discovering significant evolution patterns from satellite image time series

November 29, 2011 11:13 S0129065711003024

International Journal of Neural Systems, Vol. 21, No. 6 (2011) 475–489c© World Scientific Publishing Company

DOI: 10.1142/S0129065711003024

DISCOVERING SIGNIFICANT EVOLUTION PATTERNSFROM SATELLITE IMAGE TIME SERIES

FRANCOIS PETITJEAN∗,‡, FLORENT MASSEGLIA†,§,PIERRE GANCARSKI∗,¶ and GERMAIN FORESTIER∗,‖

∗LSIIT – UMR 7005, Pole APIBd Sebastien Brant – BP 1041367412 Illkirch Cedex – France

†INRIA Sophia Antipolis2004 Route des lucioles – BP 9306902 Sophia Antipolis – France

‡[email protected]§[email protected]

¶[email protected]‖[email protected]

Satellite Image Time Series (SITS) provide us with precious information on land cover evolution. By

studying these series of images we can both understand the changes of specific areas and discover global

phenomena that spread over larger areas. Changes that can occur throughout the sensing time can spread

over very long periods and may have different start time and end time depending on the location, which

complicates the mining and the analysis of series of images. This work focuses on frequent sequential

pattern mining (FSPM) methods, since this family of methods fits the above-mentioned issues. This

family of methods consists of finding the most frequent evolution behaviors, and is actually able to

extract long-term changes as well as short term ones, whenever the change may start and end. However,

applying FSPM methods to SITS implies confronting two main challenges, related to the characteristics

of SITS and the domain’s constraints. First, satellite images associate multiple measures with a single

pixel (the radiometric levels of different wavelengths corresponding to infra-red, red, etc.), which makes

the search space multi-dimensional and thus requires specific mining algorithms. Furthermore, the non

evolving regions, which are the vast majority and overwhelm the evolving ones, challenge the discovery

of these patterns.We propose a SITS mining framework that enables discovery of these patterns despite these constraints

and characteristics. Our proposal is inspired from FSPM and provides a relevant visualization principle.Experiments carried out on 35 images sensed over 20 years show the proposed approach makes it possibleto extract relevant evolution behaviors.

Keywords: Data mining; frequent sequential patterns; satellite image time series; land cover analysis.

1. Introduction

Change detection is an important field of remotesensing with various applications in land cover study.It has gained increasing attention due to recentimportant technological progress. The next gener-ation of satellites (e.g., V enµs, Sentinel-2 ) willactually be able to acquire image time series at

high temporal frequency. Satellite Image Time Series(SITS) are an important source of information forland cover analysis and change detection. Let usconsider, for instance, the scene illustrated in Fig. 1.Each image contains four main regions (R1 to R4)and the series show their evolution on four differentdates. In July 2007, regions R1 and R2 were mainly

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http://dx.doi.org/10.1142/S0129065711003024

November 29, 2011 11:13 S0129065711003024

476 F. Petitjean et al.

Fig. 1. Example of change in a scene involving trees and urban growth where a pattern of evolution is “trees → baresoil → urban”, verified by 50 % of regions (i.e. R1 and R2).

covered with trees, region R3 with a river and regionR4 with roofs. A global evolution to this exampleis the urbanization process, where the trees disap-pear, replaced by bare soil and finally by urban items(roads and roofs).

Extracting knowledge from time series can becrucial in domains such as prediction1 and cluster-ing.2,3 In the case of SITS, the discovery of evolutionshas important applications4–6 but requires to con-sider two important challenges related to the domainconstraints and with SITS characteristics.

First, changes in a scene might spread overa long time period (urbanization, for instance,lasts for several years and building sites do not havethe same start time and end time) or they mightcycle (such as crop rotation). Let us consider theSITS illustrated in Fig. 1. A possible region evo-lution from this series of images is the following:“trees→ bare soil→ urban”, which can be observedfor 50 % of pixels. Obviously, detecting such evo-lutions, having more than two steps, cannot relyon a naive comparison of two consecutive images.While the number of possible combinations over suchlong periods is very large, solutions to this kind ofproblem exist in knowledge discovery in the field of“frequent sequential pattern mining” (FSPM).7–9 Inthis paper, we will explore such solutions for changedetection in SITS.

Second, satellites don’t automatically givelabels (such as “tree” or “roof”) to (regions of)

pixels. Actually, their sensors measure the lightreflected by geographic areas for different wave-lengths allowing satellites to acquire images on multi-ple bands, such as “Infra-red”, “Red”, “Green”, etc.Let us now consider the images illustrated in Fig. 1,from the satellite’s point of view. Figure 2 shows thesame series of images on the red band. In the imagefrom July 2007, the red level of regions R1 and R2could be 120 (which is the response on this band ofsome trees from broad-leaved or coniferous forests).The red level of region R3 is 30 (response of theriver) and the red level of region R4 is 200 (responseof red tile roofs in the red band). Meanwhile, broad-leaved trees and conifers have different responses onthe Infra-Red band. Consequently, working with asingle band is not sufficient since the number of possi-ble values is lower than the number of possible kindsof surfaces.

This need for multiple bands is well known in geo-graphic studies. Actually, a measure on a band gen-erally takes a value in a range of 255 levels, whereasthe number of possible measures for a pixel with n

bands is 255n, allowing much more possible combina-tions. Figure 3 shows the series illustrated in Fig. 1,on the Infra-Red band. We can see, for instance, thatthe Infra-Red level is the same for red tile roofs andconcrete roads.

Understanding from SITS the evolutions thatoccurred on a scene (i.e., geographic area) is a com-plex task. Data mining methods are usually used

Fig. 2. The scene of Figure 1 from the “Infra-Red band” point of view.

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Discovering Significant Evolution Patterns from SITS 477

Fig. 3. The scene of Figure 1 from the “Infra-Red band” point of view.

rather than image/video processing methods, sinceSITS are very different from videos: (1) the imagesare not regularly sampled, (2) sensed values are abso-lute physical states of the land cover and (3) theseries are much shorter. Many differences are thenfollowing in the conception of dedicated methods.The irregular sampling prevents making any assump-tions on the content of previous/next images (framesin video). Moreover, satellite image time series aregiving a physical state of the land cover; the methodhas thus to map the changes (as well as in videos),but also to characterize it in order to provide asemantic meaning to the thematic expert. Further-more, video-dedicated methods have to be extremelyfast in order to be able to process the data flow. Wetake here the opposite direction: the aim is to mapand characterize the change with as few assumptionsas possible.

Our goal is to extract patterns of evolutions fromthese images and, to that end, we need to take intoaccount the whole sets of dates and bands. To date,the problem of mining frequent sequences from SITShas been investigated for a single band.10,11 Unfor-tunately, as explained above, a single band doesnot make it possible to distinguish between differ-ent kinds of surfaces. Let us consider a pattern dis-covery process on the red band, for instance. In thiscase we will extract region evolutions where broad-leaved trees and conifers have the same value butred tile roofs and concrete roads don’t. Therefore, apossible frequent pattern would be “conifers/broad-leaved trees → bare soil”. We can observe that“urban” does not appear since its surfaces take val-ues in very different ranges on the red band. Actu-ally, in the red band, the values of urban surfaceare divided into “roofs” (R1) and “roads” (R2).The expert would then interpret this pattern anddecide that it can be reformulated as “trees → bare

soil”. Unfortunately, this pattern corresponds to thebeginning of the urbanization process and we couldnot extract any information about “urban” surfacesin this analysis performed on the red band. Theproblem remains with other bands such as Infra-Red, for instance, where a possible pattern wouldbe “bare soil/conifer→ roof/road”, reformulated bythe expert as “bare soil → urban”. In this case, wefind the urban part of the pattern but it does notcontain enough information about the trees (becausethey have different values on the Infra-Red band).Furthermore, the expert needs time for this inter-pretation (since the information on each step of thepattern is not sufficient) and the reformulation isnot always reliable (because each value may cor-respond to multiple possible surfaces, making theresult ambiguous).

Therefore, in order to find complete, significantand relevant patterns, we need to consider the wholeset of bands. In our example, taking multiple bandsinto account would allow us to extract the patternillustrated in Eq. (1).

(conifers/hardwoods︸︷︷︸Red

)→ ( soil︸︷︷︸Red

, soil/conifers︸︷︷︸Infra-Red

)

→ (roof/road︸︷︷︸Infra-Red

) (1)

This pattern can easily be interpreted as“trees → soil → urban”, which corresponds to ourgoal. It has two important advantages (1) it is moreinformative than the patterns extracted on a singleband and (2) it is not ambiguous. As we will describelater, there is a strong correspondence between thisproblem and the problem of frequent sequential pat-tern mining (FSPM).7–9,12 However, the use of fre-quent pattern mining methods for SITS analysis is

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not straightforward, since the non evolving regions,which are the vast majority, are overwhelming theevolving ones.

This article introduces a new strategy for theextraction of evolution patterns from SITS. Theproposed method addresses the two problems ofstudying multi-spectral series and of extracting themain evolution behaviors from predominantly staticbehaviors. In this paper, we propose to analyze ascene with satellite images on important time peri-ods. Our approach will be tested over 35 images and aperiod of 20 years, which represents 28 million sensedvalues.

This paper is organized as follows. In Sec. 2 wegive an overview of existing works in SITS analy-sis. Section 3 gives the main definitions of frequentsequential patterns and Sec. 4 describes the prepro-cessing of SITS for the discovery of such patterns.In Sec. 5 we propose a FSPM method devoted toSITS, along with a visualization principle, and ourresults are described in Sec. 6. Finally, we concludethis paper in Sec. 7.

2. Related Works: SITS Analysis

Change detection in a scene allows the analysis,through observations, of land phenomenon with abroad range of applications such as the study of land-cover or even the mapping of damages following anatural disaster. These changes may be of differenttypes, origins and durations.

In the literature, we find three main fami-lies of change detection methods. Bi-temporalanalysis, i.e., the study of transitions, can locateand study abrupt changes occurring between twoobservations. Bi-temporal methods include imagedifferencing,13 image ratioing14 and change vectoranalysis (CVA).15 A second family of mixed meth-ods, mainly statistical methods, applies to two ormore images. They include methods such as post-classification comparison,16 linear data transforma-tion (Principal Component Analysis and MaximumAutocorrelation Factor),17 image regression or inter-polation18 and frequency analysis (e.g., Fourier,wavelet).19 Finally, we find methods designedtowards image time series and based on radiomet-ric trajectory analysis.20 Whatever the type ofmethods used in order to analyze satellite image timeseries, there is a gap between the amount of data

representing these time series, and the ability of algo-rithms to analyze them. First, these algorithms areoften dedicated to the study of a change in a scenefrom bi-temporal representation. Second, even if theycan map change areas they are not able to charac-terize them. As for multi-date methods, their resultsare usually hard to interpret and do not characterizethe change.

FSPM7,8 is for its part intending to extract pat-terns of evolution in a series of symbols. These meth-ods enable to identify sets of sequences that hadthe same underlying evolution. Furthermore, theyare able to characterize this evolution, by extractingthe pattern shared by this set of sequences. Extract-ing frequent sequences from SITS was introducedin Ref. 10, 11. The authors study the advantagesof such sequences in two applications: weather andagronomics. However, their proposal is restricted tothe mining of series of images where each pixel takea value on a single band only. Our proposal, asexplained in Sec. 4, applies to images where the pix-els take values on tuples, each value correspondingto a separate band. These characteristics, along withthe large number of images, will have important con-sequences on the patterns, on their relevance and onthe complexity of their discovery.

3. Mining Frequent SequentialPatterns

Sequential patterns are extracted from large sets ofrecords. These records contain sequences of valuesthat belong to a specific set of symbols, as stated byDefinition 1 (inspired by the definitions of Ref. 7).

Definition 1. Let I = {i1, i2, . . . , im}, be a set ofm values (or items). I is a vocabulary of all possi-ble values for an item. Let I = {t1, t2, . . . , tn}, bea subset of I. I is called an itemset and is noted(t1; t2; . . . ; tn). A sequence s is a non-empty list ofitemsets noted 〈s1, s2, . . . , sn〉 where sj is an itemset.A data sequence is a sequence in the dataset beinganalyzed.

Definition 2 shows the conditions for the inclu-sion of two sequences. In other words, s1 is includedin s2 if each itemset of s1 is included in an itemset ofs2 with the same order. This definition is illustratedby examples 1 and 2.

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Definition 2. Let s1 = 〈a1, a2, . . . , an〉 ands2 = 〈b1, b2, . . . , bm〉 be two sequences. s1 is includedin s2 (s1 ≺ s2) if and only if ∃i1 < i2 < · · · < in inte-gers, such that a1 ⊆ bi1 , a2 ⊆ bi2 , . . . , an ⊆ bin .

Example 1. Let us consider the sequence s1 =〈(3)(4; 5)(8)〉. s1 is made of 3 ordered itemsets.The first itemset of s1 contains only one item (i.e.(3)). The second itemset of s1 contains two items(i.e. (4; 5) an itemset of size two that containsthe items 4 and 5). The third itemset of s1 con-tains only one item (i.e. (8)). Let us now con-sidered the sequence s2 = 〈(7)(3; 8)(9)(4; 5; 6)(8)〉.Then, s1 is included in s2 (i.e s1 ≺ s2) since (3) ⊆(3; 8), (4; 5) ⊆ (4; 5; 6) and (8) ⊆ (8). Meanwhile, thesequence s3 = 〈(3; 8; 9)(4; 5)〉 is not included in s2

since (3; 8; 9) is not included in an itemset of s2.

Example 2 gives an illustration of how sequentialpattern mining may be applied to SITS as a modelof their content and meaning.

Example 2. Let us consider s1 from Example 1.Say that 3 stands for “low IR”, 4 corresponds to “lowR”, 5 to “Average NDVI” and 8 to “high NDVI”.Then, if a pixel has a series of values that correspondsto s1, it should be regarded has a pixel having a lowvalue of infra-red in an image, followed by a low valueof red and an average value of NDVI in a next imageand lastly by a high value of NDVI in a next image.

In this paper, the main characteristic for sequen-tial pattern extraction will be the frequency of thepatterns. This notion is based on the number ofoccurrences of a pattern, compared to the total num-ber of sequences, as stated by Definition 3. Finally,for simplicity in the results, only the longest patternsare kept (C.f. Definition 4).

Definition 3. A data sequence sd supports asequence s (or participates in the support of s) ifs ≺ sd. Let D be a set of data sequences. The sup-port of s in D is the fraction of data sequences in D

that support s:

support(s) =|{sd ∈ D | s ≺ sd}|

|D| (2)

Let minSupp be the minimum support value,given by the end-user. A sequence having supporthigher than minSupp is frequent.

Table 1. Sample database.

s1 〈(a)(b;d)(c; e; x)〉s2 〈(a; b)(b)(c; f ; x)〉s3 〈(n; o)(p)(q; s)〉s4 〈(a; g)(b;m)(c; g;x)〉

Example 3. Let us consider the toy database givenin Table 1 that contain four sequences. If the end-usergives a minimum support of 50%, then a sequencewill be frequent if it is supported by (i.e. includedin) at least two sequences of this database. This isthe case of 〈(a)(b)(c; x)〉, which is included in threesequences among three, thus having a support of 75%(above the minimum support given by the user).

Definition 4. Let FD be the set of frequentsequential patterns in D. In a set of sequences,a sequence s is maximal if s is not contained inany other sequence. Let LD be the set of maximalsequences of FD. LD is the set of maximal frequentsequential patterns in D.

4. Data Preparation (SITS)

We want to analyze images from SITS databases,such as Kalideos.a In our experiments, for instance,we have extracted a series of 35 images Spot-1,Spot-2 and Spot-4 (the scenes are located in thesouth-west of France) as illustrated by Fig. 4. Theseimages are described by three attributes, correspond-ing to three spectral bands: Near Infra-Red (NIR),Red (R) and Green (G). To these bands, we add afourth one, corresponding to the Normalized Differ-ence Vegetation Index (NDVI). It is the most usedindex in remote sensing since it makes it possible todistinguish between several spectral signatures. TheNDVI is calculated as follows for a pixel p:

NDVI(p) =NIR(p)− R(p)NIR(p) + R(p)

.

Since the images from these databases areacquired by different sensors, the comparison ofradiometric levels of a pixel (x, y) from one imageto another calls for corrections. The value of eachpixel has to be adjusted. First, we need to make surethat pixel (x, y) in a series covers the very same geo-graphic area in every image. Then, some corrections

ahttp://kalideos.cnes.fr

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Fig. 4. Extract of the Satellite Image Time Series from the Kalideos database. c© CNES 2010 – Distribution Spot Image.

are performed in order to reduce the impact of thedifference of the solar lighting and of the differenceof composition of the atmosphere. These correctionsmake it possible to guarantee that a geographic area(x, y) can be studied throughout the image serieswith comparable values.

This section formalizes the construction of thesequences from the image series. The idea is to studythe evolution of atomic geographic areas identified bytheir (x, y) coordinates. In this way, the image seriesprovide, for each sensed area (x, y), a 4-dimensionalseries of values (Definitions 5 and 6. Then, the valueshave to be quantified in order to be studied by thefrequent pattern mining method (Definition 7).

Notations. Let us first introduce severalnotations:

• [[X, Y ]] = [X, Y ] ⊂ N: the set of integer values fromX to Y ;

• {{a, a, c}} refers to the multi-set theory, i.e., setswith possibly multiple occurrences of elements;

• refers to the union operator for multi-sets, e.g.,{{a}} {{a}} = {{a, a}};

• Inb (x, y) denotes the reflectance level (sensed

value) of the nth image in the bth band.

Definition 5. Let Simage = 〈I1, . . . , IN 〉 be aseries of N images of width W and eight H. Let Bbe the number of bands in the images. Each multi-valued (with multiple bands) image In (n ∈ [1,N ])can be seen as a function:

In : [[1,W ]]× [[1,H]]→ ZB

(x, y) → Ik(x, y) =B∏

b=1

Inb (x, y)

(3)

with∏

be the Cartesian product.

Definition 6. Let Sv be the dataset built from thevalues of the image series. Sv is the set of sequencesdefined as:

Sv = {{〈I1(x, y), . . . , IN (x, y)〉|x ∈ [[1,W ]],

y ∈ [[1,H]]}} (4)

Finally, a discretization step is necessary on thevalues of bands for a sequential pattern extraction.Actually, this step will lower the total number ofitems during the mining step. To that end, we firstcreate B datasets Db (1�b�B): one dataset per bandfrom all images. Then, in each such dataset, wegather together all the values of the correspond-ing attribute. It makes it possible to have a com-mon discretization of the values between all images.In this way, the first slice of NIR is for exam-ple the same whatever the considered quantifiedimage. Formally, B datasets of non-temporal data arecreated as:

Db = ∀(x, y) ∈ [[1,W ]]× [[1,H]] :N⊎

n=1

Inb (x, y) (5)

Then, to each dataset Di, we apply the K-means

algorithm21 in order to obtain K groups of values.Although all discretization techniques can be usedhere, the K-means algorithm was chosen instead ofa usual histogram equalization, since the K-means

algorithm is more adaptive to the distribution ofthe data. Therefore, every attribute (NIR, R, G orNDVI) is divided into K groups. For readability, wename these groups ordered by the value of the cen-troid (i.e., the mean value of the group). For instance,the cluster NIR1 corresponds to the first slice of NIR,i.e., the cluster of NIR with lowest values, NIRK

corresponds to the last slice of NIR, i.e., the clus-ter of NIR with highest values. Definition 7 details

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the creation of S: the dataset ready for sequentialpattern extraction, from the values of Sv.

Definition 7. Let Clus be the function associ-ating the B values of a pixel (i.e., a quadruplet(nir, r, g, ndvi)) with their B slices computed by theK-means algorithmb. The dataset S from the dis-cretized values of the image series is then defined as:

S = {{〈Clus(I1(x, y)), . . . , Clus(IN (x, y))〉|x ∈ [[1,W ]], y ∈ [[1,H]]}} (6)

We are thus provided, for each pixel, witha sequence of discrete values as, for exam-ple: (NIR1; R6; G3;NDVI 16) → · · · → (NIR12;R3; G14;NDVI 19) where (NIR1; R6; G3;NDVI 16)means that the value of that pixel in the first imageis in the first slice of near infra-red, in the sixth sliceof red, in the third slice of green and in the 16th sliceof NDVI .

5. Extracting and Visualizing Patternsfrom SITS

The preprocessing steps described in Sec. 4 pro-vide us with a series of images where each pixel isdescribed on a tuple of values. Let us consider theseries of three images simply reduced to 4 pixels (p1to p4) illustrated by Fig. 5. We want to extract fre-quent evolutions from these images. In other wordsif there exists a large enough set of pixels with thesame “behavior” (i.e. these pixels have the same evo-lution), then this behavior must be discovered. Letus mention that the pixels’ position is not a criterionhere (our goal is not to extract pixels because of ashape). Our goal is to extract significant schemes inthe evolution of a set of pixels. Each pixel in this fig-ure is described on 3 values (corresponding to bandsB1 to B3). With a minimum support of 100 %, thereis no frequent pattern in these images (no “behav-ior” corresponding to the whole set of pixels). Witha minimum support of 50 %, however, we find twofrequent behaviors:

(1) 〈(B1, white; B2, white) (B1, gray; B2, red)〉.This behavior matches the sequences of valuesof pixels p2 on images 1 and 2 (or 3) and p3 onimages 1 (or 2) and 3.

Fig. 5. A series of three images, with four pixelsdescribed on three bands.

(2) 〈(B1, white; B2, white) (B1, white; B2, white)〉.This behavior matches the sequences of valuesof pixels p1 (let p1 be white on all images) B2on on images 1 and 2 and p3 on images 1 and 2.

Let us note that, in the illustration above, patternsmay be frequent even despite a lag in the images thatsupport them (according to Definition 3).

5.1. Mining sequential patterns fromimage time series

As described above, our goal is to extract sequen-tial patterns from these image time series. How-ever, the pixels that support the evolution patternsembedded in SITS are not the vast majority. In thiscontext, the sequential patterns having the highestsupport will usually correspond to “non-evolutionpatterns.” In other words, the sequences in these veryfrequent patterns contain the same frequent item,repeated several times. An example of such a pat-tern being 〈(trees)(trees)(trees)〉 meaning that, inthis series, most pixels have a value correspondingto trees on images i, j and k (with i < j < k).This would correspond to 〈(B1, white; B2, white)(B1, white; B2, white)〉 in our previous illustration,which means that the white value was found in thetwo attributes B1 and B2 at the same time, and thatanother same state was found afterwards.

However, such patterns are not really informa-tive. In SITS, patterns with the highest supportalways correspond to geographic areas that did notchange and these areas are the vast majority. How-ever, as illustrated in the introduction, mining fre-quent sequential patterns from satellite images canreveal pixels having the same evolution (since thesepixels will be characterized by the same pattern).Therefore, the challenge is to extract patterns having

bFor instance, a pixel taking values (12; 35; 200; 100) in an image could have Clus(12; 35; 200; 500) = (NIR1; R2;

G16;NDVI 11).

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a minimum threshold, while avoiding the enumera-tion of non-evolving patterns. There are several naiveapproaches for solving this problem, but they all haveimportant drawbacks.

A first idea would be to remove the pixels thatnever show any change. This is not enough, sincethe non-evolution part of the remaining pixels mightstill cause the same problems. Actually, a pixel mightnot show any change for years, and some importantchanges at the end of the series.

Another approach would consist in lowering theminimum support until we obtain patterns that cor-respond to evolutions (since the areas of changesare minority). This would lead to extracting pat-terns having lower support (say, above 2 %). Unfor-tunately, in this case, the whole (and large) setof frequent values will flood the process with anintractable number of candidate and frequent pat-terns with two important consequences. First, theresults are difficult, or even impossible, to obtain(because of the combinatorial complexity associ-ated with frequent pattern extraction). Second, evenif the results could be obtained, they would bedifficult to read and very few would be relevantbecause they would contain a lot of non-evolutionpatterns.

A third idea would consist of a candidate gen-eration process where two successive occurrences ofthe same value is forbidden. Indeed, such a solutionwould be motivated by the need to get rid of non-evolution patterns (such as 〈(trees)(trees)(trees)〉).Unfortunately, this solution is not sufficient becausenon-evolution patterns can take values on the wholeset of available bands. Let us consider, for instance,the pattern 〈(tree, b1)(tree, b2)(tree, b3)〉. This pat-tern does not contain any repetition of the samevalue since (tree, b1) corresponds to the response oftrees on band 1, (tree, b2) on band 2 etc. Therefore,since (tree, b1) �= (tree, b2) �= (tree, b3), this patternwould be considered.

Finally, a naive and extreme solution wouldconsist in the total removal of highly frequentitems from the candidate generation process. Inthis case, we could ignore important patterns thatshould be considered. Let us consider the casewhere (tree, b1) and (tree, b2) are very frequent,whereas the item (urban, b3) is barely frequent.

Then, the pattern 〈(tree, b1)(tree, b2)〉 would beignored (which is correct) but also the pattern〈(tree, b1; tree, b2)(urban, b3)〉 (which must not bethe case).

Considering the number of naive (and unrealistic)solutions, we propose a frequent pattern extractionalgorithm based on PSP8c with a targeted adjust-ment: the candidate generation step discards pat-terns having two contiguous highly frequent items.To that end, we introduce a maximum threshold(described in Definition 8) in addition to the min-imum support of Definition 3. Besides the ability todiscard non-evolution patterns, this principle signif-icantly reduces the combinatorial complexity asso-ciated with FSPM, as we explain in the followinganalysis.

Definition 8. Let maxSupp be a maximumfrequency threshold and H be the set ofitems having support larger than maxSupp(H = {i ∈ I | support(i) > maxSupp}). Let minSuppbe a minimum frequency threshold and FI be the setof items having support larger that minSupp. Thereduction factor h that applies to FI according tomaxSupp is given by h = |FI |/|FI \H |.

It is straightforward to show that a lower num-ber of combinations leads to a lower number ofpossible frequent sequential patterns. However, withTheorem 1 we provide a formal comparative anal-ysis of the upper bounds on the number of possi-ble frequent sequences when successive occurrencesof highly frequent items are considered and whenthey are discarded. For simplicity (and to makethe upper bounds easier to read while remainingreliable) we consider that the empty itemset is notdiscarded from the set of possible k-itemsets, ∀k.Therefore, the sequences containing one or moreoccurrences of the empty itemset are not discardedfrom the following reasoning. For instance, sequenceslike 〈(1)()(4)()(2)〉, which contains two occurrencesof the empty itemset, are not considered is this rea-soning. Actually, the impact of this itemset (andits repetitions) on the possible number of sequencesis negligible compared to the exponential numberof potential combinations involved by sequentialpatterns. Furthermore, our reasoning is limited to

cHowever, this principle may apply to any sequential pattern mining algorithm.

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sequences of pair length (though it can be generalizedto any length of sequences).

Theorem 1. Let n be the number of frequent itemsand h be the reduction factor introduced in Defini-tion 8. Let St be the set of possible sequences oflength t and P[0..t] =

⋃tj=0 St be the set of possible

sequences of length 0 to t. Let S′t be the set of possible

sequences of length t with no contiguous successiveoccurrences of frequent items and P ′

[0..t] =⋃t

j=0 S′t

be the set of possible such sequences of length 0 tot. Then |P[0..t]| =

∑tj=0 2nt and |P ′

[0..t]| =∑t

j=02nt

ht2

(and |P ′[0..t]| < |P[0..t]|).

Proof. First, let us recall some properties of sequen-tial patterns. |PI|, the number of possible itemsetswith n frequent items is given by the number ofitemsets having length 0 plus the number of item-sets having length 1, etc. The number of itemsetshaving length k is given by Ck

n (i.e. the number ofchoices of k items among n possible items in I).Therefore, |PI| =

∑nk=0 Ck

n = 2n (or (2n − 1) ifwe discard the empty itemset). |St|, the number ofpossible sequences of length t is given by |St| = 2nt.Actually, the number of possible combinations for St

is given by:

2n × 2n × · · · × 2n

︸︷︷︸t times

(7)

And |P[0..t]|, the number of possible sequential pat-terns of length 0 to t is given by:

|P[0..t]| =∣∣∣∣∣∣

t⋃j=0

St

∣∣∣∣∣∣=

t∑0

2nt (8)

Second, let us give an upper bound on the number ofsequential patterns when two items of H cannot becombined. |PI\H |, the number of possible itemsetsi such that minSupp < support(i) < maxSupp isgiven by |PI\H | = |PI|

h = 2n

h . The number of pos-sible combinations for St, the sequences of length t

with no contiguous successive occurrences of itemsfrom H is given by:

2n × 2n

h× 2n × 2n

h× · · · × 2n × 2n

h︸︷︷︸t times

(9)

Therefore, |S′t| = (2n × 2n

h )t2 = 22n t

2

ht2

= 2nt

ht2

and thenumber of possible such sequential patterns of length

0 to t is given by:

|P ′[0..t]| =

∣∣∣∣∣∣

t⋃j=0

S′t

∣∣∣∣∣∣=

t∑0

2nt

ht2

(10)

The importance of h is significant regarding thenumber of possible itemsets. However, when it comesto sequential patterns, it becomes crucial. Figure 6gives the difference between the upper bounds onitemsets and sequential patterns with 8 to 10 fre-quent items, a reduction factor (h) between 1.5and 5, and a sequential pattern length (t). Withh = 1.5, the maximum number of itemsets (left dia-gram in Fig. 6) drops from approximately 1000 to600, which is expected to have a limited influenceon the frequent itemset discovery process. Mean-while, the maximum number of sequential patterns ofsize t (right diagram) shows a very large theoretical

0

100

200

300

400

500

600

700

800

900

1000

1100

8 9 10

Num

ber

of p

ossi

ble

item

sets

Number of frequent items

Originalh=1.5

h=2h=3h=5

0

2e+14

4e+14

6e+14

8e+14

1e+15

1.2e+15

8 9 10

Num

ber

of p

ossi

ble

sequ

ence

s of

leng

th t=

5

Number of frequent items

Originalh=1.5

h=2h=3h=5

Fig. 6. The theoretical influence of h is much more sig-nificant on the number of possible sequential patternsthan it is on the number of possible itemsets.

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difference, growing up to an order of magnitude forh = 1.5 (and even more for higher values of h).

5.2. Towards a generic learningapproach for SITS mining

The new principle of FSPM presented in the lastsubsection is inspired from the constraints of SITS.The resulting algorithm PSPSITS is detailed inAlgorithm 1. It is based on PSP, while providinga new threshold value standing for the maximumfrequency.

The main modification of PSP that should benoticed in PSPSITS is the usage of a maximum sup-port in the learning step, where the candidates ofeach level are generated. We do not give the details ofthis candidate generation algorithm because, abovethis particular algorithm, an important characteris-tic of our work is that it can apply to any FSPMmethod. Our theoretical background shows thatthe number of generated candidates, during the

Algorithm 1 PSPSITS

Input: D //a database of sequencesInput: I //the set of items in D

Input: minSupp //the minimum supportInput: maxSupp //the maximum supportOutput: k //the maximum length of a frequentOutput: L0,k //the set of frequent sequences of

length 0 to k

k← 1//Extract the frequent items of D

Lk ← {i | i ∈ I, support(i) > minSupp}//Index the highly frequent items of D

H ← {i | i ∈ I, support(i) > maxSupp}k← k + 1Ck ← Learning(Lk−1, H)repeat

//Check the support of each candidate D

CountSupport(Ck, D)//Extract the frequent onesLk = FreqSeq(Ck)k← 1//Generate the next step candidatesif Lk �= ∅ then

Ck ← Learning(Lk−1, H)end if

until Lk = ∅

learning phase, gives a very low number of can-didate patterns that have to be checked over thedata (compared to traditional learning principles).This is due to the fact that candidates contain-ing two consecutive items having support greaterthan maxSupp (see algorithm PSPSITS in Algo-rithm 1) will be rejected. This principle is veryeasy to apply to any FSPM algorithm by modify-ing the learning step. For instance, in GSP,7 thelearning step is based on a pattern matching prin-ciple that checks the subsequences of each couple offrequent sequences. When the subsequence obtainedby removing the first item of the first sequencematches the subsequence obtained by removing thelast item of the second sequence, then a candidate isgenerated. For instance, let s1 = 〈(a)(bc)(d)〉 ands2 = 〈(bc)(d)(e)〉 be two frequent sequences hav-ing length 4, then we can generate the candidatesequence c = 〈(a)(bc)(d)(e)〉 (because the subse-quences of s1 and s2 do match). If we want to applyour filter on the consecutive highly frequent items,all we have to do is to carefully generate the candi-dates of size 2 and 3. Afterwards, the property of nothaving two consecutive highly frequent itemsets in asequence will be automatically propagated withoutany additional effort.

5.3. Visualizing Sequential PatternsExtracted from SITS

Our goal, in this section, is to propose a visualizationapproach designed towards the synthetic representa-tion of sequential patterns in SITS.

The (usually very) large number of patternsextracted by means of data mining algorithms is animportant issue, with numerous and relevant contri-butions in the literature.22–24 In the case of patternsextracted from SITS, an important advantage is thatour individuals are pixels. The main idea is thatevery sequence (as introduced in Definition 7) builtfrom the SITS corresponds to a geographic locationidentified by its coordinates in aH×W space (withHand W be the height and width of the images in theseries). Therefore, we can build aH×W image, wherethe data sequences that participate to the supportof one or more frequent sequential patterns will behighlighted (since each data sequence corresponds toa pixel in that space). Moreover, by assigning grad-ual levels to the pixels, we can see the percentageof frequent sequences they support. To the best of

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our knowledge, this is the first visualization princi-ple intending to give a synthetic representation of theset of frequent sequential patterns extracted from aseries of images. In Definition 9, we define the “con-tribution frequency” of a data sequence in a set ofsequential patterns. The larger the number of pat-terns supported by a data sequence, the higher itscontribution frequency.

Definition 9. Let FS be the set of frequentsequential patterns in S. The contribution fre-quency of a sequence sx,y = 〈Clus(I1(x, y)), . . . ,Clus(In(x, y))〉 in FS is defined as:

C(sx,y) =|{sF ≺ sx,y | sF ∈ FS}|

|FS | (11)

Definition 10. Let C(sx,y) be the contribution fre-quency of a sequence sx,y. The contribution fre-quency image is the image gathering the contributionfrequency of all sequences identified by their coordi-nates (x, y). IC is defined as:

IC : [[1,W ]]× [[1,H]]→ [0, 1] ⊂ R

(x, y) → IC(x, y) = C(sx,y)(12)

Given Definition 9, C(sx,y) can be computed foreach sequence sx,y (i.e., for each pixel (x, y)). Def-inition 10 gives the characteristic of the image rep-resenting the contribution of each pixel to a set ofsequential patterns. Once this synthetic representa-tion is obtained, the expert is provided with a valu-able guide that allows him to identify in one viewingthe kind of surfaces involved in the set of extractedpatterns. For instance, if most of the highlighted pix-els correspond to areas covered with fields, then theexpert may browse the set of extracted patterns withphenomena such as “crop rotation” in mind.

6. Experiments

Our 35 images have a resolution of 202,500 pix-els (450 × 450). Once preprocessed (as described inSec. 4), each pixel takes values on four attributes(NIR, R, G, NDVI) and our data contain a total of28 million values in the series. The values have beenquantified according to Sec. 4, attribute by attribute.Table 2 links the quantified values to the sensed ones.By applying the method described in Sec. 5 to theSITS illustrated in Fig. 4, we obtain a set of patterns

Table 2. Correspondence between sensed val-ues and quantified ones.

Quantified Mean reflectance

value NIR R G NDVI

1 60 5 7 02 93 8 11 0.283 111 10 13 0.384 121 13 16 0.455 128 16 18 0.496 132 18 20 0.547 137 22 21 0.578 144 26 22 0.619 152 28 25 0.6410 162 29 27 0.6711 174 30 30 0.7012 184 32 33 0.7213 194 35 35 0.7514 203 39 37 0.7815 214 42 40 0.8116 227 47 46 0.8417 242 56 56 0.8718 263 69 78 0.9019 293 93 138 0.9220 343 157 160 0.96

that needs to be interpreted by the experts. In thissection, we provide some results and their interpreta-tion in the two first subsections. The last subsectionis then devoted to the study of the maximum supportand its efficiency regarding the time response.

6.1. Visualization of Data MiningPatterns on Image Time Series

Examples of different IC computed for different FS

are displayed on Fig. 7. These results aim at visuallydetermining the content of the set of extracted pat-terns. Not surprisingly, the lower the minimum sup-port, the lower the number of sequences supportingthe extracted patterns (series of states of evolutions).Actually, with low supports, the extracted patternsget more and more specific and correspond to fewerpixels. Besides this obvious observation, Fig. 7(a)shows that the pixels supporting the patterns withsuch low supports are mainly located in industrialzones. Figure 7(c) contains a little more sequencessupporting the set of sequential patterns, mostly cor-responding to urban areas and reclaimed land. As toFig. 7(e), it corresponds to urban areas and wet veg-etation (swamps) in dark gray. This image providesus with another type of information: the constant

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(a) IC with parameters (0.005, 0.3) (b) IC with parameters (0.01, 0.4) (c) IC with parameters (0.02, 0.3)

(d) (d) IC with parameters (0.05, 0.5) (e) IC with parameters (0.07, 0.3) (f) IC with parameters (0.07, 0.5)

Fig. 7. IC images for different extractions with parameters (minimum support,maximum support). 256 gray levels (fromblack to white) are used in order to illustrate the contribution frequencies. White pixels correspond to high C(sx,y).

gray-level of urban areas indicates that these areasare often supporting patterns all together.

6.2. Exploring the Evolution Patterns

The images built on the principle described in theprevious subsection (and illustrated in Fig. 7) pro-vide the experts with valuable intuitive informationon the kind of patterns contained in the data min-ing result for a minimum and a maximum support.For each individual pattern in this result, we canretrieve the pixels whose series of values contain thepattern. These pixels may than be visualized (high-lighted) as illustrated by Fig. 8(b) (right). In thisfigure, each color corresponds to a pattern selectedfrom our SITS. We also report an image from theseries (left) for visualization purpose. This work issupported by the French Space Agency and theseresults where considered by experts in the domain.Here is the geographic explanation of these patterns.

The pattern 〈(IR,1) (NDVI,20)〉 is representedby the green dots in Fig. 8(b). It corresponds toswamps (wetlands) in the SITS and their cyclic

behavior. During winter, swamps are almost cov-ered with water, resulting in a low infra-red level(slice 1) since water does not reflect light a lot.During summer, these swamps are not covered withwater any more and light is reflected by the vegeta-tion. Due to its high chlorophyll concentration (dueto high irrigation), vegetation in summer has a veryhigh level in NDVI (slice 20).

The orange dots represent the pattern 〈(R,17)(R,18; NDVI,3)〉. It corresponds to urban areasthat get denser (the number of residences hasgrew). Actually, urban areas (residences) have a highresponse in the red band. The level at the beginningof the pattern (slice 17) is highly likely to be thesign of an urban area. The following level (slice 18)shows an urban densification (slices 17 and 18 areseparated by a radiometric increase of nearly 15 %),confirmed by a low level of NDVI (corresponding toalmost no vegetation). Extracting this kind of pat-terns is interesting because the urbanization behav-ior spreads over long time period, with different starttime and end time. In this SITS, the building of newhouses mostly starts in 1995 and was not finished

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(a) June 3 2006: image selected from the SITS (b) Illustration of our result with three selected

patterns (one color per pattern).

Fig. 8. A sample of our results (right) compared for visualization purpose to one image from the series (left).

in the end of the series. Moreover, the urbaniza-tion time can vary from one area to another. Theextracted pattern concentrates all building behav-iors, demonstrating the relevance of frequent pat-terns mining methods for the analysis of satelliteimage time series.

The pattern 〈(NDVI,2) (G,20) (NDVI,1)〉 isrepresented by the purple dots. This pattern cor-responds to a densification of industrial areas(e.g. increase in the number of warehouses). In fact,industrial areas have high response in the green band

0.6 0.59 0.58 0.57 0.56 0.55 0.54 0.53 0.52 0.51 0.5

minimum support

0

10

20

30

40

50

60

70

|FI\H|

Originalmax support = 0.8max support = 0.75max support = 0.7

0.6 0.59 0.58 0.57 0.56 0.55 0.54 0.53 0.52 0.51 0.5

minimum support

0

2000

4000

6000

8000

10000

12000

14000Time (s)

Originalmax support = 0.8max support = 0.75max support = 0.7

(a) Number of items in |FI\H| (b) Time responses

Fig. 9. Study of the number of items and time responses.

and show very low values of NDVI. Furthermore, thesignificant decrease of NDVI shows that vegetationalmost disappeared from these areas. Lastly, themaximum level of green is typical of flat roofs (e.g.corrugated iron) of industrial areas.

6.3. Efficiency of the Filters

The maximum support introduced in Subsec. 5.1 hasan important influence on the number of frequentitems that are not allowed in successive combinations

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(and, therefore, on the number of sequential patternsthat are evaluated during the process). In Fig. 9(left) we report the number of frequent items thatcan be combined into candidate sequential patterns.For instance, when the minimum support is 50%,the number of frequent items (“Original”) is 63, butwith a maximum support of 75%, the number ofitems i having support falling between 50% and 75%drops to 37 (which corresponds to a reduction fac-tor h = 1.7). We report in Fig. 9 (right) the timeresponses with a maximum support of 80%, 75% and70%, and we compare it to an extraction with nomaximum support (“Original”). We can observe thatthe influence of the reduction factor h is very impor-tant on the time response, as expected and describedin Subsec. 5.1 and Fig. 6.

This principle is well suited for SITS mining,since it discards non-evolution patterns and it out-performs by at least one order of magnitude the pro-cess attempting to extract the whole set of frequentsequential patterns. For instance, in our experiments,we observed that with a minimum support of 50%and no maximum support, the extraction needs sev-eral hours. With a minimum support of 10%, weestimate that it should take several weeks. How-ever, with a minimum support of 10% and a max-imum support of 50%, the process completes withinminutes.

7. Conclusion

Sequential patterns are well suited for evolution dis-covery and description. These patterns are extractedfrom series of values and they give the frequent suc-cessive values that are embedded in the series. Theintrinsic properties of sequential patterns extractionalgorithms fit the specific characteristics of satelliteimage time series: they are robust to meteorologicalnoise, they tolerate irregular sampling of images andare able to extract behaviors of different length andwith different start and end times. However, when itcomes to extracting evolution patterns from SITS,the challenge is to filter out the very large num-ber of non-evolution patterns, because of which theexperts could be overwhelmed. In this paper, we haveproposed (i) a framework for evolution pattern min-ing from SITS, (ii) a principle intended to filter outthe non-evolution patterns and enable the mining

process and (iii) a visualization technique that makesit possible to locate areas of evolution in oneviewing.

Experiments carried out on a particular datasetshowed that FSPM methods allow us to extractrepeated, shifted and distorted temporal behaviors.The flexibility of these methods makes it possible tocapture complex behaviors from multi-source, noisyand irregularly sensed data. However, this flexibil-ity could also have some drawbacks. In fact, thisflexibility could lead to skipping minority statesof evolution, even if they are characteristic of aclass. Moreover, the acquisition dates of the imageswere not used as an information for these experi-ments. Finally, this kind of methods does not providedirectly a whole classification of the sensed area, i.e.,there is not a class associated to each sensed area.

Considering these issues, we believe this workopens up a number of research directions. First, fur-ther experiments should be conducted in order todemonstrate the genericity of the proposed approach.Second, developed filters could include more sophis-ticated rules in order to integrate the knowledge ofthematic experts. Third, when dealing for instancewith agronomical classes, the date of the differentstates of evolution has to be used in order to dis-criminate one class from another. More work is stillneeded to integrate the use of the date of the imagesin a flexible way. Finally, inferring a clustering ofthe data from extracted patterns remains an impor-tant issue for both theoretical and applied researchdirection.

Finally, this work shows new interesting leads forsequential pattern extraction when the number ofpossible combinations does not allow to calculate thefrequent sequences. This may have important conse-quences in applications where the data does not varymuch and we want to find patterns with a frequencythreshold that is not very high while remaining sig-nificant. This is the case, for instance, in Web UsageMining, where the most frequent behaviors usuallycorrespond to non informative patterns (e.g. “20%of users click on homepage followed by location”).In data mining, it is well known that depending onsome characteristics of the data, one has to choosethe approach that best corresponds. Our study showsa new approach to consider, depending on the dataand their characteristics.

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Acknowledgments

This work was supported by the CNES (FrenchSpace Agency) and Thales Alenia Space. Theauthors would like to thank M. Lanselle for his care-ful reading of the article.

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discovering significant evolution patterns from satellite image time series

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