# An Assessment of Polynomial Regression Techniques for the Relative Radiometric Normalization (RRN) of High-Resolution Multi-Temporal Airborne Thermal Infrared (TIR) Imagery

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## Abstract

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^{2}) of TABI-1800 airborne data, visual and statistical results show that both new non-linear techniques improved radiometric agreement over the previously evaluated linear techniques, with the new fully-automated method, NCSRS-based polynomial regression, providing the highest improvement in radiometric agreement between the master and the slave images, at ~56%. This is ~5% higher than the best previously evaluated linear technique (NCSRS-based linear regression).

## 1. Introduction

## 2. Methods

#### 2.1. Study Area and Dataset

^{2}) portion of The City of Calgary, Alberta, Canada, composed of two adjacent TABI-1800 TIR flight lines (Figure 1A), each ~0.9 km wide by 39 km long with ~30% overlap between them. The City of Calgary is situated approximately 80 km east of the front ranges of the Canadian Rockies mountain range and, as a modern metropolitan center, is composed of a variety of urban landscape features (Figure 1B). The TABI-1800 (Thermal Airborne Broadband Imager) is an H-res TIR airborne sensor that has a swath-width of 1800 pixels, which it collects in a single channel (3.7–4.8 µm spectral range). It has an instantaneous field of view (IFOV) of 0.405 milliradians and a field of view (FOV) of ±40 degrees with a 14-bit dynamic range. The sensor’s radiometric accuracy is 0.05 °C, and it is able to collect data at 90–100 frames per second. The data for this project were acquired between 2:00 and 3:00 am on 13 May 2012, at a 50-cm spatial resolution and were ortho-rectified using a 10-m DEM (digital elevation model). The reported (horizontal) geometric accuracy of the dataset is ±1 m.

**Figure 1.**(

**A**) The City of Calgary map, displaying the location of the two TABI-1800 flight lines used in this study. (

**B**) An example of TABI-1800 imagery (at 50 cm pixels) within the study area, detailing the urban complexity resulting from roads, buildings, trees, green space, etc. Bright locations are warm, and dark locations are cool.

#### 2.2. Relative Radiometric Normalization

#### 2.2.1. Histogram Matching

**Figure 2.**A hypothetical example of radiometric normalization using the histogram matching technique. DN (digital number).

#### 2.2.2. Pseudo-Invariant Feature (PIF)-Based Polynomial Regression (PIF_Poly)

**Figure 3.**A scatterplot of pseudo-invariant features (PIFs) selected within the overlap of the master and the slave images. These PIFs represent a combination of four land cover classes (grass, river water, rooftop and road) and are shown modeled with a sixth order polynomial trend line (black). Poly, polynomial.

#### 2.2.3. No-Change Stratified Random Sample (NCSRS)-Based Linear Regressions

**Figure 4.**A scatterplot of no-change stratified random samples (NCSRS) with a sixth order polynomial trend line (black) and a linear trend line (red dotted) between the master and the slave images.

#### 2.2.4. No-Change Stratified Random Sample (NCSRS)-Based Polynomial Regression

#### 2.3. Validation of the Results Using Root Mean Square Error (RMSE)

## 3. Results and Discussions

#### 3.1. Visual Assessments

- ▪
- HM appears to perform very well for road and water, but performs only moderately well for grass and rooftop.
- ▪
- PIF_Poly performs well for road and water and moderately well for grass, but it does not perform well for rooftop.
- ▪
- NCSRS_Lin performs very well for water and moderately well for road, grass and rooftop.
- ▪
- NCSRS_Poly performs very well for road and water and well for grass and rooftop. Though subjective, we further suggest that grass and rooftop visually appear best modeled by this method.

**Figure 5.**Visual examples of four different relative radiometric normalization methods applied along the mosaic join line of four different land cover types (grass, road, water and rooftop). PIF_Poly—pseudo-invariant feature-based polynomial regression; NCSRS_Lin-no-change stratified random sample-based linear regression.

**Figure 6.**A visual example of how relative radiometric normalization techniques decrease the radiometric variability between flight lines. (

**A**) A sample area from the master image. (

**B**) The same area from the slave image. Pseudo-colored absolute image difference (

**C**) between the master and the uncorrected slave image and between the master and the normalized salve images resulting from (

**D**) HM, (

**E**) PIF_Poly, (

**F**) NCSRS_Lin and (

**G**) NCSRS_Poly.

#### 3.2. Statistical Analysis

**Table 1.**The overall RMSE of four different land cover types, for each of the four different relative radiometric normalization methods evaluated in this study. Bold values represent the lowest RMSE of each class and overall RMSE calculated for each image.

Land Cover Type | RMSE (°C) | ||||
---|---|---|---|---|---|

Slave | HM | PIF_Poly | NCSRS_Lin | NCSRS_Poly | |

Grass | 0.420 | 0.236 | 0.227 | 0.193 | 0.163 |

Road | 0.201 | 0.097 | 0.128 | 0.122 | 0.123 |

Rooftop | 0.586 | 0.436 | 0.452 | 0.371 | 0.322 |

Water | 0.216 | 0.106 | 0.108 | 0.130 | 0.113 |

Overall ^{*} | 0.356 | 0.194 | 0.210 | 0.173 | 0.159 |

^{*}Mean of RMSEs of all selected test samples for different land cover types.

^{2}of 0.84. Thus, it represents the best performing normalization method, followed by NCSRS_Lin (Figure 7D). Conversely, while the HM method improves the slope between the master and the slave (indicating that the radiometric agreement is supposed to improve), the intercept is slightly increased, and in the case of PIF_Poly, the intercept is further increased (meaning that the radiometric agreement is supposed to be decreased).

**Figure 7.**(

**A**) A comparison of the scatterplot between the original master and the slave images and after applying four normalization methods: (

**B**) HM; (

**C**) PIF_Poly; (

**D**) NCSRS_Lin; and (

**E**) NCSRS_Poly. The thin blue lines describe the data trend line, while the red dashed lines show the expected trend(s) at perfect radiometric agreement.

#### 3.2.1. A Comparison of Automatic vs Manual Methods

#### 3.2.2. An Assessment of Computation Time

**Table 2.**Computation time of four different relative radiometric normalization (RRN) methods evaluated in this study.

RRN Method | Computing Time (min) |
---|---|

Histogram Matching | 2.14 |

PIF_Poly | 4.7^{ *} |

NCSRS_Lin | 1.4 |

NCSRS_Poly | 4.7 |

^{*}Represents the computation time required after the manual collection of PIFs.

#### 3.2.3. A Comparison of Linear vs Polynomial Methods

**Figure 8.**A comparison of linear (LIN) and polynomial (Poly) regression-based radiometric normalization using the same no-change stratified random samples (NCSRS).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Mustafizur Rahman, M.; Hay, G.J.; Couloigner, I.; Hemachandran, B.; Bailin, J.
An Assessment of Polynomial Regression Techniques for the Relative Radiometric Normalization (RRN) of High-Resolution Multi-Temporal Airborne Thermal Infrared (TIR) Imagery. *Remote Sens.* **2014**, *6*, 11810-11828.
https://doi.org/10.3390/rs61211810

**AMA Style**

Mustafizur Rahman M, Hay GJ, Couloigner I, Hemachandran B, Bailin J.
An Assessment of Polynomial Regression Techniques for the Relative Radiometric Normalization (RRN) of High-Resolution Multi-Temporal Airborne Thermal Infrared (TIR) Imagery. *Remote Sensing*. 2014; 6(12):11810-11828.
https://doi.org/10.3390/rs61211810

**Chicago/Turabian Style**

Mustafizur Rahman, Mir, Geoffrey J. Hay, Isabelle Couloigner, Bharanidharan Hemachandran, and Jeremy Bailin.
2014. "An Assessment of Polynomial Regression Techniques for the Relative Radiometric Normalization (RRN) of High-Resolution Multi-Temporal Airborne Thermal Infrared (TIR) Imagery" *Remote Sensing* 6, no. 12: 11810-11828.
https://doi.org/10.3390/rs61211810