NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping
Abstract
:1. Introduction
2. Methods
2.1. LogNNet Model for NNetEn1D Calculation
2.2. Method for Two-Dimensional NNetEn2D Calculation with Circular Kernels
2.3. Method for Two-Dimensional NNetEn2D Calculation with Square Kernels
2.4. Artificial Test Image
2.5. Image Preprocessing Methods
2.5.1. Removing the Constant Component of the Brightness of the Image
2.5.2. Image Rotation
2.6. Main Steps for Estimating the NNetEn2D of Images
- Carrying out image preprocessing if necessary,
- Select kernel type (CIR_R, SQCi_R, SQRo_R, SQCo_R),
- Choose parameters R, S, DL, and division of the image area into circular kernels,
- Selecting the number of epochs Ep for calculating NNetEn2D and techniques for filling the matrix (W1M_1-W1M_6),
- Calculation of NNetEn2D in each spherical kernel,
- Calculate the resulting entropy for each pixel as the average of all NNetEn2D from all kernels using that pixel.
3. Results
3.1. Research Results on the Test Image
3.1.1. Effects of Kernel Radius and Number of Epochs on NNetEn2D Variance
3.1.2. NNetEn2D Distribution Examples for Different Kernels
3.1.3. Effects of Image Rotation on the NNetEn2D Distribution
3.1.4. Effects of Removing Constant Component on the NNetEn2D Distribution
3.2. Results of the Study on Sentinel-2 Images
3.2.1. Effects of Data Preprocessing on the NNetEn2D Distribution
3.2.2. Effect of Image Rotation on the NNetEn2D Distribution
3.2.3. NNetEn2D Distribution of Sentinel-2 Images
3.3. Research Results on Aero-Magnetic Images
4. Discussion
- Parallelize the calculation of the product of a matrix and a vector in steps 1 and 3; this can increase speed up to 10–100 times.
- Organize a parallel calculation of the entropy for each image kernel at step 12. For the example shown in the Table 6, the acceleration will be 324 times.
- Reduce the number of training images in step 7.
- Reduce the number of test images in step 10.
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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S | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Rmin | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 7 |
R | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
N | 5 | 13 | 29 | 49 | 81 | 113 | 149 | 197 | 253 |
R | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
N | 9 | 25 | 49 | 81 | 121 | 169 | 225 | 289 | 361 |
PCP (%) | R = 1 | R = 2 | R = 3 | R = 4 | R = 5 | R = 6 | R = 7 |
---|---|---|---|---|---|---|---|
kernel | PCP (%) for NNetEn2D when rotated by 45° | ||||||
CIR_R | 18.9 | 12.1 | 22.3 | 14.2 | 9.6 | 9.9 | 11.6 |
SQCi_R | 12.2 | 16.1 | 21.6 | 14.7 | 15.7 | 29.3 | 25.3 |
SQRo_R | 27.0 | 42 | 26.2 | 37.2 | 45.8 | 24.1 | 27.4 |
SQCo_R | 34.9 | 101.9 | 107.2 | 113.2 | 169.7 | 105.7 | 98.3 |
PCP (%) for NNetEn2D when rotated by 90° | |||||||
CIR_R | 12.9 | 10.5 | 10.9 | 8.7 | 4.3 | 3.9 | 6.1 |
SQCi_R | 8.1 | 11.4 | 12.5 | 7.3 | 7.6 | 3.2 | 3.2 |
SQRo_R | 34.0 | 45 | 33.4 | 51.6 | 57.2 | 43.9 | 48.6 |
SQCo_R | 115.9 | 322.7 | 164.4 | 301.1 | 704.9 | 163.5 | 180.9 |
PCP (%) | R = 1 | R = 2 | R = 3 | R = 4 | R = 5 | R = 6 | R = 7 |
---|---|---|---|---|---|---|---|
kernel type | PCP (%) for NNetEn2D when rotated by 45° | ||||||
CIR_R, Ep = 4 | 18.9 | 12.1 | 22.3 | 14.2 | 9.6 | 9.9 | 11.6 |
CIR_R, Ep = 20 | 18.9 | 13.4 | 9.3 | 17.1 | 8.4 | 9.2 | 18 |
CIR_R, Ep = 100 | 18.9 | 9.5 | 7.4 | 10.4 | 7.8 | 4.1 | 12.2 |
PCP (%) for NNetEn2D when rotated by 90° | |||||||
CIR_R, Ep = 4 | 12.9 | 10.5 | 10.9 | 8.7 | 4.3 | 3.9 | 6.1 |
CIR_R, Ep = 20 | 12.9 | 13.8 | 6.3 | 10.1 | 3.1 | 1.5 | 1.5 |
CIR_R, Ep = 100 | 12.9 | 16.3 | 6.9 | 7.6 | 1.5 | 1.2 | 4 |
Stage Number | Stage Description | Vector of Computational Cost C = (N(±), N(*), N(/), N(exp)) |
---|---|---|
1 | Multiplication of the W1 matrix by the Y vector in the reservoir (see Figure 1) | C1 = (19,625, 19,625, 0, 0) |
2 | Normalization of the vector Sh | C2 = (100, 0, 25, 0) |
3 | Forward method of the output neural network, multiplication of the matrix W2 by the vector Sh | C3 = (260, 260, 0, 0) |
4 | Normalization of the vector Sout | C4 = (10, 0, 10, 10) |
5 | Back-propagation method | C5 = (280, 540, 0, 0) |
6 | LogNNet training on one image | C6 = C1+ C2+ C3+ C4+ C5 C6 = (20,275, 20,425, 35, 10) |
7 | LogNNet training using 60,000 MNIST images | C7 = C6·60,000 C7 = (1.2165 × 109, 1.2255 × 109, 2.1 × 106, 6 × 105) |
8 | LogNNet training using Ep = 4 epochs | C8 = C7·Ep C8 = (4.866 × 109, 4.902 × 109, 8.4 × 106, 2.4 × 106) |
9 | LogNNet testing on one image | C9 = C1+ C2+ C3+ C4 C9 = (19,995, 19,885, 35, 10) |
10 | LogNNet testing using 10,000 MNIST images | C10 = C9·10,000 C10 = (1.9995 × 108, 1.9885 × 108, 3.5 × 105, 1.0 × 105) |
11 | NNetEn2D entropy calculation for one kernel | C11 = C8+ C10 C11 = (5.0660 × 109, 5.1009 × 109, 8.75 × 106, 2.5 × 106) |
12 | Calculation of entropy for one image sized 99 × 99 pixels, with parameters S = 6, R = 5. 324 circular kernels are needed to cover the entire image | C12 = C11·324 C12 = (1.6414 × 1012, 1.6527 × 1012, 2.835 × 109, 8.1 × 108) |
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Velichko, A.; Wagner, M.P.; Taravat, A.; Hobbs, B.; Ord, A. NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping. Remote Sens. 2022, 14, 2166. https://doi.org/10.3390/rs14092166
Velichko A, Wagner MP, Taravat A, Hobbs B, Ord A. NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping. Remote Sensing. 2022; 14(9):2166. https://doi.org/10.3390/rs14092166
Chicago/Turabian StyleVelichko, Andrei, Matthias P. Wagner, Alireza Taravat, Bruce Hobbs, and Alison Ord. 2022. "NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping" Remote Sensing 14, no. 9: 2166. https://doi.org/10.3390/rs14092166
APA StyleVelichko, A., Wagner, M. P., Taravat, A., Hobbs, B., & Ord, A. (2022). NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping. Remote Sensing, 14(9), 2166. https://doi.org/10.3390/rs14092166