# Correcting Position Error in Precipitation Data Using Image Morphing

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Definitions

**image registration**is to determine a spatial mapping T such that, $\forall (x,y)\in D$,

**Image warping**is the distortion of an image based on a spatial transformation of the domain. Warping can be used to transform an image into another one by using the spatial mapping T obtained from the registration method. The mapping T is gradually applied to the original image u as follows:

**Cross-dissolving**only acts on the intensity. It fades two images u and v into each other:

**Image morphing**combines warping and cross-dissolving to account for both the spatial distortion and the difference in intensity:

#### 2.2. Automatic Registration

#### Algorithm

**Smoothing**of the images u and v: the images are smoothed by convolution with a 2D-Gaussian$$\begin{array}{c}\hfill {G}_{2D}\left(\right)open="("\; close=")">x,y\\ =& \frac{1}{2\pi {\sigma}^{2}}exp\left(\right)open="("\; close=")">-\frac{{x}^{2}+{y}^{2}}{2{\sigma}^{2}},\hfill \end{array}$$The cost function J is often non-convex with respect to T and so can have several local minima. The smoothing combined with the hierarchy of grids reduce the local minima problem. They ensure that the large-scale features are fitting first, hence avoiding local minima.After the smoothing, the two fields are normalized such that their maximum is the same. The images obtained after smoothing and normalization are noted as ${\tilde{u}}_{i}$ and ${\tilde{v}}_{i}$.**Initialization**: solving the minimization problem on grid ${D}_{i}$ requires a first guess ${T}_{i}^{\mathrm{fg}}$. For $i=1$, ${T}_{1}^{\mathrm{fg}}$ is set to zeros, that is, no deformation. For $i=2,\dots ,I$, the mapping ${T}_{i-1}^{\ast}$ obtained by solving the minimization problem on grid ${D}_{i-1}$ is interpolated into the grid ${D}_{i}$ and used as the first guess ${T}_{i}^{fg}$.**Optimization**: The actual minimization problem to be solved is based on the smoothed fields, that is, ${J}_{o}\left(T\right)=\parallel {\tilde{v}}_{i}-{\tilde{u}}_{i}\circ (I+T)\parallel $. Contrary to Reference [32], we solved the minimization problem for all the nodes at the same time.There is a number of inequality constraints on this minimization problem, due to our requirements of invertibility. An iterative barrier approach is used to transform this constrained minimization problem into an unconstrained one [34,35]. In the barrier approach, the minimization is applied to a penalized cost function ${J}_{p}\left(T\right)=J\left(T\right)+\beta {\sum}_{h}{C}_{h}\left(T\right)$, where ${C}_{h}$ are the constraint functions and $\beta $ the barrier coefficient (over which we iterate when the constraints are not respected). The constraints and the minimization method are described with more details in Appendix A.

#### 2.3. Dealing with Irregularly Spaced Observations

## 3. Study Cases

#### 3.1. Synthetic Case for Algorithm’s Validation

#### 3.2. Southern Ghana Case

#### 3.2.1. Precipitation Datasets

#### 3.2.2. Data Pre-Processing

## 4. Results

#### 4.1. Synthetic Cases

#### 4.1.1. Convergence

#### 4.1.2. Validation

#### 4.2. Southern Ghana Case

#### 4.2.1. Convergence

#### 4.2.2. Validation

## 5. Discussion

#### 5.1. Convergence

#### 5.2. Validation

#### 5.3. Applications

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Optimization Step of the Automatic Registration Algorithm

#### Appendix A.1. Constraints

#### Appendix A.2. Minimization Method

## Appendix B. Sensitivity Study for the Regulation Coefficients C_{1}, C_{2} and C_{3}

**Table A1.**MAE of the warped (${u}_{\mathrm{warp}}$) and morphed ${u}_{\mathrm{morp}}$ signals compared to the target field (v) for the different sensitivity runs (for $I=4$).

Original | Only ${\mathit{C}}_{1}$ | Only ${\mathit{C}}_{2}$ | Only ${\mathit{C}}_{3}$ | All Coef. $\times 5$ | |
---|---|---|---|---|---|

${C}_{1}$ | 0.1 | 0.1 | 0 | 0 | 0.5 |

${C}_{2}$ | 1 | 0 | 1 | 0 | 5 |

${C}_{3}$ | 1 | 0 | 0 | 1 | 5 |

- MAE ${u}_{\mathrm{morph}}$(mm/h) | 0.0496 | 0.0520 | 0.0535 | 0.0515 | 0.0396 |

- MAE ${u}_{\mathrm{warp}}$(mm/h) | 0.1371 | 0.1363 | 0.1368 | 0.1366 | 0.1374 |

**Figure A1.**Sensitivity study: C

_{1}= 0.1 and C

_{2}= C

_{3}= 0. (

**a**) Mapping T obtained from the automatic registration and (

**b**) its effect on the pixel grid. (

**a**) Mapping T. (

**b**) Pixel grid D

_{n}before (in red) and after (in blue) distortion by the mapping T.

**Figure A2.**Sensitivity study: C

_{2}= 1 and C

_{1}= C

_{3}= 0. (

**a**) Mapping T obtained from the automatic registration and (

**b**) its effect on the pixel grid. (

**a**) Mapping T. ((

**b**) Pixel grid D

_{n}before (in red) and after (in blue) distortion by the mapping T.).

**Figure A3.**Sensitivity study: C

_{3}= 1 and C

_{1}= C

_{2}= 0. (

**a**) Mapping T obtained from the automatic registration and (

**b**) its effect on the pixel grid. (

**a**) Mapping T. (

**b**) Pixel grid D

_{n}before (in red) and after (in blue) distortion by the mapping T.

**Figure A4.**Sensitivity study: C

_{1}= 0.5 and C

_{1}= C

_{2}= 5. (

**a**) Mapping T obtained from the automatic registration and (

**b**) its effect on the pixel grid. (

**a**) Mapping T. (

**b**) Pixel grid D

_{n}before (in red) and after (in blue) distortion by the mapping T.

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**Figure 3.**Study domain (red rectangle) and the TAHMOstations available within the domain (white dots).

**Figure 4.**IMERG-Late (background) and TAHMO (circles) accumulated rainfall between 18:00:00 and 19:00:00, within the study domain (dotted line).

**Figure 5.**TAHMO measurements (circles) and TAHMO kriged (background) within the study domain (dotted line) for the selected event.

**Figure 6.**Fields u (IMERG-Late) and v (TAHMO) after pre-processing and used as inputs. The dotted line delimits the study domain and the solid line the extended one, with the zero-padding area in-between. (

**a**) Field u (IMERG) after pre-processing. (

**b**) Field v (TAHMO) after pre-processing.

**Figure 7.**(

**a**) Mapping T obtained from the automatic registration (with C

_{1}= 0.1 and C

_{2}= C

_{3}= 1) and (

**b**) its effect on the pixel grid, for the synthetic case. (

**a**) Mapping T. (

**b**) Pixel grid D

_{n}before (in red) and after (in blue) distortion by the mapping T.

**Figure 8.**Morphing results: (

**a**) original signal u, (

**b**) morphed signal u

_{morph}, (

**c**) absolute error between the morphed signal and the target signal v and (

**d**) error between the same two signals (using a different scale). (

**a**) u. (

**b**) u

_{morph}. (

**c**) $|{u}_{\mathrm{morph}}-v|$. (

**d**) u

_{morph}− v.

**Figure 9.**(

**a**) Mapping T obtained from the automatic registration (with C

_{1}= 0.1 and C

_{2}= C

_{3}= 1) and (

**b**) its effect on the pixel grid, for the southern Ghana case. (

**a**) Mapping T. (

**b**) Pixel grid D

_{n}before (in red) and after (in blue) distortion by the mapping T.

**Figure 11.**Warped signal ${u}_{\mathrm{warp}}$ at the stations location. The stations that differ from the TAHMO measurements are numbered.

**Figure 12.**Rainfall (in mm/h) at 18 TAHMO stations according to TAHMO, IMERG-Late, IMERG-Final and the warped signal (${u}_{\mathrm{warp}}$).

**Table 1.**Optimization results after each step $i=0,\dots ,5$ for the synthetic case. The number of iterations needed for the barrier approach ($\beta $ iterations) and for the L-BFGS-B method are given separately. The total number of iterations correspond to the sum of the L-BFGS-B iterations for each $\beta $ iteration. The cost function ${J}_{p}$ is evaluated before and after optimization (i.e., for the first guess ${T}_{i}^{\mathbf{fg}}$ and the ‘optimal’ grid ${T}_{i}^{\ast}$). The latter is also separate into three terms, the mapping error (${J}_{m}$), the background error (${J}_{b}$) and the penalization term (not shown here because of its value close to zero).

i | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Number of iterations | |||||

- Total | 81 | 996 | 1308 | 502 | 443 |

- $\beta $ iter. | 1 | 3 | 1 | 1 | 2 |

- L-BFGS-B iter. for each $\beta $ | 81, $\beta $ = 1 | 956, $\beta $ = 1 | 1308, $\beta $ = 1 | 502, $\beta $ = 1 | 263, $\beta $ = 1 |

36,$\beta $ = 10 | 180, $\beta $ = 10 | ||||

4, $\beta $ = 100 | |||||

Cost function | |||||

${J}_{p}\left({T}_{i}^{\mathbf{fg}}\right)$ | 213.169 | 54.722 | 40.568 | 24.278 | 16.371 |

${J}_{p}\left({T}_{i}^{\ast}\right)$ | 30.058 | 18.408 | 19.193 | 14.790 | 13.191 |

${J}_{o}\left({T}_{i}^{\ast}\right)$ | 27.552 | 16.352 | 17.847 | 13.399 | 11.734 |

${J}_{b}\left({T}_{i}^{\ast}\right)$ | 2.507 | 2.056 | 1.345 | 1.391 | 1.457 |

**Table 2.**MAE (in mm/h) of the morphed (${u}_{\mathrm{morph}}$) and warped (${u}_{\mathrm{warp}}$) signals obtained at different steps i, for the synthetic case.

i | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

MAE$\left(\right)$ | 0.678 | 0.161 | 0.062 | 0.050 | 0.036 |

MAE$\left(\right)$ | 0.531 | 0.281 | 0.147 | 0.137 | 0.133 |

**Table 3.**Optimization results after each step $i=0,\dots ,4$ for the southern Ghana case. The number of iterations needed for the barrier approach ($\beta $ iterations) and for the L-BFGS-B method are given separately. The total number of iterations correspond to the sum of the L-BFGS-B iterations for each $\beta $ iterations. The cost function ${J}_{p}$ is evaluated before and after optimization (i.e., for the first guess ${T}_{i}^{\mathbf{fg}}$ and the ‘optimal’ grid ${T}_{i}^{\ast}$). The latter has also been separated into three terms, the mapping error (${J}_{m}$), the background error (${J}_{b}$) and the penalization term (not shown here because of its value close to zero).

i | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Number of iterations | ||||

- Total | 10 | 60 | 433 | 1463 |

- $\beta $ iter. | 1 | 1 | 1 | 5 |

- L-BFGS-B iter. for each $\beta $ | 10, $\beta $ = 1 | 60, $\beta $ = 1 | 433, $\beta $ = 1 | 1430, $\beta $ = 1 |

16, $\beta $ = 10 | ||||

9, $\beta $ = 100 | ||||

4, $\beta $ = 1000 | ||||

4, $\beta $ = 10,000 | ||||

Cost function | ||||

${J}_{p}\left({T}_{i}^{\mathbf{fg}}\right)$ | 0.0516 | 0.0938 | 0.1068 | 0.3602 |

${J}_{p}\left({T}_{i}^{\ast}\right)$ | 0.0421 | 0.0424 | 0.0483 | 0.0679 |

${J}_{m}\left({T}_{i}^{\ast}\right)$ | 0.0168 | 0.0058 | 0.0060 | 0.0252 |

${J}_{b}\left({T}_{i}^{\ast}\right)$ | 0.0253 | 0.0376 | 0.0422 | 0.0427 |

**Table 4.**MAE and RMSE of the warped (${u}_{\mathrm{warp}}$) signal compared to the kriged TAHMO field (v) and to the gauge measurements, obtained at different steps i, for the southern Ghana case.

i | Before | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Kriged field $\left(\right)$ | |||||

- MAE (mm/h) | 0.1748 | 0.0968 | 0.0936 | 0.0886 | 0.0865 |

- RMSE (mm/h) | 1.6476 | 0.9171 | 0.9496 | 0.8976 | 0.8565 |

Gauge measurements | |||||

- MAE (mm/h) | 2.0687 | 1.4365 | 1.4434 | 1.3849 | 1.3619 |

- RMSE (mm/h) | 8.7053 | 5.5057 | 5.4984 | 4.9349 | 4.6690 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Le Coz, C.; Heemink, A.; Verlaan, M.; ten Veldhuis, M.-c.; van de Giesen, N.
Correcting Position Error in Precipitation Data Using Image Morphing. *Remote Sens.* **2019**, *11*, 2557.
https://doi.org/10.3390/rs11212557

**AMA Style**

Le Coz C, Heemink A, Verlaan M, ten Veldhuis M-c, van de Giesen N.
Correcting Position Error in Precipitation Data Using Image Morphing. *Remote Sensing*. 2019; 11(21):2557.
https://doi.org/10.3390/rs11212557

**Chicago/Turabian Style**

Le Coz, Camille, Arnold Heemink, Martin Verlaan, Marie-claire ten Veldhuis, and Nick van de Giesen.
2019. "Correcting Position Error in Precipitation Data Using Image Morphing" *Remote Sensing* 11, no. 21: 2557.
https://doi.org/10.3390/rs11212557