# Gap-Filling of Landsat 7 Imagery Using the Direct Sampling Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Direct Sampling Method

_{x}is the data event consisting of the pattern of values made by the n closest neighbors of x: {x

_{1}, x

_{2},…, x

_{n}}. For each unknown pixel x, N

_{x}is made of the values of Z at the neighboring locations, {Z(x

_{1}), Z(x

_{2}), …, Z(x

_{n})}, as well as the lag vectors that describe the relative positions between location x and its neighborhoods, defined as L = {h

_{1}, h

_{2}, …, h

_{n}} = {x

_{1}− x, x

_{2}− x, …, x

_{n}− x}. The Direct Sampling method acquires the values of unknown pixels Z(x) by directly sampling the training image in a random manner, attempting to find one pixel y in the training image with a replicate N

_{y}that is similar to N

_{x}[31]. As soon as a pixel y is obtained with a neighborhood similar to N

_{x}, the sampling process stops and the value of y is assigned to x.

_{x}is defined in (b). Figure 1c is a training image used to fill the target pixels. The data event (b) is used to search the training image randomly. The search process continues until one replicate N

_{y}that can exactly match the defined data event N

_{x}is found in the training image. Once the matched replicate N

_{y}is found (Figure 1d, check marks), the search process stops and the value of y is assigned to the target pixel x, as shown in Figure 1e. The new pixel value is then considered known and the procedure (a)–(d) is repeated to simulate the value of another unknown pixel. Each time, the previously simulated pixels are considered in subsequent data events. The order in which pixels are simulated is defined by a random path which is different for each realization.

_{x}, N

_{y}) to choose the replicates for filling gap pixels. The distance is a metric used to assess the similarity between the replicate N

_{y}found in the training image and the data event N

_{x}. When the distance is less than a given threshold t, N

_{y}is regarded similar as N

_{x}and the value of y is assigned to x. The distance can be computed in different ways (further discussion on this can be found in Mariethoz et al. [31]).

_{y}found in the training image is shown in Figure 3b with pixels marked in red. The distance is computed by calculating a weighted average of the univariate distance of each variable, as shown in Equation (3):

_{k}are weight given to each variable summing to 1, and ƞ

_{k}is the normalization constant for each variable.

#### 2.2. Experiment Design

#### 2.3. Validation of Gap-Filling Results

- RMSE

- MSA

_{2}is the number of examined bands (n

_{2}= 6 here), and the unit of MSA is degree (°). It is possible that one specific band can be perfectly predicted but other bands are filled with poor results, therefore MSA is used to assess the conjunct spectral fidelity of multiple bands instead of a single band. A smaller MSA generally means a more accurate prediction.

- rRMSE

- R
^{2}

^{2}is an index used to describe the degree of consistency between the predicted and actual values ranging in the interval [0,1]. A value of 1 indicates that the predicted values can linearly fit the actual values perfectly, while a value of 0 means there is no linear relationship between the predicted and actual values.

- MdAPE

## 3. Results

#### 3.1. Comparison of Univariate and Bivariate Simulations

#### 3.2. Comparison of Multi-Temporal Influence of Training Imagery

#### 3.3. Impact of Heterogeneity on Reconstructions

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The Direct Sampling approach and its use of a training image to predict a missing pixel in the target image. The procedure follows a number of steps, including (

**a**) definition of the simulation grid, where the pixel x represents the pixel to be filled, and the blue, red, yellow pixels represent pixels with known values; (

**b**) definition of the data event; (

**c**) use of a training image to provide information to fill the gaps, with a search employing the defined data event from (

**b**) until; (

**d**) the simulation data event is identified in the training image; and (

**e**) the missing target value corresponding to the identified training value is assigned the first data point matching the data event.

**Figure 2.**Flowchart of the Direct Sampling method using a single training image. In the flowchart, x and y are the locations of the pixels in target image and training image respectively. Z(x) and Z(y) are the values at corresponding locations; d is the distance between two data events N

_{x}and N

_{y}, which can be denoted as d{N

_{x}, N

_{y}}. f is the maximum search proportion of scanned training image and t is the user-defined threshold.

**Figure 3.**Selection of neighborhood pixels for bivariate simulations, with (

**a**) target images and (

**b**) training images of the same two variables. Colored pixels are informed, and gray pixels are unknown. The pixel x is the target pixel to be predicted at the current step. Neighborhood pixels of each variable are selected, composing the data event N

_{x}. Replicate N

_{y}of the data event N

_{x}is marked in red in (

**b**) and will be used to compute the distance.

**Figure 4.**Target images for each of the studied regions, both with and without imposed gaps, for (

**a**) desert; (

**b**) sparse agricultural; (

**c**) dense farmland; (

**d**) urban; (

**e**) braided river and (

**f**) coastal area.

**Figure 5.**ETM+ images for the six studied regions and the different temporal images used in analysis (see Table 1), corresponding to (

**a**) desert; (

**b**) sparse agricultural; (

**c**) dense farmland; (

**d**) urban; (

**e**) braided river and (

**f**) coastal area.

**Figure 6.**Mean realizations and the zoomed images for the nine test cases conducted over R3 (dense farmland). “Actual” is the actual image. “U-” and “B-” represents univariate and bivariate simulation. “D1”–“D4” represent the input images used for filling gaps, and “U-D0” is the univariate simulation without using an additional training image. The areas marked with red square in the mean realizations are magnified to the right of each pair. The location of the gap trace is denoted with an arrow in the mean realization of U-D1, and a red rectangular is marked in all zoomed images at the same location for the edge shape comparison in Section 3.2.

**Figure 7.**Scatter plots of reflectance values of band 4 for bivariate cases, with (

**a**–

**d**) corresponding to B-D1, B-D2, B-D3 and B-D4. X axis reflects the actual reflectance while the Y axis is the predicted reflectance. R2 is the coefficient of determination.

**Figure 8.**rRMSE values of bands 1–4 of cases U-D0, B-D1 and B-D4 across six tested regions, with B1-B4 corresponding bands, R1 desert, R2 sparse agricultural, R3 dense farmland, R4 urban area, R5 braided river and R6 coastal area.

**Table 1.**Band designations of bands 1–5 and 7 of Landsat Enhanced Thematic Mapper Plus (ETM+) images.

Landsat 7 | Wavelength (mm) | Resolution (m) |
---|---|---|

Band 1 | 0.45–0.52 | 30 |

Band 2 | 0.52–0.60 | 30 |

Band 3 | 0.63–0.69 | 30 |

Band 4 | 0.77–0.90 | 30 |

Band 5 | 1.55–1.75 | 30 |

Band 7 | 2.09–2.35 | 30 |

**Table 2.**The collection dates for the target images and the four input images used for informing the gap-filling process.

R1 | R2 | R3 | R4 | R5 | R6 | |
---|---|---|---|---|---|---|

Desert | Sparse Agricultural | Dense Farmland | Urban Area | Braided River | Coastal Area | |

Path | 169 | 38 | 43 | 43 | 137 | 115 |

Row | 40 | 37 | 34 | 34 | 44 | 78 |

Target image | 28 August 2002 | 5 July 2002 | 8 July 2002 | 8 July 2002 | 31 October 2002 | 17 July 2002 |

Date 1 | 13 September 2002 | 19 June 2002 | 24 July 2002 | 24 July 2002 | 16 November 2002 | 1 July 2002 |

Date 2 | 29 September 2002 | 3 June 2002 | 9 August 2002 | 9 August 2002 | 2 December 2002 | 18 August 2002 |

Date 3 | 15 October 2002 | 18 May 2002 | 25 August 2002 | 25 August 2002 | 18 December 2002 | 30 May 2002 |

Date 4 | 6 April 2002 | 11 February 2002 | 2 March 2002 | 2 March 2002 | 5 March 2002 | 11 March 2002 |

Case Name | Description |
---|---|

U-D1 | Univariate case with D1 (two-week distance) as training image |

U-D2 | Univariate case with D2 (four-week distance) as training image |

U-D3 | Univariate case with D3 (six-week distance) as training image |

U-D4 | Univariate case with D4 (longer than four-month distance) as training image |

U-D0 | Univariate case with non-gap area in target image as training image. No additional image introduced |

B-D1 | Bivariate case, with one variable the reflectance value in the target image and another the reflectance value of D1 (two-week distance) |

B-D2 | Bivariate case, with one variable the reflectance value in the target image and another the reflectance value of D2 (four-week distance) |

B-D3 | Bivariate case, with one variable the reflectance value in the target image and another the reflectance value of D3 (six-week distance) |

B-D4 | Bivariate case, with one variable the reflectance value in the target image and another the reflectance value of D4 (longer than four-month distance) |

Univariate | Bivariate | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

U-D1 | U-D2 | U-D3 | U-D4 | U-D0 | B-D1 | B-D2 | B-D3 | B-D4 | ||

RMSE | B1 | 0.0181 | 0.0174 | 0.0175 | 0.0189 | 0.0140 | 0.0096 | 0.0099 | 0.0104 | 0.0123 |

B2 | 0.0235 | 0.0224 | 0.0228 | 0.0261 | 0.0177 | 0.0125 | 0.0129 | 0.0136 | 0.0162 | |

B3 | 0.0315 | 0.0315 | 0.0323 | 0.0385 | 0.0282 | 0.0194 | 0.0196 | 0.0211 | 0.0265 | |

B4 | 0.0496 | 0.0504 | 0.0504 | 0.0490 | 0.0431 | 0.0373 | 0.0341 | 0.0391 | 0.0398 | |

B5 | 0.0463 | 0.0469 | 0.0474 | 0.0592 | 0.0447 | 0.0317 | 0.0307 | 0.0344 | 0.0390 | |

B7 | 0.0458 | 0.0452 | 0.0464 | 0.0544 | 0.0403 | 0.0306 | 0.0307 | 0.0325 | 0.0381 | |

MSA (°) | 7.7439 | 7.6745 | 7.6508 | 8.7928 | 6.0092 | 4.7069 | 4.5985 | 4.9973 | 5.6135 | |

rRMSE | B1 | 0.1348 | 0.1356 | 0.1323 | 0.1433 | 0.1026 | 0.0728 | 0.0737 | 0.0766 | 0.0904 |

B2 | 0.1939 | 0.1868 | 0.1830 | 0.1965 | 0.1463 | 0.1027 | 0.1054 | 0.1108 | 0.1284 | |

B3 | 0.3225 | 0.3245 | 0.3252 | 0.3295 | 0.2960 | 0.1950 | 0.1971 | 0.2041 | 0.2512 | |

B4 | 0.2389 | 0.2651 | 0.2512 | 0.2458 | 0.2474 | 0.1303 | 0.1274 | 0.1411 | 0.1615 | |

B5 | 0.9492 | 1.1717 | 1.3101 | 1.0687 | 2.2184 | 0.6958 | 0.7058 | 0.6629 | 0.7829 | |

B7 | 9.7379 | 9.8900 | 10.2649 | 11.3710 | 7.5240 | 1.4503 | 2.3615 | 2.0421 | 1.9940 | |

R^{2} | B1 | 0.3457 | 0.3797 | 0.3894 | 0.2675 | 0.5813 | 0.8064 | 0.7938 | 0.7748 | 0.6859 |

B2 | 0.3627 | 0.4267 | 0.4345 | 0.3391 | 0.6158 | 0.8127 | 0.7990 | 0.7795 | 0.6895 | |

B3 | 0.5708 | 0.5685 | 0.5556 | 0.4728 | 0.6428 | 0.8338 | 0.8288 | 0.8067 | 0.6910 | |

B4 | 0.7134 | 0.7076 | 0.7105 | 0.7194 | 0.7811 | 0.8365 | 0.8628 | 0.8198 | 0.8132 | |

B5 | 0.6544 | 0.6496 | 0.6462 | 0.5485 | 0.6642 | 0.8342 | 0.8438 | 0.8049 | 0.7470 | |

B7 | 0.5777 | 0.5833 | 0.5694 | 0.4786 | 0.6576 | 0.8049 | 0.8044 | 0.7823 | 0.7000 | |

MdAPE (%) | B1 | 9.9127 | 10.2607 | 9.9215 | 11.1535 | 4.7540 | 3.7660 | 3.6599 | 3.9347 | 4.3658 |

B2 | 13.8214 | 13.2522 | 13.1244 | 15.0335 | 6.6289 | 5.2183 | 5.1734 | 5.6107 | 6.1724 | |

B3 | 18.1135 | 18.3791 | 18.748 | 24.4024 | 11.7130 | 9.0637 | 9.0774 | 9.5551 | 11.3443 | |

B4 | 9.0571 | 9.2563 | 9.4933 | 8.7362 | 5.9232 | 5.3764 | 5.4635 | 5.9204 | 6.4058 | |

B5 | 11.9067 | 11.7296 | 12.0812 | 16.5120 | 8.0976 | 6.8696 | 6.5745 | 7.2466 | 7.9052 | |

B7 | 20.8301 | 19.5497 | 20.8953 | 26.7857 | 13.8738 | 11.2427 | 11.0730 | 11.7867 | 13.3628 |

**Table 5.**Summary statistics of band 4 for the target images and filled results of six studied regions.

Case Name | Mean | St. Dev. | Skew | |
---|---|---|---|---|

Desert | Actual | 0.4891 | 0.0125 | −0.0980 |

U-D0 | 0.4891 | 0.0116 | −0.2304 | |

B-D1 | 0.4890 | 0.0117 | −0.1899 | |

B-D4 | 0.4890 | 0.0119 | −0.1363 | |

Sparse agricultural | Actual | 0.3782 | 0.0413 | 2.1496 |

U-D0 | 0.3782 | 0.0400 | 2.2386 | |

B-D1 | 0.3783 | 0.0402 | 2.2373 | |

B-D4 | 0.3783 | 0.0404 | 2.2537 | |

Dense farmland | Actual | 0.3132 | 0.0910 | −0.1222 |

U-D0 | 0.3134 | 0.0879 | −0.1246 | |

B-D1 | 0.3138 | 0.0891 | −0.2066 | |

B-D4 | 0.3134 | 0.0886 | −0.1754 | |

Urban area | Actual | 0.2714 | 0.0576 | 0.8012 |

U-D0 | 0.2717 | 0.0530 | 0.8884 | |

B-D1 | 0.2714 | 0.0553 | 0.8240 | |

B-D4 | 0.2710 | 0.0538 | 0.8818 | |

Braided river | Actual | 0.1854 | 0.0901 | −0.3713 |

U-D0 | 0.1858 | 0.0887 | −0.4269 | |

B-D1 | 0.1848 | 0.0888 | −0.4063 | |

B-D4 | 0.1853 | 0.0889 | −0.4196 | |

Coastal area | Actual | 0.1147 | 0.1267 | 0.6625 |

U-D0 | 0.1150 | 0.1247 | 0.6331 | |

B-D1 | 0.1145 | 0.1263 | 0.6482 | |

B-D4 | 0.1153 | 0.1266 | 0.6347 |

© 2016 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**

Yin, G.; Mariethoz, G.; McCabe, M.F.
Gap-Filling of Landsat 7 Imagery Using the Direct Sampling Method. *Remote Sens.* **2017**, *9*, 12.
https://doi.org/10.3390/rs9010012

**AMA Style**

Yin G, Mariethoz G, McCabe MF.
Gap-Filling of Landsat 7 Imagery Using the Direct Sampling Method. *Remote Sensing*. 2017; 9(1):12.
https://doi.org/10.3390/rs9010012

**Chicago/Turabian Style**

Yin, Gaohong, Gregoire Mariethoz, and Matthew F. McCabe.
2017. "Gap-Filling of Landsat 7 Imagery Using the Direct Sampling Method" *Remote Sensing* 9, no. 1: 12.
https://doi.org/10.3390/rs9010012