Gilles Simon
email: gsimon@loria.fr

Antoine Fond
email: afond@loria.fr

Marie-Odile Berger
email: berger@loria.fr

A-Contrario Horizon-First Vanishing Point Detection Using Second-Order Grouping Laws

Keywords: Horizon line, Vanishing point detection, A-contrario model, Perceptual grouping, Gestalt theory, Man-made environments

We show that, in images of man-made environments, the horizon line can usually be hypothesized based on a-contrario detections of second-order grouping events. This allows constraining the extraction of the horizontal vanishing points on that line, thus reducing false detections. Experiments made on three datasets show that our method, not only achieves state-of-the-art performance w.r.t. horizon line detection on two datasets, but also yields much less spurious vanishing points than the previous top-ranked methods.

Introduction

Accurate detection of vanishing points (VPs) is a prerequisite for many computer vision problems such as camera self-calibration [START_REF] Wildenauer | Robust camera self-calibration from monocular images of Manhattan worlds[END_REF], single view structure recovery [START_REF] Lee | Geometric reasoning for single image structure recovery[END_REF], video compass [START_REF] Kosecka | Video compass[END_REF], robot navigation [START_REF] Lu | Automatic building exterior mapping using multilayer feature graphs[END_REF] and augmented reality [START_REF] Fond | Facade Proposals for Urban Augmented Reality[END_REF], among many others. Under the pinhole camera model, a VP is an abstract point on the image plane where 2-D projections of a set of parallel line segments in 3-D space appear to converge. In the Gestalt theory of perception [START_REF] Desolneux | From Gestalt Theory to Image Analysis: A Probabilistic Approach[END_REF], such a spatial arrangement of perceived objects is called a grouping law, or a gestalt. More specifically, as a 2-D line segment (LS) is in itself a gestalt (grouping of aligned points), a VP is qualified as a second-order gestalt [START_REF] Desolneux | From Gestalt Theory to Image Analysis: A Probabilistic Approach[END_REF].

In this paper, we are interested in VP detection from uncalibrated monocular images. As any two parallel lines intersect in a VP, LSs grouping is a difficult problem that often yields a large number of spurious VPs. However, many tasks in computer vision, including the examples mentioned above, only require that the vertical (so-called zenith) VP and two or more horizontal VPs (hVPs) are detected. In that case, a lot of spurious VPs may be avoided by first detecting the zenith and the horizon line (HL), and then constraining the hVPs on the HL. The zenith is generally easy to detect, as many lines converge towards that point in man-made environments. However, until recently, the HL was detected as an alignment of VPs, in other words, a third-order gestalt. This led to a "chickenand-egg" situation, that motivated e.g. the authors of [START_REF] Tretyak | Geometric image parsing in man-made environments[END_REF], to minimize an overall energy across the VPs and the HL, at the expense of a high computational cost. Following [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF], we show that, as soon as the HL is inside the image boundaries, this line can usually be detected as an alignment of oriented LSs, that is, a second-order gestalt (at the same perceptive level as the VPs). This comes from a simple observation, that any horizontal LS at the height of the camera's optical center projects to the HL regardless of its 3-D direction (red LSs in Fig. 1). In practice, doors, windows, floor separation lines but also man-made objects such as cars, road signs, street furniture, and so on, are often placed at eye level, so that alignments of oriented LSs around the HLs are indeed observed in most images from urban and indoor scenes. Going one step further than [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF], we effectively put the HL detection into an a-contrario framework. This transposal along with other improvements allows us to obtain top-ranked results in terms of both rapidity of computation and accuracy of the HL, along with more relevant VPs than with the previous top-ranked methods.

Related Work & Contributions

There is a vast body of literature on the problem of VP detection in uncalibrated images. [START_REF] Kosecka | Video compass[END_REF] use an Expectation-Maximization (EM) algorithm, which iteratively estimates the coordinates of VPs as well as the probabilities of individual LSs belonging to a particular vanishing direction. Although EM is often sensitive to initialization, a very rough procedure is used for this step. Several attempts have been made to obtain a more accurate initialization. [START_REF] Tardif | Non-iterative approach for fast and accurate vanishing point detection[END_REF] estimate VP hypotheses in the image plane using pairs of edges and compute consensus sets using the J-linkage algorithm. The same framework is used in [START_REF] Xu | A minimum error vanishing point detection approach for uncalibrated monocular images of man-made environments[END_REF], though a probabilistic consistency measure is proposed, which shows better performance. [START_REF] Wildenauer | Robust camera self-calibration from monocular images of Manhattan worlds[END_REF] present a RANSAC-based approach using a solution for estimating three orthogonal VPs and focal length from a set of four lines, aligned with either two or three orthogonal directions.

All these methods have been compared on the same datasets (DSs), York Urban (YU) [START_REF] Denis | Efficient edge-based methods for estimating manhattan frames in urban imagery[END_REF] and Eurasian Cities (EC) [START_REF] Tretyak | Geometric image parsing in man-made environments[END_REF] (see Section 5) and with the same protocol of [START_REF] Tretyak | Geometric image parsing in man-made environments[END_REF]. It is difficult to establish a ground truth (GT) for the VPs, as selecting relevant VPs among hundreds e.g. in an urban scene is a subjective task. For that reason, the evaluation in [START_REF] Tretyak | Geometric image parsing in man-made environments[END_REF] is focused on the accuracy of the HL. It is easy to show that the HL is orthogonal to the zenith line (ZL), which is the line connecting the principal point (PP) and the zenith (Fig. 1). The HL can then be found by performing a 1-D search along the ZL, and a weighted least squares fit, where the weight of each detected VP equals the number of corresponding lines. The horizon error is then defined as the maximum Euclidean distance between the estimated HL and the GT HL within the image boundaries, divided by the image height. To represent this error over a DS, a cumulative histogram of it is plotted. The plots are reported in Fig. 5. [START_REF] Vedaldi | Self-similar sketch[END_REF] also proposed to report a numerical value as the percentage of area under the curve (AUC) in the subset [0, 0.25] × [0, 1]. These values are also reported in Fig. 5. It can be seen that each new method improves the accuracy, from 74.34% AUC for YU and 68.62% for EC with the earliest method of [START_REF] Kosecka | Video compass[END_REF] to 93.45% for YU and 89.15% for EC with the state of the art method in 2013 [START_REF] Xu | A minimum error vanishing point detection approach for uncalibrated monocular images of man-made environments[END_REF].

A-Contrario Methods. Some authors proposed to detect meaningful VPs in the sense of the Gestalt Theory. This theory was translated by Desolneux et al. into a mathematics and computer vision program [START_REF] Desolneux | From Gestalt Theory to Image Analysis: A Probabilistic Approach[END_REF]. According to the Helmholtz principle, which states that "we immediately perceive whatever could not happen by chance", a universal variable adaptable to many detection problems, the Number of False Alarms (NFA) was defined. The NFA of an event is the expectation of the number of occurrences of this event under a white noise assumption. From this variable, a meaningful event in the phenomenological sense can be detected as a so-called -meaningful event, namely an event whose NFA is less than . Most problems of computer vision have been solved efficiently by simply setting to 1. When ≤ 1, the event is said meaningful. The Helmholtz principle has been applied to the VP detection problem in [START_REF] Almansa | Vanishing point detection without any a priori information[END_REF] and [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF]. In [START_REF] Almansa | Vanishing point detection without any a priori information[END_REF], a practical application of the Santaló's theory [START_REF]Integral Geometry and Geometric Probability[END_REF] is used to partition the infinite image plane into a finite family of so-called vanishing regions. Meaningful VPs are then detected from large votes of lines meeting in a vanishing region, thus producing low NFA. Although this method was only qualitatively assessed, it presents interesting matter for the building of our own method, and especially the use of the Santaló's theory. In [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF], a point alignment detector based on the Helmholtz principle [START_REF] Lezama | A Contrario 2D Point Alignment Detection[END_REF] is used twice: in the image domain, to group LSs into more precise lines, and in dual domains where converging lines become aligned points. This method achieved state-of-the-art accuracy in 2014 (94.51% for YU, 89.20% for EC).

Horizon-First VP Detection. Horizon-first VP detection was simultaneously introduced in two recent works [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF][START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], both based on the same principle: propose candidate HLs, score them, and keep the best. In [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], a deep convolutional neural network (CNN) is used to extract global image context and guide the generation of a set of candidate HLs. For each candidate, they identify VPs by solving a combinatorial optimization process (see Section 5.3) followed by a constrained nonlinear optimization. The final score for each candidate HL is based on the consistency of the lines in the image w.r.t. the selected VPs. This method achieved state-of-the-art accuracy in 2016: 94.78% for YU, 90.80% for EC. In section 5, we closely compare their results with ours and discuss the strengths and weaknesses of each approach. In [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF] (our previous work) the ZL is first obtained using a brute force algorithm. Centroids of the LSs orthogonal to the ZL are then projected to the ZL and candidate HLs are taken at the peaks of a histogram of the obtained coordinates. A decreasing density sampling is then performed along each candidate HL. Each sample point is scored by the number of LSs consistent with that point, based on an angular consistency measure, and the VPs are found as peaks in the score curve. The final score for each candidate HL is finally the sum of the best two scores at the peaks (or the best score in case there is only one peak). This method is fast in execution and easy to implement, but middle rank in terms of accuracy (90.40% for YU, 85.64% for EC).

Contributions. In this paper, we build on the advances of several of these works, and especially [START_REF] Almansa | Vanishing point detection without any a priori information[END_REF][START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF][START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], to obtain a novel and more accurate HL and VP detection algorithm. In particular: (i) as in [START_REF] Almansa | Vanishing point detection without any a priori information[END_REF][START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF] we put the method into a mathematically, well founded a-contrario framework. However, by fractioning the 2-D search of meaningful VPs into three 1-D searches of meaningful events (ZL, HL and VPs), we avoid computationally expensive processes encountered using the previous a-contrario approaches; (ii) the ZL itself is obtained based on the Helmholtz principle. One benefit of doing so is to allow considering several orientations for the candidate HLs, therefore succeeding where other methods fail in cases where the vertical of the scene is masked by another near-vertical direction; (iii) As in [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], a set of candidate HLs is sampled around the meaningful HLs. We use a Gaussian mixture model (GMM) for that step, whose modes are the offsets of the meaningful HLs. This step significantly improves the accuracy w.r.t. [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF], where no other candidates than the peaks of the histogram are considered; (iv) as in [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF] and [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], VPs are hypothesized along the candidate HLs. However, thanks to the Helmholtz principle, we get more meaningful VPs. Moreover, our procedure does not require using any consistency measure, which have proven to be biased [START_REF] Xu | A minimum error vanishing point detection approach for uncalibrated monocular images of man-made environments[END_REF] and/or expensive to compute. Thanks to these improvements, our approach is top-ranked in terms of accuracy on the usual DSs (95.35% on YU, 91.10% on EC), without compromising neither the easiness of implementation, nor the efficiency of computation. Our method is actually even faster than [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF]. It is also much faster than [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF] and slightly faster than [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], the two previous state-of-the-art methods. In the next section, we describe how ZLs and HLs are hypothesized based on the Helmholtz principle. VP detection along the candidate HLs and candidate scoring are presented in section 4. Experimental results are finally provided and discussed in section 5.

Candidate Horizon Lines

A-Contrario Zenith Line Detection

As mentioned in introduction, the ZL L z is the line connecting the PP and the zenith VP. An initial guess of this line is obtained based on the fact that the vertical LSs in the scene are aligned with L z when passing through the PP, and near-parallel to L z in a narrow strip around the PP (Fig. 1). This yields a second-order parallelism gestalt, which can be detected by finding the maximal meaningful modes (MMMs) [START_REF] Desolneux | From Gestalt Theory to Image Analysis: A Probabilistic Approach[END_REF] of an orientation histogram. More specifically, our procedure for detecting hypothesized ZLs consists of the following steps (Fig. 2): (i) a set of M LSs with orientations θ i ∈ [0, π[ are detected using the LSD algorithm [START_REF] Grompone Von Gioi | LSD: a Line Segment Detector[END_REF], (ii) LSs far from the PP (|l T i c| > d P P , with l i the homogeneous coordinates of the LSs normalized so that li 2 1 + li 2 2 = 1 and c the homogeneous coordinates of the PP) or far from being vertical in the image (|θ i -π/2| < θ v ) are discarded (Fig. 2-A1 shows the LSs remaining at the end of this step), (iii) a L z -bin orientation histogram of the remaining LSs is built (Fig. 2-A2) and the MMMs of this histogram are computed (blue bins in Fig. 2-A3); the middle orientations of the highest bins of the MMMs are chosen as rough estimates of the hypothesized ZLs (colored circles in Fig. 2-A3), (iv) for each estimate, a set of candidate vertical LSs is selected by thresholding the angles between all image LSs and the estimate (|θ i -θ Lz | < θ z , with θ Lz ∈ [0, π[ the orientation of L z (Fig. 2-B1, the LSs are drawn using the same color as the corresponding circles in Fig. 2-A3); the intersection point of these LSs (in direction of the colored dashed lines in Fig. 2-B2) and a set of inlier LSs are obtained using a RANSAC algorithm; finally, the intersection point (the hypothesized zenith VP) is refined from the set of inliers, based on SVD.

Step (iv) is the same as in [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF]. MMMs are computed using the large deviation estimate of the NFA 1 , with

p(a, b) = (b -a + 1)/L (1)
(L = L z ) the prior probability for a LS to have its orientation in a bin between [a, b] (a uniform distribution is used as null hypothesis). In most cases, only one MMM is detected. However, it can happen, as in Fig. 2, that several modes are obtained (a mean of 1.71 MMMs is obtained in our experiments on YU, 1.66 on EC) while the mode with highest NFA does not correspond to the expected direction. A benefit of using an a-contrario approach here, is that all hypotheses can be used to generate candidate HLs, so that the correct solution can still be found in such difficult cases (Fig. 2-B2, the GT HL is drawn in dashed yellow, the estimated HL in cyan). This is a key improvement in comparison with [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF] and [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], where only one candidate is obtained at that stage, leading to incorrect results in such cases (e.g. with [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] in Fig. 2-B3). Rarely, a histogram has no MMM. In that case, the vertical direction of the image is taken as an initial guess for the ZL, and refined according to step (iv). . It is said to be a Meaningful Mode (MM) if it is a MI and if it does not contain any MG. Finally, an interval I is a Maximal Meaningful Mode if it is a MM and if for all MMs J ⊂ I, H(J) ≤ H(I) and for all MMs J I, H(J) < H(I).

A-Contrario Horizon Line Detection

The detection of the HL is based on following geometric properties (Fig. 1): (i) the HL is perpendicular to the ZL, (ii) any horizontal LS at the height of the camera's optical center projects to the HL regardless of its 3-D direction.

From these properties we get that all horizontal LSs at height of the optical center in the scene accumulate on a line in the image plane, perpendicular to the ZL. This yields a second-order alignment gestalt, which is detected by finding the MMMs of an offset histogram. More specifically, our method for detecting the HL is as follows (Fig. 1): (i) LSs far from being perpendicular to the ZL (||θ i -θ Lz | -π/2| < θ h ) are discarded, (ii) the centroids of the remaining LSs are orthogonally projected on the ZL and their offsets are computed relative to the projection of the PP, (iii) a L h -bin offset histogram is generated and the MMMs of this histogram are computed (red bins in Fig. 1). Again, though more rarely than for the ZL, this procedure can yield several MMMs (a mean of 1.03 MMMs is obtained on YU, 1.06 on EC). The centers of the highest peaks of the N init MMMs are all considered as candidate HLs (blue dashed line in Fig. 1).

Line Sampling

This estimate of the HL can be inaccurate in some cases, due to the histogram binning and, sometimes, to some offsets between the position of the accumulated LSs and the HL. Following the approach used in [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], we tackle this issue by sampling additional candidate HLs perpendicularly to the ZL, around the initial candidates. In [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], the offset probability density function (PDF) used for this sampling is a Gaussian model, fit from the CNN categorical probability distribution outputs. As we can have several initial candidates, we use a Gaussian mixture model (GMM) where the modes are the offsets of the initial candidates and the standard deviations are identically equal to σH, with H the image height and σ provided in Tab. 1. We draw S -N init additional candidates, equally divided between the N init initial candidates. In the case where no MMM is found, we have no a priori knowledge on the position of the HL along the ZL. The offsets of the S candidate HLs are then sampled linearly between [-2H, 2H].

Candidate Vanishing Points

All S candidate HLs are assessed against the success of detecting VPs along the line. Let us assume a line candidate L with polar coordinates (θ, ρ) is indeed the HL. Then, intersecting all image LSs (extended indefinitely beyond their endpoints) with L should lead to an accumulation of intersection points around the VPs (Fig. 3-A,B). In the same spirit as previously, these accumulations can be detected by finding the MMMs of a coordinate histogram of the intersection points. However, the prior probability for the coordinates along the HL is not uniform, leading to incorrect or inaccurate MMMs if p(a, b) is taken as in Eqn. ( 1) (e.g. Fig. 3-B, the MMM, shown in red, is very large and its highest bin does not correspond to a VP). In this section, we provide the prior (null hypothesis) suited to this problem and describe how the VPs and the HL are finally obtained.

Null-Hypothesis

For simplicity, we shall consider the image domain as a circle C of center O and radius 1 (Fig. 3-A). The polar coordinates of the detected LSs are assumed uniformly distributed over this domain. The prior probability p(a, b) can then be derived from a result obtained by Luis A. Santaló in the late 1970s [START_REF]Integral Geometry and Geometric Probability[END_REF]:

If K 1 , K 2 are two bounded convex sets in the plane (which may or may not overlap) and L 1 , L 2 the lengths of the boundaries ∂K 1 , ∂K 2 , the probability that a random chord of K 1 intersects K 2 is p = Li-Le L1 , where L e is the length of the external cover C e of K 1 and K 2 , and L i is the length of the internal cover

C i of K 1 and K 2 if K 1 ∩ K 2 = ∅, or L i = L 1 + L 2 if K 1 and K 2 overlap 2 .
This result is applied to our problem as follows. Let O be the orthogonal projection of O onto the candidate HL L and let X be a point on L at a signed distance x from O (Fig. 3, right). We use

K 1 = C (L 1 = 2π) and K 2 = [O X] (L 2 = 2|x|
). The probability of a LS meeting L between O and X depends on whether or not L meets C.

Case 1: C ∩ L = ∅ (Fig. 3, top-right). Let A, B (resp. C, D) be the points of contact of the tangents to the circle C from point O (resp. X). We have: 2 The external cover Ce is the boundary of the convex hull of K1 ∪ K2. It may be intuitively interpreted as a closed elastic string drawn about K1 and K2. The internal cover Ci can also be considered realized by a close elastic string drawn about K1 and K2 and crossing over at a point between K1 and K2. See [START_REF]Integral Geometry and Geometric Probability[END_REF] for details.

L i = XO + O B + BD + DA + AC + CX, p = L i -L e L 1 = BD + AC 2π = EF π ,
where denotes a counterclockwise arc of C, and E, F are the intersection points of the circle C with lines (OO ) and (resp.) (OX) 3 . Finally:

p(x) = 1 π tan -1 x ρ . (2)
It may be noticed that this expression is similar to the inverse of the sampling function s(k) = L tan(k∆θ) used in [START_REF] Simon | A Simple and Effective Method to Detect Orthogonal Vanishing Points in Uncalibrated Images of Man-Made Environments[END_REF], though the term ρ is also involved here.

Case 2: C ∩ L = ∅. In that case, we have p = (L 1 + L 2 -L e )/L 1 with L e depending on whether X is inside or outside the circle C. In the sub-case where X is inside the circle, L e = L 1 and

p(x) = x π , (3)
which is independent from ρ. In the sub-case where X is outside the circle (Fig. 3, bottom-right), L e = L1 -AB + AX + BX and p = (2|x| + 2 tan -1 (AX) -2AX)/2π, where A, B denote the points of contact of the tangents to the circle C from point X. This yields to:

p(x) = 1 π x + tan -1 x 1 + ρ 2 -1 x 2 -x 1 + ρ 2 -1 x 2 . ( 4
)
Finally, given a coordinate histogram of the intersection points and given a bin range [a, b], the prior probability p(a, b) is given by:

p(a, b) = p(r(b)) -p(l(a)), (5)
where l(a), r(a) denote the min and (resp.) max values of the histogram bin a. 5): both VPs are correctly detected. However, the coordinates of the intersection points can be large, depending on the orientations of the detected LSs w.r.t. the HL. For a given bin width, this results in an arbitrary and potentially very large number of bins, yielding poor time performance for the MMM detection. For that reason, we rather use the following approach: (i) the coordinates of the intersection points are transformed using the function p(x), yielding new coordinates, theoretically uniformly distributed (except at the VPs) between -1/2 and 1/2, (ii) a histogram with a fixed number L vp of bins is computed from the new coordinates and the MMMs of this histogram are detected using the prior probability p(a, b) provided by equation ( 1), with L = L vp . The histogram and MMMs obtained by following this procedure are shown in Fig. 3-D. Both VPs are still detected, while the histogram is much more compact (46 bins against 3630) for the same accuracy (30 bins) inside the image domain. The accuracy may be worse outside the image domain but, as a counterpart, the propagated error e.g. on the inferred 3-D vanishing directions, decreases as the distance between the PP and the VP increases 4 . Finally, an initial set of candidate VPs are extracted at the centers of the highest bins of the MMMs. These candidate VPs are refined using an EM-like algorithm similar to the one used in [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF]. This algorithm relies on the consistency measure

A-Contrario VP Detection and Line Scoring

f c (v i , l j ) = max(θ con -| cos -1 (v i l j )|, 0)
, where l j is a LS whose consistency with a VP v i is measured. At the end of this procedure, we select the two highest weighted VPs {v i } best (or one if there is only one candidate) and compute the score of the candidate HL as {vi} best {lj } f c (v i , l j ).

It is important to notice that the consistency measure is used to refine the VPs, but not to detect them. This is a great difference in comparison with [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], where the consistency measure is used both to detect and refine the VPs, yielding more spurious VPs (see Section 5). Moreover, our 1-D search of the VPs has several advantages over the previous a-contrario approaches [START_REF] Almansa | Vanishing point detection without any a priori information[END_REF][START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF] that operated in 2-D space. With regard to [START_REF] Almansa | Vanishing point detection without any a priori information[END_REF], we avoid computationally expensive local maximization of meaningfulness as well as filtering of spurious vanishing regions, due to artificial mixtures of different segment orientations. With regard to [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF], we avoid highly combinatorial point alignment detection in the dual space, along with tricky parameters tuning (sizes of rectangles, local windows, boxessee [START_REF] Lezama | A Contrario 2D Point Alignment Detection[END_REF] for details).

Experimental Results

Implementation

The source code of our method is available at https://members.loria.fr/GSimon/v/.

Algorithm parameters are provided in Tab. 1. Those were tuned manually using a few images from the DSs. We used the same number of line samples, S = 300, as in [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF]. The PP is assumed at image center. In order quantify the parameters' sensitivity, we did the , 2, respectively, and leaving the other parameters unchanged (the first 20 images of YU and EC were used). For each parameter, we report the relative decrease from the maximum to the minimum AUC obtained over the 9 runs (last row). The consistency thresholds θz, θ h , and particularly θcon (also used in [START_REF] Xu | A minimum error vanishing point detection approach for uncalibrated monocular images of man-made environments[END_REF]) are the most sensitive parameters. The number of bins in the histograms (Lz, L h , Lvp) are not very sensitive, though L h is more sensitive than the other two. dpp, θv and σ are not sensitive. The number of samples S is not as sensitive as one might expect (from S = 150 to S = 600, the AUC increases from 93.7% to 94.3%).

Accuracy of the Horizon Line

Computation of the HL was first evaluated on the two usual DSs: (i) York Urban (YU) [START_REF] Denis | Efficient edge-based methods for estimating manhattan frames in urban imagery[END_REF], consisting of 102 images of resolution 640 × 480, taken indoor and outdoor and mostly following the Manhattan world assumption, and (ii) Eurasian City (EC) [START_REF] Tretyak | Geometric image parsing in man-made environments[END_REF], consisting of 114 images of resolution 1920 × 1080, including scenes from different parts of the world, more varied viewpoints, and poorer fit to the Manhattan assumption. Example results are provided in Fig. 4, first and second rows (resp. YU and EC). We show the images where the horizon error is the lowest (column 1), the highest (column 5), and at the 25th, 50th and 75th percentiles (columns 2, 3, 4, resp.). The table in Fig. 5 shows the performance of our method, based on the cumulative histogram of the horizon error and the AUC (Section 2). We achieve state-of-the-art performance on both DSs. On YU, we improve upon the previous best of Zhai et al. [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] by a relative improvement ∆AU C = (AU Cnew -AU C old )/(1 -AU C old ) = 10.9%. This is a significant improvement, especially considering their improvement relative to the previous state of the art [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF] was 5%. On EC, the relative improvement upon the previous best is 3.3%. To further investigate our results, we replaced our PDF-based sampling method by a linear sampling between [-2H, 2H]. The new AUC are shown in the table of Fig 5 ("Linear samp"). The accuracy is similar to that with our sampling PDF and higher to that with the PDF of [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF]. This signifies that YU is a easy DS (two large sets of parallel lines are detected in most images), that does not require fine sampling as long as it covers the range [-2H, 2H] with sufficient density. This tends to attribute the improvement of accuracy w.r.t. to [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] to our scoring procedure. It indeed appears that the method of [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] gets much more spurious VPs than ours on both YU and EC (see Sec. 5.3 below). By contrast, the best result obtained by our method on EC may be interpreted slightly differently, as here both our sampling PDF and the one of [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] improve the accuracy compared to that with a linear sampling, so that both sampling and scoring of the candidate HLs contribute to our performance.

Our method was then evaluated on Horizon Lines in the Wild (HLW), a DS introduced recently by Zhai et al. [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], and consisting of 2018 images of various resolutions. This DS is not only larger but also much more challenging than the previous ones. also groups of people, statues occupying a large part of the image, and so on. Furthermore, the roll and tilt angles of the camera have very large range of values, often leading to HLs far from the image boundaries, and ZL angles out of the assumed range (e.g. Fig. 4-C5). Example results and AUCs obtained with our method are shown in Fig. 4-Row C, and (resp.) in the third column of the table in Fig. 5. The approach of Zhai et al. outperforms our method on that DS and we get a relative decrease of 9.1% w.r.t. them. The AUC with a linear sampling is much lower than with our PDF (a relative decrease of 19.4%), which indicates that sampling plays a crucial role on this DS. To closely compare our PDF with the one of [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] and establish which parameters of the PDFs, among the modes and the spreads, are the most critical, we tested both methods using only one sample (S = 1), namely the mode of the GMM with highest NFA with our method, and the center of the PDF with the method of Zhai et al. The results are shown in the last two rows of the table in Fig. 5. The AUC with our method is now quite the same as with [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF]. This indicates that, in HLW, the spread of the sampling is the key element of the difference in performance between [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] and our method. In [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF], σ is re-estimated each frame from the CNN output, while we take a constant, empirical value in our method. A way to improve our results may be to consider the NFAs of the candidate HLs as uncertainty measures, that may be used to generate more relevant values of σ.

The predictive power of the CNN is interesting in bad images where analytical vision fails, assuming a large DS of similar examples is provided, along with a GT, to the learning process. By contrast, our method may provide accurate results in some images where the CNN fails due to insufficient representation in the learning DS. For instance, Fig. 6-A1 shows an example of an image acquired in an industrial environment. Our method succeeds in predicting the HL, refining it and getting meaningful VPs (Fig. 6-A1), while the method of Zhai et al. poorly estimates the sampling PDF and finally the HL (Fig. 6-A2).

Relevance of the Vanishing Points

Fig. 6-B1,B3,C1,C3 show some example VPs (represented by the LSs consistent with them) obtained by using our method. Performance w.r.t. the previous two best of [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF][START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] was measured by counting the number of good and spurious VPs obtained on the YU and EC DSs. We chose to use these two DSs, as those are representative of different resolutions (low and high) and get higher accuracy regarding HL detection. In our experiment, a "good VP" is a VP that indeed corresponds to a set of parallel, horizontal lines, while a spurious VP can be of two kinds: "spurious VPs" that correspond to fortuitous convergences of non parallel lines, and "split VPs", issued from undesirable splittings of parallel, horizontal lines normally corresponding to the same VP. In the latter case, one "good VP" plus one "split VP" per added VP are counted. Fig. 7-Left shows the total number of good VPs, spurious VPs and split VPs obtained on the two DSs for each method. Our method is the most relevant regarding the three criteria. We obtain the highest number of good VPs, very few spurious VPs and no split VP at all, whatever the DS is.

The method in [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] detects slightly less good VPs than ours (a mean of 2.11 per image-p.i. on the two DSs, against 2.14 with our method) but much more spurious VPs, about one for 2 good VPs, against one for 23 with our method. It also obtains a non-negligible number of split VPs (one for 29 good, against 0 with our method). These relatively poor results are mainly due to the approach used by [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] to initialize VPs along the candidate HLs. This approach consists in randomly selecting a subset of LSs {lj} and computing their intersection with the HL. An optimal subset of VPs vi is extracted from the intersections, so that the sum of weights v i l j fc(vi, lj) is maximal, while ensuring no VPs in the final set are too close. A distance threshold between two VPs has therefore to be fixed, which can lead to split LSs into several groups while they correspond to the same VP (e.g. the blue and yellow LSs on the building's facade in Fig. 6-B2). Moreover, random selection of LSs can prevent detecting a VP represented by few LSs (e.g. the VP consistent with the yellow LSs in Fig. 6-C1, not found in Fig. 6-C2). Finally, as another threshold has to be fixed for the consistency measure, any set of LSs that meet accidentally "near" the same point on the HL can generate a spurious VP (e.g. the yellow LSs in Fig. 6-C2). All these threshold problems are inherently handled when using our a-contrario framework. While also relying on an a-contrario framework, the method in [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF] gets poor results regarding the detected VPs: the lowest number of found VPs (1.80 p.i.), the second highest number of spurious VPs (one for 3 good) and the highest number of split VPs (one for 4 good). The low number of good VPs (see e.g. the VPs consistent with the orange LSs in Fig. 6-B3 and C3, not found in Fig. 6-B4 and C4, resp.) may be explained by the fact that a VP can appear as meaningful along the HL, but not in the whole image dual domain. The high number of spurious VPs (e.g. the VPs consistent with the cyan, green, red and yellow LSs Fig. 6-C4) is mainly due to accidental intersections of LSs, that appear more frequently in the whole image dual domain than on the HL. Finally, the high number of split VPs is mainly due to the fact that aligned points in the dual domain (meeting LSs in the primal domain) can be scattered in the direction orthogonal to the alignment, producing several meaningful alignments with slightly different orientations (Fig. 6-A4&B4). Using our method, LSs corresponding to the same VP can meet the HL at coordinates scattered along the HL, but generally in contiguous bins of the coordinate histogram, so that those are fused in a single MMM (Fig. 6-A3&B3).

Computation Times

The method was implemented in Matlab and run on a HP EliteBook 8570p laptop with I7-3520M CPU. Computation times are given in Fig. 7-Right. Our method is faster than the previous methods whose code is available. Moreover, contrary to e.g. [START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF], it is only slightly affected by increases in the image size, which generally yield larger numbers of LSs. Indeed, our method is in O(L 2 z + L 2 h + S(L 2 vp + M )), therefore only linearly affected by the number of LSs.

Conclusion

As soon as one wishes to detect Manhattan directions, hVPs and/or the HL in an image, which are common tasks in computer vision, our experimental results show that horizon-first strategies are definitely faster and more accurate than all previous methods. In particular, our method achieves state-of-the-art performance w.r.t. HL detection on two over three DSs. Moreover, it provides more relevant VPs than the previous two state-of-the-art approaches, which can be of great interest for any practical use of the VPs (e.g. finding the Manhattan directions). Finally, it performs well in any kind of environment, as soon as man-made objects are visible at eye level. The method Zhai et al. [START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] stays, however, alternate that may be more suited to specific environments, learned from large DSs, especially when the later condition is not met.

Fig. 1 .

 1 Fig. 1. The horizon line can be detected as a meaningful alignment of image line segments orthogonal to the zenith line.

1

  Let L be the number of bins of the histogram, M the number of data, r(a, b) the density of data with values in a bin between [a, b], and p(a, b) the prior probability for a data to have its value in a bin between [a, b]. An interval [a, b] is said to be a Meaningful Interval MI (resp. a Meaningful Gap MG) in the large deviation sense if r(a, b) > p(a, b) (resp. r(a, b) < p(a, b)) and its relative entropy H([a, b]) is greater than 1 M log L(L+1) 2

Fig. 2 .

 2 Fig. 2. A-contrario detection of the zenith line (see the text).

Fig. 3 .

 3 Fig. 3. Left: each line segment gives rise to an intersection point with the horizon line (A). The modes of a coordinate histogram of these intersections (in red and yellow) should appear at the positions of the vanishing points. Different results are shown (B,C,D) depending on the choice of the null hypothesis and the way the histogram is built. Right: computation of p depending on whether the line meets the circle or not.

Figure 3 -

 3 Figure 3-C shows an example of the PDF r(x) = ∂p ∂x (x), obtained for a line L in case 2 (purple curve). In this figure, the red and yellow MMMs are obtained using p(a, b) provided by equation (5): both VPs are correctly detected. However, the coordinates of the intersection points can be large, depending on the orientations of the detected LSs w.r.t. the HL. For a given bin width, this results in an arbitrary and potentially very large number of bins, yielding poor time performance for the MMM detection. For that reason, we rather use the following approach: (i) the coordinates of the intersection points are transformed using the function p(x), yielding new coordinates, theoretically uniformly distributed (except at the VPs) between -1/2 and 1/2, (ii) a histogram with a fixed number L vp of bins is computed from the new coordinates and the MMMs of this histogram are detected using the prior probability p(a, b) provided by equation[START_REF] Almansa | Vanishing point detection without any a priori information[END_REF], with L = L vp . The histogram and MMMs obtained by following this procedure are shown in Fig.3-D. Both VPs are still detected, while the histogram is much more compact (46 bins against 3630) for the same accuracy (30 bins) inside the image domain. The accuracy may be worse outside the image domain but, as a counterpart, the propagated error e.g. on the inferred 3-D vanishing directions, decreases as the distance between the PP and the VP increases 4 . Finally, an initial set of candidate VPs are extracted at the centers of the highest bins of the MMMs. These candidate VPs are refined using an EM-like algorithm similar to the one used in[START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF]. This algorithm relies on the consistency measure

3 CFig. 4 .

 34 Fig. 4. Horizon lines obtained at the 1st, 25th, 50th, 75th and 100th percentiles of the horizon error (Col. 1-5, resp.) for YU (Row A), EC (Row B) and HLW (Row C). The GT HL is shown in yellow dashed line, the MMMs in blue dashed lines and the estimated HL in cyan solid line. The horizon error is displayed on the top-left corner of each image result. LSs participating to a VP are shown using one color per VP.

Fig. 5 .

 5 Fig. 5. Performance results w.r.t. HL detection.

Fig. 6 .

 6 Fig.6. Qualitative comparisons between our method (Col. 1) and the method of Zhai et al.[START_REF] Zhai | Detecting vanishing points using global image context in a non-manhattan world[END_REF] (Col. 2) on the one hand, and between our method (Col. 3) and the method of Lezama et al.[START_REF] Lezama | Finding vanishing points via point alignments in image primal and dual domains[END_REF] (Col. 4) on the other hand. Plotting conventions are as in Fig.4.

Fig. 7 .

 7 Fig. 7. Left: performance results w.r.t. VP detection. Right: computation times in sec.

Table 1 .

 1 Algorithm parameters. First row: parameters' values (W is the image width). Second row: parameters' sensitivity.

	d P P W/8	θv 22.5 •	θz 10 •	Lz 45	θ h 1.5 •	L h 64	σ 0.2	S 300	Lvp 128	θcon 1.5 •
	0.0%	0.0%	-14.2% -7.2% -13.2% -12.4% 0.0% -11.4% -6.4% -28.7%
	following experiment. For each parameter p, we run our method 9 times, multiplying
	p by 1 2 , 5 8 , 6 8 , 7 8 , 1, 5 4 , 6 4 , 7 4

  Most of the photos look like holiday photos, showing man-made environments, but

	1							1
	0.9							0.9
	0.8							0.8							YU	EC	HLW
	0.6 0.7			Kosecka et al. (74.34%) Tardif (84.62%)		0.7 0.6			Kosecka et al. (68.62%) Tardif (81.33%)	N. images Ours	102 95.35 91.10 54.43 103 2018 S = 300
	0.4 0.5			Tretyak et al. (88.43%) Simon et al. (90.40%) Vedaldi et al. (91.99%)		0.4 0.5			Tretyak et al. (86.06%) Simon et al. (85.64%) Vedaldi et al. (81.84%)	Lin. samp. 95.24 88.42 45.59 Zhai et al. 94.78 90.80 58.24
	0.1 0.2 0.3			Wildenauer et al. (93.14%) Xu et al. (93.45%) Lezama et al. (94.51%) Zhai et al. (94.78%) Ours (95.35%)		0.1 0.2 0.3			Ours (91.10%) Wildenauer et al. (86.15%) Xu et al. (89.15%) Lezama et al. (89.20%) Zhai et al. (90.80%)	S = 1 70.60 72.78 45.88 Zhai et al. 75.89 68.49 45.41 Ours
	0	0	0.05	0.1	0.15	0.2	0.25	0	0	0.05	0.1	0.15	0.2	0.25

BD = F D-F B = CF -F B = CE+EF -F B = AE-AC+EF -F B = EB-AC+EF -F B ⇐⇒ AC + BD = EB + EF -F B = EF + EF = 2EF .

As the angle θ between the optical axis and a vanishing direction is arc-tangential in the distance d between the VP and the PP, the propagated error ∂θ/∂d is inversely proportional to d 2 .


Monalisa Pal
email: monalisap90@gmail.com

Amr Alzouhri Alyafi

Sanghamitra Bandyopadhyay

Stéphane Ploix
email: stephane.ploix@grenoble-inp.fr

Patrick Reignier
email: patrick.reignier@inria.fr

Enhancing Comfort of Occupants in Energy Buildings

Keywords: Differential Evolution, energy management, multi-objective optimization, Pareto-optimality, smart buildings

As buildings contribute significantly towards global energy consumption, it is essential that the occupants receive the best comfort without utilising further energy. This work treats building, environment and the occupants as a system, which presents the context and the occupants also provide their comfort criteria to a black box for yielding the a schedule of actions (opening/closing of doors/windows) for optimal comfort. The physical state of an office, situated in France, is recorded over a span of 100 days. This data is utilised by a physical model of the building to simulate the indoor ambience based on random sets of user actions from which an optimal schedule is obtained, representing equally best trade-off among minimal thermal and CO2 based air quality dissatisfaction. Results indicate that adopting the proposed schedule of user actions can efficiently enhance the occupants comfort.

Introduction

Considering the ever-growing energy demand and the depletion of non-renewable energy resources, it is important to limit the energy usage in buildings which constitute roughly about 40% of the global energy supply. Thus, it is imperative to satisfy the demands of the building occupants without increasing the present rate of energy consumption in buildings. Occupant, on the other hand, being an corresponding author.

integral part of the building system, can affect the indoor environmental condition through their actions. By intelligently utilising these actions, positive effects can be brought upon the indoor environment. Thus, guiding the occupant's actions, like opening and closing of doors and windows, for significant period of time, can help achieve better comfort in energy building at the same cost of energy consumption.

Attempts have been considered to meet the comfort demands of the occupants by improving building construction techniques and adding insulation to walls and ceilings. Building regulations also play an important role in the overall (both global and local) energy management. However, obtaining positive impacts of occupant's actions in the energy consumption can help to manage their own comfort. It is therefore important to assist the occupants with an optimal energy plan in order to explain that their expectations on comfort can be attained by themselves to some extent.

The approach used in this work considers an office, situated in Grenoble Institute of Technology, France, fitted with 27 sensors, for collection of data like temperature, solar illuminance, wind speed, humidity, moisture, CO 2 concentration, etc., to construct the physical context. Also a physical model of the office is used which based on the physical context of outside environment and neighbouring corridors, and a random set of actions can simulate the indoor ambience. The indoor temperature and CO 2 concentration are responsible for thermal and air quality dissatisfaction of the occupants. This work uses a multi-objective optimization algorithm viz. Differential Evolution to obtain the schedule of user actions that can lead to minimal thermal and air quality dissatisfaction. On presenting the occupants with this optimal schedule, they can compare it with their previous schedule and adopt the new schedule, after trial, if they find significant improvement of comfort.

Depending on the weather changes, the optimal schedule is variable. However, rarely, there are day to day changes in the outside temperatures and CO 2 concentrations. Hence, learning from past day's environmental conditions and occupant's actions, the proposed optimal set of actions can be adopted for next day.

Rest of the paper is organized as follows. Section 2 describes the information flow to yield the schedule of actions corresponding to the trade-offs between minimal thermal and air-quality dissatisfactions. Section 3 discusses the results to assess the efficacy of the proposed approach. Section 4 presents the conclusion while directing towards future research.

Experimental framework

This work presents optimal schedules of actions (opening/ closing of doors/ windows) which the occupants can adopt to achieve better thermal and CO 2 based air quality comfort at no extra energy expenditure. As the proposed scheme aims at improving occupant's comfort without increasing energy consumption, this can be considered as an energy management scheme. of the work is shown in Fig. 1. In this section, the function of each of these modules is explained in details along with the overall interconnection of the modules to explain the information flow. It is to be mentioned that the environment, buildings and the occupants represent the smart building system where the environment and buildings contribute to the physical context/state, and the occupants, on the other hand, generate actions and provide their comfort criteria to a black box. Finally, the occupants compare their actions with the optimal actions generated by the black box in order to learn the scope of improvement in their comfort. As the internal operation of the optimizer is unknown to the occupant, it has been labeled as a black box.

Building and Environment

Through an array of sensors, the physical state of the outdoor, indoor and neighbouring zones are recorded. Here, physical state refers to temperature, CO 2 concentration, humidity, etc. The inertia of these physical quantities from the outside environment and neighbouring zones like corridors, staircase, etc. influence the physical quantities inside the room of the occupant. Hourly samples of such quantities are recorded in the database for future reference. Here, as a testbed, an office room at Grenoble Institute of Technology, France, where four researchers work, are fitted with 27 sensors for recording the physical state/context and usual schedule of actions. This acts as the smart building system for the proposed work.

Occupants

Occupants are the integral part of the entire system. On one hand, they provide the optimization criteria like comfort, energy consumption, etc. for obtaining an optimal schedule of actions and on the other hand, their actions influence the system which in turn affects the optimization criteria. In absence of controllable HVAC (Heating, Ventilation and Air Conditioning) system, the only actions of the occupants which can influence the indoor ambient conditions (and hence, their comfort) are opening and closing of doors and windows. The occupants can compare the optimal schedules with their usual schedule and adopt the proposed schedules as per need. For example, if the difference in comfort is negligible the occupant might not prefer to change their schedule, whereas if the occupant observes significant improvement in comfort, the occupant is expected to adopt the proposed schedules of actions.

After the basic building block viz. the smart building system, and its purpose, has been explained, the next step is to obtain the optimal schedules of actions. The optimizer module, as explained next, helps in searching for best trade-offs among the objectives provided by occupants.

Optimizer

The role of the optimizer is to yield a few optimal schedules of actions. The essential specifications for the optimization module are as follows: the representation of the solution vector in order to decode the result, the optimization algorithm, the objectives and their relation to the solution vector, the stopping condition and the algorithmic parameters.

A solution of the optimization problem represents a set of actions. The allowed actions for the occupants are opening and closing of doors and windows. As the data are recorded in an office environment, the actions are noted over 12 working hours i.e. from 8am to 8pm. Hence, the solution is represented by a 24-dimensional binary vector where the first 12 entries imply opening/closing (open = 1, close = 0) of windows and the later 12 entries are for opening/closing of doors for each of the 12 working hours, respectively.

Given the environmental context of the room, the primary objective of the work is to obtain various schedules, such that each of the schedules represent trade-offs among several conflicting objectives like minimizing thermal dissatisfaction, minimizing CO 2 based air quality dissatisfaction, minimizing humidity based air quality dissatisfaction, minimizing energy consumption, etc. Due to the presence of multiple objectives, a multi-objective version of an optimization algorithm viz. Differential Evolution [START_REF] Robič | Demo: Differential evolution for multiobjective optimization[END_REF][START_REF] Pal | Many-objective feature selection for motor imagery eeg signals using differential evolution and support vector machine[END_REF] has been employed.

Assuming a physical context (indoor and outdoor environmental variables like CO 2 concentration, temperature, etc.) and a set of actions (opening/closing of doors/windows) as inputs to the smart building system (outdoor environment and building with occupants), it outputs some effects (like thermal and air quality comfort). A physical model [START_REF] Amayri | Estimating occupancy in an office setting[END_REF][START_REF] Scanu | Methodologie pour la comparaison de structures de modeles simplifies[END_REF] representing this system has been used which can simulate the effects corresponding to random sets of actions, in the same context as obtained from the database. These random sets of actions generate various sets of effects. Several best trade-offs among the effects (generating the Pareto-Front) are chosen. Then, the occupants can compare the schedule of actions corresponding to these trade-offs with their usual schedules to analyze the difference. Hence, evaluation of objectives is a two-step process. The first step uses the physical model of the office to evaluate indoor environmental variables depending on a true context and a randomly assumed schedule of actions. The model is represented by Eq. ( 1) and (2) where the variables are defined in Table 1. Some of these variables represent sensor measurements whereas the remaining ones are learned by repeated simulation of the physical model to match the office room. The second step evaluates effects (here, thermal and CO 2 based air quality dissatisfaction) from the simulated indoor physical variables. These effects are shown in Eq. ( 4) and ( 5) which represent thermal and air quality dissatisfaction at the i-th hour, respectively. The objectives are formulated keeping in mind that the indoor temperature is preferred between 21 • C to 23 • C and the indoor CO 2 concentration is preferred between 400 ppm to 1500 ppm. The purpose of the optimizer is to optimize (minimize) the effects (dissatisfaction levels) as formulated by Eq. [START_REF] Pal | Many-objective feature selection for motor imagery eeg signals using differential evolution and support vector machine[END_REF].

T in = R R i τ + R 1 R out + ζ W R W T out + R 1 R n + ζ D R D T n (1)
V dC in dt = -Q out 0 (t) + Q cor 0 (t) + ζ W (t)Q W + ζ D (t)Q D C in + Q out 0 (t) + ζ W (t)Q W C out + (Q cor 0 (t) + ζ D (t)Q D ) C cor + S CO2 × n(t) (2) Minimize: D(actions) = [d 1 (T in ), d 2 (C in )] = 12 i=1 d i 1 12
,

12 i=1 d i 2 12 (3)
where,

d i 1 (T in ) =      21-Tin 21-18 if T in < 21 0 if 21 ≤ T in ≤ 23 Tin-23 26-23 if T in > 23 (4)
d i 2 (C in ) = 0 if C in ≤ 400 Cin-400 1500-400 if C in > 400 (5) actions = ζ 1 W , ζ 2 W , • • • , ζ 12 W , ζ 1 D , ζ 2 D , • • • , ζ 12 D
Like most evolutionary optimization algorithms, the multi-objective version of Differential Evolution is executed for a predetermined number of generations by which the optimization algorithm is expected to have converged. The description of various parameters and their values for which best results are obtained are noted in Table 1.

At the end of the optimization algorithm, a few optimal schedules of actions are generated. The user can choose among these schedules based on their preference among the multiple objectives of optimality by comparing their usual schedule with the chosen schedule.

Comparing Schedule of Actions

Based on any schedule of actions, the physical model can generate the corresponding indoor ambience. The indoor ambience based on the usual schedule of the occupant is in the available from the database. Occupants can compare the simulated ambience (what best could have happened) to their usual ambience (what had actually happened) and understand the difference in effects to gain better comfort.

After the description of the experimental setup, the next section discusses various results in order to validate the proposed approach.

Result and discussion

This section analyses the performance of the proposed approach using the various results. Dissatisfaction levels as obtained from various simulated schedules are shown in a scatter plot. From these, the set of Pareto-optimal schedules obtained using optimization techniques are marked. Next the variation of average indoor temperature and CO 2 concentration, obtained from optimal schedules, are compared with usual schedules. Finally, the variation of net dissatisfaction resulting from the proposed optimal schedule is shown to validate the optimization performance. The experimental data is collected for 100 days (1st April, 2015 to 9th July, 2015) and the analysis is conducted in 10 groups of 10 days each as shown in Table 2.

Pareto-front and optimal schedules

A set of Pareto-optimal solutions is obtained for every working day in the experimental duration. Depending on the occupant's preference, any one of these schedules can be chosen as the preferred optimal schedules. Considering equal preference for both the objectives, the solution nearest to reference point (at the minima for thermal and air-quality dissatisfaction i.e. at (0, 0)) is considered as the best schedule. The solution at the end of the Pareto-front are also analysed further for comparison because these represent the best schedules with respect to each objectives (minimal thermal or CO 2 based air quality dissatisfaction), independently. Hence, occupant's usual schedule is compared with three other schedules: best schedule, schedule for optimal thermal comfort and schedule for optimal air quality comfort. Using the context from 16th April, 2015, the thermal versus air quality dissatisfaction corresponding to various simulated schedules is shown Fig. 2. It also shows the Pareto-front and the three optimal schedules of interest, along with the usual level of dissatisfaction.

Comparison of physical variables for different schedules

As there are very less day to day changes in environmental conditions, the average of physical variables of 10 days (working hours only) are considered for comparison of the various schedules. Variations in indoor temperature and CO 2 concentrations due to various schedules are presented in Fig. 3 and 4 for all the 10 groups of 10 days each as mentioned in Table 2. From the figures, it can be noticed that the difference between usual and proposed indoor physical variables (temperatures and CO 2 concentrations) is more during earlier days of experiment. In summer (group 7 to 10), it is difficult to maintain the physical variables in preferred ranges just by varying the schedule of opening/closing of doors/windows. Hence, in extreme cases where outdoor physical variables (Table 2) are higher than preferred ranges, HVAC system is needed to regulate the indoor physical variables. It is also to be noted that the best schedule usually presents higher temperature than the schedules for optimal thermal comfort. Similar observation is also noted for indoor CO 2 concentration. This is due to the fact the the best schedule presents a trade-off solution whereas the other proposed schedules are optimal with respect to one objective at a time.

Performance analysis of optimization algorithm

Optimization algorithm, in this case, presents a set of solutions, called the Pareto-optimal solution. The dissatisfaction values corresponding to these solutions create the Pareto-Front. To assess the convergence of the best schedule with respect to the ideal optimal solution, the city-block distance (sum of absolute difference) [START_REF] De Souza | Clustering of interval data based on cityblock distances[END_REF] between the corresponding dissatisfaction levels are measured. This distance represents the net dissatisfaction which is to be minimized. The parameters required for this performance metric viz. the reference point, is noted in Table 1. Box plots for the distribution of net dissatisfaction for the best sched- ule of the Pareto-front is noted in Fig. 5 for all the 10 groups which shows that the median is close to 0 and the range of values is very less towards the earlier group and becomes higher for later groups. For a general idea of the range of dissatisfaction values, the theoretical variation of dissatisfaction are plotted against indoor temperature and indoor CO 2 concentration in Fig. 5 which follows from Eq. ( 4) and ( 5). As noted from these plots, a combined dissatisfaction value around 2 or higher indicates that the indoor physical variables are not within preferred ranges. This implies that the proposed optimization approach is reproducible and efficient in yielding optimal schedules for this application during those period when HVAC system is not needed and thus, managing energy efficiently in the studied scenario.

Conclusion

The objective of this work is to demonstrate an approach which considers the occupant's schedule of actions as the only controllable parameter of thermal and CO 2 based air quality comfort and yields several schedules of actions which represent the best trade-offs between thermal and air quality dissatisfaction.

Among the several schedules, the schedule corresponding to the equally best compromise of air quality and thermal comfort is analysed further. This optimal schedule can lead to efficient energy management. This work also addresses the need of HVAC system which arises when the environmental variables (here, temperature and CO 2 concentration) are too high from the preferred values such that change in schedule of actions can negligibly influence the comfort level of the occupants.

However, just presenting the optimal schedule might not convince the occupants to change their schedule. In order to gain occupant's trust on the system, the effect of small changes in action and the internal working of the black box are to be explained in non-technical terms. This can lead to better comfort without depending on external devices like HVAC system. The authors are working on the explanations and adding more context to improve the parameterization of overall comfort. On the other hand, when several devices including HVAC system (if required) are operational, their energy consumption and environmental impact form other important effects to be considered along with the comfort of the occupants. This forms open area of research along this domain of energy management in smart buildings.

Fig. 1 .

 1 Fig. 1. General Schema of Energy Management in Smart Buildings.

Fig. 2 .

 2 Fig. 2. Pareto-Front and Schedules of Interest.

Fig. 4 .

 4 Fig. 4. Variation in Indoor CO2 Concentrations Resulting from Different Schedules in the Experimental Duration.

Fig. 5 .

 5 Fig. 5. Variation in Net Dissatisfaction for the Best Schedule in the Experimental Duration.

  The general schema

	Environment Buildings	Sensors	Set of Actions
	Occupants Compare using Usual Actions Di erence Analyze Physical Model	Database Objectives Optimal Actions	Context OPTIMIZER Simulate with Physical Model Evaluate Objectives
		Black Box

Table 1 .

 1 Algorithmic Parameters and their Values.

	Module	Parameters Explanation	Values
	Physical	ζW , ζD	Status of window(W), door(D) Open = 1, Close = 0
	Model	Tin, Tn, Tout Temperatures of indoor,	From database
			adjacent corridor, outdoor
		RD, RW	Thermal resistances of door(D), From database
			window(W)
		Ri, Rn, Rout Resistance of walls, adjacent	From database
			corridor, outdoor
		R	Equivalent resistance	Ri Rout Rn
				RW (when ζW = 1)
				RD(when ζD = 1)
		τ	Thermal coefficient representing From database
			building inertia
		V	Volume of the room (office)	From database
		Cin, Cout, Ccor CO2 concentrations indoor,
			outdoor, in adjacent corridor	From database
		Q out , Q cor	Air speed outdoor, in corridor From database
		QW , QD	Air speed through window(W), From database
			door(D)
		SCO 2	Breath production of CO2	From database
			per occupant
		n(t)	Number of occupants at time t From database
	Differential N P	Population size	20
	Evolution Gmax	Maximum generations	300
	(Optimizer) F	Scale Factor	Randomly choosen
				between 0 and 2
		CR	Crossover Rate	0.8
		r	Reference point for ranking	Ideal point i.e.
			and decision making	(0,0)

Table 2 .

 2 Groups of Experimental Data Recorded during Working Hours (8am to 8pm).

	Group Period	Mean Outdoor	Outdoor CO2
			Temperature ( • C) concentration (ppm)
	1	April 1 to April 10, 2015	10.1528
	2	April 11 to April 20, 2015	17.8809
	3	April 21 to April 30, 2015	17.1548
	4	May 1 to May 10, 2015	20.7708
	5	May 11 to May 20, 2015	20.1111	395
	6	May 21 to May 30, 2015	15.9167
	7	May 31 to June 9, 2015	24.8571
	8	June 10 to June 19, 2015	21.5119
	9	June 20 to June 29, 2015	23.5417
	10	June 30 to July 9, 2015	28.7812

Acknowledgments. This study has been supported by the Indian side of the project sanctioned vide DST-INRIA/ 2015-02/ BIDEE/ 0978 by the Indo-French Centre for the Promotion of Advanced Research (CEFIPRA -IFCPAR).

This work benefits from the support of the INVOLVED ANR-14-CE22-0020-01 project (http://www.agence-nationale-recherche.fr/?Projet=ANR-14-CE22-0020) of the French National Research Agency, ANR: Agence Nationale de la recherche, which aims at implementing new occupant interactive energy services, like MIRROR, WHAT-IF and SUGGEST, into a positive energy building constructed at Strasbourg in France by Elithis.