spatial–Spectral Collaborative Representation. *Electronics*. 2023; 12(18):3777.
https://doi.org/10.3390/electronics12183777

**Chicago/Turabian Style**

Zheng, Guifeng, Xuanrui Xiong, Ying Li, Juan Xi, Tengfei Li, and Amr Tolba.
2023. "Hyperspectral Image Classification Using Geodesic Spatial–Spectral Collaborative Representation" *Electronics* 12, no. 18: 3777.
https://doi.org/10.3390/electronics12183777