Accuracy analysis of remote sensing change detection …€¦ · Accuracy analysis of remote sensing change detection by rule-based rationality evaluation with post-classiﬁcation

Accuracy analysis of remote sensing change detection by rule-basedrationality evaluation with post-classification comparison

H. LIU

Department of Geography, Beijing Normal University, Beijing 100875,PR China; e-mail: [email protected]

and Q. ZHOU

Department of Geography, Hong Kong Baptist University, Kowloon Tong,Kowloon, Hong Kong, PR China; e-mail: [email protected]

(Received 8 April 2002; in final form 24 March 2003 )

Abstract. Accuracy assessment for remote sensing classification is commonlybased on using an error matrix, or confusion table, which needs reference, or‘ground truthing’, data to support. When undertaking change detection usingnumerous multi-temporal images, it is often difficult to make the accuracyassessment by the ‘traditional’ method, which typically requires simultaneouscollection of reference data. In this study, we propose a new approach byarguing change rationality with post-classification comparison. Multi-temporalLandsat TM images were classified for land use in an urban fringe area ofBeijing, China and the post-classification comparison of these classified imagesshows change trajectories through the time series. These change trajectories werethen analysed by assessing their rationality against a set of logical rules toseparate cases of ‘real land use change’ and possible classification errors. Theanalysis results show that the overall accuracy for land use change in the urbanfringe area was 86%, with a fuzziness of 7%. Although it is argued that theuncertainty still exists on classification accuracy assessed by this method, itnevertheless provides an alternative approach for more reasonable assessmentwhen ideal simultaneous ‘ground truthing’ is not available.

1. Introduction

In urban fringe areas, land use and land cover changes take place rapidly as a

consequence of economic development and population growth. The characteristics

of spatial land use patterns are multiform, miscellaneous and changing

(Charbonneau et al. 1993). The geography of urban growth offers a graphic

depiction of the interplay between economics, political systems and the environ-

ment (Masek et al. 2000). Thus, it becomes important to monitor change in this

area frequently at time periods determined by various stages of economical

development.

Remote sensing images from Landsat and SPOT satellites have been used in

land use and land cover change detection for years (Green et al. 1994, Kwarteng

and Chavez 1998, Li and Yeh 1998, Masek et al. 2000, Ji et al. 2001, Weng 2001,

Zhang 2001). Numerous remote sensing change detection methods have been

International Journal of Remote SensingISSN 0143-1161 print/ISSN 1366-5901 online # 2004 Taylor & Francis Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/0143116031000150004

INT. J. REMOTE SENSING, 10 MARCH, 2004,

VOL. 25, NO. 5, 1037–1050

developed, such as image differencing, vegetation index differencing, selective

principal components analysis, direct multi-date unsupervised classification and

post-classification comparison (Gong 1993, Lunetta and Elvidge 1999).

Remote sensing change detection in urban fringe areas has two aspects, namelymonitoring the urban growth area, or urban expansion, and detecting the land use

and land cover type shifting. Many remote sensing applications for urban growth

detection are based on the method that uses images acquired at the beginning and

end of study periods (Fung and Ledrew 1988, Green et al. 1994, Kwarteng and

Chavez 1998, Li and Yeh 1998, Ridd and Liu 1998, Mas 1999, Morisette et al.

1999, Chan et al. 2001, Ji et al. 2001, Weng 2001). We may call this the ‘two-time

comparison method’, by which the classification accuracy is typically assessed using

an error matrix against simultaneous reference data, such as aerial photographs andfield checking (Li and Yeh 1998). Some studies analyse the overall accuracy in more

detail (Pilon et al. 1988, Green et al. 1994, Mas 1999, Kwarteng and Chavez 1998),

some with alternative methodological approaches such as ‘consistency checking’

(Gong and Mu 2000).

Beyond the two-time comparison method, multi-temporal comparison studies

were also reported (i.e. use of more than two multi-temporal images; Martin 1989,

Charbonneau et al. 1993, Michener and Houhoulis 1997, Petit et al. 2001).

Charbonneau et al. (1993) used Landsat MSS images of 1972, 1979 and 1982 tomonitor the urbanization process in Montreal, Canada. To study the dynamics of

urban growth in Washington, DC metropolitan area, Masek et al. (2000) used MSS

data of 1973, TM images of 1985, and SPOT images of 1991 and 1996 together.

The ‘traditional’ error matrix and kappa coefficient method were used in

assessing these multi-temporal comparison studies where the reference data were

available. Mertens and Lambin (2000) estimated classification accuracy for each

image, using an independent sample of 262 observations from the field campaigns

and the low-altitude aerial photographs. They reported high classification accuracy

results of 97%, 97% and 98% for 1973, 1986 and 1991 images, respectively. Whenthe simultaneous reference data were not available, some supporting methods were

proposed (Masek et al. 2000, Petit et al. 2001).

Most ‘traditional’ accuracy assessment methods are developed for the one point

in time (OPIT) thematic mapping (Biging et al. 1999). When multi-temporal

classification results are assessed, it would be difficult to confirm the overall

accuracy of the whole study period. For change detection based on the multi-

temporal classification method, the accuracy of each classified image has to be

somehow aggregated as the overall assessment on accuracy of the whole-periodresult, typically by simple statistics such as average, or through an error

propagation model (Martin 1989, Charbonneau et al. 1993, Michener and

Houhoulis 1997, Mertens and Lambin 2000, Carmel et al. 2001, Petit et al.

2001). When the number of multi-temporal images increases, analysis on derived

time-series data has been reported (Alves and Skole 1996), but without the accuracy

assessment on the result.

The objectives of this study, therefore, are (a) to develop an alternative

approach to assess accuracy of remote sensing change detection in an urban fringearea by analysing change rationality in post-classification comparison according to

given land use shifting rules; and (b) analysing the overall accuracy of the whole

study period when only limited simultaneous reference data are available. Five

multi-temporal Landsat TM images were classified for land use in an urban fringe

area of Beijing, China and the post-classification comparison of these classified

1038 H. Liu and Q. Zhou

images shows change trajectories through the time series. These change trajectories

were then analysed by assessing their rationality against a set of logical rules to

separate cases of ‘real land use change’ and possible classification errors.

2. Study area and data

This study was undertaken at Chaoyang District located in the eastern part of

Beijing (figure 1). The district covers an area of 455.2 km2, with a population over

1.3million—the highest of all districts in Beijing. The majority of the district is

characterized as urban fringe with the most rapid urban expansion in the past two

decades. From 1978, the area of agricultural land decreased sharply: 18.8% from

1982 to 1989 and 6.5% from 1989 to 1992. Residential and industrial land, on the

other hand, increased about 28.2% from 1982 to 1989, and 14.9% from 1989 to

1992 (Cao and Cai 1993).

Landsat TM images were acquired on 2 October 1984, 21 April 1988, 6 May

1991, 28 August 1994 and 16 May 1997. The land use map, compiled in 1991 and

Figure 1. The study area (Chaoyang District), located in the urban fringe area of Beijing,China.

Accuracy analysis of remote sensing change detection 1039

based on field surveys at the scale of 1 : 50 000, was used for the accuracy

assessment. The master scene (1991) was geometrically corrected and registered to

the land use map, using 36 selected Ground Control Points (GCP) and second-

order polynomial transformation with nearest neighbour resempling. The other

scenes were then registered to the master scene by image-to-image registration.

3. Methodology

In order to achieve the objectives of this research, our approach towards

accuracy assessment of change detection is based on the widely used post-

classification comparison method (Lillesand and Kiefer 2000). Using the post-

classification comparison, land cover change is identified where the classification

categories are found to be different between two or more image dates. The

comparison results (i.e. pixels with ‘change’ status) are then assessed by a rule-based

analysis based on both ‘change/no change’ and ‘from-to’ (land use shift) rationality.

The methodology can be outlined as below:

1. The master scene (1991) image was selected and independently classified

using two classifiers.

2. The one-time accuracy assessment on both of the classified images was

undertaken using the common error matrix analysis with simultaneous

reference data (land use map). By comparing the assessment results, the

‘preferred’ classifier was then selected for image classification of other multi-

temporal images.

3. Using the ‘preferred’ image classifier, the four other multi-temporal images

were classified and integrated with the master image. The pixels with the

‘change’ status through the study period were then identified by the post-

classification comparison.

4. A test for sampling was conducted to determine the number of samples

required for the rule-based rationality analysis.

5. The rule-based rationality analysis was then applied to the ‘change’ pixels to

separate the ‘real change’, ‘fuzzy’ and ‘classification error’ pixels so as to

derive statistics as the basis for the accuracy assessment of the change

detection.

3.1. Image classification

Two classifiers were employed in this study, namely, the maximum likelihood

(MLC) and artificial neural network (NNC) classifiers, of which the former is the

most commonly used. One major difference between the two classifiers is the

number and purity requirement on the training areas (Atkinson and Tatnall 1997,

Kanellopoulos and Wilkinson 1997). The NNC needs fewer and less pure seeding

data in comparison to that of MLC.

PCI 6.0 remote sensing image processing software was used to accomplish the

classifications. 4835 pixels were selected as the training data on the 1991 TM image.

The BP model of NNC was applied in the classification, with a structure of 6, 32

and 7, which refer to input, nods number and output, respectively. After

classification, the post process was applied to both results from MLC and NNC to

aggregate classified categories in order to match those of the land use map. Five

land cover classes were finally mapped, namely water, vegetable garden, forest,

farmland, and built-up area.


3.2. One-time accuracy assessment on the image classification

The one-time classification error matrix was constructed for the land cover

classification on the 1991 image with the 1991 land use map as the reference data. A

total of 1212 samples were randomly selected over the study area on the land use

map. With a higher overall accuracy of 3% than that of MLC, NNC was chosen to

be the ‘preferred’ classifier with an overall accuracy of 79.6% and the kappa

coefficient of 0.696 (table 1 and table 2).

3.3. Multi-temporal image classification and change trajectory establishment

Thus NNC was applied to classify the remaining four multi-temporal images.

Together with the 1991 image, the five multi-temporal classified images were used to

establish the land cover change trajectory for each pixel from 1984 to 1997. For

each pixel, therefore, a change trajectory can be identified as either ‘no change’, or

‘shifting’ between the interested cover types.

3.4. Sampling test

Before taking samples from the classified images for analysing the change

rationality, the reliability of sampling was tested. Two tests were undertaken,

namely the number of samples and checking of the wrap points. Samples were

taken from randomly generated locations and wrap points were checked.

The frequency distribution of the cover type change times during the five

monitoring dates was employed to check the stability of the sampling. When the

frequency distribution became stable with increasing sampling points, a total of

Table 1. Accuracy assessment of the artificial neural network classification of the 1991 TMimage.

Land usecategories

Produceraccuracy (%)

Useraccuracy (%)

Averageaccuracy (%) Kappa

Farmland 79.9 88.0 84.0 0.80Water 53.5 72.9 63.2 0.70Vegetable garden 64.2 53.0 58.6 0.48Forest 75.0 34.1 54.6 0.33Built-up area 89.0 84.6 86.8 0.74

Overall accuracy: 79.6%; kappa coefficient: 0.7%.

Table 2. Confusion matrix of the artificial neural network classification of the 1991 TMimage.

Categories

Reference data (land use map)

Farmland WaterVegetablegarden Forest

Built-uparea

Totalpixels

Farmland 374 21 12 3 15 425Water 13 62 2 2 6 85Vegetable garden 25 10 70 0 27 132Forest 12 8 3 15 6 44Built-up area 44 15 22 0 445 526Total pixels 468 116 109 20 449 1212


3171 samples, which constitute about 0.6% of the total study area, were then

selected (table 3).

3.5. Rule-based rationality evaluation

The spatio-temporal land use shifting pattern has been an active research field

of land use change detection (Roy and Tomar 2001, Weng 2001). While the

landscape is regularly monitored by remote sensing, the concept and methodology

of change trajectory has been developed. From the point of view of change

detection, the change trajectory was defined as trends over time among the

relationships between the factors that shape the changing nature of human–

environment relations and their effects within a particular region (Kasperson et al.

1995). The trajectory of land cover change refers to successions of land cover types

for a given sampling unit over more than two observations (Mertens and Lambin

2000, Petit et al. 2001).

In an urban fringe area, most land use and land cover changes are the

consequence of urban growth. Based on this understanding, an assumption can be

made that change to ‘built-up area’ from other land use types is irreversible. A set

of rules, therefore, can be made to evaluate the rationality of detected change

trajectory over every sample so that cases of ‘real change’, ‘fuzzy’ or ‘classification

error’ can be identified.

If we let t denote the number of detected categorical changes over the five

monitoring dates from 1984 to 1997, we have:

0¡t¡4 ð1Þwhere the extreme case of t~0 refers to no change, while t~4 refers to the fact that

the category of the sample has changed for every detection period.Six rules have been applied to determine the rationality of the change trajectory

for each sample. Let A denote the case that ‘the pixel is correctly classified’, and B

denote the case that ‘the pixel is in a fuzzy state’. When A is rejected, it means ‘the

pixel is not correctly classified’, thus it is the classification error. Land cover types

are represented as Ci (where C1~‘water’, C2~‘vegetable garden’, C3~‘forest’,

C4~‘farmland’, and C5~‘built-up area’). The detected change trajectory between

the cover types is denoted as T(Ci …) where, for example, T(C4, C5) means ‘change

from farmland to built-up area’.

For each sampled pixel, the rules are applied in the sequential order as below

and their overall logic flow structure is shown as in figure 2.

Table 3. Results of the sampling stability test.

Changingtimes

Random samples

3171 points 2500 points 2000 points 1000 points 500 points

Pixels % Pixels % Pixels % Pixels % Pixel %

0 1362 43.0 1067 42.7 864 43.2 430 43.0 207 41.41 746 23.5 604 24.2 471 23.6 228 22.8 119 23.82 342 10.8 265 10.6 206 10.3 106 10.6 59 11.83 535 16.9 420 16.8 345 17.3 181 18.1 86 17.24 186 5.9 145 5.8 114 5.7 55 5.5 29 5.8


Rule I: IF t~0 THEN accept A.

Rule II: IF t~1 AND T(Ca, Cb) THEN accept A. (a|b; a|5)

Rule III: IF t~1 AND T(C5, Cb) THEN reject A. (b|5)

Rule IV: IF t~2 AND T(Ca, Cb, Ca) THEN accept A. (a|b)

Rule V: IF t~2 AND T(Ca, Cb, Cc) THEN accept B (a|b|c; a|5; b|5)

Rule VI: IF tw2 AND dƒ1 THEN accept A ELSE reject A.

where d denotes the distance between the sample and the nearby ‘unchanged’ area

identified by Rule I, measured by the number of pixels.

The meaning of Rule I is obvious. If the pixel is classified as the same land cover

type for all monitoring dates, then there is no change and the pixel is regarded as

correctly classified.

The built-up area of Beijing doubled from 1984 to 1997. This happened when

the other land cover types were transformed to built-up area and this process could

not be reversed. Rule II, therefore, states that if once-only change is detected from

one cover type (except built-up area) to another, then the change is regarded as a

‘true’ case so that the pixel is correctly classified. Rule III, on the other hand,

defines that if the reverse process were detected (i.e. once-only change from built-up

area to another cover type), the change would be unlikely so that the pixel is not

correctly classified.

Rule IV addresses one-time error of multi-temporal remote sensing image

Figure 2. The structure of the rule-based rationality evaluation.


classification. If a pixel was found changed from one cover type (Ca) to another

(Cb) and then back to its origin (i.e. Ca), it is regarded as one-time classification

error case (i.e. Cb was the incorrect class). This one-time error does not affect the

final result of change detection, so that the pixel is regarded as correctly classified as

Ca.

Rule V specifies the case where the land cover changed two times to different

cover types during the study period. This is possible where land cover could shift

from, for example, farmland to vegetable garden to built-up area. On the other

hand, it is also uncertain whether the trajectory shows the ‘true’ multiple land cover

change or it is simply caused by classification error. We therefore consider this pixel

as a ‘fuzzy’ case with an uncertain land cover class. Note that we exclude the cases

of the ‘reverse change’ (i.e. from built-up area to another cover type) from Rule V

as it is regarded as error (Rule III).

Rule VI considers the ‘marginal pixels’ that changed frequently between cover

types and located adjacent to unchanged areas. This is most likely the consequence

of misregistration in geometric image rectification (Townshend et al. 1992, Stow

1999). The distance from the sample to the nearby unchanged area (d ) is therefore

measured. If d is within one pixel, then the sample is the ‘marginal pixel’, thus it is

regarded as a correct case and has the same cover type as that of adjacent

unchanged area.

Finally the ‘error’ state (i.e. reject A) is assigned to those that fail to pass every

given rule.

4. Results

4.1. Image classification

Figure 3 shows the results of classification for each acquisition date. During the

entire study period from 1984 to 1997, the built-up area expanded from 26.7% to

55.9% of the total district, i.e. doubling in 14 years. On the other hand, farmland

and vegetable garden decreased from 51.1% and 15.5% to 26.3% and 6.7%,

respectively. Forest area decreased slightly but the area of water bodies increased

sharply, mainly due to the increase of commercial fish ponds in the district (table 4).

By overlapping the multi-temporal classified images, we derived an urban expansion

map shown as figure 4.

4.2. Change trajectory test: stage 1

The rule-based assessment of change trajectory over the five-time multi-

temporal image classification results can be categorized into two stages. Stage 1 is

composed of Rule I and II, providing an initial assessment on the classification

accuracy. The application of these rules distinguishes samples into two groups,

namely, ‘correct’ and ‘uncertain’ (table 5). Among 3171 tested samples, 1362

samples remained unchanged during the entire study period, accounting for 43% of

the total; while 704 samples were found changed irreversibly (i.e. only once) from

other cover types to built-up area, constituting 22% of the total. Thus, the results

show that the overall classification accuracy was at least 65%, leaving 35% of

samples remaining as ‘uncertain’ to be further tested.

4.3. Change trajectory test: stage 2

The objective of the Stage 2 test is to further identify ‘correct’, ‘incorrect’ and

‘fuzzy’ classification samples among the remaining 1105 ‘uncertain’ samples from


Stage 1 test, by applying Rules III, IV, V and VI (table 6). The results show that 43

(3.9%) of ‘uncertain’ samples were found as ‘reverse change’ cases (i.e. ‘true’ for

Rule III), so that they are identified as ‘incorrect’. Rule IV (one-time error cases)

has found 100 samples accounting for 9% of total ‘uncertain’ samples. Rule V

(multiple changes between cover types other than built-up area) identified 242

samples (21.8%) indicating the ‘fuzzy’ cases. For the ‘marginal pixel’ cases (Rule

VI), 591 (53.5%) samples were found within one pixel distance to the unchanged

area (see locations of these sample points in figure 4).

5. Discussion5.1. Overall accuracy of the five-time multi-temporal land cover classification

The overall accuracy of the five-time rationality evaluation is shown in table 7

by compiling the results from the above change trajectory tests. From table 7, the

overall test for classification accuracy shows that 86.9% of samples were correctly

Figure 3. Land cover classification of the 1984, 1988, 1991, 1994 and 1997 images.

Table 4. The percentage of land use categories for each image acquisition date.

Acquisition date Built-up area Farmland Forest Vegetable garden Water

1984 26.7 51.1 3.7 15.5 3.01988 39.8 39.9 2.7 10.7 6.81991 46.0 34.9 2.7 8.2 8.21994 49.9 32.0 3.0 6.9 8.31997 55.9 26.3 2.9 6.7 8.2


classified, 5.5% were incorrectly classified, and 7.6% were in the fuzzy states. It is

therefore concluded that the overall accuracy of the multi-temporal land cover

classification was at least 86.9%, assuming the worst scenario (i.e. all ‘fuzzy’

samples are errors).

Figure 4. Urban expansion of Chaoyang District of Beijing. Note most samples where landcover type shifting was greater than two were one pixel distant from the unchanged area.

Table 5. Change trajectory test: stage 1 results.

Pixel number Percentage

Total samples 3171 100Correct (total) 2066 65.2Correct (Rule I) 1362 43.0Correct (Rule II) 704 22.2Uncertain 1105 34.8


5.2. Rules

Rules applied in this study are based on the characteristics of land use change in

the urban fringe area of Chaoyang District in Beijing, and the focus of this study

(i.e. urban expansion). In the urban fringe area, urban growth is the most

significant spatio-temporal change in land use, which shows some predictable

change trajectories (e.g. other land use types to built-up area). The evaluation of

accuracy rationality is based on these trajectories. It is argued that the land use

trajectories can be various in different study areas and thus the rules for this study

may not apply. On the other hand, this approach can be useful to other

applications of change detection such as deforestation detection and monitoring,

where land cover change trajectories are well known (Mertens and Lambin 2000).

5.3. Influence of image registration accuracy

In this study, image geometric rectification was undertaken before classification

and rationality evaluation. The multi-temporal image-to-image registration was

controlled in an allowable range (average rms~0.56 with the maximum of 0.8).

There was still potential registration error as it was hard to keep the geometric

correction error below half a pixel for the entire image, although some ‘registration

noise’ reduction methods may be applied (Gong et al. 1992). This potential

registration error, therefore, was considered with Rule VI, discussed above, which

identified more than 18% of the total samples located in one pixel distance to the

unchanged area mask and classified them into correctly classified cases. Although it

is recognized that the spatial registration error is unavoidable and the accuracy

shown in this study is quite acceptable, whether the ‘marginal’ samples show the

Table 6. Change trajectory test: stage 2 results.

Pixel number Percentage of test samples

Total ‘uncertain’ samples 1105 100Rule III 43 3.9Rule IV 100 9.0Rule V 242 21.8Rule VI True 591 53.5

False 130 11.8

Table 7. Overall rule-based rationality evaluation results.

Rules Pixel numbers %

Correct Rule I 1362 43.0Rule II 704 22.2Rule IV 100 3.2

Rule VI: True 591 18.6Total 2757 86.9

Incorrect Rule III 43 1.4Rule VI: False 130 4.1

Total 173 5.5

Fuzzy Rule V 242 7.6

Total 3171 100


‘true’ land use change or the ‘false’ cases caused by image registration inaccuracy

obviously represents an uncertainty.

5.4. Overall evaluation of the method

In this study the change trajectory test revealed an overall accuracy of 86.9%

with an uncertainty of 7.6% (see §5.1). Given the 20.4% classification error found by

the one-time accuracy assessment on the classification of the 1991 image against the

reference data, and 5.5% error found through the trajectory test, it is estimated that

the overall propagated error of the overall change detection study should be within

the range of 21.1% (fuzzy cases as ‘correct’) to 22.5% (fuzzy cases as ‘incorrect’).

This estimation would be significantly different from that obtained using the

method proposed by Lunetta and Elvidge (1999), by which the overall change

detection accuracy is approximated by multiplying the accuracies of individual

classifications. We believe, however, the trajectory test will provide a more realistic

scenario since the one-time classification error may be cancelled out by other multi-

temporal image classifications, provided that there are sufficient observation dates

to ensure the confidence of the test.

6. Conclusion

In this study, 3171 samples were selected to evaluate the accuracy of change

detection using the rule-based rationality evaluation. Six rules based on the

recognized land use change trajectory in the urban fringe area of Chaoyang District

in Beijing were established and applied to the results of post-classification

comparison of multi-temporal TM images. Using the change trajectory test,

samples showing locally recognized land use change trend (e.g. change from other

land use to built-up area) could be identified and classified as ‘true’ change cases.

The results also showed that more than half of the ‘uncertain’ samples were caused

by the slight image registration errors that were within the allowable range. The

trajectory test showed that the overall change detection accuracy was 86.9%, with

5.5% error and 7.6% in ‘fuzzy’ states. This gives an overall error assessment of

22.5% or less, given the 20.4% one-time classification error.

Ideally the error assessment for change detection using the post-classification

comparison method should follow the ‘traditional’ error assessment method, i.e.

assessing errors using simultaneous reference data for each image classification, and

then evaluating the overall accuracy using an error propagation model. However,

since in reality it is almost impossible to obtain simultaneous reference data for

long-term change detection in most application cases, an alternative approach has

to be engaged. This study demonstrates a promising method employing a rule-based

change trajectory test to the error assessment. Our approach is in some way close to

that of ‘consistency checking’ proposed by Gong and Mu (2000), but in the context

of consistency checking in time instead of in space. Although there are still

questions to be answered such as whether the rules have sufficient coverage of all

possible scenarios or if they are applicable to other cases, this approach has

nevertheless been shown to be a realistic and practical method for error assessment

with confidence.


AcknowledgmentsThis study was supported by RGC Project HKBU 2086/01P, Hong Kong

Croucher Chinese Visitorships, Hong Kong and by the Foundation for University

Key Teachers of the Ministry of Education of China.

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1050 Accuracy analysis of remote sensing change detection

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Accuracy analysis of remote sensing change detection …€¦ · Accuracy analysis of remote sensing change detection by rule-based rationality evaluation with post-classiﬁcation