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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

More than 20 techniques have been developed to de-noise time-series vegetation index data from different satellite sensors to reconstruct long time-series data sets. Although many studies have compared Normalized Difference Vegetation Index (NDVI) noise-reduction techniques, few studies have compared these techniques systematically and comprehensively. This study tested eight techniques for smoothing different vegetation types using different types of multi-temporal NDVI data (Advanced Very High Resolution Radiometer (AVHRR) (Global Inventory Modeling and Map Studies (GIMMS) and Pathfinder AVHRR Land (PAL), Satellite Pour l’ Observation de la Terre (SPOT) VEGETATION (VGT), and Moderate Resolution Imaging Spectroradiometer (MODIS) (Terra)) with the ultimate purpose of determining the best reconstruction technique for each type of vegetation captured with four satellite sensors. These techniques include the modified best index slope extraction (M-BISE) technique, the Savitzky-Golay (S-G) technique, the mean value iteration filter (MVI) technique, the asymmetric Gaussian (A-G) technique, the double logistic (D-L) technique, the changing-weight filter (CW) technique, the interpolation for data reconstruction (IDR) technique, and the Whittaker smoother (WS) technique. These techniques were evaluated by calculating the root mean square error (RMSE), the Akaike Information Criterion (AIC), and the Bayesian Information Criterion (BIC). The results indicate that the S-G, CW, and WS techniques perform better than the other tested techniques, while the IDR, M-BISE, and MVI techniques performed worse than the other techniques. The best de-noise technique varies with different vegetation types and NDVI data sources. The S-G performs best in most situations. In addition, the CW and WS are effective techniques that were exceeded only by the S-G technique. The assessment results are consistent in terms of the three evaluation indexes for GIMMS, PAL, and SPOT data in the study area, but not for the MODIS data. The study will be very helpful for choosing reconstruction techniques for long time-series data sets.

Analysis of Normal Difference Vegetation Index (NDVI) time-series data is becoming increasingly important for ecological research on environmental dynamics and climate change [

Several methods have already been developed to reconstruct NDVI time series. They can be grouped into five types: (1) threshold methods, including the best index slope extraction technique (BISE) [

Comparisons of these techniques have shown that each has its own advantages and drawbacks [

Although many studies have focused on NDVI noise reduction techniques comparison, there are still some aspects that need to be improved and completed. First, previous work has been restricted by a small selection of the available noise reduction techniques, and the majority of the literature compares two or three techniques, typically, single novel technique is compared to one or two widely-known techniques [

The main purpose of the present study was to support other basic studies by comparing eight techniques for representing time-series NDVI data to support other basal studies. These techniques include the A-G, M-BISE, CW, D-L, IDR, MVI, S-G, and WS techniques. Several studies have indicated that the A-G and D-L are superior to other fitting techniques and filters [

Four widely used long time-series NDVI data sets were used in this study (

The Heihe River Basin, in the middle of the Hexi Corridor of Gansu Province, is the second largest inland river basin in China. It is located between 97°1′–102°0′E and 37°7′–42°7′N, with an area of approximately 143,000 km^{2}. The elevation in the Heihe River Basin ranges from 5500 m in the south to 1000 m in the north [

Four long time-series NDVI data sets of remote sensing were used in this study, and detailed information about the data sets is shown in

The 2001 Vegetation Map of China [

Eight de-noising techniques were selected for this research, and the main objective of each technique is shown in

The parameters used for the techniques greatly affect the reconstruction results. As described by Atkinson

By comparison, we used 0.1 as the threshold for the M-BISE technique and a 20% multi-year average for the MVI filter threshold. For S-G, m (the half-width of the smoothing window) was 4, d (the degree of the polynomial) was 6, and the IDR threshold was 0.02. These values are similar to their authors’ recommended. For WS, λ was 15 because the entire research area contains single-season vegetation types [

Three statistical indicators were used to evaluate the performance of each noise reduction technique: (1) the root mean square error (RMSE); (2) Akaike’s Information Criterion (AIC); and (3) Bayesian Information Criterion (BIC).

RMSE is a well-accepted absolute goodness-of-fit indicator for continuous response variables that describes the difference between the observed and predicted values in the appropriate units [

The Akaike’s Information Criterion (AIC) [

For M-BISE, MVI, and IDR required only one free parameter. For S-G and CW, two and three free parameters were needed, respectively. For the A-G and D-L techniques, seven and six parameters were needed, respectively. For WS, the free parameter is 9.37 when λ is 15 according to Atkinson

Due to space limitations, only the results for one year (2000) are shown in the figure. It was observed that most de-noising techniques were effective for reconstructing high-quality NDVI time-series data sets. The results closely resemble the profiles of vegetation growth, especially for crops, grass, shrubs, and meadows. However, some obvious differences were found in the reconstructed NDVI time-series created with the eight techniques. For example, the WS technique produced the smoothest fitted curve, but it showed a lower maximum NDVI than other techniques (

The RMSE, the AIC, and the BIC were calculated at the pixel level for the eight techniques for all of the data sets after reconstruction. All of the homogeneous pixel samples for each vegetation type were used to calculate the mean value of the three indexes to further analyze the fit of the techniques. To test the performance of the eight techniques at the regional scale, the RMSE was used as an example to display the differences between the techniques in the study area.

The technique performance differs based on the three evaluation indexes. For crop, desert, grassland and meadow, the optimal techniques are S-G, WS, S-G, and CW, respectively, in terms of the three evaluation indexes. However, the best technique is different for shrub in terms of the three evaluation indexes, the optimal technique is S-G for RMSE and AIC, while it is WS for BIC. At the same time, the worst technique is also different in terms of the three evaluation indexes, for the five vegetation types both RMSE and BIC indicate that IDR perform worst, while it is D-L for AIC. For all the eight techniques, the accuracy differs for different vegetation types. For example, the WS technique performs better than the D-L technique for shrub and desert areas, but the situation is reversed for other three vegetation areas. Overall, the S-G, CW and WS techniques are effective for the GIMMS NDVI time series data. The IDR, M-BISE, and MVI techniques perform worse than other techniques for most vegetation types. Evaluation indexes can affect the assessment results for some vegetation types.

There are a few differences between the results for the GIMMS NDVI data and the PAL NDVI data (

For the SPOT VGT NDVI data, the S-G technique is also the best reconstruction technique for shrub, crop and meadow, and the WS technique outperforms the other six techniques for desert (

As shown in

Overall, the S-G, CW, and WS techniques show better reconstructed effects than the other techniques, and the IDR technique shows generally poor performance in terms of RMSE, AIC, and BIC for most vegetation types in the Heihe River Basin (

Except for the S-G and the IDR techniques, the performance of other techniques is relatively unstable, meaning that they change for different vegetation types and data sources. For example, for SPOT VGT NDVI data sets, the CW and the WS are effective techniques for all vegetation types, performing worse than the S-G technique (desert and grassland excepted) and better than the rest five techniques (

As ground reference measurements are challenging to obtain due to the medium/coarse resolution of the imagery, the problem of developing a robust, accurate and fast filter is amplified by the difficulty of obtaining reference measurements to use for validation [

Generally, the results of the three indexes are consistent in most cases, especially for worst-performing technique. For this technique, the assessment results for all vegetation types for all four NDVI data sets are in complete agreement (

The performance is different between the eight techniques even under the same vegetation type and data source according to the above results (

Vegetation type is also a factor affecting the techniques’ performance. For the five vegetation types in our study, the performances of the techniques for each type are unstable for most of the NDVI data. Similar findings were found by the research of Atkinson

As this study was focus on the Heihe River Basin, the vegetation types were relatively simple; most of the study area was covered by desert. Our findings and conclusions may not be as applicable to areas with multiple growing seasons (e.g., subtropical zones). However, our comparison of the performances of eight reconstruction techniques was conducted systematically and comprehensively using four NDVI data sets covering five different vegetation types, and we determined obtained the optimum technique for GIMMS, PAL, SPOT, and MODIS data for the different vegetation types. These findings will be of great reference and actual using values for choosing de-noise techniques and their parameter values.

In this study, the performances of eight de-noised techniques were compared for different vegetation types represented in four NDVI data sets in the Heihe River Basin, and the following results can be observed: the S-G, CW, and WS techniques perform better than other techniques for almost all vegetation types according to the RMSE, the AIC, and the BIC. The IDR, M-BISE, and MVI techniques performed worse than the other techniques for most vegetation types using the four sensor data sets. The best technique varies with vegetation types and NDVI data sources. However, the S-G performs best in most situations, the CW and WS techniques are next to it. The assessment results are consistent among the three evaluation indexes for most situations, but subtle differences exist for some vegetation types; and comprehensive consideration of several indexes is very helpful to decide the best technique for certain situations.

This work was supported by the National Natural Science Foundation of China (grant number: 91125004), the Knowledge Innovation Program of the Chinese Academy of Sciences (grant number: KZCX2-EW-312), and the Chinese State Key Basic Research Project (grant number: 2009CB421305). We would like to thank Atzberger C. (Joint Research Centre of the European Commission), Julien Y. (University of Valencia), and Zhu W.Q. (Beijing Normal University) for providing the programs of the Whittaker smoother (WS), the iterative interpolation for data reconstruction (IDR) and the changing-weight filter (CW), respectively. We also thank the three anonymous reviewers for their valuable comments and suggestions on how to improve the manuscript.

Mingguo Ma outlined the research topic and study concept, assisted with manuscript writing and coordinated the revision activities. Shuzhen Jia and Haibo Wang were involved in data collection. Xufeng Wang and Wenping Yu performed the data comparison analysis. Liying Geng performed the data comparison analysis, results interpretation, manuscript writing, and coordinated the revision activities.

The authors declare no conflict of interest.

The vegetation types and the distribution of the sample points used for the comparison of the reconstruction techniques in the Heihe River Basin.

Eight techniques fitted to GIMMS NDVI time-series data acquired from homogeneous pixels of (

Eight techniques fitted to PAL NDVI time-series data acquired from homogeneous pixels of (

Eight techniques fitted to SPOT VGT NDVI time-series data acquired from homogeneous pixels of (

Eight techniques fitted to MODIS NDVI time-series data acquired from homogeneous pixels of (

Root mean square error (RMSE) values of GIMMS NDVI for the eight techniques: (

Root mean square error (RMSE) of the PAL NDVI for the eight techniques: (

Root mean square error (RMSE) of the SPOT NDVI data for the eight techniques: (

Root mean square error (RMSE) of MODIS NDVI for the eight techniques: (

Long time-series Normalized Difference Vegetation Index (NDVI) data sets used in the study.

| ||||
---|---|---|---|---|

GIMMS | January 1982–December 2006 | 15 d/8 km | AVHRR | Cold and Arid Regions Science Data Center [ |

Pathfinder | July 1981–July 1994 |
10 d/8 km | AVHRR | Cold and Arid Regions Science Data Center [ |

SPOT VEGETATION (S10) | April 1998–April 2013 | 10 d/1 km | SPOT VGT | VITO Earth observation [ |

MOD13A2 | February 2000–February 2013 | 16 d/1 km | Terra/MODIS | The Land Processes Distributed Active Archive Center [ |

Summary of the NDVI time-series reconstruction techniques selected for comparison.

Modified-best index slope extraction (M-BISE) | Compares the current term value with the previous and the next term within a predefined sliding window, and replaces these values with the mean value of the previous and the next values if the percentage difference is greater than a predefined threshold. | [ |

Asymmetric Gaussian function-fitting (A-G) | Fits local, nonlinear functions at intervals around the local maxima and minima, then merges these into a global function describing the full NDVI time series. | [ |

Double logistic function fitting (D-L) | Uses six parameters to model the NDVI time series with a double logistic function. These parameters are the winter NDVI (wNDVI), maximum NDVI (mNDVI), two inflection points, one as the curve rises (S) and one as it drops (A), and the rate of increase or decrease (mS and mA) of the curve at the inflection points. | [ |

Savitzky-Golay filtering (S-G) | Based on a simplified least-squares-fit convolution for smoothing and computing derivatives of a set of consecutive values (a spectrum). The convolution can be understood as a weighted moving average filter with weighting given as a polynomial of a certain degree. The weight coefficients, when applied to a signal, perform a polynomial least-squares fit within the filter window. This polynomial is designed to preserve higher moments within the data and to reduce the bias introduced by the filter. | [ |

Mean value iteration filtering (MVI) | Iteratively compares each date with the average of the dates before and after it, replacing the date with this average if the difference is above a certain threshold. The maximum difference date value will be removed in an iteration process. Iteration will stop when all differences are less than the threshold. | [ |

Whittaker smoother (WS) | Based on penalized least squares, fits a discrete series to discrete data and penalizes the roughness of the smooth curve. In this way, it balances the reliability of the data and roughness of the fitted data. | [ |

Iterative interpolation for data reconstruction (IDR) | Creates an alternative NDVI time series by computing the mean of the immediately preceding and following observations and comparing it to that of the original time series, replacing the original data with the alternative time-series data if the maximum difference between the alternative and original time series is above a certain threshold. | [ |

Changing-weight filtering (CW) | Filters an NDVI time series with a three-point changing-weight filter and replaces the local maximum/minimum points in a growth cycle. | [ |

RMSE, AIC, and BIC values for the eight techniques of Global Inventory Modeling and Map Studies (GIMMS) NDVI time-series data from January 1982 to December 2006 (best-fitting technique shown in bold).

^{a} |
^{b} | ||||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | shrub | 0.0071 | 0.0081 | 0.0067 | 0.0104 | 0.0082 | 0.0061 | ||

crop | 0.0123 | 0.0174 | 0.0093 | 0.0114 | 0.0214 | 0.0153 | 0.0122 | ||

desert | 0.004 | 0.0045 | 0.0033 | 0.0038 | 0.0058 | 0.0047 | 0.0033 | ||

grassland | 0.0128 | 0.0166 | 0.009 | 0.0118 | 0.0199 | 0.0152 | 0.0126 | ||

meadow | 0.0162 | 0.0192 | 0.0152 | 0.0238 | 0.0175 | 0.0102 | 0.0165 | ||

| |||||||||

AIC | shrub | −2330 | −2206 | −2559 | −1421 | −2407 | −2170 | − |
−2600 |

crop | −1549 | −1192 | −1833 | −802 | −1653 | −1262 | − |
−1619 | |

desert | −2889 | −2791 | −3111 | −1968 | −2929 | −2728 | −3080 | − | |

grassland | −1441 | −1184 | −1851 | −759 | −1545 | −1247 | − |
−1493 | |

meadow | −1161 | −996 | − |
−573 | −1241 | −1074 | −1696 | −1161 | |

| |||||||||

BIC | shrub | −6137 | −6040 | −6384 | −6219 | −5717 | −6004 | −6393 | − |

crop | −5357 | −5026 | −5658 | −5465 | −4780 | −5096 | − |
−5448 | |

desert | −6697 | −6624 | −6936 | −6740 | −6304 | −6562 | −6909 | − | |

grassland | −5249 | −5018 | −5676 | −5357 | −4800 | −5081 | − |
−5322 | |

meadow | −4968 | −4829 | − |
−5053 | −4581 | −4908 | −5525 | −4990 |

Root mean square error (RMSE), the Akaike’s Information Criterion (AIC), and the Bayesian Information Criterion (BIC). The smaller the value, the better the fit;

Techniques: Asymmetric Gaussian (A-G) technique, the modified-best index slope extraction (M-BISE) technique, the changing-weight filter (CW) technique, the double logistic function (D-L), the iterative interpolation for data reconstruction (IDR) technique, the mean value iteration (MVI) technique, the Savitzky-Golay filter (S-G) technique, and the Whittaker smoother (WS) technique.

RMSE, AIC, and BIC values for the eight techniques for the Pathfinder AVHRR Land (PAL) NDVI time-series data from July 1981 to December 2000 (except August 1994–December 1994) (best-fitting technique shown in bold).

^{a} |
^{b} | ||||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | shrub | 0.0143 | 0.019 | 0.0137 | 0.0131 | 0.0251 | 0.0197 | 0.0103 | |

crop | 0.0198 | 0.027 | 0.0152 | 0.0181 | 0.033 | 0.024 | 0.0148 | ||

desert | 0.0066 | 0.0099 | 0.0073 | 0.0062 | 0.0139 | 0.0101 | 0.0052 | ||

grassland | 0.0196 | 0.0269 | 0.0173 | 0.0182 | 0.0334 | 0.0249 | 0.0149 | ||

meadow | 0.0228 | 0.0277 | 0.0178 | 0.0209 | 0.0331 | 0.026 | 0.0174 | ||

| |||||||||

AIC | shrub | −1528 | −1128 | −1545 | −1656 | −788 | −1087 | −2036 | − |

crop | −950 | −580 | −1310 | −1075 | −316 | −698 | − |
−1377 | |

desert | −2514 | −2038 | −2393 | −2587 | −1675 | −2053 | −2828 | − | |

grassland | −985 | −574 | −1147 | −1087 | −289 | −662 | − |
−1392 | |

meadow | −777 | −536 | −1102 | −900 | −309 | −606 | − |
−1174 | |

| |||||||||

BIC | shrub | −5962 | −5589 | −5997 | −6094 | −5249 | −5547 | −6492 | − |

crop | −5383 | −5041 | −5761 | −5513 | −4777 | −5159 | − |
−5833 | |

desert | −6948 | −6499 | −6844 | −7025 | −6136 | −6514 | −7284 | − | |

grassland | −5419 | −5035 | −5599 | −5525 | −4750 | −5123 | − |
−5848 | |

meadow | −5211 | −4997 | −5554 | −5338 | −4770 | −5067 | − |
−5630 |

Root mean square error (RMSE), the Akaike’s Information Criterion (AIC), and the Bayesian Information Criterion (BIC). The smaller the value, the better the fit;

Techniques: Asymmetric Gaussian (A-G) technique, the modified-best index slope extraction (M-BISE) technique, the changing-weight filter (CW) technique, the double logistic function (D-L), the iterative interpolation for data reconstruction (IDR) technique, the mean value iteration (MVI) technique, the Savitzky-Golay filter (S-G) technique, and the Whittaker smoother (WS) technique.

RMSE, AIC, and BIC values for the eight techniques for SPOT VGT NDVI time-series data from April 1998 to April 2013 (best-fitting technique shown in bold).

^{a} |
^{b} | ||||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | shrub | 0.0111 | 0.0129 | 0.0089 | 0.0111 | 0.0131 | 0.0121 | 0.0078 | |

crop | 0.0133 | 0.0142 | 0.0091 | 0.0135 | 0.0144 | 0.0145 | 0.0099 | ||

desert | 0.0043 | 0.005 | 0.0039 | 0.004 | 0.0057 | 0.0049 | 0.0033 | ||

grassland | 0.0128 | 0.0146 | 0.0101 | 0.0125 | 0.0147 | 0.0144 | 0.0084 | ||

meadow | 0.014 | 0.0157 | 0.0108 | 0.0138 | 0.0159 | 0.0146 | 0.0095 | ||

| |||||||||

AIC | shrub | −1716 | −1563 | −1897 | −1745 | −1518 | −1605 | − |
−2121 |

crop | −1369 | −1271 | −1719 | −1354 | −1270 | −1248 | − |
−1698 | |

desert | −2504 | −2361 | −2607 | −2590 | −2211 | −2377 | −2801 | − | |

grassland | −1376 | −1238 | −1605 | −1408 | −1235 | −1253 | −1828 | − | |

meadow | −1292 | −1165 | −1541 | −1321 | −1144 | −1229 | − |
−1725 | |

| |||||||||

BIC | shrub | −5083 | −4957 | −5281 | −5116 | −4911 | −4998 | − |
−5510 |

crop | −4737 | −4664 | −5103 | −4725 | −4663 | −4641 | − |
−5087 | |

desert | −5871 | −5754 | −5992 | −5962 | −5604 | −5770 | −6190 | − | |

grassland | −4744 | −4631 | −4989 | −4779 | −4628 | −4646 | −5217 | − | |

meadow | −4659 | −4558 | −4926 | −4692 | −4537 | −4622 | − |
−5113 |

Root mean square error (RMSE), the Akaike’s Information Criterion (AIC), and the Bayesian Information Criterion (BIC). The smaller the value, the better the fit;

Techniques: Asymmetric Gaussian (A-G) technique, the modified-best index slope extraction (M-BISE) technique, the changing-weight filter (CW) technique, the double logistic function (D-L), the iterative interpolation for data reconstruction (IDR) technique, the mean value iteration (MVI) technique, the Savitzky-Golay filter (S-G) technique, and the Whittaker smoother (WS) technique.

RMSE, AIC, and BIC values for the eight techniques for the NDVI time-series data of Terra MOD13A2 from February 2000 to February 2013 (best-fitting technique shown in bold).

^{a} |
^{b} | ||||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | shrub | 0.0138 | 0.0143 | 0.0119 | 0.0141 | 0.019 | 0.016 | 0.0122 | |

crop | 0.0182 | 0.0217 | 0.0182 | 0.0194 | 0.0294 | 0.0243 | 0.0181 | ||

desert | 0.0043 | 0.0041 | 0.0032 | 0.0035 | 0.0044 | 0.0041 | |||

grassland | 0.0142 | 0.0181 | 0.0147 | 0.0147 | 0.0219 | 0.0195 | 0.0136 | ||

meadow | 0.017 | 0.0175 | 0.0142 | 0.0169 | 0.0234 | 0.02 | 0.0156 | ||

| |||||||||

AIC | shrub | −1454 | −1466 | −1600 | −1538 | −1394 | −1455 | − |
−1654 |

crop | −1648 | −1730 | −1884 | −1786 | −1694 | −1737 | −1923 | − | |

desert | −1301 | −1309 | −1460 | −1385 | −1241 | −1293 | − |
−1507 | |

grassland | −1701 | −1758 | −1903 | −1824 | −1715 | −1764 | −1950 | − | |

meadow | −1701 | −1769 | −1910 | −1826 | −1722 | −1773 | −1952 | − | |

| |||||||||

BIC | shrub | −2365 | −2221 | −2255 | −2326 | −2202 | −2270 | − |
−2283 |

crop | −2095 | −1990 | −1994 | −2070 | −1885 | −2072 | − |
−1999 | |

desert | − |
−2755 | −2804 | −3008 | −2801 | −2779 | −2861 | −2941 | |

grassland | −2107 | −2016 | −2047 | −2096 | −1969 | −2072 | − |
−2061 | |

meadow | − |
−1939 | −1943 | −2085 | −1873 | −1972 | −2095 | −2020 |

Performance orders of the eight techniques for different data sources in terms of RMSE, AIC, and BIC.

^{a} | |||||||||
---|---|---|---|---|---|---|---|---|---|

GIMMS NDVI | shrub | WS | S-G | CW | D-L | A-G | M-BISE | MVI | IDR |

Crop | S-G | CW | D-L | WS | A-G | MVI | M-BISE | IDR | |

desert | WS | CW | S-G | D-L | A-G | M-BISE | MVI | IDR | |

grassland | S-G | CW | D-L | WS | A-G | MVI | M-BISE | IDR | |

meadow | CW | S-G | D-L | WS | A-G | MVI | M-BISE | IDR | |

| |||||||||

PAL NDVI | shrub | WS | S-G | D-L | CW | A-G | M-BISE | MVI | IDR |

Crop | S-G | WS | CW | D-L | A-G | MVI | M-BISE | IDR | |

desert | WS | S-G | D-L | A-G | CW | MVI | M-BISE | IDR | |

grassland | S-G | WS | CW | D-L | A-G | MVI | M-BISE | IDR | |

meadow | S-G | WS | CW | D-L | A-G | MVI | M-BISE | IDR | |

| |||||||||

SPOT VGT NDVI | shrub | S-G | WS | CW | D-L | A-G | MVI | M-BISE | IDR |

Crop | S-G | CW | WS | A-G | D-L | M-BISE | IDR | MVI | |

desert | WS | S-G | CW | D-L | A-G | MVI | M-BISE | IDR | |

grassland | WS | S-G | CW | D-L | A-G | MVI | M-BISE | IDR | |

meadow | S-G | WS | CW | D-L | A-G | MVI | M-BISE | IDR | |

| |||||||||

MODIS NDVI | shrub | S-G | WS | CW | A-G | D-L | M-BISE | MVI | IDR |

Crop | S-G | WS | CW | A-G | D-L | M-BISE | MVI | IDR | |

desert | WS | S-G | CW | D-L | A-G | M-BISE | MVI | IDR | |

grassland | S-G | WS | A-G | CW | D-L | M-BISE | MVI | IDR | |

meadow | S-G | CW | WS | D-L | A-G | M-BISE | MVI | IDR |

Techniques: Asymmetric Gaussian (A-G) technique, the modified-best index slope extraction (M-BISE) technique, the changing-weight filter (CW) technique, the double logistic function (D-L), the iterative interpolation for data reconstruction (IDR) technique, the mean value iteration (MVI) technique, the Savitzky-Golay filter (S-G) technique, and the Whittaker smoother (WS) technique. The techniques shown at the beginning of the row perform better than those at the end.