JOURNAL OF LA GLoG: Laplacian of Gaussian for Spatial ... · patterns over time. Nevertheless, the...

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1

GLoG: Laplacian of Gaussian for Spatial PatternDetection in Spatio-Temporal Data

Luis Gustavo Nonato, Member, IEEE, Fabiano Petronetto, and Claudio Silva, Member, IEEE

Abstract—Boundary detection has long been a fundamental tool for image processing and computer vision, supporting the analysis ofstatic and time-varying data. In this work, we built upon the theory of Graph Signal Processing to propose a novel boundary detectionfilter in the context of graphs, having as main application scenario the visual analysis of spatio-temporal data. More specifically, wepropose the equivalent for graphs of the so-called Laplacian of Gaussian edge detection filter, which is widely used in imageprocessing. The proposed filter is able to reveal interesting spatial patterns while still enabling the definition of entropy of time slices.The entropy reveals the degree of randomness of a time slice, helping users to identify expected and unexpected phenomena overtime. The effectiveness of our approach appears in applications involving synthetic and real data sets, which show that the proposedmethodology is able to uncover interesting spatial and temporal phenomena. The provided examples and case studies make clear theusefulness of our approach as a mechanism to support visual analytic tasks involving spatio-temporal data.

Index Terms—Data Filtering, Data Transformation, Feature Detection.

F

1 INTRODUCTION

Feature extraction and transformation comprise fun-damental steps in the visualization and visual analyticpipeline. In the particular case of spatio-temporal data,such preprocessing steps are of critical importance, as manyvisual analytic tools rely on features to enable meaningfulvisualizations of patterns and trends hidden on the data [1],[2], [3]. Over the years, a number of techniques devotedto extract and processing features from spatio-temporaldata have been proposed, ranging from simple aggrega-tion schemes [4] to sophisticated topological [5] and tensordecomposition methods [6]. Those techniques are designedto capture specific properties or phenomena present in thedata. For instance, topological methods look for locationswhere data assume extreme values while techniques basedon tensor decomposition aim to decompose the data so asto identify regions and time slices with common properties.

Identifying locations where data changes abruptly haslong been the goal of many feature extraction methods.Regions of sharp variation of the data typically correspondto locations where data transitions from one ”state” to an-other, thus being of great relevance for analytical purposes.Feature extraction methods able to detect local changesin a signal have been extensively used in fields such asimage processing and computer vision [7], being callededge detection methods. In fact, edge detection has beenthe building block of methods designed for recognizingand segmenting objects, enhancing details, and performingfeature preserving denoising in images and videos. Despitethe importance, few has been done towards developing

• Gustavo Nonato is with University of Sao Paulo - Brazil.E-mail: [email protected].

• Fabiano Petroneto is with Federal University of Espırito Santo - Brazil.E-mail: [email protected].

• Claudio Silva is with New York University.E-mail: [email protected].

techniques similar to edge detection to assist visual ana-lytic tasks, mainly in applications involving data definedin unstructured domains. A reason for this gap is that mostedge detection techniques developed in the context of imageprocessing rely on mathematical tools such as derivativesand convolution, which can hardly be defined on non-regular grid-like domains.

This work brings an alternative to the issue pointedabove, proposing an operator able to identify abruptchanges in a signal defined on the nodes of a graph. Morespecifically, we propose a novel filter, called GLoG, whichis the counterpart for graphs of the so-called Laplacianof Gaussian edge detection method (also known as Marr-Hildreth operator) widely used in image processing [8]. Ourapproach relies on the theory of Graph Signal Processing(GSP) [9], which provides a solid mathematical frameworkfor adapting and extending well known tools from theclassical signal processing field to the more general contextof graphs [10], [11]. The proposed GLoG filter can identifyspatial locations of abrupt changes in a signal, uncoveringregions (boundaries) where the signal changes its proper-ties. Moreover, we build upon the GLoG filter to highlighttime intervals where “expected” and “unexpected” bound-aries take place. In other words, we rely on the GLoG filterto define the concept of entropy of time slices, from which wederive a temporal entropy diagram. The latter allows the visualidentification of time instances where observed boundariesare likely to happen (lower entropy time instants) as wellas moments where observed boundaries are less expected(higher entropy time instants). The resulting analysis makeseasier the visual identification of expected and unexpectedpatterns over time. Nevertheless, the Gestalt law of prox-imity [12] states that groups of points spatially close toeach other are pre-attentively perceived as a common setof abstract features. The boundaries extracted from a signalfit naturally in this concept, turning out a valuable visualresource to reveal spatial patterns and trends.

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We show the effectiveness of our approach in syntheticand real data sets containing information about taxi tripsin Manhattan, NYC, and geo-referenced criminal activitiesin the city of Sao Paulo, Brazil. The provided examplesand case studies make clear the usefulness of the proposedboundary detection and entropy diagram as basic tools tosupport the visual analysis of spatio-temporal data.

In summary, the main contributions of this work are:• GLoG (Graph Laplacian of Gaussian), a boundary detec-

tion filter that is the graph counterpart of the so-calledLaplacian of Gaussian filter typically used in imageprocessing. As far as we know, this is the first time aboundary detection filter is proposed in the context ofGraph Signal Processing;• the concept of entropy of time slices and associated

entropy diagram, which enables the visual identificationof expected and unexpected spatial phenomena overtime;• two case studies that attest the usefulness and potential

of the proposed methodology to support visualizationassisted analysis of spatio-temporal data. These casestudies shows the potential of GLoG operator as a fea-ture extraction method in the context of spatio-temporaldata.

We emphasize that the proposed GLoG filter should not beseen as a competitor of other feature extraction techniques.In fact, the GLoG aims to detect a specific property (abruptchange) of a signal, which is not captured by existingmethods. Therefore, GLoG can be combined with othertechniques to produce features that captures a wide varietyof traits present in spatio-temporal data.

2 RELATED WORKThe literature about visualization assisted spatio-temporaldata analysis is extensive, ranging from georeferenced time-varying data analysis [1], [13] to dynamic networks [14],[15]. The different methodologies have been reviewed andorganized in a number of surveys [16], [17], [18], [19] andbooks [4], [20]. To better contextualize our contribution, wefocus on techniques that rely on feature extraction and datatransformation mechanisms to leverage the visualization ofspatial and temporal phenomena present in the data. Wegroup spatio-temporal feature extraction and/or data trans-formation techniques in three categories: temporal, spatial,and spatio-temporal.Temporal methods aim to uncover patterns by transformingthe temporal component of the data associated in eachspatial locations. In visual analysis, transformations such astemporal aggregation [21], [22] and moving averages [23]figure among the most common temporal approaches. Thereare also alternatives that rely on prediction mechanismsto detect unexpected temporal events, which are high-lighted during visualization [24]. Classical signal processingmethods have also been employed to extract features fromtime-series associated to specific spatial locations so as tomake the visual identification of similar temporal patternsan easier task [25]. When regions made up of severallocations have to be handled, spatial data aggregation isfirstly employed to combine multiple time-series into asingle one, which is then processed to identify patterns andtrends [26]. Aggregation tends to attenuate high-frequency

events, hampering the identification of patterns associatedto abrupt variations of the signal in particular locations. Infact, aggregation can be seen as low-pass filter applied priorto a feature extraction mechanism. The approach proposedin this work naturally combines low pass filter and featureextraction in a single operator.

Spatial transformation schemes aim to uncover spatial pat-terns present in the data. More specifically, given a timeslice (which can correspond to aggregated data from a timeinterval), spatial methods typically split the spatial domainin subregions, extracting and/or emphasizing features ineach subregion. Subregions can be defined based on den-sity information [27] or by grouping spatial locations withsimilar content [1], [28], [29]. A large number of meth-ods have been proposed to extract features from spatiallocations to support visualization tasks. Some examplesare the interchangeable matrices [30], which uncover co-occurring spatial events, and topological approaches [3], [5],which identify spatial locations that bear “peculiar” behav-ior. Topic modeling has been exploited as a mechanism tocharacterize spatial locations [31]. Graph signal processingtools such as windowed graph Fourier transform have beenemployed to extract features from spatial locations [32].Most of the spatial methods described above are designedto extract and identify patterns in fixed time slice, resortingto visualization as a main analytical resource to understandpatterns and their dynamics over time. The boundary detec-tion approach proposed in the current work can be seen asa spatial method, although the derived entropy diagram isa temporal visual tool.

Spatio-Temporal transformation techniques operate ontemporal and spatial dimensions so as to extract mean-ingful information that is emphasized during visualization.A typical alternative is to compute separate temporal andspatial features, resorting to visual analytic tools to unifythe spatial and temporal analysis. This is the methodologyimplemented in the MobilityGraphs technique [33], wherespatial or temporal signatures are used to cluster trajecto-ries in order to reduce data complexity and visual clutter.Another common use of independent spatial and temporaltransformations is the use of spatial signatures to identifyregions of interest and temporal features to analyze thebehavior of such regions over time [34]. Some techniquesdynamically extract temporal and spatial features accord-ing to user specified queries, enabling complex analysis oftime-stamped trajectories [35], [36], frequent trips [37], andspatial/temporal traffic patterns [38], [39].

More elaborate techniques operate on spatial and tem-poral information simultaneously, allowing for identifyingspatial, temporal, and spatio-temporal patterns. Featurevectors made up of temporal and spatial attributes is analternative that have been applied in several scenarios [40].Optimization procedures designed to detect periodical pat-terns in spatial events have also been employed to visuallyidentify tends in epidemiological data [41]. Spatio-temporalwavelet-based signatures [11] have turned out effectiveto simultaneously characterize spatial locations and theirtemporal behavior. More recently, transformations based ontensor decomposition are being used as a resource to assistthe visualization of spatio-temporal data [2], [6].


Fig. 1. Method pipeline: our methodology apply GLoG filter at each graph signal time slice to compute edge node configurations and define entropydiagram.

The proposed feature extraction and transformationtechniques bear a number of properties not simultaneouslypresent in any existing technique. Similarly to topologicalmethods for spatio-temporal analysis [3], [5], the proposedapproach is scale invariant, thus producing identical spatial“signatures” for signals that only differ by a scaling factor.However, the capability of generating spatial signatures onregions of large variation of the signal is a differential ofour approach when compared to topological schemes. Topo-logical approaches are able to point out the location wherea signal reaches its extremal values, but it demands verysophisticated methods (like Morse-Smale complexes com-putation) to account for where the underlying phenomenonis transitioning, a trait naturally captured by our approach.Moreover, the proposed methodology allows for transform-ing spatial patterns into a single scalar value (entropy)that indicates the degree of randomness of each time slice.Entropy values are much easier to interpret than, for ex-ample, patterns derived from tensor decomposition, whichtypically demand sophisticated visualization resources tobecome meaningful [2], [6].

3 EDGE NODE CONFIGURATION AND ENTROPYDIAGRAM TO SUPPORT VISUAL ANALYTIC TASKS

The proposed spatio-temporal analytic tool comprises twomain steps, i) the detection of boundary nodes in each timeslice and ii) the computation of entropy for the time slices,as illustrated in Figure 1. The former relies on the GLoGoperator described in Section 3.2 while the computation ofentropy derives from the concept of edge node probabilitydiscussed in Section 3.3. Before presenting both concepts,though, we describe the mathematical foundations of GraphFourier Transform (GFT) and Graph Filtering, which are thebasis of our approach.

3.1 Graph Signal Processing: Basic ConceptsThis section presents basic concepts of Graph Signal Pro-cessing from which our approach derives. A more detaileddiscussion about basic concepts of Graph Signal Processingcan be found in the work by Shuman et al. [9].

Graph Fourier Transform. We denote by G = (V,E,w) agraph made up of a node set V = {τ1, τ2, . . . , τn}, an edgeset E = {(τi, τj), τi, τj ∈ V, i 6= j}, and a weight functionw : E → R that associates a non-negative scalar to eachedge in G. In our context, G is assumed to be connected, thatis, for every pair of nodes there is a sequence of adjacentedges connecting those nodes.

The weighted adjacency matrix of G, denoted by A = (aij),is the matrix satisfying aij = w(τi, τj) if (τi, τj) ∈ E andaij = 0 otherwise. This matrix is used to define the (non-normalized) graph Laplacian, which is given by L = D − A,where D = diag(d1, d2, . . . , dn) is a diagonal matrix withentries di =

∑j aij and n is the number of nodes in V . The

graph Laplacian is a real, symmetric, and semi-positive def-inite matrix, which ensures a complete set of orthonormalreal eigenvectors u`, with corresponding non-negative realeigenvalues λ`, ` = 1, 2, . . . , n. Moreover, zero is alwaysan eigenvalue of L whose corresponding eigenvector is aconstant vector.

The eigenvalues and eigenvectors of the graph Laplacianplay a similar role as frequencies and basis functions in theclassical Fourier theory. More specifically, eigenvalues closerto zero correspond to low frequencies while large eigen-values correspond to high frequencies. Moreover, eigenvec-tors associated to small eigenvalues tend to have a lessoscillatory behavior than eigenvectors associated to largeeigenvalues. A more detailed discussion about the relationbetween the spectrum of Laplacian matrices and Fouriertheory can be found in the work by Shuman et al. [42] andDal Col et al. [10].

A signal defined on the nodes of G is a function f :V → R that associates a scalar f(τi) to each node τi ∈ V .Denoting the eigenvalues and corresponding eigenvectorsof the Laplacian matrix of G by λ` and u`, ` = 1, . . . , nrespectively, and assuming the eigenvalues are sorted innon-decreasing order, 0 = λ1 < λ2 ≤ . . . ≤ λn (thefirst strict inequality is due to the assumption that G isconnected), the Graph Fourier Transform (GFT) of a signal f ,denoted by f : Λ → R, where Λ is the spectral domain (setof eigenvalues), is defined as:

f(λ`) = 〈u`, f〉 =n∑

j=1

u`(τj)f(τj), (1)

Given the GFT f , the original signal f can be recoveredvia the inverse Graph Fourier Transform (iGFT), which isdefined as:

f = iGFT (f) =n∑

`=1

f(λ`)u` (2)

If we denote by U the (orthogonal) matrix with columnsgiven by the eigenvectors u`, the GFT and iGFT can beobtained via matrix multiplication as follows:

GFT iGFT

f = U>f f = Uf (3)

Spectral Filtering. A graph spectral filter h : Λ → R is afunction defined in the spectral domain that associates ascalar value h(λ`) to each eigenvalue λ` ∈ Λ. The GFTf : Λ → R can be seen as a particular instance of a graphspectral filter.

A graph spectral filtering of a signal f is defined as:

fh = f · h (4)

where f is the GFT of f , h is a graph spectral filter and · isthe element-wise multiplication. Using the matrix notation


Fig. 2. Noisy step function (left) f(τi) = 1 + ε in the yellowish nodesand f(τi) = ε in the purplish, where ε is a random noise. Gaussiansmoothing with σ = 1 (top right) and σ = 3 (bottom right). The two plotson the left shows the shape of the Gaussian filter related.

defined in Eq. (3) and some algebraic manipulation one canobtain the filtered version f of f in the graph domain bycomputing:

f = UHU>f (5)

where H is a diagonal matrix with entries h(λ1), . . . , h(λn).The design a proper filter h is application dependent. A

particularly useful filter in our context is the Gaussian filtergiven by:

hg(λ) =1

σ√

2πexp

(−λ22σ2

)(6)

where λ is the independent variable and σ is a parameter.The Gaussian filter is a low-pass filter that preserves lowfrequencies while attenuating the higher ones. Figure 2illustrates the effect of applying hg to a noisy step functionwith two different values of σ.

3.2 GLoG for Boundary DetectionThe boundaries (or edges) of a signal correspond to thelocations of abrupt change in the signal. Many differentapproaches have been proposed to identify edges in the con-text of image processing and computer vision [8]. Amongthe existing edge detection methods, the Laplacian of Gaus-sian (LoG) figures among the most important ones, mainlydue to its solid mathematical foundation and optimalitycriteria [43].GLoG filter. The classical LoG is a filter built based on twomain principles: 1) boundaries take place on locations wherethe first derivative (or gradient) of a signal is maximum, orequivalently, the locations where the Laplacian of the signalis zero, which is called the zero-crossings of the Laplacian;2) zero-crossings may be caused by noise, so a smoothingfilter must be applied to the signal before computing theLaplacian. The chosen smoothing filter is the Gaussianfilter, as it provides optimal localization in the space andfrequency domains [43].

σ = 1 σ = 2 σ = 3

Fig. 3. GLoG filter applied to the noisy step function depicted in Figure 2(left). Blue and red colors correspond to nodes where the GLoG filter isnegative and positive, respectively.

In mathematical terms, the classical LoG filter can bedefined as

LoG(f) = ∇2(G ∗ f) = ∇2G ∗ f (7)

where ∇2 is the Laplacian operator, G is the Gaussianfunction, f is the signal, and ∗ is the convolution operator.The right most expression is a consequence of the derivativerule for convolution.

There are two main issues for adapting Eq.(7) in thecontext of graphs: how to define the convolution operatorand how to compute ∇2G on a graph structure. The convo-lution operator is not well defined in graph domains, as itdemands a shift mechanism that can not be directly definedon graphs. However, convolution becomes multiplication inthe spectral domain, thus, with the help of Graph SignalProcessing theory, the convolution between two functions fand g can be defined as [9]:

f ∗ g = iGFT (f · g) (8)

where · is the element-wise multiplication, f and g are theGFT of f and g (defined in Eq. (1)), and iGFT is the inverseGFT given in Eq. (2). Using Equations (7) and (8), we candefine the Graph Laplacian of Gaussian (GLoG) filter as:Definition (GLoG):

GLoG(f) = iGFT (∇2G · f). (9)

In order to make Eq. (9) of practical use, we mustproperly define ∇2G. We know that the classical 2D Fouriertransform of ∇2G is given by −4π2ω2 exp(−σ2ω2) [43]. Ifwe interpret the variable ω, which is a complex variable inthe classical Fourier transform theory, as a real variable inthe graph spectral domain, then ∇2G can be given by thefollowing graph spectral filter:

∇2G(λ) = −4π2λ2 exp(−σ2λ2). (10)

where σ is a user defined parameter and λ is the indepen-dent variable in the graph spectral domain. Defining ∇2G asa graph spectral filter makes the definition and computationof GLoG feasible, being one of the contributions of thiswork.

Having defined ∇2G(λ), the GLoG filter (Eq. (9)) be-comes a band-pass filter in the spectral graph domain.Moreover, in the particular case where G is a regular 2Dgrid, the GLoG matches the LoG filter as defined in classicalsignal processing.

Figure 3 shows the result of the GLoG filter whenapplied to the noisy step function depicted in Figure 2(left) using three different values of σ. Blueish and reddish


Fig. 4. GLoG for boundary detection (Section 3.2). From left to right, signal defined on a linear graph, GLoG of the signal, zero-crossing edges, thestrongest edges, and the edge nodes obtained by algorithm (highlighted in red over graph signal).

µ+ 2std

µ+ std

Fig. 5. Edge node configuration obtained by keep zero-crossing pairswhose score is greater than the average µ plus (left) 1 and (right) 2 timesthe standard deviation std of the scores in the GLoG signals depictedin Figure 3 middle, respectively.

colors correspond to negative and positive values of theGLoG respectively. Notice that the larger the value of σ thesmoother the result of the GLoG filter is.

Extracting the Strongest Edge Nodes. The result of ap-plying a GLoG filter to a signal is another signal definedon the nodes of the graph, which we denote by GLoGf

to make clear its dependence of the input signal f . Apair of adjacent nodes (τi, τj) ∈ E is a zero-crossing pairif GLoGf (τi)GLoGf (τj) < 0. The nodes belonging to azero-crossing pair are called edge nodes. We can associatethe score |GLoGf (τi) − GLoGf (τj)| to each zero-crossingpair, the larger the score the stronger the signal variation isin that pair, which we call strong edge nodes (or simply edgenodes). “Weak” pairs can be filtered out by considering onlyzero-crossing pairs whose score is among the largest ones.After computing the stronger edge nodes one can generatea binary signal fe where fe(τi) = 1 if τi is an edge node andfe(τi) = 0 otherwise. The signal fe is called an edge nodeconfiguration of f . Figure 4 illustrates all the steps involvedin the computation of edge nodes of a signal f .

Figure 5 shows edge node configurations for zero-crossing pairs whose score is greater than the mean plus 1.0and 2.0 times the standard deviation of the zero-crossingpair score distribution. Such edges nodes have been com-puted from the GLoG signal for σ = 2 depicted in Figure 3.Notice that, as expected, the higher the threshold the smallerthe number of edge nodes.

Computational Aspects. The pseudocode for the GLoGfilter computation can be stated as in Algorithm 1. Themethod get_edge_nodes compute edge nodes by firstidentifying the edge node pairs, that is, pairs of adjacentnodes where g changes its sign. The edge node pairs whosescore is larger than a threshold (strong edge node pairs)are returned by the method. In our implementation, thethreshold is the third quartile of the edge node pair scoredistribution.

Algorithm 1 GLoG(L,f )1: Compute the eigenvector matrix U and eigenvalues Σ

from the Laplacian L2: f = U>f # GFT, Eq. (1)3: g = f · ∇2G # ∇2G as in Eq. (10)4: g = Ug # iGFT, Eq. (2)5: e = get_edge_nodes(g) # find edge-nodes6: return e

The spectral filtering can be computed via Chebyshevpolynomial approximation [44], which avoids the computa-tion of the whole set of eigenvalues and eigenvectors of L,thus making possible to handle large graphs in reasonablecomputational times. If Chebyshev polynomial approxima-tion is used, only the largest eigenvalue has to be computedin step 1 of the algorithm and step 2 and 3 are merged in asingle one (see [44] for details). Our implementation makesuse of Chebyshev polynomial approximation.

3.3 Edge Node Probability and Time Slice Entropy

In this section we will show how the boundary detectionmethodology described in the previous section can be usedto assist spatio-temporal data visualization. Lets assume atime series is associated to each node of a graph G, thatis, given a set of time slices T = {t1, t2, . . . , tm}, there isa function f : V × T → R that associates a scalar valuef(τi, tj) to each node τi in the time slice tj .

Edge Node Probability. The GLoG filter (9) can be appliedto each time slice tj in order to identify the edge nodes intj . Assuming that only the strongest edge nodes are kept


in each time slice (score larger than the third quartile of thescore distribution), one can estimate the probability pe(τi)of a node τi being an edge node as follows:

I(τi, tk) =

{1 if τi is an edge node in time slice tk0 otherwise

pe(τi) =1

m

m∑k=1

I(τi, tk)

(11)where m is the number of time slices. The probability ofa node τi not being an edge node is given by 1 − pe(τi).Notice that the probability pe(τi) is simply the number oftime slices where τi appears as an edge node divided by thetotal number of time slices.

Time Slice Entropy. The following function

p(I(τi, tk)) =

{pe(τi) if I(τi, tk) = 11− pe(τi) if I(τi, tk) = 0

(12)

computes how probable the observed configuration of anode τi (in time slice tk) is in terms of it being or not anedge node. The entropy of the edge node configuration intime slice tk is given by:

Definition (Entropy):

E(tk) = −n∑

i=1

p(I(τi, tk)) log p(I(τi, tk)) (13)

The entropy measures the degree of randomness of a timeslice, the larger the entropy the more unpredictable the timeslice is in terms of its edge node configuration. Time slicewhere the edge node configuration is of low probabilitypresents larger entropy. Therefore, by simply plotting theentropy over time, which we call entropy diagram, one canvisualize time slices where the signal presents unexpected(high entropy) or predictable (low entropy) edge node con-figurations. The definitions of entropy and entropy diagramare another contribution of this work. The spatial distribu-tion of edge nodes in each time slice is also an importantfeature that can be used to understand a signal.

3.4 Synthetic time-varying data

In order to illustrate how edge nodes and entropy informa-tion can help in the visual analysis of spatio-temporal data,we have manufactured a data set by randomly placing 600points within a 2D unitary square, connecting the pointsusing the Delaunay triangulation. The points correspondto the graph nodes and the Delaunay links to the graphedges. All graph edges are assumed to have unitary weight.A spatio-temporal signal f : V × T → R is associated to thenodes of the graph. In each time slice the signal f is equal to1 for nodes lying inside a circle with radius 0.1 and centeredin the position (0.5+δx, 0.5+δy), otherwise, f is set to 0. Thepair δx and δy correspond to a random perturbation in thecenter of the circle, each ranging in the interval [−0.05, 0.05].The random perturbation prevents the signal to be exactlythe same in each time slice. A random noise in the interval[−0.1, 0.1] is also added to f in each time slice. One hundredtime slices are generated using the procedure describedabove, however, in 12 time slices the center of the circle isfurther translated towards the top right or bottom left corner

Fig. 6. Three time slices of the synthetic signal and corresponding edgenode configurations (botton right inset). The bottom right image showsedge nodes (red circled nodes) jointly with signal.

of the unitary square domain, forcing an abrupt change inthe signal in those time slices. Figure 6 shows the signalgenerated as described above. Top left image correspondsto a time slice where the circle is centered close to (0.5, 0.5)while bottom left and top right images correspond to timeslices where the center of the circle has been shifted towardsthe top right or bottom left corner of the unitary square. Foreach time slice, the corresponding edge node configurationis illustrated in the inset on the bottom right.

Figure 7 presents the entropy diagram of the signal fdescribed above. Notice that the 12 time slices where thecenter of the circle is displaced towards the top and bottomcorners of the unit square are revealed, as they have largerentropy values when compared to the remaining time slices.Moreover, since the edge node configuration can be seen asa binary vector in an n-dimensional space, we can clusterthe high-dimensional vectors in order to identify time sliceswith similar edge node configuration. The color of the dotsin Figure 7 indicates the cluster each time slice belongs to(we have clustered edge node configuration vectors in threegroups using a simple k-means). The red group correspondsto time slices where the center of the circles have beenshifted towards the bottom left of the unit square, the greengroup are the time slices where the centers moved towardsthe top right corner. Blue group gathers time slices wherethe center of the circles are close to (0.5, 0.5).

Although simple, the synthetic example discussed inthis section illustrates the potential of the GLoG filter andentropy diagram as visual analytic tools for spatio-temporaldata. Such usefulness will become more clear in the casestudies presented in the following section.


Fig. 7. Entropy over time. The three patterns revealed correspond to 6 time slices where the signal is shifted towards the top right corner, 6 wherethe signal is shifted to the bottom left corner, and the remaining ones correspond to time slices where the signal is centered close to (0.5, 0.5). The12 time slices holding abrupt changes in the signal are clearly revealed by the entropy diagram.

Fig. 8. Total number of taxi pick ups (left), edge nodes probability pe(center), and enhanced total taxi pick up visualization (right).

4 CASE STUDIES

In this section we apply our methodology in two case stud-ies involving real data. In both studies we strongly rely onthe proposed GLoG filter and associated entropy diagramto assist the visual analysis and identification of spatio-temporal patterns and phenomena. The first case studyaccounts for the analysis of taxi pick up during one week indowntown Manhattan - NYC. The second case study dealswith crime data in the city of Sao Paulo - Brazil.

4.1 NYC Taxi Data AnalysisThe graph domain in this case study is downtown Man-hattan street network, where street intersections correspondto graph nodes and the street blocks connecting the in-tersections make up the set of graph edges. The graph ofdowntown Manhattan is made up of 4694 nodes and 6350edges in each time slice. The Taxi data set [38] contains infor-mation about one week of taxi pick up in each intersectionof downtown Manhattan from August 11th to August 18th,2013. The data was aggregated into half-hour intervals ineach node, giving rise to 336 time slices. The GLoG filter isapplied to each time slice independently.

Figure 8 left shows the total number of taxi pick upin each node of the graph (the sum, over the 336 timeslices, of the number of taxi pick ups in each node). A fewnodes are highlighted in the neighborhood of Penn Station,a major train and metro station hub in NYC, and a fewblocks uptown, close to Port Authority, the main bus stationterminal in Manhattan. Since the number of taxi pick upsin those two regions are much larger than in other areasof the city, other important locations are overshadowed,becoming difficult to visualize regions where the numberof taxi pick ups is relevant but not so large as in theneighborhood of Penn Station and Port Authority. Figure 8

middle shows the probability pe(τi) (see Eq. 11) of each edgenode τi. Since the GLoG filter is scale invariant, it reveals anumber of nodes that are not highlighted in Figure 8 left,all corresponding to nodes with high probability of beingedge nodes. Nodes presenting a not so high signal intensitybut with high probability of being an edge node are usuallyassociated to interesting spatial events that deserve to beanalyzed. Figure 8 right corroborates this claim, showingan enhanced total taxi pick up visualization generated byadding the probability pe to the normalized total number oftaxi pick ups. The combined visualization is the equivalentof what is called “image enhancement” in the context ofimage processing [7]. As annotated in Figure 8 right, be-sides Port Authority (A), the whole neighborhood of PennStation, including Korea Town (B), Times Square (C), GrandCentral Station (D), the luxurious hotel neighborhood (E),and the region that concentrates most of the consulates inNY (F) are also visible in the enhanced visualization. Evensmaller spots like the Meatpacking District (G) and the 9/11memorial (H) show up after the enhancement process. Theregion indicated by (K) is also interesting to analyze, as itcorresponds to Jersey St., a very quiet two-block street inthe middle of Soho where the number of taxi pick ups ismuch smaller if compared against its neighborhood. Noticethat the enhanced visualization not only reveals importantareas in Manhattan but also locations with almost no pickups, which can indicate quiet spots or issues in the traffic asbloched streets and accidents.

Figure 9 shows the taxi pick up entropy diagram. Noticethat entropy behaves similarly in the weekdays, reachingits maximum after midnight, dropping down at the dawn,presenting a local maximum right after 6am, increasingconsistently along the day. Weekend days also present maxi-mum entropy at dawn, however, the gradual increase alongthe day is not observed. In contrast, an abrupt growth is ob-served in late evening on Saturday while entropy decreasesalong the day on Sunday. Therefore, the entropy diagramshows that taxi pick up becomes more random at dawn,specially in the weekend.

Edge node configuration associated to each time slicehave been clustered in seven different groups using k-means(as explained in Section 3.4). The number of clusters hasbeen defined by analyzing the silhouette coefficient [45],which indicated the number 7 as the “ideal” number ofclusters to balance quality and number of groups. It ispossible to see that the groups nicely split the behavior oftaxi pick up according to the periods of the day, indicatingthat the edge node configuration is capturing patterns of


Fig. 9. Taxi pick up entropy diagram. Entropy tends to increase along the day in weekdays, reaching its maximum around midnight. On Saturday,the gradual increase along the day is not observed, but an abrupt growth in the late evening is. On Sunday, entropy tends to decrease along theday. Therefore, taxi pick up is more random at dawn, specially in the weekends. Edge node configuration (see Figure 10) of each time slice havebeen clustered in seven groups, revealing the different behavior of taxi pick up along the day.

Fig. 10. Example of edge node configuration along the day in weekdays. The edge node configuration reveals the dynamics of the city, with aconcentration of edge nodes in midtown in the early morning, extending to east side in the early afternoon, and moving to downtown in the evening.

Fig. 11. Enhanced taxi pick up visualization for some of the time slicesdepicted in Figure 10.

taxi pick up. Moreover, the particular behavior taking placeduring the weekend is also captured by the edge nodeconfiguration. Figure 10 shows randomly chosen examplesof edge node configurations from 6 out of the 7 groups.The Early Morning group concentrates edge nodes mostlyin the neighborhood of main public transportation hubssuch as Penn Station, Port Authority, Grand Central, and

other smaller path train and metro station terminals. TheMorning group shows that taxi pick up spread to north-east of downtown Manhattan, in the area of luxurioushotels and embassies. In the Afternoon group, one can seethat taxi pick up start to move downwards to 14th street,with a concentration around Union Square park. WeekdayEvening is characterized by a concentration of edge nodesin Soho and Meatpacking district, which are well knowncommercial areas with famous stores markets and bars. Inthe Late Evening group, edge nodes spread to East Village,an area with many bars and restaurants that is known byits nightlife. Edge nodes in the Dawn group are mostlyconcentrated in the south part of the island, presenting areduction on the number of edge nodes in the northeastarea. As pointed out by the entropy diagram, late night anddawn have larger entropy, meaning that the pattern of taxipick up is more random during those periods.

The visualization of edge node configuration patternsdepicted in Figure 10 reveals the dynamics of the city, wherepeople tend to arrive in the public transportation terminalsduring the morning, staying more concentrated in the northand northeast area during the day, migrating to the southand east side of Manhattan in the evening and late nightto enjoy the bars and restaurants located on those areas.Figure 11 shows the enhanced visualization of taxi pickup in six time slices shown in Figure 10, one for eachgroup of edge node configuration. Notice the agreementbetween the enhanced visualization of taxi pick up and the


Fig. 12. Total number of passerby robbery from 2007 to 2017 (left). Theprobability pe(τi) of each node being an edge node (right).

patterns of edge node configuration, showing that the edgenode configuration can be an interesting alternative whenanalyzing the behaviour of a signal overtime, since evensmall variation on the signal can be identified as edge nodes.

4.2 Crime DataIn this case study we apply the proposed methodology toanalyze crime data in the city of Sao Paulo - Brazil. Thedata is available for downloading in the Sao Paulo OpenData repository (http://dados.prefeitura.sp.gov.br) and itprovides, among others, information about date and time ofeach criminal event, the type of crime, and the census regionwhere the crime event took place. In this study we focusedon eleven years of data, from 2007 to 2017, accounting onlyfor passerby robbery as crime type. Since the geolocation ofeach criminal activity is given by census regions, we havebuilt a graph where nodes correspond to the census regionand graph edges correspond to links connecting censusregions that geographically intersect each other. Only censusregions at a distance of less than five kilometers from thecenter of Sao Paulo are considered, resulting on a graph with3, 805 nodes and 12, 483 edges. We have applied a monthaggregation to the data, resulting in 132 time slices.

Figure 12 shows a visualization of the total number ofpasserby robbery along the years (left) and the probabilitype(τi) (see Eq. 11) of each node τi being an edge node (right)(purple colors indicate larger probabilities). As in the caseof NYC taxi pick up, the number of passerby robbery in thecenter of Sao Paulo is much larger than in other regions,overshadowing nodes and regions where the number ofpasserby robbery is not in the same magnitude as in thecity center, but still important to be identified and analyzed.

The entropy diagram of Sao Paulo passerby robbery isdepicted in Figure 13. Three high and three low entropytime slices are chosen to be investigated in detail (high-lighted in the entropy diagram). High entropy time slicesare associated to (edge node) patterns that are less likelyto happen. In the context of crime data, this means thatcrimes are taking place in “unexpected” locations. In fact,an edge node configuration with high entropy tends tobear a considerable number of low probability (pe) edgenodes. Therefore, by analyzing low probability edge nodesthat show up in high entropy time slices one can identifylocations where the number of passerby robbery changesabruptly, which can indicate the action of gangs or emergingof juvenile offenders in those location or even a sudden re-duction on crime rates, which is also interesting to analyze.

The images on the right column in Figure 14 shows lowprobability (probability lower the 0.5) edge nodes presentin the high entropy time slices highlighted in Figure 13.Notice that a number of locations and neighborhoods farfrom the city center are evident. By analyzing the numberof crime events in those locations we noticed that they donot present a large number of passerby robbery but, in thepointed high entropy time slices, the number of criminalevents increases substantially. In contrast, the edge nodeconfiguration associated to low entropy time slices tend toreveal (edge node) patterns that are probable to happen,thus indicating neighborhoods where criminal activities areexpected. Left column in Figure 14 shows high probabilityedge nodes present in the low entropy time slices indicatedin Figure 13. It is easy to see that the most probable edgenodes are located in downtown Sao Paulo, indicating afrequency of passerby robbery in that area. The analysisabove shows the potential of GLoG as a basic tool to supportvisual analytic tasks, mainly in applications where scaleinvariance is a basic requirement, as is the case of crimeanalysis.

5 DISCUSSION AND LIMITATIONS

The case studies presented in Section 4 shows the effec-tiveness of the proposed GLoG filter and entropy diagramto support the visual analysis of spatio-temporal data. Theentropy diagram reveals the degree of randomness of eachtime slice, allowing an intuitive visual identification of pre-dictable and unpredictable time slices in terms of their edgenode configuration.

The edge node configuration is, by itself, a human in-terpretable signature from which one can understand thespatial behavior of a signal. With the help of clusteringschemes, edge node configuration can uncover time inter-vals where a signal presents similar behavior, thus helpingusers to understand spatial phenomena over time.

Our methodology demands essentially two parameters,σ, which controls the strength of the Gaussian filter, andthe threshold used to filter out weak edge node pairs. Thelatter is defined in terms of the standard deviation of thedistribution of edge node pair scores. In the case studies wefixed σ = 3 and the edge node pair threshold in the thirdquartile, as those values resulted on interesting analysisand visual patterns. However, those parameters impact inthe edge node configuration and thus in the entropy andcluster analysis. The larger the σ and the threshold smalleris the number of edge nodes tend to be present in the edgenode configuration. Although we have not experienced anydifficulties to set those parameters in the two case studiesdiscussed in Section 4, that might not always be the case,and a fine tuning can be necessary depending on the appli-cation.

Regarding computational times, the whole process, in-cluding the computation of the largest eigenvalue (involvedin the Chebyshev approximation), GLoG filtering, edgenode pairs identification and threshold, takes only a fewseconds, which is pretty reasonable. For instance, the wholeprocess took less than 5 seconds in both case studies.

The proposed methodology opens a wide set of futuredirections. For instance, it would be interesting to inves-tigate techniques able to track edge nodes over time in

http://dados.prefeitura.sp.gov.br


Fig. 13. Entropy diagram of Sao Paulo passerby robbery. The highlighted picks correspond to some of the highest and lowest entropy time slices.

Fig. 14. Sao Paulo crime: High probability edge nodes in low entropytype slices (left ) and low probability edge nodes in high entropy timeslices (right). Left column, purple points indicate high probability edgenodes, which are locations where criminal activity is common. Rightcolumn, blue points indicate low probability edge nodes, indicatingneighborhoods where the criminal activity is unexpected.

order to visualize how the spatial “boundaries” evolves,allowing a better understanding of the dynamics of the data.Another interesting problem is how to build upon boundarydetection to design spatial segmentation techniques thatrespect the edge nodes. This kind of segmentation has beensuccessfully employed in image processing and computervision. Properly segmenting spatial locations based on datais an major problem in visualization that can greatly benefitfrom the methodology proposed in this work. The GLoGfilter can also be combined with other feature extraction

mechanisms such as topological methods, allowing the anal-ysis of ”extremal” and transitional data locations. Moreover,the proposed methodology can also be applied to problemsbeyond geo-spatial data, as for example high-dimensionaldata analysis, mainly through dimensionality reduction.

6 CONCLUSION

In this work we have introduced a novel methodology toprocess and visualize spatio-temporal data based on theconcept of boundary detection and entropy diagram. Theproposed methodology strongly relies on spectral filteringfrom graph signal processing theory, which enables theprecise definition of a Laplacian of Gaussian boundarydetection filter that operates in graph domains. Edge nodescomputed by the proposed machinery can be used as featurevectors as well as to define the notion of entropy of timeslices, allowing the identification of low and high probabletime slices. Visual analytic tasks involving synthetic andreal data sets show the effectiveness and usefulness of thepropose methodology to support visualization applications.Moreover, the proposed methodology can also be combinedwith other feature extraction methods so as to capturedifferent facts of time-varying data, opening a number ofpossibilities for future work.

ACKNOWLEDGMENTS

This work was supported by: the Moore-Sloan Data Sci-ence Environment at NYU; NASA; NSF awards CNS-1229185, CCF-1533564, CNS-1544753, CNS-1626098, CNS-1730396, CNS-1828576; 302643/2013-3 CNPq-Brazil and2016/04391-2 and 2014/12236-1 Sao Paulo Research Foun-dation (FAPESP) - Brazil. C. T. Silva is partially supportedby the DARPA MEMEX and D3M programs. Any opinions,findings, and conclusions or recommendations expressed inthis material are those of the authors and do not necessarilyreflect the views of DARPA and Sao Paulo Research Foun-dation.

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