# What Can Multifractal Analysis Tell Us about Hyperspectral Imagery?

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## Abstract

**:**

## 1. Introduction

#### 1.1. Fractals

#### 1.2. Multifractals

## 2. Data

## 3. Methodology

## 4. Results

#### 4.1. Degree of Multifractality for Radiance

#### 4.2. Degree of Multifractality for Reflectance

## 5. Discussion

#### 5.1. Interpretation of the Multifractal Results

#### 5.1.1. Influence of Atmospheric Absorption

_{2}absorption. In the NIR zone, three subzones can be clearly observed (bands 41–63, 64–84 and 85–103). This division may be caused by the absorbance of solar radiation by H

_{2}O vapour at the edges. The values for mountains reveal two fragments where there is a sudden increment (bands 63–65 and 81–86). These parts of the electromagnetic spectrum are where absorption caused by H

_{2}O is observed. Atmospheric correction causes smoothing of the Δ curve for the mountain landscape in these bands. It could suggest that the Δ parameter may be used to evaluate atmospheric correction quality. Moreover, in the bands adjacent to these zones (in SWIR 1 and SWIR 2), we observed increased values only for agriculture and mountain landscapes. It may be related to the fact that these two landscape samples include plants. After atmospheric correction, Δ for these bands also seems to be smoothed.

#### 5.1.2. Landscape Types and Dimensionality Reduction

#### 5.1.3. Size of Images and Spatial Resolution

#### 5.2. Comparison with Other Characteristics

#### 5.2.1. Correlation with Statistical Moments

^{2}) to evaluate the complementarity between Δ and statistical moments. In the first assessment, values of Δ for all bands for the four landscape types in reflectance images were combined. Low values of r

^{2}were obtained with four moments: mean (0.003), standard deviation (0.225), skewness (0.114) and kurtosis (0.034). In the next step, each landscape type was analysed separately (Table 6). In this case, we observed the highest coefficient values for urban landscape, notably for the mean (0.834) and skewness (0.858). This may be because the urban sample has the most complex structure and contains a dense mixture of dark and bright objects (see Figure 1c). Coefficient values were much lower for the three other landscape types.

#### 5.2.2. Comparison with Fractal Dimension

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Landscape Type | Original Image (Radiance) | Original Image (Reflectance) |
---|---|---|

Agriculture | f130522t01p00r11rdn_e | f130522t01p00r11_refl |

Mountains | f130522t01p00r12rdn_e | f130522t01p00r12_refl |

Urban | f130612t01p00r11rdn_e | f130612t01p00r11_refl |

Water | f130606t01p00r10rdn_e | f130606t01p00r10_refl |

**Figure A1.**Values of first four statistical moments (from top: mean, variance, skewness and kurtosis) determined for radiance (left column) and reflectance (right column) data for four landscape types: water (blue), mountains (black), agriculture (green) and urban (red).

## Appendix B

**Table A2.**Values of the Δ estimation for radiance and reflectance data. Bold font indicates the lowest and the highest Δ values for specific landscape types.

Radiance | Reflectance | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

VIS | NIR | A 1 | SWIR 1 | A 2 | SWIR 2 | All | VIS | NIR | SWIR 1 | SWIR 2 | All | ||

Agriculture | Min | 0.006 | 0.039 | 0.047 | 0.068 | 0.075 | 0.157 | 0.006 | 0.032 | 0.039 | 0.068 | 0.157 | 0.032 |

Mean | 0.088 | 0.070 | 0.073 | 0.106 | 0.197 | 0.264 | 0.137 | 0.151 | 0.072 | 0.101 | 0.262 | 0.151 | |

Max | 0.210 | 0.109 | 0.159 | 0.188 | 0.423 | 0.498 | 0.498 | 0.497 | 0.110 | 0.171 | 0.497 | 0.497 | |

Mountains | Min | 0.001 | 0.024 | 0.033 | 0.052 | 0.164 | 0.095 | 0.001 | 0.031 | 0.025 | 0.053 | 0.096 | 0.025 |

Mean | 0.031 | 0.037 | 0.116 | 0.070 | 0.349 | 0.135 | 0.092 | 0.100 | 0.034 | 0.064 | 0.129 | 0.079 | |

Max | 0.084 | 0.076 | 0.222 | 0.138 | 0.581 | 0.278 | 0.581 | 0.206 | 0.047 | 0.083 | 0.208 | 0.208 | |

Urban | Min | 0.002 | 0.034 | 0.024 | 0.071 | 0.032 | 0.082 | 0.002 | 0.045 | 0.041 | 0.058 | 0.061 | 0.041 |

Mean | 0.060 | 0.048 | 0.083 | 0.087 | 0.048 | 0.134 | 0.080 | 0.126 | 0.056 | 0.089 | 0.134 | 0.098 | |

Max | 0.104 | 0.055 | 0.484 | 0.103 | 0.062 | 0.172 | 0.484 | 0.198 | 0.064 | 0.108 | 0.176 | 0.198 | |

Water | Min | 0.0003 | 0.004 | 0.004 | 0.004 | 0.011 | 0.001 | 0.0003 | 0.003 | 0.005 | 0.005 | 0.002 | 0.002 |

Mean | 0.002 | 0.005 | 0.081 | 0.005 | 0.079 | 0.007 | 0.014 | 0.005 | 0.006 | 0.006 | 0.006 | 0.006 | |

Max | 0.004 | 0.005 | 0.768 | 0.006 | 0.234 | 0.030 | 0.768 | 0.025 | 0.006 | 0.006 | 0.022 | 0.025 |

**Table A3.**Errors in the Δ estimation for radiance data. Bold font indicates the highest mean value within a zone; green represents the lowest mean error for a specific landscape type; and red, the highest.

VIS | NIR | A 1 | SWIR 1 | A 2 | SWIR 2 | All | ||
---|---|---|---|---|---|---|---|---|

Agriculture | Min | 0.0004 4% | 0.0031 8% | 0.0036 7% | 0.0051 7% | 0.0060 7% | 0.0125 8% | 0.0004 4% |

Mean | 0.0070 7% | 0.0056 8% | 0.0061 8% | 0.0083 8% | 0.0240 13% | 0.0245 9% | 0.0124 8% | |

Max | 0.0192 9% | 0.0086 8% | 0.0132 13% | 0.0161 9% | 0.0464 29% | 0.0581 12% | 0.0581 29% | |

Mountains | Min | 0.0001 5% | 0.0008 2% | 0.0009 2% | 0.0034 4% | 0.0065 4% | 0.0047 3% | 0.0001 2% |

Mean | 0.0026 9% | 0.0027 7% | 0.0061 5% | 0.0044 6% | 0.0244 7% | 0.0075 6% | 0.0059 7% | |

Max | 0.0068 15% | 0.0087 11% | 0.0150 8% | 0.0071 8% | 0.0458 9% | 0.0156 7% | 0.0458 15% | |

Urban | Min | 0.0001 8% | 0.0080 18% | 0.0030 8% | 0.0062 9% | 0.0012 4% | 0.0031 4% | 0.0001 4% |

Mean | 0.010417% | 0.010822% | 0.029219% | 0.016319% | 0.0096 20% | 0.024718% | 0.016019% | |

Max | 0.0158 24% | 0.0131 24% | 0.2524 52% | 0.0185 23% | 0.0167 28% | 0.0347 22% | 0.2524 52% | |

Water | Min | 0.0000 17% | 0.0006 16% | 0.0006 14% | 0.0005 13% | 0.0021 19% | 0.0001 10% | 0.0000 10% |

Mean | 0.0003 17% | 0.0008 17% | 0.0224 24% | 0.0009 16% | 0.028833% | 0.0012 17% | 0.0039 18% | |

Max | 0.0006 23% | 0.0008 17% | 0.2124 34% | 0.0009 17% | 0.0846 46% | 0.0083 28% | 0.2124 46% | |

All | Mean | 0.0052 13% | 0.0050 14% | 0.0159 14% | 0.0075 12% | 0.0217 18% | 0.0145 12% |

**Table A4.**Modulus of the difference between error values of Δ calculated for reflectance and radiance data. Average values.

Landscape Type | VIS | NIR | SWIR 1 | SWIR 2 |
---|---|---|---|---|

Agriculture | 0.007 | 0.0003 | 0.001 | 0.001 |

304% | 5% | 5% | 2% | |

Mountains | 0.004 | 0.001 | 0.001 | 0.001 |

542% | 32% | 12% | 7% | |

Urban | 0.009 | 0.003 | 0.003 | 0.002 |

494% | 32% | 21% | 6% | |

Water | 0.001 | 0.0002 | 0.0001 | 0.00005 |

440% | 25% | 11% | 6% | |

All | 0.005 | 0.001 | 0.001 | 0.001 |

442% | 21% | 12% | 5% |

## Appendix C

**Figure A2.**Fractal dimension determined for radiance images of water (blue), mountain (black), agriculture (green) and urban (red) landscapes.

**Figure A3.**Fractal dimension determined for reflectance images of water (blue), mountain (black), agriculture (green) and urban (red) landscapes.

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**Figure 1.**Selected spectral bands of four Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) images used in the analyses: (

**a**) agriculture, (

**b**) mountains, (

**c**) urban and (

**d**) water.

**Figure 2.**Degree of multifractality (Δ) determined for radiance images of water (blue), mountain (black), agriculture (green) and urban (red) landscapes. The whole spectrum is divided according to ranges presented in Table 3.

**Figure 3.**Degree of multifractality calculated for radiance and reflectance images of water (blue), mountain (black), agriculture (green) and urban (red) landscapes, divided into four zones according to Table 3.

**Figure 4.**Degree of multifractality calculated for radiance images of different sizes: 512 × 512, 256 × 256, 128 × 128 and 64 × 64 pixels, and four landscape types: agriculture (green), mountains (black), urban (red) and water (blue).

**Table 1.**Examples of local and global (multi)fractal descriptions applied to hyperspectral data analysis.

Paper | Sensor/Dataset | Number of Bands | Image Size | Parameters Used/Method | |
---|---|---|---|---|---|

Global description | Qiu et al. (1999) [20] | AVIRIS ^{1}/MalibuAVIRIS/LA | 224 224 | 614 × 512 614 × 512 | Fractal Dimension/isarithm method and triangular prism method |

Myint et al. (2003) [21] | ATLAS (5 classes) | 15 | 17 × 17 33 × 32 65 × 65 | Fractal dimension/isarithm, triangular prism and variogram method | |

Su et al. (2008) [29] | OMIS ^{2}/Beijing | 64 | 536 × 512 | Fractal dimension/double blanket method | |

Krupiński et al. (2014) [22] | AVIRIS (4 classes) AVIRIS/Malibu AVIRIS/LA | 224 224 224 | 512 × 512 512 × 512 512 × 512 | Fractal dimension/differential box counting method | |

Local description | Combrexelle et al. (2015) [30] | Hyspex/Madonna AVIRIS/Moffit Field | 160 224 | 256 × 256 64 × 64 16 × 16 | Coefficients of the polynomial describing multifractal spectrum/wavelet leader multifractal formalism |

^{1}Airborne Visible/Infrared Imaging Spectrometer,

^{2}Operational Modular Imaging Spectrometer.

**Table 2.**Examples of (multi)fractal spectral curve descriptions applied to hyperspectral data analysis.

Paper | Sensor/Dataset | No. of Bands | No. of Classes | Parameters Used/Method | |
---|---|---|---|---|---|

Spectral Curve Description | Dong et al. (2008) [17] | HYPERION | 138 of 242 | 5 | Fractal dimension/blanket method |

Ghosh et al. (2008) [15] | AVIRIS/Moffit Field | 224 | 4 | Fractal dimension/adapted Hausdorff metric | |

Ghosh et al. (2008) [38] | AVIRIS/Moffit Field | 30 128 | 5 | Fractal dimension/adapted Hausdorff metric | |

Junying et al. (2008) [39] | MAIS ^{1}OMIS | 176 of 220 | 4 4 | Fractal dimension/step measurement method | |

Ziyong et al. (2010) [40] | HYPERION | 191 of 210 12 | - | Fractal dimension/modified blanked method | |

Hosseini et al. (2012) [41] | HYDICE ^{2}/WashingtonF210 | 191 of 210 12 | 6 9 | Fractal dimension/Hausdorff metric | |

Mukherjee et al. (2012) [14] | HYDICE AVIRIS/Indian Pine AVIRIS/Cuprite | 188 of 210 200 of 224 197 of 224 | 5 9 of 16 14 | Fractal dimension/power spectrum method | |

Mukherjee et al. (2013) [19] | HYDICE AVIRIS/Indian Pine AVIRIS/Cuprite | 188 of 210 200 of 224 197 of 224 | 5 9 of 16 14 | Fractal dimension/variogram method | |

Mukherjee et al. (2014) [18] | AVIRIS/Indian Pine AVIRIS/Cuprite | 200 of 224 197 of 224 | 9 of 16 14 | Fractal dimension/Sevcik’s method, power spectrum method, variogram method | |

Li et al. (2015) [42] | PHI ^{3}/FangluAVIRIS/Indian Pines | 64 200 of 224 | 6 16 | 4 parameters related to multifractal spectrum | |

Wan et al. (2017) [43] | AVIRIS/Indian Pines AVIRIS/KSC | 200 of 224 176 of 224 | 9 of 16 13 | Holder exponent, multifractal spectrum features | |

Krupiński et al. (2019) [44] | CASI ^{4}/University of Houston | 144 | 15 | 6 parameters related to multifractal spectrum/multifractal detrended fluctuation analysis |

^{1}Modular Airborne Imaging Spectrometer,

^{2}Hyperspectral Digital Imagery Collection Experiment,

^{3}Pushbroom Hyperspectral Imager,

^{4}Compact Airborne Spectrographic Imager.

Zone Name | VIS | NIR | A 1 | SWIR 1 | A 2 | SWIR 2 |
---|---|---|---|---|---|---|

Bands | 1–40 | 41–103 | 104–114 | 115–152 | 153–168 | 169–224 |

Wavelength (nm) | 366–724 | 734–1313 | 1323–1423 | 1433–1802 | 1811–1937 | 1947–2496 |

**Table 4.**Modulus of the difference between Δ values for reflectance and radiance data. Average absolute and relative (%) values. Relative values refer to radiance data.

Landscape Type | VIS | NIR | SWIR 1 | SWIR 2 |
---|---|---|---|---|

Agriculture | 0.074 | 0.003 | 0.007 | 0.004 |

233% | 4% | 4% | 1% | |

Mountains | 0.062 | 0.005 | 0.004 | 0.009 |

524% | 11% | 8% | 5% | |

Urban | 0.063 | 0.007 | 0.000 | 0.002 |

383% | 15% | 5% | 2% | |

Water | 0.003 | 0.001 | 0.001 | 0.0003 |

446% | 25% | 11% | 6% | |

All | 0.049 | 0.005 | 0.004 | 0.004 |

394% | 9% | 7% | 3% |

**Table 5.**Modulus of the difference between values of Δ calculated for radiance data for original and resampled data. Average values.

Mean Δ Differences | Mean Error Differences | |||||
---|---|---|---|---|---|---|

Landscape Type | 256 × 256 | 128 × 128 | 64 × 64 | 256 × 256 | 128 × 128 | 64 × 64 |

Agriculture | 0.007 | 0.007 | 0.012 | 0.003 | 0.008 | 0.014 |

4% | 4% | 10% | 2% | 5% | 11% | |

Mountains | 0.006 | 0.012 | 0.015 | 0.001 | 0.002 | 0.002 |

6% | 14% | 18% | 1% | 2% | 2% | |

Urban | 0.023 | 0.033 | 0.037 | 0.008 | 0.012 | 0.014 |

29% | 44% | 52% | 4% | 7% | 8% | |

Water | 0.005 | 0.008 | 0.011 | 0.002 | 0.004 | 0.004 |

21% | 44% | 76% | 2% | 2% | 5% | |

All | 0.010 | 0.015 | 0.019 | 0.003 | 0.005 | 0.009 |

15% | 26% | 39% | 2% | 3% | 6% |

**Table 6.**Correlation coefficients (r

^{2}) between the degree of multifractality and four moments calculated for reflectance for the four landscape types after removal of outliers.

Landscape Type | Mean | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|

Agriculture | 0.452 | 0.003 | 0.012 | 0.508 |

Mountains | 0.664 | 0.075 | 0.038 | 0.012 |

Urban | 0.834 | 0.500 | 0.858 | 0.762 |

Water | 0.255 | 0.180 | 0.039 | 0.008 |

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**MDPI and ACS Style**

Krupiński, M.; Wawrzaszek, A.; Drzewiecki, W.; Jenerowicz, M.; Aleksandrowicz, S. What Can Multifractal Analysis Tell Us about Hyperspectral Imagery? *Remote Sens.* **2020**, *12*, 4077.
https://doi.org/10.3390/rs12244077

**AMA Style**

Krupiński M, Wawrzaszek A, Drzewiecki W, Jenerowicz M, Aleksandrowicz S. What Can Multifractal Analysis Tell Us about Hyperspectral Imagery? *Remote Sensing*. 2020; 12(24):4077.
https://doi.org/10.3390/rs12244077

**Chicago/Turabian Style**

Krupiński, Michał, Anna Wawrzaszek, Wojciech Drzewiecki, Małgorzata Jenerowicz, and Sebastian Aleksandrowicz. 2020. "What Can Multifractal Analysis Tell Us about Hyperspectral Imagery?" *Remote Sensing* 12, no. 24: 4077.
https://doi.org/10.3390/rs12244077