Optimizing the binary discriminant function in change detection ...

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RSE-07112; No of Pages 16

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Remote Sensing of Environment xx (2008) xxx – xxx www.elsevier.com/locate/rse

Optimizing the binary discriminant function in change detection applications Jungho Im a,⁎, John R. Jensen b , Michael E. Hodgson b a

Department of Environmental Resources and Forest Engineering, State University of New York, College of Environmental Science and Forestry, United States b Department of Geography, University of South Carolina, United States Received 5 June 2007; received in revised form 10 January 2008; accepted 12 January 2008

Abstract Binary discriminant functions are often used to identify changed area through time in remote sensing change detection studies. Traditionally, a single change-enhanced image has been used to optimize the binary discriminant function with a few (e.g., 5–10) discrete thresholds using a trialand-error method. Im et al. [Im, J., Rhee, J., Jensen, J. R., & Hodgson, M. E. (2007). An automated binary change detection model using a calibration approach. Remote Sensing of Environment, 106, 89–105] developed an automated calibration model for optimizing the binary discriminant function by autonomously testing thousands of thresholds. However, the automated model may be time-consuming especially when multiple change-enhanced images are used as inputs together since the model is based on an exhaustive search technique. This paper describes the development of a computationally efficient search technique for identifying optimum threshold(s) in a remote sensing spectral search space. The new algorithm is based on “systematic searching.” Two additional heuristic optimization algorithms (i.e., hill climbing, simulated annealing) were examined for comparison. A case study using QuickBird and IKONOS satellite imagery was performed to evaluate the effectiveness of the proposed algorithm. The proposed systematic search technique reduced the processing time required to identify the optimum binary discriminate function without decreasing accuracy. The other two optimizing search algorithms also reduced the processing time but failed to detect a global maxima for some spectral features. Published by Elsevier Inc. Keywords: Systematic search technique; Change detection; Optimization; Binary discriminant function; Hill climbing; Simulated annealing

1. Introduction Monitoring environmental change is a very important research topic for those responsible for managing natural resources and mitigating adverse affects of human activities. Examples of environmental change include land cover change (e.g., from forest to built-up), vegetation change (e.g., migration of species), hazardous waste site monitoring (e.g., subsidence in a landfill), and disaster monitoring (e.g., hurricane wind induced damage). Remote sensing technology has been demonstrated to be an accurate and efficient method for environmental change detection purposes (e.g., Coppin & Bauer, 1996; Im & Jensen, 2005; Im et al., 2007; Jensen et al., 1995; Lu et al., 2004). ⁎ Corresponding author. E-mail address: [email protected] (J. Im).

While detailed “from-to” change information is required for many change detection applications, simple binary change detection (i.e., change or no-change information) is often sufficient. Traditional binary change detection using remotely sensed data is based on a simple calibration (e.g., Lunetta et al., 2002; Morisette & Khorram, 2000). The calibration works on one resultant change image (i.e., a difference image) and a few discrete thresholds. The manual calibration approach is based on the selection of the threshold that yields the highest change detection accuracy. The optimum threshold(s) can be identified using a trial-and-error manual method or an automated generateand-test threshold selection procedure (Im et al., 2007). The manual trial-and-error approach has several limitations, including 1) it is time-consuming and labor intensive due to the manual processing, 2) it uses only one image (i.e., a difference image) in the analysis, and 3) it tests just a few (e.g., 5–10) discrete thresholds.

0034-4257/$ - see front matter. Published by Elsevier Inc. doi:10.1016/j.rse.2008.01.007 Please cite this article as: Im, J., et al., Optimizing the binary discriminant function in change detection applications, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.01.007

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J. Im et al. / Remote Sensing of Environment xx (2008) xxx–xxx

Im et al. (2007) reported an automated calibration model that removed the limitations of the traditional approach. Using this method, a continuum of thresholds can be autonomously tested, not just a few discrete thresholds. Also, multiple variables (e.g., multiple input images) can be used together in the calibration process. The automated generate-and-test approach for examining candidate thresholds removes the time-consuming labor of the manual method. The automated calibration model outperforms traditional binary change detection methods. Unfortunately, the calibration model by Im et al. (2007) still has a major drawback when multiple variables (i.e., change-enhanced images) are calibrated. Because the autonomous model adopted an exhaustive generate-and-test search strategy, the calibration processing time increases in proportion to the number of input images and the threshold distance increment (e.g., 1/1000 of the whole domain). For example, if a 1/1000 increment and one image are used with the model, one thousand iterations are needed to calibrate. For one input image, the calibration requires only a few minutes. If a 1/1000 increment and two input images are used with the model, then one million iterations are required (i.e., 10002 = 1,000,000). Skilled users can reduce the processing time by specifying a smaller range between the minimum and maximum candidate thresholds for the search process. However, the use of multiple input images with very small increments between candidate thresholds may still result in prohibitively high processing times. Optimization algorithms can be used to reduce high processing times associated with search space problems. When adopting an optimization algorithm to find optimum threshold(s) in a search space, a major issue would be how effectively the algorithm detects a global maxima without becoming trapped in local maxima. There are numerous optimization algorithms developed in the computer science with several adopted for GIScience search problems (Hodgson, 1989, 1992). Few algorithms have been adopted for remote sensing problems (Hardin & Thomson, 1992) where the search space is less well understood. The spectral search space for change thresholds is not smooth and continuous as it is derived from contingency tables using a ‘small’ (e.g., hundreds) set of reference points. In theory, if the number of reference points was large then the Kappa search space surface would be approximately smooth. The advanced algorithms based on derivative functions (e.g., first or second derivatives) of smooth continuous surface are not well suited for the Kappa search space. The objectives of the study were 1) to develop a systematic search (SS) algorithm suitable to identify optimum threshold(s) with the automated calibration model, 2) to evaluate the SS algorithm, and 3) to compare the SS algorithm to the other two optimizing algorithms and to the exhaustive search (ES) algorithm in terms of accuracy and efficiency using real-world satellite imagery. This study investigated two popular heuristic optimizing algorithms, hill climbing (HC) and simulated annealing (SA), to find optimum threshold(s). This study also developed a SS algorithm, optimized for change detection studies using single or multiple change-enhanced images. It adopts a basic concept of HC, but removes the limitations of HC such as the local maxima problem. The ES algorithm, proposed

in Im et al. (2007), was used to detect a global maxima in a search space and used as the “reference data.” The results from the three different optimizing algorithms were compared to those from the ES algorithm. 2. Optimum threshold search algorithms for change detection 2.1. Hill Climbing (HC) algorithm HC is an optimization algorithm which searches for optimum value(s) of variable(s) by systematically testing nearby values in the search space until the optimum (e.g., Kappa accuracy) is found (e.g., Huang & Shibasaki, 1995; Jacobson et al., 2006). The algorithm is heuristic in that it will find the global maxima if no local maxima exist — without testing all possibilities. The search logic tests new local values and ‘moves’ uphill to higher values for each subsequent test. The major weakness of the HC algorithm is that it easily fails to find a global maxima when the search space contains local maxima (Russell & Norvig, 2003). There are several parameters in the HC algorithm to be predetermined such as the increment (fixed or varying) and stop criteria. The HC algorithm used in this study follow the steps: 1) Start with random value(s) of variable(s) and return a Kappa accuracy. 2) Add an increment (10% of the range of each variable) to each variable for each direction and return a Kappa accuracy. 3) Move to the location yielding a higher accuracy and repeat step 2; if no location yields a higher accuracy, reduce an increment and repeat step 2. 4) Stop when the increment is less than 0.1% of the range of each variable. A varying increment was adopted because it was more computationally efficient and is better at avoiding local maxima. 2.2. Simulated Annealing (SA) algorithm SA was originally developed in 1983 to solve highly nonlinear problems (Kirkpatrick et al., 1983). SA has been widely used in diverse fields including economics, oceanography, and computer sciences (e.g., Goffe et al., 1994; Kruger, 1993; Li et al., 2004). Simulated annealing is a random search optimization algorithm, which approaches a global maximization (or minimization) problem in a similar way that liquids freeze or metals recrystalize in the process of annealing and the search for a minimum in a more general system (Busetti, 2003). The process starts at high temperature and slowly cools so that the system at any time is in thermodynamic equilibrium. The system becomes more ordered as the cooling process continues. The process stops when the system is in a stable ground state at very low temperature (T = ~ 0). In the SA process, the parameters of an initial temperature and the rate of cooling of the temperature are critical. If an initial temperature is too low, SA may be trapped in local maxima (or minima). If the cooling rate is too slow, SA can find the global

Please cite this article as: Im, J., et al., Optimizing the binary discriminant function in change detection applications, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.01.007

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maxima (or minima) but with considerably higher processing time. The major advantage of the SA algorithm is the ability to avoid local maxima (or minima) compared to other optimization algorithms (e.g., HC). SA accepts parameters (e.g., thresholds) producing better solution (e.g., higher Kappa accuracy) at each iteration. However, it also occasionally accepts parameters not yielding a better solution based on the Metropolis criterion, which uses a Boltzmann's probability distribution as a temperature function (Penn, 2002). If the probability is greater than

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a random number in to domain 0 to 1, the parameters are accepted. Fig. 1 summarizes the SA process flow designed for this study. The acceptance criteria of new thresholds can be expressed as: Accept if

DKappa N 0 Probabilityð exp ðDKappa=Td ÞÞ N Ran½0; 1

ð1Þ

where T is a current temperature and d is the average step size.

Fig. 1. Process flow diagram of the SA algorithm used in the study. Please cite this article as: Im, J., et al., Optimizing the binary discriminant function in change detection applications, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.01.007

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2.3. Exhaustive Search (ES) algorithm An ES algorithm was adopted in an automated calibration model for binary change detection (Im et al., 2007). The ES was used to identify a global maxima in a Kappa search space and utilized as reference data to compare the results from the optimizing algorithms (i.e., HC, SA, and SS). Fundamentally, the ES algorithm tests all possible values in the search space. Details on the algorithm are found in Im et al. (2007). 2.4. Systematic Search (SS) algorithm A SS algorithm was designed to reduce the processing time for calibrating threshold(s) when conducting binary change detection. The basic idea in the SS algorithm is to dynamically adjust the search direction and increment to identify optimum threshold(s) that yield the highest Kappa Coefficient accuracy. The SS algorithm adopted basic concepts of the HC algorithm. However, SS removes major weaknesses of HC and was optimized for change detection applications. The proposed SS algorithm adopts a heuristic search algorithm. More feasible threshold(s) are determined using a series of if–then functions. The only assumption for the proposed algorithm is that the search space is globally unimodal but may contain numerous smaller local maxima. Our preliminary testing of Kappa search spaces suggests that most changeenhanced images (e.g., band-differenced images) produced globally unimodal Kappa search spaces with many small bumps in the surfaces depending on reference data and imagery. In other words, the search spaces can be viewed as “hills” at a distance. However, small bumps can be found in a close view of the spaces. This might be a major reason that HC would be trapped in such small bumps and be likely to fail to find a global maxima. The SS algorithm focuses on how to efficiently avoid the small bumps to find a global maxima. Fig. 2 illustrates how SS works with a single variable to reach an optimum threshold (i.e., the global maxima) in a onedimensional search space. It follows rules that are similar to a line maximization function (Masters, 1993). The search starts with threshold one near the minimum (e.g., 5% of the original

domain), then moves to threshold two with an increment of 10% of the original domain. If threshold two exhibits a higher Kappa value, the search continues by increasing the value of threshold two (i.e., ‘climbing the hill’). If the current step increment is 10% of the range the search will continue using the 10% increment (Fig. 2). Threshold five exhibits a Kappa smaller than the previous threshold tested. Now it is known that a maxima (i.e., the highest Kappa accuracy) exists between thresholds four and five. The search domain (i.e., start and end) is then reset to the value of four and five. The step increment (10% of search space) is reset resulting in smaller steps as the search domain is smaller. The optimum threshold search continues as before. The search stops when the incremental increase in Kappa is less than a specified amount (e.g., 0.001% accuracy increase). This search method can be used with multiple input change images. However, the approach may easily be ‘trapped’ in a higher-dimensional search space with large local maxima. A modified approach suitable for solving the local maxima problem in a higher-dimensional search space was developed. The key concept behind the modified search is the ability to search in eight directions when two variables are used (i.e., the four cardinal directions plus the four diagonal directions). The original SS only searches in orthogonal directions as in the HC algorithm. Fig. 3 illustrates how SS works in two-variable space. The actual Kappa surface isolines using two input images (i.e., a neighborhood correlation and slope images) are shown in Fig. 3a. The global maxima is in the middle right of the Kappa surface with relatively smooth isolines. A profile through the two-image Kappa search space from x to y illustrates the small local maxima present. The search through this space begins in the center of the search space with nine candidate search points located one-third of the distance of the search domain in each direction (Fig. 3b). First, three thresholds for each variable domain, which are located a third, a half, and two-thirds of each domain, are extracted. The nine (i.e., for a two-variable search space) candidate search point threshold combinations are examined. The new search center is the candidate search point with the highest accuracy (e.g., upper right corner in the example). New domains are set for both images (Fig. 3c). The new domains are two-thirds of the original domains and dependent

Fig. 2. Schematic diagram of the SS algorithm in a one-dimensional search space. Iteration numbers appear in the squares. Please cite this article as: Im, J., et al., Optimizing the binary discriminant function in change detection applications, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.01.007


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Fig. 3. Schematic diagram of the SS algorithm in a two-dimensional search space.

on the threshold combination selected in the previous round. The search for the global maxima Kappa continues as before with the new candidate set of nine search points (Fig. 3d). The search domain is narrowed at each step until it converges at the threshold combination with the highest accuracy (i.e., global maxima). This searching concept can be extended into higherdimensional search spaces (e.g., a three-variable search space). The number of candidate search points at each step is 3n, where n is the number of input images. For example, a fourdimensional search space problem results in just 34 (81) threshold combinations to be tested for each search step. This SS strategy can use candidate search points in any number of directions (i.e., not just 45° angles). Also, the search domain may be narrowed by increments other than 1/3 increments. Three candidate search points per input image was decided upon based on empirical testing with several of the datasets reported in Im et al. (2007). Four or more threshold search points per variable may produce marginally better results (i.e., easy to identify a global maxima), but will increase processing time.

2.4.1. Stop criteria and local maxima problem One might ask “When does the SS algorithm stop searching?” Two criteria were adopted for the SS algorithm. The first one is the parameter of maximum iterations, specified by a user. When the search reaches this parameter (i.e., maximum iterations), it stops searching. Another is “jump functions” (described below) which are applied after the same Kappa accuracy (the same highest accuracy per round for two or higherdimensional search space) is obtained for three consecutive search iterations. The proposed SS algorithm is also a heuristic algorithm and, in theory, is not guaranteed to identify the global maxima. It is possible the identified ‘maximum’ Kappa is one of many local maxima (Fig. 4). How can the search strategy avoid, or at least, minimize problems with such local maxima? The SS algorithm utilizes “jump functions” to avoid the local maxima problem. The “jump functions” are a defined base on the current domain in a one-dimensional search space, and a combinatorial function of the current domains and the optimum thresholds for single variables in a two- or higher-


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Fig. 4. Examples of local maxima and jump functions.



dimensional search space (Fig. 4b). The jump function is defined as: if n ¼ 1; if n N 1;

Jn Un Ln Jn

¼ CVn Fci CDn ¼ CVn þ cinþ CDn ¼ CVn cin CDn ¼ Ln þ 13 ; 12 ; 23 ðUn Ln Þ;

ð2Þ

where Jn is the threshold(s) from jump function for the nth variable, and CVn is the converged threshold(s). Un and Ln are the upper and lower boundaries for the jump functions in a twoor higher-dimensional search space, respectively. The variables ci are constant parameters, n is the dimensionality of the search space, CDn is the current domain for the nth variable, and i is the number of jump functions for each direction in one domain. Five different constants were adopted for each direction in a one-dimensional search space. Five different sizes of new domains (with Un and Ln boundaries) were set for a two or higher-dimensional search space. The constant parameters (ci) are symmetrical to the converged threshold for a onedimensional search space. However, they are not symmetrical with the converged threshold(s) for higher-dimensional search spaces. If an optimum threshold for one variable in a onedimensional search space is greater than a converged threshold for that variable in a multi-dimensional search space, the jump functions will test 50% more thresholds in the lower part of the variables (i.e., abs(cin+) b abs(cin−)), and vice versa (Fig. 4b). Such weights were assigned due to the tendency that optimum threshold(s) yielding a global maxima are generally located around the optimum threshold for each variable in a onedimensional search space. For example, if a global maxima is located at (0.68, 0.58) in a two-dimensional search space and the optimum thresholds for the single variables 1 and 2 are 0.7 and 0.6, respectively (Fig. 4b) then the systematic search technique converges to a local maxima of (0.65, 0.65). Then, the jump functions will test thresholds from five different domains with 50% more weights to the global maxima (e.g., 0.6–0.75 for variable 1 and 0.55–0.7 for variable 2). These new unsymmetrical domains work to escape the local maxima since 1) a global maxima tends to be located around the optimum thresholds any one variable, and 2) using symmetrical domains repeatedly test the same thresholds that were previously tested. Determining how many jump functions are necessary to escape a local maxima is one of the critical parts in the SS algorithm. As the number of jump functions increases so does processing time. The user may specify the number of jump functions. However, empirical testing with the datasets used in Im et al. (2007) suggested the following number of jump functions: • Ten thresholds (i.e., five for each direction) for a onedimensional search space • Five different domains (e.g., 3n × 5 threshold combinations to be tested: n is dimensionality) for a two- or higherdimensional search space. If no thresholds from the jump functions yield a higher accuracy than the current candidate threshold(s), the model

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stops searching. Once any candidate search point from the jump functions result in a higher accuracy than the last candidate search point, the search will recenter to the higher Kappa point and continue the converging process. The jump functions will be repeated. Fig. 5 summarizes the flow diagram of how the SS algorithm finds a global maxima. 3. Case study 3.1. Study area and data Two study sites were selected to demonstrate the performance of the systematic search technique for optimizing a binary change detection Kappa surface. Site A located in Las Vegas, NV, is a residential development which contains new roads. Site B is a commercial development which includes new parking lots located in Atlanta, GA. Two different high spatial resolution multispectral satellite image sources (i.e., QuickBird from DigitalGlobe, Inc and IKONOS from GeoEye, Inc) were used for the two study sites. Table 1 presents the characteristics of the satellite imagery. The bi-temporal panchromatic bands of each site are shown in Fig. 6. The bi-temporal image datasets for the two study sites were coregistered to a Universal Transverse Mercator (UTM) map projection in the WGS84 datum with a 0.5 pixel RMSE. Relative radiometric normalization was also performed based on regression analysis using four RCPs (radiometric control points) as pseudo invariant features for each study site. Four hundred random sample points were generated for each study site and used as reference data. Through the visual interpretation of the bi-temporal datasets, reference change information (i.e., change versus no change) was determined. Out of the 400 reference points 130 points and 123 points were identified as change areas for Site A and Site B, respectively. 3.2. Methods Four change-enhanced images were used to test the three optimization algorithms for each site: • neighborhood correlation (NC), slope (NS), and intercept (NI) images from neighborhood correlation image analysis, and • change vector magnitude (CVM) image from change vector analysis (CVA). Details about neighborhood correlation image analysis are found in Im & Jensen (2005) and Im et al. (2007, 2008). Neighborhood correlation images (i.e., correlation, slope, and intercept) between the bi-temporal images were created using a 3 × 3 neighborhood for each site. CVA can be used to determine the direction and magnitude of spectral change between bitemporal datasets (Chen et al., 2003; Malila, 1980). The resultant vector consists of two bands: 1) the total change magnitude and 2) the change direction for each pixel. The total change magnitude (CVM) is determined by calculating the Euclidean distance between end points through n-dimensional


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Fig. 5. Process flow diagram of the SS algorithm proposed in the study.

change space using the equation (Jensen, 2005; Michalek et al., 1993): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X CVM ¼ ð3Þ ðBVk2 BVk1 Þ2 k¼1

where BVk2 and BVk1 are the pixel values for each band k in dates 1 and 2. Pixels with the highest values in the change magnitude image are most likely to have experienced changes. The change direction for each pixel is specified as positive (≥ 0, including 0 to ensure that all pixels are assigned a direction) or negative (b 0) in each band, which can result in 2n possible types of changes (Michalek et al., 1993; Virag & Colwell, 1987). Johnson & Kasischke (1998) provide a detailed description of

change vector analysis. So that high values in the change magnitude images are associated with no change, the difference between a certain value (e.g., 3000) and the original change magnitude values were computed and stored as a CVM image for each site. This magnitude inversion was necessary since the model recognizes a linear typed input image where high values are associated with greater possibility of no change. Four individual variables (i.e., NC, NS, NI, and CVM) were used to compare performance from the exhaustive versus heuristic search techniques. The NC and CVM images are linear scale variables. The NS image is a ratio scale variable, and the NI image is a difference scale variable. A total of 60 calibrations including single and multiple variables were performed for each site. Fifteen calibrations used



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Table 1 QuickBird and IKONOS data characteristics Study site

Satellite data

Las Vegas, NV Site A

QuickBird Band 1 Blue 0.45–0.52 Band 2 Green 0.52–0.60 Band 3 Red 0.63–0.69 Band 4 NIR 0.76–0.89 Band 5 PAN 0.45–0.90 IKONOS Band 1 Blue 0.45–0.52 Band 2 Green 0.52–0.60 Band 3 Red 0.63–0.69 Band 4 NIR 0.76–0.90 Band 5 PAN 0.45–0.90

Atlanta, GA Site B

Spectral range (µm)

Spatial resolution (m)

Radiometric Acquisition date Acquisition time Sensor Sensor resolution (bits) (GMT) azimuth (°) elevation (°)

2.4 × 2.4 2.4 × 2.4 2.4 × 2.4 2.4 × 2.4 0.6 × 0.6 4×4 4×4 4×4 4×4 1×1

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each optimization algorithm (i.e., HC, SA, and SS) and the remaining fifteen used the ES algorithm. The ES algorithm utilized a search increment of 1/1000 of the image value domain for one- and two-dimensional search spaces. Calibrations with more than two images (i.e., three- and four-dimensional search spaces) using ES were performed using the stepwise method suggested in Im et al. (2007) in order to reduce enormous computational processing time (refer to Fig. 12). The first calibration was performed with a large increment (e.g., a tenth). Next, new narrow domains for each variable that may yield higher accuracy were selected. The second calibration was then performed with a smaller increment (e.g., 1/1000 of the image value domain). This stepwise process may be repeatedly performed if there were multiple narrowed domains for each variable. The case study focused on answering two research questions: 1) do the three optimizing algorithms result in accuracies at least as high as the ES algorithm, assuming that ES always find a global maxima in a search space, and 2) how

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May 10, 2002

18:26

113.8

76.1

May 18, 2003

18:19

164.2

75.2

May 18, 2000

16:08

23.1

86.0

June 12, 2002

16:37

351.7

73.3

much do the three optimizing algorithms improve temporal processing requirements, especially for problems with multiple variables. 3.3. Results and discussion The four change-enhanced image variables for Site A (i.e., NC, NS, NI, and CVM) are shown in Fig. 7. The original domain and search increment information for each image used in the calibration analysis are presented in Table 2. Optimum threshold(s) and Kappa accuracy using each search algorithm with single and multiple images for Site A (i.e., QuickBird data) are summarized in Table 3. The HC algorithm found global maxima (i.e., optimum thresholds yielding the highest Kappa accuracy) in only 20% of the calibrations, which resulted in an average 2.5% Kappa difference from the results of ES. SA produced better results than the HC algorithm. The SA algorithm found global maxima in 33% of the fifteen calibrations

Fig. 6. Bi-temporal panchromatic bands of the study sites: QuickBird (Site A) in a) and b) and IKONOS (Site B) in c) and d). Please cite this article as: Im, J., et al., Optimizing the binary discriminant function in change detection applications, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.01.007

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Fig. 7. Four change-enhanced images used to evaluate the optimizing algorithms (i.e., NC, NS, NI, and CVM) for Site A.

with an average 0.4% Kappa difference. The SS algorithm was well behaved in search spaces and produced the best results among the three optimizing algorithms. SS identified global maxima in the 80% of the calibrations with only 0.1% Kappa difference. The HC algorithm was not especially successful using multiple variables, where the search spaces have much more small local maxima than the single variable search space. The SA algorithm had also difficulty in finding global maxima in multiple variable domains. The SA algorithm identified maxima was close to global maxima, but was still trapped in numerous small local maxima near the global maxima. On the other hand, the SS algorithm worked well in the multiple domains. The SA and SS algorithms can be improved by adjusting several parameters such as the cooling rate in SA and the jump function in SS. However, these adjustments often result in a substantial increase in processing time. Table 2 Domain information of each variable and the 1/1000 increment for the ES algorithm Site

Variable

Site A

NC NS NI CVM NC NS NI CVM

Site B

Minimum − 0.2987 − 0.2605 0 459.6 − 0.8423 − 3.5915 0 21.7

Maximum

Domain

1/1000 increment

1 1 1554.9 2997.4 1 1 2379.6 1994.2

1.2987 1.2605 1554.9 2537.8 1.8423 4.5915 2379.6 1972.5

1.2987 × 10− 3 1.2605 × 10− 3 1.5549 2.5378 1.8423 × 10− 3 4.5915 × 10− 3 2.3796 1.9725

Fig. 8 illustrates the observed annealing schedule of one of the SA calibrations using two image inputs (i.e., NC and NS) for Site A. It showed rapid fluctuations in Kappa values at first and then converged to a high Kappa as temperature cooled. A graphical depiction of the calibration results using the ES and SS algorithms with the single image for Site A is depicted in Fig. 9. Variation in Kappa accuracy caused by jump functions in the systematic search technique can be easily found in the graphical results (e.g., see around 40–45 iterations in Fig. 9f). All calibrations with single variables using the SS algorithm required less than 50 iterations (b30 s), compared to 1000 iterations using the ES algorithm. Not surprisingly, the HC algorithm also required a small number of iterations using single variables (b 30). But the SA algorithm required a larger number of iterations (380) since a relatively slow cooling rate (i.e., 0.8) was used. The overall shape of the one-dimensional search spaces with thresholds are found in Fig. 9a, c, e, and g. Fig. 10 depicts the graphical calibration results using the SS algorithm with two images together for Site A. Calibration with multiple input images does not start until calibration with each single image is completed and identifies the optimum threshold for each single image (Fig. 10a). The optimum thresholds for single images are necessary since they are used in the jump functions in the calibration with the multiple input images. The SS algorithm was successful in identifying the global maxima for all image combinations except two cases: NC and CVM; NI and CVM. The differences in Kappa accuracy between the techniques for these two cases were approximately 1%. The performance of the jump functions in calibration with multiple



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Table 3 Optimum threshold(s) and Kappa accuracy using each search algorithm for Site A QuickBird data Type

Single variable

Two variables

Three variables

Four variables

Variable

NC NS NI CVM NC NS NC NI NC CVM NS NI NS CVM NI CVM NC NS NI NC NS CVM NC NI CVM NS NI CVM NC NS NI CVM

ES

HC

SA

SS

OT⁎

Kappa

OT

Kappa

OT

Kappa

OT

Kappa

0.858 0.732 161.2 2639.5 0.852 0.666 0.858 254.5 0.830 2510.9 0.732 254.5 0.732 2250 218.7 2561.0 0.852 0.666 261.6 0.852 0.666 2322.5 0.830 301.6 2510.2 0.731 254.8 2275.4 0.852 0.667 254.8 2277.5

0.8945 0.8970 0.7631 0.8786 0.9542

0.8681 0.7353 161.4 2615.6 0.1809 0.7200 0.8728 662.1 0.4524 2615.0 0.7322 412.8 0.2219 2614.3 215.1 2410.9 0.8311 0.5575 448.5 0.7997 0.5070 2326.6 0.4776 208.0 2278.1 0.6680 1002.3 998.6 0.2044 0.6647 991.7 1992.6

0.8888 0.8970⁎⁎ 0.7631 0.8782 0.8966

0.860 0.735 151.0 2640.4 0.831 0.609 0.872 758.0 0.845 2423.9 0.736 393.1 0.730 2202.9 212.6 2644.8 0.856 0.632 448.5 0.833 0.598 861.7 0.842 659.7 2572.8 0.732 1197.1 2089.1 0.852 − 0.014 399.9 2426.4

0.8945 0.8970 0.7555 0.8786 0.9537

0.858 0.736 161.1 2643.9 0.859 0.667 0.858 269.8 0.854 2538.8 0.733 306.8 0.733 2293.7 211.7 2574.4 0.852 0.667 282.3 0.852 0.667 2351.1 0.856 307.5 2524.6 0.733 307.5 2293.7 0.852 0.667 282.3 2351.1

0.8945 0.8970 0.7631 0.8786 0.9542

0.8945 0.9484 0.9029 0.9088 0.9077 0.9542

0.9542

0.9484

0.9088

0.9542

0.8893 0.8782 0.9029 0.8782 0.8889 0.9359

0.9359

0.8643

0.8945

0.9006

0.8893 0.9480 0.9029 0.9085 0.8920 0.9482

0.9537

0.9372

0.9088

0.9482

0.8945 0.9430 0.9029 0.9088 0.8966 0.9542

0.9542

0.9430

0.9088

0.9542

⁎OT: Optimum Threshold. ⁎⁎Kappa values in bold indicate that the heuristic search algorithms yielded as high accuracy as the exhaustive search algorithm.

input images is shown in Fig. 10a. The pattern of Kappas with increasing iterations is relatively different from the input images used. However, they showed a very similar pattern to identify a global maxima including the converging process and jump functions. The more input images used in the calibration, the greater the iterations, which increase processing time (Figs. 9

Fig. 8. Variations of temperature and a Kappa function using two variables: NC and NS for Site A.

and 10). However, note that the SS algorithm required only 1616 iterations to find a global maxima with four images during the calibration (1–2 min). The HC and SA algorithms also required only 146 (HC) and 1520 (SA) iterations with four images. Fig. 11 illustrates how a Kappa function varies as iterations increase with three optimizing algorithms using two images (NC and NS). The HC algorithm always required the smallest number of iterations (i.e., shortest processing time). When using less than four images, the SS algorithm required less processing time than the SA algorithm. However, as the number of the images increased, the SS algorithm took slightly more time than the SA algorithm because of the value of several parameters, such as a jump function and search direction. The optimum threshold(s) and Kappa accuracy for Site B is shown in Table 4. Similar to the calibration results for Site A, the HC algorithm was the quickest, but yielded the poorest Kappa accuracy among three optimization algorithms. The HC algorithm found global maxima in only 13% of the calibrations, which resulted in an average 2.5% Kappa difference from the results of ES (i.e., global maxima). The SA algorithm identified the global maxima in 27% of the fifteen calibrations with an average 0.7% Kappa difference. The SS algorithm


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Fig. 9. Graphical calibration results using the ES and SS algorithms for single variables for Site A. Note that the x-axis in a, c, e, and g represents thresholds while the x-axis in b, d, f, and h represents iterations.



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Fig. 10. Graphical calibration results using the SS algorithm with two variables for Site A.

Fig. 11. Variations of a Kappa function as iterations increase with each optimizing algorithm using NC and NS variables for Site A. Please cite this article as: Im, J., et al., Optimizing the binary discriminant function in change detection applications, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.01.007

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Table 4 Optimum threshold(s) and Kappa accuracy using each search algorithm for Site B IKONOS data Type

Single variable

Two variables

Three variables

Four variables

Variable

NC NS NI CVM NC NS NC NI NC CVM NS NI NS CVM NI CVM NC NS NI NC NS CVM NC NI CVM NS NI CVM NC NS NI CVM

ES

HC

SA

SS

OT⁎

Kappa

OT

Kappa

OT

Kappa

OT

Kappa

0.685 0.522 247.5 1666.7 0.678 0.521 0.685 400 0.678 1624 0.540 449 0.521 1659 338 1667.1 0.678 0.530 400 0.478 0.545 1550 0.480 395 1646 0.521 400 1670 0.478 0.521 395 1550

0.8215 0.8196 0.7588 0.8129 0.8770

0.7566 0.5297 240.2 1699.1 0.6894 − 0.3868 0.6878 430.1 0.6110 1583.6 0.5426 469.4 0.5224 1601.3 339.0 1574.5 0.6977 − 1.9624 804.5 0.4859 − 0.4568 1656.3 0.4924 744.4 1570.0 0.5347 880.6 821.5 0.4053 − 0.6658 471.0 1660.5

0.8022 0.8186 0.7482 0.8033 0.8215

0.688 0.523 260.3 1669.4 0.699 0.568 0.721 452.4 0.474 1664.2 0.530 460.4 0.498 1662.5 288.7 1669.5 0.715 0.579 1637.6 − 0.331 0.548 1591.6 0.401 404.1 1663.9 0.532 1486.7 1574.5 − 0.380 0.490 1559.0 1665.8

0.8215 0.8196 0.7555 0.8129 0.8604

0.689 0.527 247.8 1669.7 0.689 0.528 0.689 412.9 0.483 1667.5 0.525 402.3 0.537 1667.4 316.4 1665.4 0.678 0.530 405.1 0.493 0.528 1582.3 0.689 412.9 1548.8 0.537 419.5 1667.4 0.479 0.527 409.1 1548.8

0.8215 0.8196 0.7588 0.8129 0.8714

0.8529 0.8653 0.8328 0.8919 0.8615 0.8770

0.8924

0.8742

0.8919

0.8924

0.8529⁎⁎ 0.8382 0.8328 0.8851 0.8344 0.8223

0.8615

0.8474

0.8262

0.8603

0.8419 0.8552 0.8328 0.8851 0.8467 0.8604

0.8909

0.8667

0.8846

0.8851

0.8529 0.8615 0.8328 0.8919 0.8572 0.8770

0.8924

0.8714

0.8919

0.8924

⁎OT: Optimum Threshold. ⁎⁎Kappa values in bold indicate that the heuristic search algorithms yielded as high accuracy as the exhaustive search algorithm.

outperformed the two other optimizing algorithms with an 80% detection of global maxima and only a 0.1% Kappa difference. A combination of NC and NS images yielded the highest Kappa accuracy (0.954) for Site A. A combination of NC, NS, and CMV resulted in the highest Kappa accuracy (0.892) for Site B. Additional images (i.e., more than two images for Site A and more than three images for Site B) did not yield higher accuracies. There is no rule for the optimum number of images to use in creating a change map with the highest Kappa accuracy. However, it has been empirically shown that two or three images are generally sufficient for identifying a global maxima using the automated calibration model (Im et al., 2007). Processing time was also recorded for each calibration in order to evaluate the efficiency of the three optimization algorithms. The entire processing time was recorded for the calibrations using the ES algorithm with one or two images. However, the processing time was not recorded for the calibrations with three or four image combinations due to the enormous computational demands of ES (e.g., one billion iterations for the three image combinations). Instead, an average unit processing time (i.e., one search threshold evaluation) was measured and the expected search processing time for the complete search was

computed based on the total number of required iterations for the three and four image combinations. Actual calibrations using the ES algorithm with more than two images based on the stepwise method required between 48 and 72 h. A comparison of the efficiency between the four search algorithms is shown in Fig. 12. The processing time recorded for the ES calibration was the same for Sites A and B as it is only a function of the number of reference points and images. On the other hand, the calibrations using the SA or SS algorithm resulted in different processing times for each input sample. Thus, the averaged processing time was used for the plot. For comparison, the processing time when using the ES algorithm with a 1/100 search space increment was also plotted. A larger increment for the ES algorithm will dramatically reduce calibration time especially when multiple change images are used; however, the use of a larger search increment may result in the failure to identify a global maxima. All three optimization algorithms reduced calibration time considerably, compared to the ES algorithm (Fig. 12). The stepwise method with the ES algorithm could reduce the processing time but is somewhat labor intensive. Nevertheless, even the stepwise search method took much more time than the heuristic algorithms (e.g., greater than



15

maxima at the cost of processing time. The SS algorithm was well behaved in a Kappa search space producing the lowest false detection among the three optimizing algorithms. The superiority of the SS algorithm compared to the other two optimization algorithms is based on the ability of modeling a globally “hill-shaped” search surface and avoiding local minima using a jump function. Although the SA algorithm is also good at avoiding local minima, the SA algorithm uses a random search not considering the shape of a Kappa surface. The calibration model with the SS algorithm is distributed to the public without charge in two different versions. One is a standard version with the same parameter values used in the study. The other is an advanced version that allows users to select some of the parameters including the number of jump functions and the number of thresholds extracted per image. The models can be obtained for free upon request to the corresponding author. Fig. 12. Relationship between the number of input images and processing time with different search algorithms. Note that the y-axis represents the processing time in seconds in a logarithmic scale. Note that the processing times of the ES algorithm using three and four input images were approximated based on the unit processing time when using one and two input images.

100 times more for three images). The three heuristic algorithms converged on the optimum threshold combination in less than 100 seconds for any combination of images for both study sites. The ES algorithm with even a moderate search increment (i.e., 1/100) resulted in prohibitively long (i.e., a day or more) processing requirements when the number of images was greater than three. Not surprisingly, using smaller search increments (e.g., 1/1000) results in very long processing times when using more than one change image. As mentioned in the Section 2, many parameter values were fixed in the three heuristic search algorithms. Adjusting the parameters (e.g., slow cooling rate in SA, more jump functions in SS) may improve performance (i.e., finding global maxima) at the cost of calibration time. In the SS algorithm, two parameter values are critical in identifying a the global maxima: 1) the number of jump functions and 2) the number of thresholds extracted per change image in each round for multiple input images. Ten jump functions were used for single images and five different domain sizes were applied for multiple images in the test. Three thresholds were extracted per image in each search iteration when calibrating multiple images with the systematic search technique. More than one threshold per image can be used with the algorithm. In summary, the HC algorithm was the quickest among the three optimizing algorithms. However, it frequently failed to detect a global maxima in a search space because of many small local maxima in the search space even though it was globally “hill-shaped”. The SA algorithm showed good performance: although the SA algorithm did not always identify the global maxima, its solution was closer to an optimum solution based on the heuristic random search avoiding local maxima. A slower cooling rate increases the likelihood of identifying a global

4. Conclusion The performance of any heuristic searching algorithm is linked to the form of the search space. Specifically, if the search space is well behaved with one local maxima (i.e., the global maxima), then simple search algorithms (e.g., hill climbing) work well and are guaranteed to identify the global maxima. For search spaces with multiple local maxima, modifications to the simple heuristic methods are required, decreasing processing efficiency and optimal solutions. This study developed an efficient search algorithm (i.e., SS) suitable for calibrating the threshold(s) in a binary change study. The primary goal of the research was to develop and evaluate an algorithm to reduce the considerable processing time required for the calibration problem without decreasing change detection accuracy. The SS algorithm reduced calibration time considerably when compared to the ES algorithm, especially for multiple input change images. The SS algorithm also produced Kappa Coefficient of Agreement accuracies approximately the same as the ES algorithm. The other two optimization algorithms (i.e., HC and SA) were quite efficient in terms of calibrating time, but often failed to find the global maxima (i.e., optimum thresholds yielding the highest Kappa accuracy). Future research could examine the applicability of the SS algorithm to other types of change detection (e.g., vegetation species). Acknowledgment The authors are grateful to the Research and Development Imagery for Institutions (RADII) program by DigitalGlobe, Inc., for providing the QuickBird satellite remote sensing data. References Busetti, F. (2003). Simulated annealing overview. http://www.geocities.com/ francorbusetti/saweb.pdf, last accessed on October 12, 2007. Chen, J., Gong, P., He, C., Pu, R., & Shi, P. (2003). Land-use/land-cover change detection using improved change-vector analysis. Photogrammetric Engineering & Remote Sensing, 60(3), 287−298.


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Coppin, P. R., & Bauer, M. E. (1996). Digital change detection in forest ecosystems with remote sensing imagery. Remote Sensing Reviews, 13, 207−234. Goffe, W. L., Ferrier, G. D., & Rogers, J. (1994). Global optimisation of statistical functions with simulated annealing. Journal of Econometrics, 60(1/2), 65−100. Hardin, P. J., & Thomson, C. N. (1992). Fast nearest neighbor classification methods for multispectral imagery. The Professional Geographer, 44(2), 191−202. Hodgson, M. E. (1989). Rapid searching methods for grid interpolation. The Professional Geographer, 41(1), 51−61. Hodgson, M. E. (1992). Rapid k-neighbor identification algorithm for the sphere. Cartography and Geographic Information Systems, 19(3), 165−174. Huang, S. B., & Shibasaki, R. (1995, April). Development of genetic algorithm/ hill-climbing method for spatio-temporal interpolation. The 6th symposium on Functional Image Inf. System (pp. 81−86). lIS, Univ. of Tokyo. Im, J., & Jensen, J. R. (2005). A change detection model based on neighborhood correlation image analysis and decision tree classification. Remote Sensing of Environment, 99, 326−340. Im, J., Jensen, J. R., & Tullis, J. A. (2008). Object-based change detection using correlation image analysis and image segmentation techniques. International Journal of Remote Sensing, 29(2), 399−423. Im, J., Rhee, J., Jensen, J. R., & Hodgson, M. E. (2007). An automated binary change detection model using a calibration approach. Remote Sensing of Environment, 106, 89−105. Jacobson, S. H., McLay, L. A., Hall, S. N., Henderson, D., & Vaughan, D. E. (2006). Optimal search strategies using simultaneous generalized hill climbing algorithms. Mathematical and Computer Modeling, 43, 1061−1073. Jensen, J. R. (2005). Introductory digital image processing: A remote sensing perspective 3rd. Upper Saddle River, NY: Prentice Hall. Jensen, J. R., Rutchey, K., Koch, M., & Narumalani, S. (1995). Inland wetland change detection in the Everglades Water Conservation Area 2A using a time series of normalized remotely sensed data. Photogrammetric Engineering and Remote Sensing, 61(2), 199−209. Johnson, R. D., & Kasischke, E. S. (1998). Change vector analysis: A technique for the multispectral monitoring of land cover and condition. International Journal of Remote Sensing, 19(3), 411−426.

Kirkpatrick, S., Gelatt, C. D., Jr., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671−679. Kruger, J. (1993). Simulated annealing — A tool for data assimilation into an almost steady model state. Journal of Physical Oceanography, 23(4), 679−688. Li, X., Koike, T., & Pathmathevan, M. (2004). A very fast simulated reannealing (VFSA) approach for land data assimilation. Computers & Geosciences, 30, 239−248. Lu, D., Mausel, P., Brondizio, E., & Moran, E. (2004). Change detection techniques. International Journal of Remote Sensing, 25(12), 2365−2407. Lunetta, R. S., Ediriwickrema, J., Johnson, D. M., Lyon, J. G., & McKerrow, A. (2002). Impacts of vegetation dynamics on the identification of land-cover change in a biologically complex community in North Carolina, USA. Remote Sensing of Environment, 82, 258−270. Malila, W. A. (1980). Change vector analysis: An approach for detecting forest changes with Landsat. Proceedings, LARS Machine Processing of Remotely Sensed Data Symposium (pp. 326−336). W. Lafayette, IN: Laboratory for the Applications of Remote Sensing. Masters, T. (1993). Practical neural network recipes in C++. London, UK: Academic Press. Michalek, J. L., Wagner, T. W., Luczkovich, J. J., & Stoffle, R. W. (1993). Multispectral change vector analysis for monitoring coastal marine environments. Photogrammetric Engineering & Remote Sensing, 59(3), 381−384. Morisette, J. T., & Khorram, S. (2000). Accuracy assessment curves for satellitebased change detection. Photogrammetric Engineering and Remote Sensing, 66, 875−880. Penn, B. S. (2002). Using simulated annealing to obtain optimal linear endmember mixtures of hyperspectral data. Computers & Geosciences, 28, 809−817. Russell, S. J., & Norvig, P. (2003). Artificial intelligence: A modern approach. 2nd edition (pp. 111−114). Upper Saddle River, NJ: Prentice Hall. Virag, L. A., & Colwell, J. E. (1987). An improved procedure for analysis of change in Thematic Mapper image-pairs. Proceedings of the 21st International Symposium on Remote Sensing of Environment (pp. 1101−1110). Ann Arbor: Environmental Research Institute of Michigan.


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