# kCCA Transformation-Based Radiometric Normalization of Multi-Temporal Satellite Images

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Relative Radiometric Normalization Based on kCCA Transformation

#### 2.1. kCCA Transformation and NIFs Extraction

**V**denotes the linear combination of $\mathit{z}$ in the high-dimensional feature space. The workflow of kCCA is shown in Figure 1. Similar to the solving process of linear correlation analysis, the first step is to solve $\mathit{c}$ and $\mathit{d}$ so that the correlation coefficients of the combined variables $\mathit{U}$ and $\mathit{V}$ are maximized.

**V**in Equations (2) and (3).

#### 2.2. Fitting Non-Linear Transformation for Radiometric Normalization

## 3. Data and Results

#### 3.1. Test Data

#### 3.2. NIFs Distribution Map

#### 3.3. Derive Nonlinear Transformations from NIFs

_{r}is the average of the residuals and S

_{r}is the standard deviation of the residuals. As indicated in Table 3, the absolute values of the cubic coefficients of image0330, image0403 and image1214 are significantly larger than those of image0705 and image0717.

#### 3.4. Radiometric Normalization Results

#### 3.4.1. Clouds Pixels in the Image

#### 3.4.2. The Threshold Parameters τ for NIFs Extraction

#### 3.4.3. Quantitative Comparison of the Radiometric Normalization Results

^{CCA}is RMSE of the radiometric normalization results from the CCA-based method, RMSE

^{kCCA}is RMSE of the results from the kCCA-based transformation, and RMSE

^{H}is RMSE of the results from the histogram matching method. ${\mathsf{\rho}}^{CCA}$, ${\mathsf{\rho}}^{kCCA}$ and ${\mathsf{\rho}}^{H}$ are Pearson correlation coefficients of the radiometric normalization results from the CCA-based, kCCA-based and the histogram matching methods, respectively. ${\mathsf{\rho}}_{h}^{CCA}$, ${\mathsf{\rho}}_{h}^{kCCA}$ and ${\mathsf{\rho}}_{h}^{H}$ are histogram correlation of the radiometric normalization results from the CCA-based, kCCA-based and the histogram matching methods, respectively.

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | linear dichroism |

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**Figure 2.**The GF-1 images of the test area (Beijing, China). The GF-1 data, (

**a**–

**f**) are image0330, image 0403, image0705, image0717, image1002 and image1214, acquired in 30 March, 3 April, 5 July, 17 July, 2 October, 14 December 2013.

**Figure 3.**The distribution map of NIFs selected by the kCCA transformation. (

**a**–

**e**) are, in order, image0330, image0403, image0705, image0717 and image1214. The green points are the NIFs of each image.

**Figure 4.**The ratio of NIFs in vegetation area to the total number of NIFs. (

**a**–

**e**) refer to the result of image0330, image0403, image0705, image0717 and image1214, respectively. The blue represents the NIFs in vegetation area and the yellow is the other NIFs.

**Figure 5.**The density scatter plot map of the NIFs. The horizontal axis in the figure is the TOA value of the original image that linearly stretched from 0–1 to 0–255, and the vertical axis is that of the reference image. (

**a**–

**e**) refer to the result of image0330, image0403, image0705, image0717 and image1214, respectively.

**Figure 6.**The fitted functions of the four band for the 5 test GF-1 image-pairs. (

**a**–

**d**) refer to the results of band1, band2, Band 3 and band4 based on the proposed method. (

**e**–

**h**) are the linear regression equations of the PIFs-based method. —is the fitted function of image0330 and the reference, —is that of image0403 and the reference, —is that of image0705 and the reference, —is that of image0717 and the reference, —is that of image1214 and the reference.

**Figure 7.**The superimposed display of the NIFs and PIFs distribution map and the curve of the fitted functions based on the NIFs and PIFs described above. The horizontal axis in the figure is the TOA value of the original image, and the vertical axis is that of the reference image. (

**a**–

**e**) refer to the NIFs results of image0330, image0403, image0705, image0717 and image1214, respectively. (

**f**–

**j**) are that of PIFs results.

**Figure 8.**The comparison of the results of radiometric normalization based on the kCCA transformation, the CCA method and the histogram matching. Each image is a mosaic image. The left one is the corrected image, and the right one is the reference image. (

**a**) is the test data and the reference image; (

**b**) is the results of the CCA algorithm and the reference images; (

**c**) is the results based on NIFs extracted by the kCCA transformation; (

**d**) is the results of the histogram matching.

**Figure 9.**The results of the radiometric normalization of image0330 based on two NIFs set and the fitted curve of the two normalized results. (

**a**) is the result based on all NIFs (including the cloud-NIFs), (

**b**) is the result based on NIFs without cloud pixels, (

**c**–

**f**) are histogram of image0330. (

**g**–

**j**) are the fitted curve of four bands of image-pair (image0330 and image1002). — denotes the fitted curve using all NIFs including cloud pixels. — is the fitted curve using NIFs without the cloud pixels.

**Figure 10.**The distribution map of NIFs based on different threshold τ. The data is image0717. (

**a**–

**c**) are the distribution maps when τ = 0.95, τ = 0.90 and τ = 085. (

**d**–

**g**) are histogram of four bands of image0717. (

**h**–

**k**) are fitting curves of four bands based on the above NIFs.

Satellite Performance | Technical Capability | |
---|---|---|

Satellite Orbit | type | Solar synchronous circular orbit |

Average orbit height | 644.5 km | |

descending/ascending nod sun-synchronous | 10:30 a.m. | |

regressive period | 41 days | |

Revisit/Coverage characteristic | Revisit: 4 days for 2/8 m camera under side sway | |

Coverage: 4 days for 16 m camera; 41 days for 2/8 m camera; | ||

High resolution imaging | Spectrum/μm | Panchromatic: 0.45–0.90μm |

B1: 0.45–0.52 μm; B2: 0.52–0.59 μm; B3: 0.63–0.69 μm; B4: 0.77–0.89 μm; | ||

resolution | Panchromatic: better than 2 m Multispectral: better than 8 m | |

Swath width (km) | >60 | |

Wide imaging | Spectrum/μm | B1: 0.45–0.52 μm; B2: 0.52–0.59 μm; B3: 0.63–0.69 μm; B4: 0.77–0.89 μm; |

resolution | Better than 16 m | |

Swath width (km) | >800 |

Image Name | Image Date |
---|---|

Image0705 | 5 July 2013 |

Image0717 | 17 July 2013 |

Image1002_{reference} | 2 October 2013 |

Image1214 | 14 December 2013 |

Image0330 | 30 March 2014 |

Image0403 | 3 April 2014 |

**Table 3.**The coefficients and error ratios of regression equation a

_{0,}a

_{1}, a

_{2}and a

_{3}represent the coefficients of the constant term, the linear term, the quadratic term, and the cubic term, respectively. M

_{r}is the average of the difference between the function value and the expected value, and S

_{r}is the standard deviation.

a_{3} | a_{2} | a_{1} | a_{0} | M_{r} | S_{r} | ||
---|---|---|---|---|---|---|---|

Image 0330 | Band1 | 18.3284 | −13.3953 | 3.1748 | −0.1532 | 0.019194 | 0.000002 |

Band2 | 21.5494 | −14.4433 | 3.1393 | −0.1489 | 0.019906 | 0.000003 | |

Band 3 | 10.9105 | −7.7441 | 1.8255 | −0.0855 | 0.02473 | 0.000004 | |

Band4 | 1.4785 | −1.7238 | 0.7561 | 0.072 | 0.035371 | 0.000007 | |

Image 0403 | Band1 | −7.367 | 5.2422 | −0.5349 | 0.0748 | 0.011784 | 0.000012 |

Band2 | −3.0287 | 2.4336 | −0.0403 | 0.0372 | 0.01312 | 0.000013 | |

Band 3 | −1.8955 | 2.0456 | −0.099 | 0.0288 | 0.018398 | 0.000015 | |

Band4 | 7.3705 | −6.866 | 2.1832 | −0.046 | 0.030512 | 0.000037 | |

Image 0705 | Band1 | −0.7959 | 0.4918 | 0.194 | 0.0297 | 0.010231 | 0.000024 |

Band2 | −0.4582 | 0.0524 | 0.3273 | 0.0071 | 0.011725 | 0.00002 | |

Band 3 | 0.173 | −0.4517 | 0.4073 | 0.0041 | 0.015785 | 0.000023 | |

Band4 | 0.0878 | −0.2848 | 0.3699 | 0.0379 | 0.0341 | 0.000083 | |

Image 0717 | Band1 | −2.2136 | 2.4316 | −0.445 | 0.0895 | 0.008238 | 0.000082 |

Band2 | −0.7765 | 0.7466 | 0.1471 | 0.0187 | 0.009817 | 0.00005 | |

Band 3 | −0.0018 | 0.0113 | 0.3381 | 0.0061 | 0.013293 | 0.000033 | |

Band4 | 0.6642 | −1.2344 | 0.9188 | −0.0681 | 0.031373 | 0.000042 | |

Image 1214 | Band1 | −87.0617 | 28.3288 | −1.1949 | 0.0675 | 0.010096 | 0.000023 |

Band2 | −44.4402 | 12.7071 | 0.2623 | 0.02687 | 0.011614 | 0.000023 | |

Band 3 | −41.5801 | 14.5948 | −0.1363 | 0.0267 | 0.017183 | 0.000032 | |

Band4 | 13.8154 | −9.1873 | 2.0127 | 0.07651 | 0.026603 | 0.000044 |

$\mathit{\tau}=0.99$ | $\mathit{\tau}=0.95$ | $\mathit{\tau}=0.90$ | $\mathit{\tau}=0.85$ | |
---|---|---|---|---|

Total number of NIFs | 11466 | 27596 | 41421 | 53453 |

NIFs in vegetation area | 10563 | 25319 | 38021 | 49097 |

Ratio of NIFs in vegetation area (%) | 92.12 | 91.75 | 91.79 | 91.85 |

**Table 5.**RMSE is the RMSE of the test data and reference image, $\mathsf{\rho}$ is the Pearson correlation coefficients of the test data and reference image, ${\mathsf{\rho}}_{h}$ is histogram correlation of the test data and reference image.

RMSE | $\mathsf{\rho}$ | ${\mathsf{\rho}}_{\mathit{h}}$ | |
---|---|---|---|

image pair (imag0330, image1002) | |||

Band 1 | 0.09 | 0.32 | 0.04 |

Band 2 | 0.09 | 0.40 | 0.03 |

Band 3 | 0.12 | 0.45 | -0.05 |

Band 4 | 0.06 | 0.36 | 0.74 |

image pair (imag0403, image1002) | |||

Band 1 | 0.07 | 0.80 | 0.02 |

Band 2 | 0.07 | 0.77 | 0.05 |

Band 3 | 0.10 | 0.71 | -0.02 |

Band 4 | 0.06 | 0.51 | 0.69 |

image pair (imag0705, image1002) | |||

Band 1 | 0.15 | 0.80 | 0.07 |

Band 2 | 0.15 | 0.77 | 0.09 |

Band 3 | 0.13 | 0.78 | 0.09 |

Band 4 | 0.42 | 0.40 | 0.24 |

image pair (imag0717, image1002) | |||

Band 1 | 0.15 | 0.89 | 0.07 |

Band 2 | 0.14 | 0.87 | 0.08 |

Band 3 | 0.11 | 0.86 | 0.07 |

Band 4 | 0.38 | 0.46 | 0.23 |

image pair (imag1214, image1002) | |||

Band 1 | 0.03 | 0.81 | 0.43 |

Band 2 | 0.02 | 0.81 | 0.78 |

Band 3 | 0.02 | 0.73 | 0.75 |

Band 4 | 0.11 | 0.67 | -0.04 |

**Table 6.**RMSE

^{CCA}is the RMSE of the radiometric normalization results based on CCA algorithm, RMSE

^{kCCA}is the RMSE of the results based on NIFs extracted by the kCCA transformation, RMSE

^{H}is the RMSE of the results based on the histogram matching method. $\mathsf{\rho}$

^{CCA}, $\mathsf{\rho}$

^{kCCA}and $\mathsf{\rho}$

^{H}are the Pearson correlation coefficients of the radiometric normalization results based on the CCA algorithm, kCCA transformation and the histogram matching method. ${\mathsf{\rho}}_{h}^{CCA}$, ${\mathsf{\rho}}_{h}^{kCCA}$ and ${\mathsf{\rho}}_{h}^{H}$ are histogram correlation of the radiometric normalization results based on the CCA algorithm, kCCA transformation and the histogram matching method.

RMSE^{CCA} | RMSE^{kCCA} | RMSE^{H} | $\mathsf{\rho}$^{CCA} | $\mathsf{\rho}$^{kCCA} | $\mathsf{\rho}$^{H} | ${\mathsf{\rho}}_{\mathit{h}}^{\mathit{C}\mathit{C}\mathit{A}}$ | ${\mathsf{\rho}}_{\mathit{h}}^{\mathit{k}\mathit{C}\mathit{C}\mathit{A}}$ | ${\mathsf{\rho}}_{\mathit{h}}^{\mathit{H}}$ | |
---|---|---|---|---|---|---|---|---|---|

image pair (imag0330, image1002) | |||||||||

Band 1 | 0.02 | 0.02 | 0.02 | 0.32 | 0.42 | 0.32 | 0.58 | 0.69 | 0.98 |

Band 2 | 0.02 | 0.02 | 0.02 | 0.40 | 0.47 | 0.39 | 0.77 | 0.78 | 0.97 |

Band 3 | 0.03 | 0.02 | 0.03 | 0.45 | 0.47 | 0.41 | 0.73 | 0.75 | 0.98 |

Band 4 | 0.04 | 0.03 | 0.04 | 0.36 | 0.37 | 0.37 | 0.81 | 0.82 | 0.94 |

image pair (imag0403 image1002) | |||||||||

Band 1 | 0.01 | 0.01 | 0.01 | 0.81 | 0.81 | 0.80 | 0.95 | 0.95 | 0.98 |

Band 2 | 0.01 | 0.01 | 0.01 | 0.78 | 0.78 | 0.77 | 0.93 | 0.93 | 0.99 |

Band 3 | 0.02 | 0.02 | 0.02 | 0.71 | 0.72 | 0.71 | 0.86 | 0.93 | 0.99 |

Band 4 | 0.06 | 0.03 | 0.04 | 0.51 | 0.55 | 0.55 | 0.74 | 0.89 | 0.95 |

image pair (imag0705, image1002): | |||||||||

Band 1 | 0.02 | 0.01 | 0.01 | 0.80 | 0.83 | 0.81 | 0.92 | 0.93 | 1.00 |

Band 2 | 0.02 | 0.01 | 0.01 | 0.77 | 0.80 | 0.78 | 0.98 | 0.94 | 0.91 |

Band 3 | 0.03 | 0.01 | 0.02 | 0.78 | 0.79 | 0.77 | 0.97 | 0.91 | 0.88 |

Band 4 | 0.06 | 0.03 | 0.04 | 0.41 | 0.42 | 0.38 | 0.82 | 0.81 | 0.99 |

image pair (imag0717, image1002): | |||||||||

Band 1 | 0.01 | 0.01 | 0.01 | 0.89 | 0.89 | 0.89 | 0.94 | 0.94 | 0.99 |

Band 2 | 0.01 | 0.01 | 0.01 | 0.87 | 0.87 | 0.87 | 0.94 | 0.94 | 0.98 |

Band 3 | 0.02 | 0.01 | 0.01 | 0.86 | 0.86 | 0.86 | 0.93 | 0.93 | 0.98 |

Band 4 | 0.06 | 0.03 | 0.04 | 0.47 | 0.47 | 0.46 | 0.88 | 0.88 | 0.99 |

image pair (imag1214, image1002): | |||||||||

Band 1 | 0.01 | 0.01 | 0.01 | 0.83 | 0.85 | 0.85 | 0.81 | 0.85 | 0.85 |

Band 2 | 0.01 | 0.01 | 0.01 | 0.81 | 0.82 | 0.82 | 0.86 | 0.88 | 0.88 |

Band 3 | 0.02 | 0.02 | 0.02 | 0.74 | 0.76 | 0.74 | 0.74 | 0.79 | 0.86 |

Band 4 | 0.03 | 0.03 | 0.03 | 0.67 | 0.68 | 0.67 | 0.77 | 0.90 | 0.92 |

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**MDPI and ACS Style**

Bai, Y.; Tang, P.; Hu, C. kCCA Transformation-Based Radiometric Normalization of Multi-Temporal Satellite Images. *Remote Sens.* **2018**, *10*, 432.
https://doi.org/10.3390/rs10030432

**AMA Style**

Bai Y, Tang P, Hu C. kCCA Transformation-Based Radiometric Normalization of Multi-Temporal Satellite Images. *Remote Sensing*. 2018; 10(3):432.
https://doi.org/10.3390/rs10030432

**Chicago/Turabian Style**

Bai, Yang, Ping Tang, and Changmiao Hu. 2018. "kCCA Transformation-Based Radiometric Normalization of Multi-Temporal Satellite Images" *Remote Sensing* 10, no. 3: 432.
https://doi.org/10.3390/rs10030432