ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327

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ISPRS Journal of Photogrammetry and Remote Sensing

journal homepage: www.elsevier.com/locate/isprsjprs

An efficient stochastic approach for building footprint extraction from digitalelevation modelsO. Tournaire ∗, M. Brédif, D. Boldo, M. DuruptLaboratoire MATIS, IGN, 73 avenue de Paris, 94165 Saint-Mandé, France

a r t i c l e i n f o

Article history:Received 30 April 2009Received in revised form3 February 2010Accepted 3 February 2010Available online 12 March 2010

Keywords:Digital Elevation ModelBuilding footprintEnergetic modelingMarked point processesRJMCMC

a b s t r a c t

In the past two decades, building detection and reconstruction from remotely sensed data has beenan active research topic in the photogrammetric and remote sensing communities. Recently, effectivehigh level approaches have been developed, i.e., the ones involving the minimization of an energeticformulation. Yet, their efficiency has to be balanced by the amount of processing power required to obtaingood results.In this paper, we introduce an original energetic model for building footprint extraction from high

resolution digital elevation models (≤1 m) in urban areas. Our goal is to formulate the energy in anefficient way, easy to parametrize and fast to compute, in order to get an effective process still providinggood results.Our work is based on stochastic geometry, and in particular on marked point processes of rectangles.

We therefore try to obtain a reliable object configuration described by a collection of rectangular buildingfootprints. To do so, an energy function made up of two terms is defined: the first term measures theadequacy of the objects with respect to the data and the second one has the ability to favour or penalizesome footprint configurations based on prior knowledge (alignment, overlapping, . . . ). To minimize theglobal energy, we use a Reversible Jump Monte Carlo Markov Chain (RJMCMC) sampler coupled witha simulated annealing algorithm, leading to an optimal configuration of objects. Various results fromdifferent areas and resolutions are presented and evaluated. Our work is also compared with an alreadyexisting methodology based on the same mathematical framework that uses a much more complexenergy function. We show how we obtain similarly good results with a high computational efficiency(between 50 and 100 times faster) using a simplified energy that requires a single data-independentparameter, compared to more than 20 inter-related and hard-to-tune parameters.

© 2010 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published byElsevier B.V. All rights reserved.

1. Introduction

Building extraction and reconstruction from remotely sensedimages has been a motivating topic for many researchers in thepast years. Indeed, 3D urban models can be very useful in variousapplications: virtual tourism, models of wave propagation fortelecommunication operators or realistic environments for videogames among others. However, before providing a volumetricrepresentation of the buildings, most automatic methods need tofocus on their 2D outlines (Haala et al., 1998; Brunn and Weidner,1997; Lafarge et al., 2008). This is the problem we propose toaddress in this paper, without any prior knowledge on the buildinglocation and only loose constraints on shape.

∗ Corresponding author. Tel.: +33 1 43 98 84 29; fax: +33 1 43 98 85 81.E-mail addresses: olivier.tournaire@ign.fr (O. Tournaire), mathieu.bredif@ign.fr

(M. Brédif), didier.boldo@edf.fr (D. Boldo), melanie.durupt@ign.fr (M. Durupt).

0924-2716/$ – see front matter© 2010 International Society for Photogrammetry anddoi:10.1016/j.isprsjprs.2010.02.002

The work presented here is centered on footprint extraction.In 3D reconstruction tools, these regions of interest (ROI) usuallycome from cadastral maps, manually delineated outlines, or fromsomemanual or automatic processing on a digital elevation model(DEM, rasterized heights on a regular grid) or an orthoimage.However, such data are costly to create ‘‘by hand’’ and not alwaysavailable, may be outdated and can suffer of discrepancies withground truth. Thus it appears necessary to have a rapid, robust andreliable methodology to extract building footprints automatically.To do so, we use the framework of Ortner et al. (2007), andpropose improvements tomake it useful in an operational context,that is to say with reasonable computation times and simpleparametrization, without losing quality, robustness or generality.

1.1. State of the art

Today, a tremendous variety of approaches exists in theliterature. With the growing amount of data to process due tothe increasing resolution of data and extent of areas to cover,

Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

318 O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327

automatic methods need to be considered with a particularattention.Haithcoat et al. (2001) use size, height and shape characteristics

to differentiate buildings from other objects on a LiDAR digitalelevation model. Thresholds discriminate small objects likecars and differential geometries quantities eliminate trees. Theresult is rasterized and footprints are simplified. However,this simplification assumes orthogonality of the building maindirections, too restrictive a hypothesis in dense urban areas.With a similar approach based on simple thresholds, Alharthy

and Bethel (2002) evaluate the potential of LiDAR data for3D building extraction. This first analysis provides blobs whichneed to be converted into intersection free polygons. A coarserepresentation is then obtained using restrictive hypothesis oforthogonality of building parts.From high-resolution commercial satellite data, Shackelford

et al. (2004) involve mathematical morphology operators ina multi-scale framework for the delineation of buildings. Thismethod is however sensitive to errors, as they are propagated fromone level of analysis to the next.In an optimization framework and with LiDAR data, Wang et al.

(2006) first chain boundary points detected on LiDAR data andsimplify them to obtain an estimate of the building footprint. ABayesian Maximum a Posteriori (MAP) estimation then allows toimprove the footprint. The distance of the boundary points to thepolygon and a prior favouring particular directions of the buildingedges are embedded.In a more recent work, Frédéricque et al. (2008) populate

a database of reconstructed buildings by first detecting theirfootprints. Their approach focuses on a ROI and extracts itsskeleton. A set of rectangle hypotheses is then generated with theprincipal directions at given points of the skeleton. An iterativealgorithm then allows to obtain a simplified graph of rectangles,providing the representation of a building block as a set ofrectangles (computation time for the whole process is about 20 sper building block).In this paper, we use the marked point process framework

presented in Ortner et al. (2007). It proposes an energeticformulation and states the problem as a minimization of afunctional made up of a data attachment term and a regularizingterm. The sampling of this objective function is conducted thanksto a RJMCMC sampler (Hastings, 1970; Green, 1995) and coupledwith a simulated annealing (Salamon et al., 2002) to find itsoptimum. This last point is important because our problem liesin a high-dimensional space of unknown dimension and is alsonon-convex. Thus, this global optimization tool is well adaptedto our case as it finds the minimum of the functional withoutbeing trapped in local minima. It also allows us to find an optimalconfiguration of objects without any initialization prior on theobject localizations or even their number. External knowledge canalso be introduced, which allows the modeling of the layout of thebuildings.In our work, building footprints are extracted from digital

elevation models with various resolutions (from 10 cm to 1 mGSD). For buildings, a good geometric shape descriptor coveringthis range of resolutions is the rectangle. Even if buildings aremorecomplicated than a simple rectangle, they can be described by aunion of rectangles which is able to generate complex structuresand thus handle most of urban built areas (Fig. 1).The main drawback of the approach is its computational

efficiency. Ortner’s energy is composed of many terms and is thuscomputationally intensive. Among its many parameters, some arerelated to physical values (resolution, lengths, . . . ), some are not(weighting factors, number of profiles, . . . ). As all these parametershave to be tuned simultaneously, this is a major pitfall of theapproach. We focus in this paper on a way to simplify thisenergy, its parameters tuning and to improve the computationtime without loss of quality. This point is crucial when large areashave to be processed in a reasonable amount of time.

Fig. 1. Examples of buildings on a DEM (left) and on an image (right), 25 cm,Amiens, France. In the bottom of the images, buildings can be modeled by a groupof 3 separate rectangles; on the top, the city-hall building can be represented by aunion of rectangles (in this case, depending on the generalization, 3 or 5 rectanglesare required).© 2009, IGN

1.2. Global strategy and organisation of the paper

As aforementioned, our work is based on an already existingmethodology. Our aim is to propose an efficient way to use it andthe rest of the paper is dedicated to the resolution of this problem.First, we remind the foundations of the mathematical frameworkinvolved in the process and the optimization outline (Section 2).In Section 3, we give the essential details of Ortner’s energy andpropose our new formulation to solve the problem. Section 4presents the choices we made to implement our approach: theyconcern geometric implementation details and the data structuresinvolved. Section 5 presents a wide variety of results. They coverdifferent areas, and show the ability of the approach to deal withvarious resolutions and data sources. As input data, we will usea DEM, computed either from high resolution images or from a3D point cloud acquired with an aerial LiDAR. Finally, Section 6concludes our work and proposes future directions.

2. Mathematical background

This section presents the mathematical background on whichrelies the methodology developed in this paper. We begin withgeneralities (Sections 2.1 to 2.3) and then focus on the keypoints of the framework, namely the sampler and the optimizer(Sections 2.4 and 2.5).

2.1. Generalities

The principle of the approach presented in this work is basedon marked point processes, an object oriented approach. Let aconfiguration of objects be a collection of rectangles in imagespace, each of them representing a building part. Our aim is to findthe best configuration. To do so, an energy is built, using 2 terms:the data attachment term measures the consistency of the objectswith respect to the image, the regularization term favours specificlayouts of rectangles.Stochastic models have shown their versatility in various

domains of image processing and computer vision (van Lieshout,2000;Winkler, 2003). A particular family of stochastic approaches,namely marked point processes, originally developed in responseto various physical or biological problems, is very useful to detectobjects modeled with geometric shapes. First of all, they canhandle geometric constraints on the objects of interest and areable to manage very high dimension spaces, as this is the case

O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327 319

Fig. 2. (a) A realization of a point process onP ; (b) A realization of a marked pointprocess of rectangles on P × M; (c) An object ui of our marked point process isdescribed by its center ci = (xi, yi), its semi-major axis

→

vi and its aspect ratio ri =liLi.

in our application. They also allow an energetic formulation ofthe problem which can be seen as a functional minimization. Inremote sensing, they have been used by Lacoste et al. (2005) todetect linear structures like roads and rivers in satellite images.In a forestry context, Perrin et al. (2006) uses ellipses as objectsof a marked point process to delineate tree crowns from highresolution aerial images. Lafarge et al. (2008), using buildingfootprints as an input, proposes a stochasticmodeling for 3D urbanenvironments reconstruction.The mathematical background on which relies all those works

has already been well presented in the literature. We remind hereonly the basis of those principles. The reader interested in moredetails will find theoretical and practical aspects of marked pointprocesses in Cressie (1993) or Stoyan et al. (1996).

2.2. Definitions

Informally, a point process is a stochastic model governing thelocations of some events {xi} in a bounded set X (Fig. 2a). Moreformally, this a measurable application of a probability space into[N,P ]. Here, N represents the set of sequences of points in animage P = [0, xmax]× [0, ymax] locally finite (each finite subset ofP must contain only a finite number of points) and simple (xi 6= xjif i 6= j). Amarked point process Xt is a point processwhere each {xi}has an associated mark {mi} in a set M, thus allowing to definea geometric object ui = (xi,mi). A marked point process is thusdefined as a point process on P ×M (Fig. 2b).Building footprints in a DEM can be seen as the realization of a

marked point process of rectangles. For performance purposes (seeSection 4.3), we choose a parametrization of the rectangle whichdiffers from the classical representation (center, width, height,orientation; see Fig. 2c). Formally, it is described by its center ci =(xi, yi), a vector −→vi for the semi-major axis (i.e. ‖−→vi ‖ =

Li2 ) and

its aspect ratio ri =liLi≤ 1 where Li (resp. li) is the rectangle

dimension along (resp. across) −→vi . As we are only interested inrectangles contained in the DEM image support P , the compactsetM of the marks is:

M =

[−xmax2

,xmax2

]×

[−ymax2

,ymax2

]︸ ︷︷ ︸

−→vi

×]0, 1]︸ ︷︷ ︸ri

. (1)

2.3. From a density to an energy

Let us consider a marked point process Xt defined through itsprobability density f with respect to the law πν (.) of a Poissonprocess known as the reference process. f (.) can be defined in twoways: in a Bayesian framework, which requires to have a model ofheights in the whole DEM or through a Gibbs energy. We choosethe second form because of the complexity required to build aheight model valid for the whole area of interest. Thus, the densityf (.) of Xt is:

f (Xt) =1Ze−U(Xt ) (2)

where Z =∫Xte−U(Xt ) ensures normalization. Moreover, the

energy U (.) can be expressed as a weighted sum of an internalenergy measuring the spatial quality of the collection of objects(prior or internal field) and an external energy which qualifies thequality of the objects with respect to the underlying DEM (externalfield). The unnormalized formulation is then:

f (Xt) = e−[Uprior(Xt )+Udata(Xt )]. (3)Extracting building footprints consists in finding the configurationof objects X̃t maximizing the posterior probability X̃t =

argmax f (.). This is where the RJMCMC sampler and the simulatedannealing take place.

2.4. RJMCMC sampler

Classical MCMC methods such as Metropolis–Hastings cannothandle dimension jumps, i.e., changes in dimension betweensamples. RJMCMC, which consists in simulating aMarkov Chain onthe configuration space, is an extensionwhich allows themodelingof a scene with an unknown number of objects. Green (1995)proposed to use reversible jumps, i.e., a transition from one step ofthe chain to another is guided through a set of proposition kernels,the jumps. They enable to build a new configuration from a startingconfiguration, with the modification of an already existing object,or with adding or removing an object of the current configuration,thus involving a change in the dimension space. Algorithm 1describes the overall process (a complete description can be foundin Robert and Casella (2004)).

Algorithm 1: RJMCMC samplerInput: An initial configuration X0 (empty or random

configuration)repeat• Randomly choose a proposition kernel Qi according to adefined law• Build a new configuration Xnew from Xt w.r.t. theselected kernel• Compute the acceptance ratio R (Xt , Xnew)• Compute the acceptance rate α = min

(1, R (Xt , Xnew)

)•With probability

{α set Xt+1 = Xnew(1− α) set Xt+1 = Xt

until convergence;

2.5. Simulated annealing

The RJMCMC sampler is coupled with a simulated annealingin order to find the optimum of the density f (.). Instead of f (.),we use in the optimization process f (.)

1Tt , where Tt is a sequence

of decreasing temperatures which tends to zero as t tends toinfinity. Theoretically, convergence is guaranteed whatever theinitial configuration X0 is if the decrease of the temperature followsa logarithmic scheme. In practice this is impossible to use sucha scheme since it is too slow. Thus, it is generally replaced witha geometric decrease which gives a good solution close to theoptimal one. To estimate initial and final temperatures, we samplethe configuration space and consider the variance of the globalenergy (Salamon et al., 2002).

3. Energetic modeling

Formulation of the energies is the core of ourwork. This is one ofthemajor aspects of the simplicity and computational efficiency ofour implementation. Before presenting ourmodeling (Section 3.2),we describe howOrtner builds the terms of its energy (Section 3.1).

320 O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327

Classically, it is made up of a regularizing term (Section 3.1.1) anda data attachment term (Section 3.1.2).

3.1. Ortner’s energy

3.1.1. Regularizing termOrtner’s energy for the internal field is decomposed into 4

different interactions, computed thanks to the relative positionsof interacting rectangle corners. The first interaction regardsalignment of objects. Based on 2 thresholds, it aims at favouringsmall angular differences between neighboring objects. Theassociated energy is defined through a weighted sum of distancesbetween the corners and the angular difference. The second termis related to completion. As the data term detects discontinuitiesalong two opposite sides of the rectangle, its goal is to detectdiscontinuities in the orthogonal directions. It is also basedon the previous thresholds and favours orthogonal orientationsbetween the objects. Another term favours paving interactions,i.e. arrangements of objects along the side were the discontinuitylies. Finally, the last term aims at avoiding redundant objects, i.e.,objects with a high intersection surface.Once all these terms are defined, the final external energy is

made up of a weighted sum of the interactions. This involves 7parameters.

3.1.2. Data attachment termThe global data term is defined as a sum of 4 energies, each one

associated with a set of objects. To do so, a candidate rectangle issliced orthogonally to the main direction to study profiles. Threemeasures are then used to compute the data term. First, a hit lengthcounts the number of high gradients along profiles. A volume ratetaking into account the number of profiles, their width and lengthis also computed. Finally, a moment rate measures the averagedistance between a rectangle hypothesis and hit points.These measures are based on physical parameters and thresh-

olds. However, a lot of parameters generally imply a fine tuning.When the input data change, the parameters need to be tunedagain individually and in interaction. It is also quite uneasy toweigh the relative importance of the 3 terms in the final data term.As for the internal field, 7 parameters are involved in the compu-tation of the external field.

3.2. Proposed energy

We now present our choices to define in a simple and efficientfashion our energetic terms. To avoid parameters tuning andsensitive thresholds setting, we have drastically simplified theenergetic terms. The remaining parameters are also related tophysical quantities, which eases their tuning.

3.2.1. Regularizing termThe regularizing energy Uprior is used to favour some config-

urations of rectangles and to penalize some others. It requires thedefinition of a neighborhood system. Let us remind the general def-inition of a neighborhood. If∼ is a symmetric relation on P ×M,the neighborhood N (u, Xt) of an object u is defined as the set ofobjects in relation with u:

∀u ∈ Xt , N (u, Xt) = {v ∈ Xt | u ∼ v, u 6= v}. (4)

We choose to define the neighborhood system thanks to ageometric criterion. For an object ui, it is simply the set of objectsintersecting ui:

ui ∼ uj ⇔ ui ∩ uj 6= ∅. (5)

This choice is really natural, because a single building can berepresented by a single rectangle, that is to say, without anyinteracting object and thus an empty neighborhood. In the case

of building blocks, the arrangement of rectangles required torepresent such a structure is composed of contiguous or slightlyoverlapping objects (Fig. 1).To obtain the desired configurations, our energy is very simple

and has a single repulsive term. For a pair of neighboring objects uiand uj, it is simply their weighted intersection area Sinter:

Uprior(ui, uj

)= β.Sinter

(ui, uj

). (6)

So, this regularizing term is only repulsive. β is a weightingparameter which tunes the importance of prior energy versusdata energy (its physical meaning is discussed in Section 3.3).In our work, the attractiveness is only given through the dataattachment term. Furthermore, in order to avoid accumulationof objects where the internal energy is favourable, the externalenergy defined through Eq. (6) is perfectly designed. Indeed, thehigher the overlap area between objects is, the higher the energy is,and therefore, this kind of configuration tends to be rejected as theprocess evolves. Furthermore, intersection area is not normalizedas we want to keep a physical interpretation of the energy (andnot a proportion); this is also interesting as small objects are lessrelevant than bigger ones (small ones should take into account theposition of big ones).We think that it is not necessary to design a complex system of

interactions and, in fact, experience has proven that if the externalterm is well designed, a simple internal term is sufficient andmodels the problem quite well. For instance, it is not necessary tointroduce an alignment interaction, because the modeling of thedata term already includes this property (Section 3.2.2).Finally, for a given Xt , the global internal energy is given by the

sum of individual terms for each pair of neighboring objects:

Uprior (Xt) =∑i6=j

Uprior(ui, uj

). (7)

3.2.2. Data attachment termWe now focus on the external energy term which aims at

measuring the adequacy of objects with respect to the DEM. Thissection explains the choices we made in order to detect buildingfootprints, that is to say how we define an attractive object.As we aim at finding building footprints from DEMs, which

are rasterized height data, we are naturally looking for altimetricdiscontinuities. The higher the discontinuity is, the more likely theprobability to find the edge of a building at this place is. Fromthis, we obviously use a measure based on the DEM gradient anddefine it in order to align rectangles edges with the highest heightvariations in the DEM. Large height variations are representedby high values of the gradient magnitude. To test if a rectanglematches DEM discontinuities, we use the dot product of thegradient vector and the normal vectors attached with the edges ofthe rectangle (Fig. 3).Considering a rectangle edge segment [a, b], its outgoing

normal −→n =−→ab⊥‖ab‖ from a to b (where

⊥ denotes a π2 vector

rotation), its external energy is:

Udata (a, b) =∫‖ab‖

0

(−→n ·−→∇

(a+ t

−→ab‖ab‖

))dt (8)

where−→∇ (p) is the gradient magnitude at pixel p. This can be

reformulated, using orthogonal vectors:−→∇ ·−→n =

−→∇⊥·−→n ⊥ =

−→∇⊥·−→ab‖ab‖ .

Udata (a, b) =−→ab ·

(∫ 1

0

−→∇

(a+ t−→ab)dt)⊥. (9)

For efficiency, the integral is discretized using a nearest neighborscheme, where the (tk) correspond to the segment [a, b] crossinga pixel boundary at a point pk = a + tk

−→ab , such that tk is

O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327 321

Fig. 3. Explanation of the external energy for a segment [pji, pj+1i ] of a given

rectangle. The grid stands for the DEM pixels; the segment [pji, pj+1i ] begins at p

ji ,

ends at pj+1i and crosses 8 pixels of the DEM (represented with a dark gray outline).

increasing, t0 = 0 and tn+1 = 1.−→∇ k denotes the constant value

approximation of−→∇ along the segment [pk, pk+1]:

Udata (a, b) =−→ab ·

(n∑k=0

(tk+1 − tk)−→∇ k

)⊥ (10)

Udata (a, b) is the dot product of the edge vector−→ab and the average

gradient along the edge∫ −→∇ . It is thus linear with the edge length,

and with the mean gradient magnitude (corresponding to theaverage façade height), yielding a measure of a façade surface.Note that the integral formulation and the discretized (tk+1 − tk)weighting ensures that the measure is isotropic. It further takesinto account that the orientations of the vectors to only measurethe gradient outgoing the rectangle edges. It thus favours rectangleedges alignment with large height variations in the DEM. For anobject ui, its external energy is the sum of the external energies ofthe four individual segments [pji, p

j+1i ] subtracted from a constant

wdata:

Udata (ui) = wdata −3∑j=0

max(0,Udata

(pji, p

j+1i

)). (11)

This constant wdata is used in order to give to each rectangle aninitial weight. This weight can be considered as theminimal façadesurface required for a building to be detected and it depends onthe desired level of generalization. The data term thus becomesnegative when there is enough gradient across the rectangleedges. However, setting this parameter, as it has a straightforwardphysical interpretation, is easier than setting an obscure weightingparameter.Another notable aspect of this energy is that, using themax(0, ·)

expression, only segments of the rectangles that have a positivediscontinuity support are taken into account, i.e., the orientationof the discontinuity must match the segment outgoing normal.Finally the global external term of the energy is:

Udata (Xt) =∑ui∈Xt

Udata (ui) . (12)

3.3. Contributions of our energy definition

Before ending this section, we would like to sum up thecontribution of our work with regard to that of Ortner. Thissimply consists in comparing our energies, the overall number ofparameters to adjust and their meaning.Our regularizing term is only based on the overlapping area

of the neighboring objects. This is a real simplification since thisformulation has a physical meaning. Thus, our prior energy is onlyrepulsive and aims at avoiding superposition of objects. In fact,it is not necessary to develop a complex system of interactions,because the data term fulfills the attractive part of the globalenergy. However, as buildings are not composed of parts perfectly

aligned or perpendicular, slightly overlapping objects must notbe too much penalized. This is the purpose of the β parameterdefined in Eq. (6). It allows to tune, with respect to the surface ofthe intersection area, the level of penalization for an overlappingconfiguration. More precisely, it weighs the overlapping surfaceversus the façade surface. So a value of 10 means that a rectangleneeds 10m2 of façade to compensate 1m2 of overlap.The other part of the global energy, the data attachment term is

only based on a simple observation: building edges are orthogonalto the high gradient of the DEM. Thus, it can be simply definedas a quantity depending on the amount of gradient orthogonal toeach edge of the rectangles. Moreover, we also have simplified thedefinition and the tuning of the parameters since there is onlyone physical parameter, wdata, which allows to tune the level ofgeneralization. The necessary highlighted points in Ortner et al.(2007) for its data term are:• an object is attractive if it has enough gradient along its edges,• a non-attractive object is slightly repulsive,• the better the object is aligned with discontinuities, the betterits energy is.

In our implementation, they are all respected with an efficientto compute and simple definition of the energy. Moreover, bothterms express areas related to building: its footprint and its façadesurface. We thus do not have heterogeneous quantities which areuneasy to compare and to weigh each other.

4. Implementation

Previous sections were dedicated to an energy-based definitionof ourmarked point process for building outline detection.Wenowexpress our choices for its effective and generic implementationwhich relies on two crucial points: data-processing structure andgeometric optimization.Althoughwe are looking for an effective approach, wewant it to

be totally generic asweusemarkedpoint process in other contexts.No assumptions have been made on the geometric properties ofobjects (this code would work for segments or circles) nor on thedimensions of the exploration space (this code would work forstudying a point cloud).

4.1. Proposition kernels

We first detail the simple transition kernels we use in theRJMCMC sampler. The first ones, birth and death kernels arenecessary to change to higher or lower dimension and guaranteethat the Markov Chain visits the whole configuration space (Geyerand Møller, 1994). Birth (resp. death) simply consists in adding(resp. removing) a random object from (resp. to) the currentconfiguration.The perturbation kernel, allows to modify the mark of an

object. Although it is not necessary to ensure convergence, itsuse is strongly recommended in the optimization process toavoid consecutive birth and death steps. It not only increasescomputational efficiency but also enables to finely position objectsat the end of the process when all of them are detected but justneed to be adjusted in shape or position.Our implementation uses only 2 perturbations. The left

perturbation of Fig. 4 translates an edge of the rectangle: 2 cornersremain fixed, while the center and the aspect ratio are modified.The right perturbation combines a rotation and a similarity arounda rectangle corner: the chosen corner and the aspect ratio remainfixed whereas the center and the 3 other corners are modified.

4.2. Data-processing structure

Remember that an object configuration is made up of interact-ing objects, i.e., each object has a set of neighbors (which couldbe empty). The Markov property naturally leads to use a graph as

322 O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327

p'3

p'1= p1

p'2

p2

p2

p2p1

p0p0

p0

p3

p3

p3

Fig. 4. Examples of some perturbations of a rectangle (the original rectangle is inblack, the transformed one is in dashed red). (left) Translation of an edge. (right)Combination of rotation and similitude around one of the corner. The translationvector (see Section 4.3.1) is applied to the red corner.

Fig. 5. The neighborhood system. Each rectangle is associated to a node (red).Each rectangle has a set of neighbors connected by the graph’s edges (blue). (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)© 2009, IGN

data-processing structure to store a configuration andmaintain the∼ relation between the objects (see Fig. 5). Indeed, an object ui ofa configuration can be seen as a node, and a neighborhood relationbetween 2 objects is easily represented by an undirected edge.Indeed, we use a valued graph where each node models

an object ui of the configuration (storing its position in imagecoordinates, marks and data energy Udata(ui)), and each edgerepresents a pair of neighboring objects. Edges are undirected, the∼ relation being symmetric, and valued by the regularizing energyUprior(ui, uj).This graph allows an efficient updating of the running overall

energy when the object configuration is modified. The local graphmodification of a birth kernel amounts to create a node and aset of adjacent edges, caching all the corresponding unary andbinary energy terms. They are then readily available to decreasethe overall energywhen a node and its adjacent edges are removed(death kernel). The perturbation kernel is finally applied to thegraph by successively applying a death and a birth kernel.

4.3. Geometry

The computational bottleneck of the algorithm is its geometricroutines. We thus propose an optimized parametrization yieldingefficient computations.

4.3.1. Rectangle representationA rectangle is represented (Fig. 6) by its center point c , a vector

−→v from the center to the middle of one of its smaller sides and itsaspect ratio r (i.e. the ratio of the length l of the side orthogonal to−→v to the length L of the side parallel to−→v ).

Fig. 6. The rectangle (c,→

v , r).

Fig. 7. Intersection area of R1 and R2 is eased by considering the bounding box R3of R2 aligned with R1 .

Contrary to the usual representation with the center, anorientation angle and the two side lengths L and l, which is alsominimal, our representation avoids the time consuming squareroots and trigonometric functions, limits the use of divisions, andotherwise only use additions, subtractions and multiplications,which is the key of an efficient handling of 2D rectangles. Usingthe notations v2 = −→v · −→v and−→v ⊥ = (−vy, vx) for a π2 rotation:

corners = c ±−→v ± r−→v ⊥ and area = rv2.

4.3.2. Rectangle perturbationsThe proposed representation enables low cost perturbations

that do not require any square root or trigonometric operation.Pure translations affect only c , while combinations of rotationsand isotropic scaling only involve−→v . The proposed perturbationsinvolve translating a rectangle edge while keeping its oppositeedge fixed, and a composition of a isotropic scaling and a rotationof the rectangle around one of its corners as illustrated in Fig. 4.They are conveniently performed on a rectangle (c,−→v , r) usingthe following transformations, where (ε, ε′) ∈ {−1, 1}2 selects thefixed corner or edge:

Rotation/scaling → (c + ε−→w + ε′r−→w ⊥,−→v +−→w , r) (13)

Edge translation →ε′=−1

(c + ε(t − 1)−→v , t−→v ,

rt

)(14)

→ε′=1

(c + ε(t − 1)r−→v ⊥,−→v , rt). (15)

Note that translating an edge using one of the equations abovemayyield an aspect ratio r > 1. This is simply fixed by using the alter-native equivalent representation (c, r−→v ⊥, 1r ). The translation per-turbationmoves 2 corners by 2ε(t−1)−→v or 2ε(1−t)r−→v ⊥ and therotation/scaling one moves the 3 unfixed rectangle corners by 2−→wand/or 2r−→w ⊥. It is then straightforward to limit the magnitude ofthe corner translations. This upper bound on the corner translationmagnitude is expressed in meters, and is therefore not sensible tochanges in resolution. The experiments we carried out have shownthat these perturbations are sufficient, so that, for instance, purerotations and translations are not needed.

O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327 323

Table 1Cumulative operation counts, assuming (v2i ) and (riv

2i ) values are cached, which

saves 2 additions and 6 multiplications. The intersection test returns as soon as aseparating axis is found, so that the exact operation count among the last 4 columnsdepends on the which axis, if any, is found to be a separating axis.

Operation counts λ,µ,−→c12 −→v1 ? −→v2 ? −→v1

⊥? −→v2⊥?

| · | absolute values 2 3 4 5 6+ additions 4 7 10 13 16×multiplications 4 7 10 13 16

4.3.3. Intersection testTheproposed representation allows to efficiently testwhether a

point−→p is inside a rectangle (c,−→v , r). First, the point is expressedin a frame centered at c and oriented along −→v : x = (

−→p − c) ·−→v and y = (

−→p − c) · −→v ⊥. Then −→p is inside the rectangleif and only if |x| ≤ v2 and |y| ≤ rv2. The energy functionalneeds to check efficiently whether 2 rectangles R1 and R2 intersectand, if so, compute their intersection area. Since rectangles areconvex polygons, the Separating Axis Theorem states that they donot intersect if and only if a line supporting one of the 8 sides of the2 rectangles separates the 2 rectangles. A naive implementationtests, for each side s of a rectangle, whether the 4 points of theother rectangle are on the other side of the supporting line of s. Thesymmetries of the problem allow amore efficient implementation:computing only once the vector −→c12 = c2 − c1 between rectanglecenters and the absolute values: λ = |−→v1 · −→v2 |, µ = |−→v1 · −→v2⊥|.For instance, the corners of R2 are on the side of the line segment[c1 + −→v1 − r1−→v1⊥, c1 + −→v1 + r1−→v1⊥] opposite to R1 if and only if∀(ε, ε′) ∈ {−1, 1}2:(c2 + ε−→v2 + ε′r2−→v2⊥

)·−→v1 >

(c1 +−→v1

)·−→v1

⇔−→c12 · −→v1 > v21 + λ+ r2µ. (16)

The opposite edge of R1 yields the similar expression (−−→c12) ·−→v1 >

v21+λ+ r2µ, so that they can be tested together using the absolutevalue

∣∣−→c12 · −→v1 ∣∣. Finally, the rectangle intersection is tested usingonly 4 quantities, for separating axes along ±−→v1 , ±−→v2 , ±−→v1⊥ and±−→v2⊥:

Intersects(R1, R2)=∣∣−→c12 · −→v1 ∣∣ ≤ v21 + λ+ r2µ&∣∣−→c12 · −→v2 ∣∣ ≤ v22 + λ+ r1µ&∣∣−→c12 · −→v1⊥∣∣≤ r1v21 + µ+ r2λ&∣∣−→c12 · −→v2⊥∣∣≤ r2v22 + µ+ r1λ.

(17)

The efficiency of this rectangle intersection test is measured bythe low number of operations involved and more importantly theabsence of square roots or trigonometric operations (Table 1).

4.3.4. Intersection areaTo compute the intersection area of 2 rectangles R1 and R2,

the bounding box R3 of R2 aligned with R1 is considered. Fig. 7shows how R3 may be partitioned into 4 right triangles Ti and therectangle R2. The area of intersection between 2 aligned rectanglesR1 and R3 is trivial. Likewise, the ones between the rectangle R1and a right triangle Ti, which edges adjacent to the right angleare aligned with R1 may be computed efficiently, using techniquessimilar to the rectangle intersection test of the previous section.(R2, T1, T2, T3, T4) being a partition of R3, the intersection area ofR1 and R2 is computed as:

area(R1 ∩ R2) = area(R1 ∩ R3)−4∑i=1

area(R1 ∩ Ti). (18)

If area(R1 ∩ R3) = 0 (R1 and R3 do not intersect) or if area(R1 ∩R3)−

∑ki=1 area(R1∩ Ti) vanishes before k = 4, then the area(R1∩

Ti) terms for i > k have to be zero and their computation is

Fig. 8. Iterating over the pixels crossed by a segment.

Ene

rgy

0 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06Iterations

EnergySamples of the energy

30000

25000

20000

15000

10000

5000

0

-5000

35000

-10000

Fig. 9. The curve shows the evolution of the energy. The samples denoted by dotscorrespond to the superimposed configurations.

thus avoided, yielding a faster answer to simpler (more separated)problems.

4.3.5. Segment iterationFig. 8 illustrates that the segment length between 2 vertical

(resp. horizontal) line crossings is a constant dtx (resp. dty). Theiteration over the pixels crossed by a segment can be computedefficiently by computing various quantities as a preprocess. Forinstance, after crossing a vertical line, the next vertical line crossingis computed by adding dtx. The next line crossed is along thedirection that has the closest line crossing length t .Such a scheme is particularly interesting to compute efficiently

the data term of the energy.

5. Experimental results

In our experiments, we used aerial (50 cm GSD1; Fig. 13)satellite (50 cm GSD; Fig. 10) and LiDAR (50 cm GSD; Fig. 11)DEMs. Satellite and aerial DEMs were obtained with a graph-cut optimization approach in a multiview framework (Roy andCox, 1998; Pierrot-Deseilligny and Paparoditis, 2006). Imageswereacquired by the 16 mega-pixels Institut Géographique National(IGN) digital camera. Inter and intrastripes overlaps are about 60%.The Toposys r© LiDAR scanner was used for the acquisition of 3DLiDAR point clouds. DEMs have then been computed from the 3DLiDAR points using a simple rank filter. In this section, all givencomputation times were obtained on a linux Ubuntu 9.10 PC, withan Intel r© CoreTM 2 Duo CPU T7700 @ 2.50 GHz.

1 Ground Sampling Distance.

324 O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327

Fig. 10. Comparison of the results obtain with our strategy (left) and with Ortner’s strategy (middle). The right image is a direct comparison: in dark gray our results alone,in medium gray Ortner’s results alone, in light gray the common part of both results (see also Table 2 for a quantitative study).

Fig. 11. Comparison of the results obtain with our approach on a satellite (left) and LiDAR (middle) DEMs. The right image is a direct comparison: in dark gray our resultsalone, in medium gray the LiDAR results alone, in light gray the common part of both results (see also Table 3 for a quantitative study).

Table 2Quantitative comparison of building/non-building classification with both strate-gies. Percentages express the rate of pixel in each category w.r.t. the total numberof pixels.

Our methodologyBuilding Not building Total

Ortner’s Building 35.5% 11.2% 46.7%Methodology Not building 4.7% 48.6% 53.3%

Total 41.3% 59.8% 100%

5.1. Qualitative results

We first illustrate the evolution of the detection at differentsteps of the algorithm on a sample building (Fig. 9). At thebeginning of the process, there are many objects, not well located.500000 iterations later (a few seconds), some objects begin to findtheir correct location. Then, as long as the temperature decreases,more and more objects are accurately positioned, and finally,all objects are well detected and located, until convergence. Theassociated energy graph is presented on the same figure withhighlighting representative samples.Fig. 10 compares our results with Ortner’s ones on a bigger

area. In both cases, all building parts are well detected andrectangle edges correspond most of the time to true buildinglimits. False alarms exist in both results and are mainly due totree alignments or trees in courtyards, but, could be avoided forinstance by computing a vegetation mask. Both approaches alsoexhibit that some rectangles straddle the streets or embrace 2buildings. However, this is not prejudicial, because a further stepcan deal with these cases to obtain a more relevant descriptionof the buildings by splitting existing footprints. More generallyour results tend to be less broken up, that is to say that for agiven building part, we have only one rectangle, whereas Ortner’s

Table 3Quantitative comparison of building/non-building classification with satellite andLiDAR DEMs. Percentages express the rate of pixel in each category w.r.t. the totalnumber of pixels.

Satellite DEMBuilding Not building Total

Building 34.3% 5.6% 39.9%LiDAR DEM Not building 5.8% 54.3% 60.1%

Total 41.1% 58.9% 100%

results tend to superimpose some rectangles or divide a coherentbuilding part in several rectangles. This is due to the attractivepaving interaction that we removed from our regularization termformulation. The last point to notice concerns the undetectedbuildings. Most of the time, they are the small ones in innercourtyards: the height discontinuity between the roof and theground is too weak to be detected. They also exist in our approachwhen a building part is too small (‘‘9’’-shaped building at the centerof the image) and can be explained by a too high value ofwdata.In the same way, Fig. 11 compares the results obtain with

satellite and LiDAR DEMs on the same area as Fig. 10. The LiDARDEM comes from an aerial based point cloud (discontinuities aresharper and there is less noise on the ground). The results are notsignificantly different (see Table 3), which demonstrates the abilityof the algorithm to successfully operate on various data sources.The aforementioned problems are also encountered:missing smallbuildings, straddling of the streets, single rectangle for multiplebuildings, trees, . . .The last test is on a building with a complex shape (Fig. 12).

Indeed, it is not regular and only a few parts are rectangular, ahigher block is in the center and most of his layout is curved. Ourapproach tends towards a paving of the building, without overlap.Thus, the building cannot be fully covered by rectangles. Ortner’s

O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327 325

Fig. 12. Comparison of both approaches on a complex building (left: ours; right:Ortner’s).© 2009, IGN

Fig. 13. Result obtained with or approach on a 20 cm aerial DEM.© 2009, IGN

approach, offers a better coverage since the whole building ispaved with rectangles. This is an advantage of his prior energydefinition. However, neither of the approaches is able to handle thecurve shape of the footprint, naturally because our marked pointprocess involves rectangles.

5.2. Quantitative results

Figs. 13 and 14 are detailed areas on an aerial DEM with a20 cm GSD (St-Mandé, France). We used the result presented onFig. 13 as a basis for a comparison with a ground truth (comingfrom a digitized cadastral map) and with Ortner’s results. Bothapproaches are compared in Table 4 using classical indicators.2They show that they are equivalent. The high false alarm ratecan mostly be explained by the detection of the trees, whichcan be avoided with a vegetation mask. Indeed, we do not havesuch a mask and the ground truth is more accurate than theDEM itself. Thus, these results are relevant for comparing bothmethods, but not to evaluate the general quality of the approach.To test the reliability of the method, we have manually removedthe rectangles corresponding to trees in our result (last line ofTable 4). False alarm and quality rates are clearly improved. Thesmall decrease of the detection rate corresponds to a rectanglewhich embraced trees and small buildings in the original result.On the building block presented in Fig. 14, our approach

successfully detects its main parts. However, some small parts andsome other with a low height above the ground are undetected.This can be explained by the definition of our data energy whichmeasures the total amount of gradient outgoing the rectangleedges. The small parts that our approach does not detect do

2 detection rate = TPtotal , false alarm rate =

FPtotal and quality rate =

TPFP+total ,

where TP denotes pixel-wise true positive, TP false positive and total is the numberof building pixels in the reference.

Table 4Quantitative comparison between our work and Ortner’s approach on the resultpresented in Fig. 13. Results are clearly equivalent.

Rates (%)Detection False alarm Quality

Our methodology 86.1 34.9 63.8Ortner’s methodology 88.3 45.8 60.5

Our methodology (without trees) 83.8 14.3 73.3

Fig. 14. Result obtained with our approach on a 20 cm aerial DEM. See also Table 4© 2009, IGN

not have enough gradient. If wdata is decreased, buildings willbe detected but as an undesired aggregate of multiple smallrectangles. However, if we accept a small degree of generalization,the results are still reliable.

5.3. Timing and parameters

Table 5 compares computation times of both approaches for thepresented results. Our approach clearly provides faster results. Theratio can achieve a factor of 100 in our favour. We have also runa test on a 1 km2 area to have an idea of the computation timeneeded for processing awhole city. The computation complexity ismore dependent on the number of objects than on the image size.However, such a large area contains enough buildings to providea relevant idea of what we could expect while processing a wholecity. In our test, we used a 50 cm GSD aerial DEM (2000 × 2000pixels) on a dense urban area with almost 1100 objects. The resultwas obtained in about 50 min.In our implementation, only a few parameters have to be set.

Regarding the energies, the weighting β of the intersection areabetween 2 objects (Eq. (6)) has always been set to 10 in our tests.A higher value avoids intersection but does not change the finalresult: objects overlap less, but the general layout of the results issimilar (see Fig. 15).A more important parameter iswdata (Eq. (11)) which gives the

minimal façade surface that a building must have to be detectedby our algorithm. Indeed, it must be large enough to avoid thedetection of very small structures mainly due to noise in the DEM,or isolated trees. However, if it is too large, it is more robust to theprevious problems, but, small building parts will not be detected.The effects of different values ofwdata on the results are presentedin Fig. 16.

6. Conclusion and future works

We have presented in this paper a reliable method to extractbuilding footprints from a DEM. Our aim is to provide an efficient

326 O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327

Table 5Computation times of the compared approaches. Our methodology provides a gain factor varying from 80 to 100.

Image size ] objects Computation times ] iterationsOur methodology Ortner’s methodology Our methodology

654× 665 (Fig. 10) '100 '120 s '4 h '5 000 000435× 263 (Fig. 11) ' 90 ' 60 s N/A '5 000 000758× 1359 (Fig. 13) 30 '100 s '3 h '4 000 000978× 1149 (Fig. 14) 11 '100 s '3 h '3 500 000

Fig. 15. Influence of β parameter (left, β = 10; right, β = 100). Results are strictlyequivalent in terms of energy, even if the layout is not the same.

implementation with regards to the quality of the results, thecomputation time and the simplicity of parametrization. Resultsand their associated computation times clearly show that the goalsare reached.This efficiency relies on 2 aspects. The first one is the definition

of the energy to minimize, the second one are the choicesand the efforts made in the implementation. Regarding theenergy, we choose a very simple definition while protecting theessential aspects allowing to describe what a building footprintin a DEM is. The simplicity of our definitions leads to veryefficient computation times and simple parametrization, whichis our primary goal. The efficiency of our approach also comesfrom our implementation of the algorithm. On the one hand,we have an original parametrization of the base object of theprocess, the rectangle, allowing to transform each object with aminimumof operations and avoiding the high costs of square rootsand trigonometric functions. On the other hand, the underlyinggraph structure of a configuration allows to efficiently apply theconfiguration changes.It is also important to note that the both terms of our energy are

homogeneous in the sense that they both measure an area. This

gives our formulation an easy interpretation and avoids mixingseveral quantities such as angles, distance or areas.Extracting footprints is the first step towards building 3D

reconstruction. However, there is an intermediate step betweenthese two stages in order to obtain a fine representation of the roofridge tiles. An approach has already been presented in Lafarge et al.(2006)which consists in detecting discontinuities in the rectanglesand merging rectangles into connected polygons. However, thealgorithm is only based on the geometric properties of the objects.This could be improved by using the DEM to find polygon edgescorresponding to DEM discontinuities. It could also be interestingto improve the simulated annealing parametrization. In fact, at thebeginning of the process, when temperature is high, most of themoves are accepted. As long as the process evolves, birth and deathare less and less accepted, and perturbations become dominant.To further improve computation times and propose only relevantmodifications of the configuration at the end of the process,we could detect when the optimization only consists in finelypositioning the objects (for example when the acceptation ratioare small enough for both birth and death on the past iterations).Hence, once this step detected, we will be able to propose onlydata-driven modifications, i.e., objects close to already detectedobjects, without intersection. We think this approach could beefficient for our purpose.There is also much room for parallelisation. A multithreaded

implementation has been prototyped that concurrently proposesand evaluates modification kernels, with atomic kernel modifi-cations of the underlying graph data structure. First results arepromising and need further investigation.More theoretically, stochastic annealing or jumpdiffusion could

also be interesting orientations.

Acknowledgements

The work reported in this paper has been performed aspart of the Cap Digital Business Cluster Terra Numerica project.The authors would also like to thank J. Zerubia and M. Ortner(ARIANA/INRIA) for providing the executable binary used tocompare with our results.

Fig. 16. Influence ofwdata parameter. From left to right and top to bottom,wdata value is 25, 100, 250 and 1000 m2 . For too small a value, too many rectangles are detected(noise in the DEM), and, as the value increase, the level of generalization becomes higher.

O. Tournaire et al. / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 317–327 327

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