Yannick Wend

Kuni Zoetgnande
email: yannick.zoetgnande@univ-rennes1.fr

Geoffroy Cormier

Alain-Jérôme Fougères

Jean-Louis Dillenseger

Sub-pixel matching method for low-resolution thermal stereo images

Keywords: Thermal images, Stereo vision, Sub-pixel matching, Robust feature extraction, Phase correlation, Low-resolution

In the context of a localization and tracking application, we developed a stereo vision system based on cheap low-resolution 80×60 pixels thermal cameras. We proposed a threefold sub-pixel stereo matching framework (called ST for Subpixel Thermal): 1) robust features extraction method based on phase congruency, 2) rough matching of these features in pixel precision, and 3) refined matching in sub-pixel accuracy based on local phase coherence. We performed experiments on our very low-resolution thermal images (acquired using a stereo system we manufactured) as for high-resolution images from a benchmark dataset. Even if phase congruency computation time is high, it was able to extract two times more features than state-of-the-art methods such as ORB or SURF. We proposed a modified version of the phase correlation applied in the phase congruency feature space for sub-pixel matching. Using simulated stereo, we investigated how the phase congruency threshold and the sub-image size of sub-pixel matching can influence the accuracy. We then proved that given our stereo setup and the resolution of our images, being wrong of 1 pixel leads to a ±500 mm error in the Z position of the point. Finally, we showed that our method could extract four times more matches than a baseline method ORB + OpenCV KNN matching on low-resolution images. Moreover, our matches were more robust. More precisely, when projecting points of a standing person, ST got a standard deviation of ≈ 300 mm when ORB + OpenCV KNN gave more than 1000 mm.

Introduction

Detection of people is a crucial task in computer vision for security or safety applications (intrusion detection, pedestrian collision detection, fall detection). In this context, thermal cameras are often considered because they are robust to illumination changes, and they ensure the preservation of anonymity [START_REF] Pittaluga | Sensor-level privacy for thermal cameras[END_REF]. In some of these applications, the localization and tracking of people in the space is also a crucial issue. In this case, stereo-vision is often chosen because it produces accurate spatial information about the tracked people. Some works have already been performed on thermal stereo-vision based on high-resolution thermal cameras [START_REF] Bertozzi | Infrared stereo vision-based pedestrian detection[END_REF][START_REF] Dhua | Triangulation based technique for efficient stereo computation in infrared images[END_REF], but one of the obstacles on the democratization of thermal cameras is their cost [START_REF] Cnet | Heat seeker: Meet the thermal-imaging camera you can afford[END_REF]. Recently, a few manufacturers proposed very low-cost thermal cameras, e.g., the FLIR Lepton 2 1 . The counterpart of these cameras is their low spatial resolution (80 × 60 pixels for the Lepton 2). Such resolution has a direct impact on several steps of the traditional stereo-vision framework: stereo calibration, information extraction, information matching between the two views and triangulation. In [START_REF] Zoetgnandé | Robust low resolution thermal stereo camera calibration[END_REF], the authors proposed a robust low-resolution thermal stereo camera calibration. Matching is a prior step of 3D vision using a stereo system. The matching can be dense or sparse. Dense matching works well for textured images; unfortunately, thermal images are lacking texture. Thus, sparse matching on features extracted from the images must be considered. But given a thermal image how to characterize the information it contains?

A feature can be either global or local. Generally, global features are used in image retrieval, object detection, and classification. They are based on the fact that humans can easily recognize objects with a single glance [START_REF] Olivia | Building the gist of a scence: the role of global image feaures in recognition[END_REF]. The most common global features descriptors are Histogram of Oriented Gradient [START_REF] Bertozzi | A pedestrian detector using histograms of oriented gradients and a support vector machine classifier[END_REF][START_REF] Olmeda | Contrast invariant features for human detection in far infrared images[END_REF], Invariant moments [START_REF] Zhang | Aircraft recognition in infrared image using wavelet moment invariants[END_REF][START_REF] Dai | A feature-based image registration algorithm using improved chain-code representation combined with invariant moments[END_REF] and Co-occurrence Histograms of Oriented Gradients [START_REF] Watanabe | Co-occurrence histograms of oriented gradients for pedestrian detection[END_REF][START_REF] Iwata | Object detection based on multiresolution cohog[END_REF]. The main drawback of these features is that their extraction is difficult and computationally costly. Unlike global features, local features are easier to extract. The most common local features methods are Harris corner detector [START_REF] Harris | A combined corner and edge detector[END_REF], KLT [START_REF] Tomasi | Shape and motion from image streams: a factorization method[END_REF], phase congruency [START_REF] Morrone | Feature detection from local energy[END_REF][START_REF] Kovesi | Image features from phase congruency[END_REF], FAST [START_REF] Rosten | Faster and better: a machine learning approach to corner detection[END_REF] and BRIEF [START_REF] Calonder | BRIEF: Binary robust independent elementary features[END_REF]. In the specific context of thermal images, the features can be sometimes merely extracted by thresholds and template matching [START_REF] Liu | Pedestrian detection using stereo night vision[END_REF], using an improved version of the Harris corner detector [START_REF] Dhua | Triangulation based technique for efficient stereo computation in infrared images[END_REF] or using the Canny edge detector [START_REF] Bertozzi | Pedestrian detection by means of far-infrared stereo vision[END_REF]. However, in [START_REF] Hajebi | Sparse disparity map from uncalibrated infrared stereo images[END_REF], the authors compare some features detectors such as Harris, Canny, Difference-of-Gaussian with the phase congruency. They prove that, regarding thermal images, phase congruency can extract more features than the methods as mentioned earlier. But after having extracted features, how to exploit them in stereo vision?

The idea behind the stereo-vision is to exploit the difference between the two views, knowing more or less the position of each camera relative to the other one. Several method families are proposed in the literature for stereo computation: global methods such as dynamic programming [START_REF] Ohta | Stereo by intra-and inter-scanline search using dynamic programming[END_REF], intrinsic curves [START_REF] Brown | Advances in computational stereo[END_REF] or local ones including block matching [START_REF] Saxena | Imaging modalities to diagnose carotid artery stenosis: progress and prospect[END_REF], gradient-based optimization [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] and features matching [START_REF] Baumberg | Reliable feature matching across widely separated views[END_REF]. Among these methods, features matching seems to be the most adapted to a sparse features situation. If most of the proposed methods have been developed for visible images, other authors propose an adaptation to thermal images. In [START_REF] Dhua | Triangulation based technique for efficient stereo computation in infrared images[END_REF], after having extracted Harris corners, the authors perform features matching by computing correlation within a search window, discarding outliers, and regularizing the matched features spatially. In [START_REF] Bertozzi | Infrared stereo vision-based pedestrian detection[END_REF][START_REF] Bertozzi | Pedestrian detection by means of far-infrared stereo vision[END_REF], among several variants, the authors propose a relatively similar framework for pedestrian detection: after having extracted features with the Canny edge detector, they perform matching using cross-correlation and specialized methods adapted to a human silhouette. Features matching is also important for thermal image analysis in medical applications [START_REF] Dey | Thermal imaging in medical science[END_REF], [START_REF] Saxena | Study on methods to extract high contrast image in active dynamic thermography[END_REF]. Most of the time, Thermal images are used to analyze the variation of the temperature of the skin or an organ. So in such a condition, the extracted features must be very robust. In [START_REF] Saxena | Infrared (ir) thermography-based quantitative parameters to predict the risk of post-operative cancerous breast resection flap necrosis[END_REF], the authors propose a framework to correct motion artifact due to body movement when recovering breast images. While the amplitude of the Fourier transforms gives the pixel intensity, the phase components represent the spatial information. Given three consecutive frames I 1 , I 3 and I 3 , they combine their phase and amplitude to get a final matched image I 3 . The matching is performed using the method described in [START_REF] Reddy | An fft-based technique for translation, rotation, and scale-invariant image registration[END_REF]. In [START_REF] Hajebi | Sparse disparity map from uncalibrated infrared stereo images[END_REF], the authors, extract the features from thermal images using Log-Gabor filters bank and then perform 1D matching through epipolar lines using the Lades similarity [START_REF] Lades | Distortion invariant object recognition in the dynamic link architecture[END_REF] as a matching criterion. So, processing thermal images using Fourier-based methods is a common way.

All of these works concern thermal images with a reasonable pixel resolution (over 80 × 60). In [START_REF] Hajebi | Structure from infrared stereo images[END_REF], the authors state that the computation time of the phase congruency can be reduced by down-scaling the original from 320×240 to 160×120 and 80 × 60, but they have not explored this solution given the induced loss of efficiency. In our specific case of stereovision, the image resolution has a direct impact on the 3D reconstruction. Indeed being wrong of 1 pixel in a 4 × 4 image is not the same as being wrong of 1 pixel in a 1000 × 1000 image. The information matching between left and right views of the stereo pair is a critical phase, since small errors at this step may yield significant errors in the 3D localization as demonstrated below. The distance between a 3D point and the stereo system can be determined as follows [START_REF] Delon | Small baseline stereovision[END_REF]:

z = b/h ( 1
)
where b is the baseline of the stereo system (distance between the two cameras), the disparity function and h the distance between the scene and the camera system. Deriving 1, we have:

dz = d b/h (2)
The error d on the disparity has a direct impact on the precision in z, especially when b/h is small. The influence of the baseline on the depth precision has been well studied in the literature. In [START_REF] Maier-Hein | Optical techniques for 3d surface reconstruction in computer-assisted laparoscopic surgery[END_REF], the authors empirically show that to determine the distance of a point accurately, the latter must not be further away more than (10 -15) × baseline. Moreover, in [START_REF] Haller | Design of interpolation functions for subpixel-accuracy stereo-vision systems[END_REF], the authors stated that using their stereo setup, a disparity error larger than 0.1 pixels will result in a relative distance error of 2.5% for an object located at 60 meters. This is why, most of the time, the stereo-vision system uses a large baseline [START_REF] Pritchett | Wide baseline stereo matching[END_REF]. In this application, we chose to use a small baseline given indoor constraints. Nevertheless, some applications (as our indoor system) are constrained to use a small baseline. Is it possible to gain in 3D accuracy even in case of small baseline and low image resolution?

Many solutions exist to improve the accuracy of stereo-vision. Among them we have super-resolution [START_REF] Zoetgnande | Edge focused super-resolution of thermal images[END_REF] and sub-pixel matching [START_REF] Tian | Algorithms for subpixel registration[END_REF]. Considering sub-pixel matching, we must first define the disparity map, which "refers to the apparent pixel difference or motion between a pair of a stereo image" [START_REF] Smith | Processing of digital motion images[END_REF]. In the classical methods, the difference is expressed in integer precision. Sub-pixel matching proposes to estimate e in floating precision. There are three main methods of sub-pixel matching: interpolation, fitting, and phase correlation. In interpolation methods, the sub-pixel disparity is estimated by searching the extreme of interpolated cost volume. These methods have two main drawbacks. First, the interpolation of the cost volume is computationally costly. Second, artifacts can be introduced by the interpolation even if most of the time, the accuracy is satisfactory [START_REF] Haller | Design of interpolation functions for subpixel-accuracy stereo-vision systems[END_REF][START_REF] Miclea | New sub-pixel interpolation functions for accurate real-time stereo-matching algorithms[END_REF]. The fitting methods use disparity plane or cost volume to estimate the sub-pixel disparity. The methods using disparity plane fitting involve segmentation constraints [START_REF] Shi | High-accuracy stereo matching based on adaptive ground control points[END_REF]. The cost volume fitting methods search the extreme of a parabola representing this cost volume. These methods are fast but not accurate. Unlike these two methods, phase correlation offers both high efficiency and accuracy. It is performed using fast Fourier transforms and other supplementary approaches [START_REF] Stone | A fast direct Fourierbased algorithm for subpixel registration of images[END_REF]. The phase correlation is the normalized cross-power spectrum, so the matching corresponds to a peak, which has to be estimated [START_REF] Takita | High-Accuracy Subpixel Image Registration Based on Phase-Only Correlation[END_REF] in sub-pixel accuracy. In the state of the art, all of the methods are performed for visible images. We proposed to compute the sub-pixel disparity for thermal images using phase correlation. To our knowledge, it is the first time a sub-pixel matching is performed in such a way in thermal images.

The paper is structured as follows: Section 2 presents our framework ST. It details the implementation of the features extraction method for thermal images based on phase congruency, the stereo matching method, and the sub-pixel matching method based on phase correlation. In Section 3, we discuss, analyze, and explain these results. Finally, section 4 concludes the paper and gives perspectives.

2 Materials and method 2.1 Our stereo system Our indoor acquisition system is composed of a pair of FLIR's lepton 2 cameras (Fig. 1). The horizontal field of view is 51 • and the diagonal field of view is 63.5 • . The maximum frame rate is eight frames per second. The baseline is set to 16 mm.

As input images, we had a pair of low-resolution thermal images acquired in stereo condition I l and I r . The images are rectified, using the output of a robust calibration method [START_REF] Zoetgnandé | Robust low resolution thermal stereo camera calibration[END_REF]. The rectification step simplifies the stereo reconstruction step since that a feature lies now on the same line in I l and I r . The feature matching (and so the estimation of the disparity) is reduced to only estimate the translation δ x of the feature between the 2 images along the x direction. We have:

δ x = d i (x) + d d (x)
where d i (x) and d d (x) are respectively the integer and the decimal parts of the disparity.

Our sub-pixel matching framework is composed of three main steps (Fig. 2): 1) robust features extractions method from the low-resolution thermal images, 2) first rough stereo matching of these features in integer precision (the estimation of d i (x)) and 3) refined sub-pixel matching around the previously matched features (the estimation of d d (x)).

Features extraction

Features extraction was one of the critical points of our framework. Thermal images are characterized by a lack of texture, noisy, and since the resolution of our images is low, a robust method is necessary to extract features from such images. Besides the low-resolution aspect, other characteristics of our images had to be taken into account. Our cameras are uncooled, and thus, they are influenced by ambient temperature. The temperature drift is sometimes compensated in the camera by some corrections which introduce sudden brightness changes. More, this is camera dependent and is not synchronized, which leads to a time-varying difference in brightness between the images. For all these reasons, we chose a method based on phase congruency, which takes into account primitives that are only linked to the difference of viewpoint between the two cameras.

To extract features, we adapted the phase congruency estimation method proposed by [START_REF] Kovesi | Phase congruency: A low-level image invariant[END_REF] and its variant proposed by [START_REF] Wang | Image feature detection based on phase congruency by monogenic filters[END_REF] to our low-resolution thermal images. The 1D phase congruency is the ratio [START_REF] Morrone | Feature detection from local energy[END_REF]:

P haseCong(x) = max φ n A n cos(φ n (x) -φ) n A n (3)
Where A n represents the amplitude or energy of the n th Fourier component and φ n (x) the local phase, which can be calculated using the Hilbert transform. Unfortunately, this version of the phase congruency was noise sensitive and yielded inaccurate features localization as proved by [START_REF] Kovesi | Image features from phase congruency[END_REF]. In [START_REF] Kovesi | Phase congruency detects corners and edges[END_REF], the authors modified a bit the equation to circumvent the previous version drawbacks:

P haseCong 2 (x) = W (x) n A n [Υ n -Ψ n ] -T n A n (x) - (4)
where Υ n = cos(φ n (x) -φ(x)) ; Ψ n = sin(φ n (x)-φ(x)), W (x) is the frequency spread weighting, T is a noise threshold and is a small value (eg. 1-e3) to avoid division by 0.

The equation ( 4) can be extended to the two-dimensional image domain by applying it on several orientations θ after filtering the image by a bank of Log-Gabor filters. To reduce the computation cost due to the number of orientations and scales of the Log-Gabor filters bank, we implemented the variant proposed by [START_REF] Wang | Image feature detection based on phase congruency by monogenic filters[END_REF] using a monogenic filter instead of Log-Gabor. The monogenic signal is a Riesz transform concatenated with a 2D signal. This was possible by constructing a monogenic filter in the frequency domain [START_REF] Wang | Image feature detection based on phase congruency by monogenic filters[END_REF].

As results of the 2D extension, we get a set of P C(θ), the phase congruency at orientation θ. The features are then estimated by combining all the P C(θ). This is done by computing the following values [START_REF] Kovesi | Phase congruency detects corners and edges[END_REF]:

a = (P C(θ)cos(θ)) 2 (5) b = 2 (P C(θ)cos(θ))(P C(θ)sin(θ)) (6) c = (P C(θ)sin(θ)) 2 (7)
Combining a, b and c gives a hint of the strength of the feature. More precisely, the maximum moment M and the minimum moment m can be estimated by:

M = 1 2 (c + a + 2 b 2 + (a -c) 2 ) (8) m = 1 2 (c + a -2 b 2 + (a -c) 2 ) (9)
These moments are used to characterize the features. Higher is the maximum moment more significant will be the feature, and higher is the minimum moment more probably this feature point will be a corner. Because of the lack of information in images, we only took M into accounts. So a feature was considered significant if M was higher than a threshold γ.

As a result of this step, we produced two images I f l and I f r , which are the images of the moment M after applying the phase congruency on respectively the input images I l and I r (Fig. 2).

Stereo matching

In rectified image condition, the matching of a feature F l of I f l to its corresponding one F r in I f r is simplified to find the most similar feature along the corresponding epipolar line. As suggested by [START_REF] Hajebi | Sparse disparity map from uncalibrated infrared stereo images[END_REF], we used the Lades similarity [START_REF] Lades | Distortion invariant object recognition in the dynamic link architecture[END_REF] performed a 5 × 5 window as matching metric:

S(F l , F r ) = nb f eat j f l j f r j 2 nb f eat j f l 2 j nb f eat j f r 2 j ( 10
)
where nb f eat is the total number of features in the selected window. However, compared to their work, we took into account more matching constraints than only the similarity, epipolar constraints and left-rigth consistency:

• Uniqueness: a feature in the left image was matched with only one feature in the right image.

• Continuity: the disparity must vary smoothly.

• Ordering: for a couple of matched features f 1l ↔ f 1r , f 2l ↔ f 2r (the symbol ↔ represents the matching relationship), if f 1l is at the left of f 2l we ensured that f 1r was at the is of f 2r .

We also took into account the type of features and their orientation in the matching process.

We tried to reduce the computation time using a disparity range of d. This range can be obtained if there is prior knowledge about the scene or it can be deduced by the disparities values of the previous frame in a frame by frame framework.

Sub-pixel matching

Phase correlation is a well-known method allowing to get the translation of a signal. The translation between two images in the spatial domain is expressed in the frequency domain using the Fourier transform. Let I 1 and I 2 be two M × N images, M and N odd for mathematical simplicity. These images can be expressed like I 1 = I(x, y) and I 2 = I(x + δx, y + δy) with the translation (δx, δy). Let m and n defined as M = 2m + 1 and N = 2n + 1. By computing their respective Fourier transform F 1 (u, v) and F 2 (u, v), the normalized cross-power spectrum is given by:

R(u, v) = F 1 (u, v)F 2 (u, v) |F 1 (u, v)F 2 (u, v)| (11)
where u = -m, ..., m, v = -n, ..., n and F 2 is the complex conjugate of F 2 .

Given that the most relevant components in the phase correlation matrix are the low-frequency ones, certain authors propose to filter R by a rectangular lowpass function of size U [START_REF] Takita | High-Accuracy Subpixel Image Registration Based on Phase-Only Correlation[END_REF]. They also proved that the ratio of U M = 0.5 is the one that gives the best accuracy.

The Phase-only correlation is the inverse discrete Fourier transform of R and is defined as follows:

r(x, y) = 1 N • M u=m u=-m v=n v=-n R(u, v) exp(-2πiux/M ) exp(-2πivy/N ) (12)
where u = -m, ..., m and v = -n, ..., n. The peak position, in r, corresponds to the translation along x and y directions, but this position is in integer precision.

Some authors propose to recover the displacement in sub-pixel precision from the information contained in r. There are globally three ways to estimate the subpixel displacement accurately: the detection of the peak by local least square fitting on selected spectral components in R [START_REF] Stone | A fast direct Fourierbased algorithm for subpixel registration of images[END_REF], by fitting a closed-form analytic model to the correlation peak [START_REF] Takita | High-Accuracy Subpixel Image Registration Based on Phase-Only Correlation[END_REF], or directly from the data of r using a peak detection formula assuming a peak model [START_REF] Foroosh | Extension of phase correlation to subpixel registration[END_REF][START_REF] Nagashima | A subpixel image matching technique using phase-only correlation[END_REF]. We implemented this last principle.

The correlation peak was modeled as a cardinal sine affected by the low pass filter:

P eakM odel xy = r(x, y) ≈ α sin( V M π(x + δ x )) π(x + δ x ) × sin( V N π(y + δ y )) π(y + δ y ) (13)
where δ x ,δ y are respectively the displacements along x and y axis, α is the peak value ( 0 ≤ α ≤ 1) and V = 2U + 1.

Given that we used stereo cameras, by rectifying the images, we were able to neglect δ y . So the phase correlation function became:

P eakM odel x = r(x) α sin( V M π(x + δ x )) π(x + δ x ) (14)
The main idea was then to recover δ x , the max peak in sub-pixel accuracy, from the sample values in r. A robust solution has been described in [START_REF] Nagashima | A subpixel image matching technique using phase-only correlation[END_REF]. Let consider the case in Fig. 3, the peak model max is close to the sample x = p which has the highest values α. We also considered two other measurements located at ±d pixels away from p at x = p + d and x = p -d.

Using the model, the three points can be rewritten as:

       P eakM odel x (p -d) = r(p -d) α sin( V M π(p-d+δx)) π(p-d+δx) P eakM odel x (p) = r(p) α sin( V M π(p+δx)) π(p+δx) P eakM odel x (p + d) = r(p + d) α sin( V M π(p+d+δx)) π(p+d+δx) (15)
By combining these equations, it can be proved that:

(p -d + δ x ) r (p -d) + (p + d + δ x ) r (p + d) -2 (p + δ x ) cos V N πd r (p) = 0
which can be rewritten as

v(p, d) = δ x u(p, d) (16)
with

u(p, d) = r(p -d) + r(p + d) -2 cos V N πd r (p) and v (p, d) = 2p cos V N πd r (p) -(p -d) r (p -d) -(p + d) r (p + d)
This allows us to estimate δ x = u (p, d) -1 v (p, d). However, in order to minimize the impact of noise, several observations with different values of p and d around the highest peak in r can be used. So it is possible to select χ values of p i and d i and then get χ equations v (p i , d i ) = δ x u (p i , d i ) where i ∈ {1, 2, ..., χ}. Resolving these equations is equivalent to minimize δ x in:

δ x = χ i=1 |v (p i , d i ) -u (p i , d i )| 2 (17)
This equation can be solved using Singular Value Decomposition (SVD) [START_REF] Abdi | Singular value decomposition (svd) and generalized singular value decomposition[END_REF]. The existing phase correlation-based sub-pixel matching methods are applied on high resolution visible images (around 3000 × 3000 pixels [START_REF] Ma | A subpixel matching method for stereovision of narrow baseline remotely sensed imagery[END_REF]) using large subwindows (41 × 41 pixels) for the phase correlation estimation. We had to adapt this class of methods to our low-resolution problem. Let consider two matched points F l with coordinate (x l , y l ) and F r with coordinate (x r , y r ) respectively in I f l and in the shifted image I f r . We knew that y l = y r and d i (x) = x r -x l . Then we defined two sub-images Is l (x l , y l ) and Is r (x r , y r ) with the same windows size W around F l and F r (Fig. 2) and performed the refined phase correlation-based matching in these sub-images.

The main parameters of this method are the sub-images windows size W and the number χ of observations used for the least square estimation of the sub-pixel displacement [START_REF] Rosten | Faster and better: a machine learning approach to corner detection[END_REF]. Because of our low resolution we limited the number of observations χ to 6: p i = p ± 1 and d i ∈ {1, 2}.

Results and discussion

Datasets

In this paper, we used two datasets: a thermal infrared video benchmark for visual analysis with high-resolution infrared images [START_REF] Wu | A thermal infrared video benchmark for visual analysis[END_REF] and an own dataset (called Tvvlgo in the rest of the paper) with images acquired by our system. Tvvlgo was created by placing the stereo system in the ceil of a room, and we collected 1000, 80 pixels images pair of a person moving in a room. We also used another third image where a person is sitting in from of the cameras. The dataset we used is available online [START_REF] Zoetgnande | Dataset low-resolution thermal images[END_REF].

Feature extraction

In [START_REF] Kovesi | Image features from phase congruency[END_REF], the author proposes to use a threshold on the phase congruency moments of γ = 0.3. Because of our low-resolution images context, we feared that this threshold could not extract enough features. We tried other smaller thresholds at γ = 0.1 and γ = 0.01.

Evaluation of the number of extracted features

To validate our choice of the features extraction method, we compared the implemented phase congruency to other standard features extraction methods using a low-resolution image given by a FLIR lepton 2 (Fig. 4-a). The authors in [START_REF] Hajebi | Structure from infrared stereo images[END_REF] have already made such comparison, but they only compared the phase congruency method to Harris corner detector, SIFT, Canny, and KLT. We wanted to go further. So we have compared our approach with the three thresholds (PhaseCong(0.01), PhaseCong(0.1) and PhaseCong(0.3)) to the OpenCV [START_REF] Kaehler | Learning OpenCV 3: computer vision in C++ with the OpenCV library[END_REF] implementations of ORB, BRISK, FAST, Shi Tomasi, SURF, AGAST, GFTT, and KAZE. Our framework was implemented using C++ compiled with GCC 7. We compared the methods according to 2 criteria: the number of extracted features and the execution time. For this later measurement, all the codes were executed on an Intel Core i7-3687U CPU. To get the execution time we used the C++ API std::chrono::high resolution clock::now. For each feature detector, the feature extraction process was completed 1000 times, and we computed the mean and the standard deviation of all computation times. The extracted features can be seen in Fig. 4 and measurements are sampled in Table 1.

As illustrated by Table 1 Phase congruency can extract more features than other methods. Even using a high threshold, we could extract two times more features than Shi-Tomasi and GFTT. The counterpart of this advantage is a relatively high computation time compared to techniques such as FAST. Fortunately, the feature extraction for a pair of images took approximately 10 ms with Phase congruency, which is compatible with the 8 frames per second rate of our cameras.

These results are visually confirmed by Figure 4, where we extracted features from a low-resolution image using different feature extractors. It is noticeable that while most of the feature extractors try to extract robust features, phase congruency does the same by focusing on edges. Using the threshold γ, we could control the number of features returned by phase congruency. γ = 0.1 seemed to give a good trade-off between the sparsity and redundancy of features.

To conclude, even if Phase congruency is slower to compute than the other classical method, it gave a higher number of features and so seemed to be unavoidable in processing our low resolution and texture-less images.

Robustness to illumination change

One particularity of our cameras is that we sometimes noticed a sudden change in the brightness. It is probably a re-calibration of the sensor. The feature extraction method should so be robust to brightness changes. We evaluated this robustness using a simulated brightness change as defined by Szelinski [56, Chapter 3]: The color of feature is generated randomly.

g(x) = αf (x) + β (18)
where f(x) and g(x) are the pixel values, and α, β control respectively the contrast and the brightness.

So we took 1000 images (80x60) acquired using lepton 2, then we simulated brightness changes by varying the parameter β from 0 to 100. For each image, we computed the number of features detected for each β. Let Ω β be the number of matches for a value of brightness control β. The Figure 5 represents the variation of the Ω β according to β.

We also needed to evaluate the features extraction method in terms of redetection.

For this, first, we computed the features at β=0. Then by increasing β, we estimated this robustness by counting the percentage of features still detected at the same location as for β=0. We called this percentage features a re-detection rate. Figure 6 shows the Features re-detection rate vs. brightness change for one image of the Tvvlgo dataset [START_REF] Zoetgnande | Dataset low-resolution thermal images[END_REF].

We compared PhaseCong(0.1) with other feature extractors such as ORB, FAST, SURF, AGAST, and GFTT. Our results also confirmed those of Hajebi et al [START_REF] Hajebi | Dense surface from infrared stereo[END_REF] assessing that the features detected by phase congruency from thermal images present the advantage to be more stable than the others to illumination changes (Fig. 5 and6). Figure 5 shows that given different values of illumination, phase congruency is the feature extraction method that can extract the highest number of features. Figure 6 shows that phase congruency is the method with the best re-detection rate. While detecting more features, phase congruency is also more robust to brightness changes than the other classical methods.

Stereo matching

To evaluate our stereo matching method, we have performed a stereo matching on 15 low-resolution 80 × 60 stereo pairs provided by our cameras (Tvvglo [START_REF] Zoetgnande | Dataset low-resolution thermal images[END_REF]). The stereo matching method explained in the section 2 was applied. To verify the accuracy of the method, we manually counted the number of mismatches given the number of matches. For each feature extracted in I l , we visually verified if the matched feature in I r was inside a 5 × 5 window centered on the estimated disparity. The results showed that we had a percentage of mismatches of less than 1% through all of the 15 image pairs.

Sub-pixel matching

As mentioned in section 2.4, one of the key parameters is the size W of the subimages windows in which the phase correlation is computed. In the state of the art, the sizes of sub-images vary a lot. For images with good resolution such as 4000 × 4000, a W of 41 × 41 can be chosen [START_REF] Ma | A subpixel matching method for stereovision of narrow baseline remotely sensed imagery[END_REF]. But these window sizes must not only be set regarding the size of the images but also depends on the type of the scene and how the disparity varies through the whole image. Nevertheless, it would be unreasonable to use a W of 41 × 41 when the image size is 80 × 60.

In fact, we set 2 experiments, one on good resolution images in order to estimate the sole impact of W on the accuracy of sub-pixel matching and one on our low resolution images to prove the validity of our method according to the end application and to estimate the combined impact of the phase congruency threshold γ and W on the sub-pixel matching accuracy. We designed the following experiment to estimate the sub-pixel matching accuracy: for a specific thermal image I 1 , we created a second image I 2 by shifting the information of I 1 by ∆ pixels along the x-axis. The accuracy can be determined according to the distance between an estimated sub-pixel disparity to ∆ (the true disparity). We called it matching error. We also created a matching precision rate measurement. For this, we set a precision threshold of τ . Given an estimated disparity δ if |δ -∆| ≤ τ the match is considered as correct else as a mismatch. The matching precision rate is the ratio between the number of correct matches (relative to a level of precision τ ) to the total number of features.

For the good resolution thermal images, we used a 512 × 512 image of people coming from [START_REF] Wu | A thermal infrared video benchmark for visual analysis[END_REF]. We shifted these images by ∆ ∈ {20, 20.125, 20.250, ..., 25} pixels. The matching errors were computed on all the features for all the set of ∆. Fig. 7 shows the box plot of the root-mean-square deviation (RMSD) of the matching errors as a function of the window size W , with W ∈ {7, 9, . . . , 41}. The impact of the window size on the rate of good matching respects to a specific level of precision τ can be seen in Fig. 8. We drew this curves for 4 levels of precision: τ = {0.05, 0.01, 0.025, 0.5} pixels.

The low resolution (80×60 pixels) images were acquired with our FLIR Lepton 2 camera system ( [START_REF] Zoetgnande | Dataset low-resolution thermal images[END_REF]). For each image we shifted them by ∆ ∈ {0, 0.125, .., 30}pixels with a stride of 0.125 pixels. The size of our sub-images was in the range W ∈ {7 × 7; 9 × 9; 11 × 11}. The Figure 9 shows the impact of the window size W and the phase congruency moments threshold γ on the rate of good matching respects to a specific level of precision τ , τ ∈ {0.5, 0.25, 0.1, 0.05} pixels in our case. The value of the phase congruency moments threshold γ is directly related to the number of features extracted on the images (Table 1). However, we had to find a good tradeoff between the number and significance of the extracted features.

We studied the impact of the sub-images window size in which the phase correlation is estimated on the accuracy of the sub-pixel localization. As shown in Fig. 7, using a wide sub-images window allowed us to get better precision. From 7 × 7 to 29 × 29, the precision was increased progressively, but from 31 × 31, the gain in precision was no more very noticeable. The phase congruency magnitude represents the filtered information of the image, so using a wider window does not help necessary to get better precision. This result is also confirmed in Fig. 8 and9 on the cases were low precision was sufficient. These Figures showed the percentage of correct matches at a certain precision using different sizes of window size. For low precision (errors below τ = 0.5 or τ = 0.25 pixels), the matching rate was increased with higher sub-images window size. This behavior was shared on both high and low-resolution images. The only difference between these 2 cases is that, for the matching of low-resolution images, we had to limit the sub-images window size to 13 × 13, which is already large compared to the 80 × 60 image size.

More surprisingly, by analyzing the three Figures accurately, it can be noticed that a window 9 × 9 offered better precision than an 11 × 11 one. This is even more visible on the high precision matching rates (errors below τ = 0.1 or τ = 0.05 pixels) curves (Fig. 8-c andd and9-c andd). One explanation could be that the precision of the match did not only depend on the size of the window size but overall by the ratio between noise and relevant information in this window.

In our framework, they are 2 processes that handle the image noise: 1) the phase congruency magnitude represents filtered information of the image and 2) a low pass filter the phase correlation. In this latter case, the filter bandwidth was directly set proportional to the sub-images window size (U = 0.5W in [START_REF] Watanabe | Co-occurrence histograms of oriented gradients for pedestrian detection[END_REF]). It seemed like that for W = 9 × 9, the ratio between the useful information (on low spatial frequency in the cross-power spectrum [START_REF] Watanabe | Co-occurrence histograms of oriented gradients for pedestrian detection[END_REF]) and noise (on the high spatial frequency) reached a local maxima. If this hypothesis is confirmed, more noise is included in the phase correlation for higher W , degrading so its shape.

The expected precision also had a direct impact on the matching rate. We can see in Fig. 8 that for the high-resolution image, almost all the features (more than 99%) are matched with a precision lower than 0.5 pixels regardless of the window size. This matching rate decreased to around 90%, 55%, and 33% for respectively a precision lower than 0.25, 0.1, and 0.05 pixels. We can notice the same behavior for the low-resolution image with matching rates (in the case of the best phase congruency threshold) around 97%, 83%, 55% and 34% for respectively a precision lower than 0.5, 0.25, 0.1 and 0.05 pixels. Moreover, we noticed the same matching rate (≈ 33%), whatever the resolution of the images when a high matching precision is expected.

The choice of the phase congruency moments threshold also had an impact on the matching rates. Because of the low resolution of our images, we wanted to increase the number of features using the lower phase congruency moments threshold. This was the case, as shown in Table 1. The number of extracted features increased from 275 to 1793 when we decreased the threshold of γ from 0.3 (as recommended in [START_REF] Kovesi | Phase congruency detects corners and edges[END_REF]) to 0.01. Unfortunately, the matching rates for a specific precision, also decreased when we decreased the threshold, as we can see in Fig. 9. Decreasing the threshold brought less stable features to match. Depending on the application, a good tradeoff has to find between the number of matched features and their reliability.

To summarize, in our specific low resolution stereo thermal camera case, a subimages window size window of 9 × 9 can be a good tradeoff between the accuracy and the phase correlation computation time. With this configuration, we were able to match around 97% of the features with a precision of at least 0.5 pixels. If a higher sub-pixel matching accuracy is needed, this rate falls to 55% or 34% for a respectively a precision less than 0.1 or 0.05 pixels. In this case, an external outlier rejection process based on the image or the 3D scene content should be added during or after the matching [START_REF] Wolff | Point cloud noise and outlier removal for image-based 3D reconstruction[END_REF].

3D reconstruction

Importance of sub-pixel matching

After matching, an error on the estimation of the disparity can have high consequences for the stereo reconstruction, especially to estimate the distance of a 3D point to the cameras (Z direction) [START_REF] Haller | Design of interpolation functions for subpixel-accuracy stereo-vision systems[END_REF] accurately. In this section, we wanted to estimate the range of error along the Z direction when we estimated the disparity with an error of one pixel (in pixel resolution) and also the evolution of this range using sub-pixel disparity estimation (error of 0.5 or 0.1 pixels). For this, we chose a pair of matched points, which gave a distance Z=3350 mm after stereo reconstruction for a disparity d. If we have an error of 1 pixel (a disparity of d new = d ± 1 pixel), the estimation of Z was spread from Z = 3108 mm to 3590 mm (a range of

Evaluation of the whole framework

In this section, we tried to compare the whole framework of our method ST versus a method available in classical computer vision libraries. As a feature extractor, we chose ORB because this feature extractor has already been proved to be more robust than other [START_REF] Karami | Image matching using sift, surf, brief and orb: performance comparison for distorted images[END_REF]. For the matching, we used KNN because of the sparsity of the features and also because it estimates the disparity in sub-pixel precision. We implemented this framework (we called ORB + KNN matching) using the functions available in OpenCV. For ST, we set γ = 0.1 and for phase correlation we used a sub-image size of 9 × 9.

We applied these methods on two image pairs (Fig. 10a and Fig. 10d) we selected from the Tvvlgo dataset. In these images, the person was approximately at a distance of 3-4 meters. We compared the results of these methods according to two criteria: the numbers of matched features and the consistency of the 3D reconstruction, especially along the Z direction.

The Figure 10 represents the matches projected on the left images when using ORB + KNN (10b and 10e) and ST (10c and 10f). First of all, we can see that using ST, we have 4 times more matches than ORB + KNN: 80 vs. 24 and 71 vs. 18 respectively for the 507 th pair and the 547 th pair. It can also be noticed that the matches are better distributed over the whole human shape than using ST than using ORB + KNN. This is due to the fact that phase congruency can extract more features from thermal images than ORB, as reported in Table 1. For all the matches, we performed triangulation to estimate the 3D position of the point. We made statistics on the estimated depth Z. Figure 11 shows the boxplot of the distributions of Z for both methods. The mean and the standard deviation of these distributions are given in Table 2. The results obtained using ORB + KNN seems to be inconsistent (very wide distribution of Z in a range of about 5000 mm). On the contrary, for ST, the distribution is more compact. The median and mean values are around 3500 mm, which is consistent with the experimental conditions, and the standard deviations (256 mm and 391 mm) are more reliable concerning human body proportion. The results output by ORB + KNN can be explained by the fact that the matches are not correct, leading to very bad triangulation outputs.

Conclusion

In this paper, we proposed our method ST which is a sub-pixel stereo matching method adapted to thermal images. Thermal images have the disadvantages of being less textured compared to gray-scale or color images. Besides this lack of texture, low-cost thermal cameras can have a very low resolution (80 × 60 pixels in our case), which is detrimental to an accurate stereo reconstruction. To overcome these limitations, we proposed a framework composed of a robust feature extraction method based on phase congruency, a robust rough matching process based on Lades distance between the extracted features, and a refined sub-pixel matching process based on phase correlation. When applied to low-resolution thermal images, our feature extraction method was able to extract more features than state of the art methods. As well, our sub-pixel matching method was able to match around 97% of extracted features with an (average) error under 0.5 pixels. For about 55% of the features, the matching error was even below 0.1 pixels. Such a level of accuracy is necessary for the stereo reconstruction of a 3D scene or the 3D localization of objects. With our stereo setup, a precision bellow 0.1 pixels corresponds to a maximal error of ≈ 51 mm in the depth direction. With our framework, such a level of precision seems now achievable even for very low-resolution thermal stereo cameras. We also compared ST versus a classical method in state of the art (ORB + KNN) for 3D reconstruction. Our method showed more consistent results than ORB + KNN.

Once the sub-pixel matching is performed, it could be essential to compare our method in 3D localization using ground truth values. Such evaluation will show us how sub-pixel matching is improving 3D vision.

Figure 1 :

 1 Figure 1: The stereo system composed of two lepton 2 cameras placed.

Figure 2 :

 2 Figure 2: Sub-pixel matching framework: 1) robust features extraction; 2) rough features matching; 3) refined matching in sub-pixel accuracy.

Figure 3 :

 3 Figure 3: Peak model. In the vertical axis, the value of the phase correlation value at the point in the horizontal axis.

Figure 4 :

 4 Figure 4: Features (colored dots) extracted by several feature extractor methods.

Figure 5 :Figure 6 :

 56 Figure 5: Average number of features detected for each image by some features extractors

Figure 7 :

 7 Figure 7: Box plots of the root-mean-square deviation (RMSD) of the matching error (in pixel) vs. sub-images window size

Figure 8 :

 8 Figure 8: Impact of the sub-images window size (abscissa) on the rate of good matching (ordinate in %) respects to a specific level of precision

Figure 9 :

 9 Figure 9: Percentage of correct matches (ordinate in %) with a given level precision τ according to the sub-images window size (abscissa) and the phase congruency moments threshold (PC threshold)

  (a) Left image of the 507 th pair of Tvvlgo (b) Matches re-projected using ORB (#24) (c) Matches re-projected using ST (#80) (d) Left image of the 547 th pair of Tvvlgo (e) Matches re-projected using ORB (#18) (f) Matches re-projected using ST (#71)

Figure 10 :

 10 Figure 10: 3D points projected in the images space with Z as the value of the pixel.

Figure 11 :

 11 Figure 11: Box plot in the value of Z for our method ST and ORB + OpenCV KNN.

Table 1 :

 1 Comparison between feature extractor methods ORB, BRISK, FAST, Shi Tomasi, SURF, AGAST, GFTT, KAZE and Phase congruency

	Feature extractions methods	Number of features Execution time
	ORB (Oriented fast and Rotated Brief)	117	± 20
	BRISK (BinaryRobust Invariant Scalable Keypoints)	34	± 350
	FAST	77	± 7
	Shi Tomasi	120	± 15
	SURF (min Hessian = 300) (Speeded Up Robust Features) 14	± 100
	AGAST (Adaptive and generic corner detection based on the accelerated segment test)	84	± 29
	GFTT (Good Features to track)	120	± 32
	KAZE	33	± 123
	PhaseCong(0.01)	1734	± 480
	PhaseCong(0.01)	777	± 480
	PhaseCong(0.01)	275	± 480

Table 2 :

 2 1523.94 ± 1612.76 3648.10 ± 256.43 547 th pair 1800.85 ± 1538.92 3746.56 ± 391.44 Mean value and standard deviation of Z after triangulation. Values are in millimeters.

	ORB + OpenCV KNN (Baseline)	ST (Our method)
	507 th pair

https://flir.netx.net/file/asset/19205/original

Acknowledgments

This work was part of the PRuDENCE project (ANR-16-CE19-0015) which has been supported by the French National Research Agency (ANR).


Sharan Raj

Rangasamy Mahendren

Hélène Welemane

Olivier Dalverny

Amèvi Tongne

Unilateral behaviour of microcracks and thermal conduction properties: a homogenization approach

Keywords: Heat conduction, homogenization, microcracking, unilateral effect

Damaging effects of microcracks on the elastic properties of brittle materials (rocks, concrete, ceramics) have been extensively studied through experiments and modelling approaches. In the latter case, the homogenization (up-scaling) technique appears as an effective tool to provide the overall properties from the microstructural features of the material [1,2]. Especially for microcracks, main difficulties arise from: (i) the anisotropy induced by the oriented nature of defects, (ii) their ability to be open or closed according to tension or compression loading and to influence differently the overall response of the material. Such unilateral behaviour is typical of contact problems related to that kind of defects. Regarding the thermal properties of microcracked media, very few investigations on the effects have been done in the literature, even on the experimental point of view [3]. Some micro-macro modelling works have been proposed to derive effective properties of a given Representative Volume Element (RVE) but consider only the open state of microcracks [4]. This work aims to explore a homogenization-based approach to derive thermal properties accounting for unilateral effect. We also intend to perform numerical simulations through a finite-element modelling in order to validate these theoretical results.

Considering stationary thermal conduction, the macroscopic temperature gradient G (respectively macroscopic heat flux Q) corresponds to the average of its microscopic quantity g (resp. q) over the RVE, i.e. G = g ; Q = q . Let us consider a 3D RVE composed of a matrix weakened by randomly distributed parallel microcracks. Such a heterogeneous material shows classical matrix-inclusion topology allowing the use of averaging techniques. We can either impose uniform G or uniform Q on the outer boundary δΩ of the RVE. Taking this into account the microscopic and macroscopic properties can be linearly linked thought the equation g = A • G and q = B • Q, where A (resp. B) is the second order temperature gradient localization (resp. heat flux concentration) tensor.

The derivation of the closed-form expression for the thermal properties starts with the linear thermal problem. According to the Fourier law,

q = -λ • g and g = -ρ • q (1)
where λ and ρ are the second order thermal conductivity and resistivity tensors respectively. Therefore, the overall behaviour of the RVE can be given similar to Eq. 1 as:

Q = -λ hom • G and G = -ρ hom • Q (2)
Taking all simplifications into account, the effective properties can be given as

λ hom = λm + fc (λc -λm) • A c and ρ hom = ρ m + fc (ρ c -ρ m ) • B c (3)
where λ m (resp. λ c ) and ρ m (resp. ρ c ) are the conductivity and resistivity tensors of the matrix (resp. cracks), f c is the volume fraction and • c is the mean value over the cracks phase. Also the following relations are respected: λ m = (ρ m ) -1 , λ c = (ρ c ) -1 .


DESIGN AND PRELIMINARY STUDY OF A NEUROFEEDBACK PROTOCOL TO REDUCE DROWSINESS

Thibaut Monseigne [START_REF] Micoulaud-Franchi | EEG neurofeedback treatments in children with ADHD: an updated meta-analysis of randomized controlled trials[END_REF][START_REF]OpenViBE: An Open-Source Software Platform to Design, Test and Use Brain-Computer Interfaces in Real and Virtual Environments[END_REF] , Fabien Lotte 1,2 , Stephanie Bioulac [START_REF]OpenViBE: An Open-Source Software Platform to Design, Test and Use Brain-Computer Interfaces in Real and Virtual Environments[END_REF][START_REF] Sitaram | Closed-loop brain training: the science of neurofeedback[END_REF] , Jean-Arthur Micoulaud-Franchi [START_REF]OpenViBE: An Open-Source Software Platform to Design, Test and Use Brain-Computer Interfaces in Real and Virtual Environments[END_REF][START_REF] Sitaram | Closed-loop brain training: the science of neurofeedback[END_REF] , Pierre Philip 3,4

1 Inria Bordeaux -Sud-Ouest, Talence, France 2 LaBRI -(CNRS / Univ. Bordeaux / Bordeaux INP), Talence, France 3 SANPSY (CNRS / Univ. Bordeaux), USR 3413, Bordeaux, France [START_REF] Sitaram | Closed-loop brain training: the science of neurofeedback[END_REF] Clinique du sommeil, Hôpital Pellegrin-Tripode, Bordeaux, France E-mail: thibaut.monseigne@inria.fr NeuroFeedback (NF) consists in using electroencephalographic (EEG) measurements to guide users to perform a cognitive learning using information coming from their own brain activity, by means of a realtime sensory feedback (e.g., visual or auditory) [START_REF] Sitaram | Closed-loop brain training: the science of neurofeedback[END_REF].

Many NF approaches have been studied to improve attentional abilities, notably for Attention Deficit Hyperactivity Disorder [START_REF] Micoulaud-Franchi | EEG neurofeedback treatments in children with ADHD: an updated meta-analysis of randomized controlled trials[END_REF][START_REF] Micoulaud-Franchi | Electroencephalographic neurofeedback: Level of evidence in mental and brain disorders and suggestions for good clinical practice[END_REF]. However, to our knowledge, no NF solution has been proposed to specifically reduce drowsiness. Thus, we propose a complete EEG-NF solution to train users to self-regulate an EEG marker of drowsiness. This marker is based on a ratio of beta over theta/alpha power in Cz electrode. In addition to this EEG marker of drowsiness, we also carefully selected and designed the duration, the sequencing, the objective evaluation metrics and the visual and audio feedback to use in for each NF session.

Preliminary study with five healthy subjects showed that three of them could learn to self-regulate this EEG marker with a relatively short number of NF sessions (up to 8 sessions of 40 min). Clinical trials with sleep-deprived subjects are expected to begin in 2019 to study possible cognitive and clinical benefits of this self-regulation. The implementation of this NF solution is available for free [START_REF] Micoulaud-Franchi | EEG neurofeedback treatments in children with ADHD: an updated meta-analysis of randomized controlled trials[END_REF] , with the OpenViBE platform [START_REF]OpenViBE: An Open-Source Software Platform to Design, Test and Use Brain-Computer Interfaces in Real and Virtual Environments[END_REF], under the AGPL-3.0 license.


Emotional Mapping -towards a geographical explanation of drug use

Keywords: drug user, urban, space, emotion, mapping

The article highlights how place plays an important role in explaining drug consumption. Within a perspective rooted in critical cartography and social geography, we propose emotional mapping as a method of conceiving urban lived spaces of drug users in European cities. The method provides insights in the spatial experiences and emotions regarding urban space. Our geographical explanation relies on two concepts from social geography. At a macro scale, structural phenomenon of socio-spatial exclusion in European metropolis explain how neoliberal urban policies produce spatial eviction. At a micro scale, personal emotional appropriation of urban places explains individual patterns of drug use. The data presented in this paper is part of an ongoing research process within the DRUSEC (Drug Use and Urban Security) research consortium.

Introduction

In Europe, most of the open drug scenes have similar spatial and social characteristics.

European cities are structured around historically dense inner cities with a high intensity of activities and functions such as housing, tourism and consumption, and a wide diversity of social groups. Inner-city areas, e.g. main station areas, are the places where inhabitants, tourists, partying people and homeless drug users cohabit. Inner city places are highly conflictual, which -especially in the growing dynamics of gentrification -potentially marginalise more and more drug users in public spaces.

Today, drug use is widely understood as a socially embedded phenomenon that is constructed and determined through the social environment. The specificity of the urban geographical approach is to consider the relation between places, spaces and drug use and to understand the spatial embeddedness of drug use [START_REF] Duff | Towards a theory of drug use contexts: Space, embodiment and practice[END_REF][START_REF] Moreno | Using Space: Critical Geographies of Drugs and Alcohol[END_REF][START_REF] Parkin | Habitus and drug using environments[END_REF]. As Potter (2018, p.13) points out, taking drugs "are physical events that happen in specific locations": consumption settings, place and context do matter. Our aim is to provide geographical explanations of the question "why do marginalised drug users do drugs in the public places of the city?". Thus, we don't aim at merely localising consumption places, but at asking "where takes place the life of the marginalised drug users?" Inspired by social geography, critical cartography and critical criminology, we chose a theoretical and methodological framework able to consider marginalised drug users as the subjects of their narratives and spatial representations. We consider their everyday lives as wholes and their complex -and emotional -relationships to the urban spaces.

We developed a mapping interview methodology called "emotional mapping" in order to consider the users knowledge which is beyond the cartographic practices of official (police) statistics, media representations and other institutional instruments of knowledge production.

The development and first results of the "emotional mapping" presented in this article are part of a French-German research consortium that focuses on drug use in public settings and urban security (DRUSEC) and is a work in progress. The interviews we rely on in this paper took place in Nuremberg and Munich (cities without Drug Consumption Room, from now on DCR) as well as in Berlin and Frankfurt (cities with DCR).

From "maps in minds" 1 to a map on a paper

As geographers, we highlight the spatiality of social practices. Space is not only a location, background or context, but also a meaningful dimension of social practices.

In this sense, drug consumption has its own geographies -on a world scale, regional scale or on the scale of places -which are embedded in economic, political and social practices [START_REF] Potter | Introduction: Drugs, place, space and time[END_REF]. In this paper, we focus at the scale of urban places within the city. Emotional mapping is a qualitative methodology, focused on the marginalised drug user as an urban subject and as such it differentiates itself strongly from other drug mapping practices. Our aim is to find a method able to draw how marginalised drug users -the particular group we focus on in this paper -have specific routines and emotions in home and shelters, places to make money, dealers, substitutions medicine and other resources. At the end of the research process, the interpretation of interviews and maps should provide an overview on the drug geographies of cities from the position of marginalised users

Mapping drugs in the city: positivist approaches and fear maps

There is increasing research interest in spatial approaches to understanding drug issues, including mapping drugs at an urban scale: Spatial approaches understand space mostly as a physical phenomenon which enables quantitative and spatial modelisations, based on epidemiological or geocriminological theories [START_REF] Gruenewald | Geospatial Analyses of Alcohol and Drug Problems: Empirical Needs and Theoretical Foundations[END_REF][START_REF] Gorman | Linking Places to Problems: Geospatial Theories of Neighborhoods, Alcohol and Crime[END_REF]. Such positivist approaches reduce the complexity of the social [START_REF] Belina | Kriminalitätskartierung -Produkt oder Mittel neoliberalen Regierens. [Crime Maps -Products or Means of neoliberal Governement[END_REF], e.g. when spatial drug research takes neither the everyday lives of marginalised drug users, nor broader urban structures into account. So, epidemiological and geocriminological theories fail to understand drug use practices.

Positivists approaches are concerned by the question of the data they rely on.

Available data for mapping drug practices generally derive from the work of different institutions, such as police (arrests), harm reduction and social workers (number of clients), pharmacists, etc. The maps based on these data reflect not drug practices but the respective work of the institutions [START_REF] Germes | Cartographies policières. La dimension vernaculaire du contrôle territorial[END_REF].

Other drug mapping practices focus on perceived safety, perceived risk or the fear as immediate emotion, often conflating them [START_REF] Hinkle | Emotional Fear of Crime vs. Perceived Safety and Risk: Implications for Measuring "Fear" and Testing the Broken Windows Thesis[END_REF]. In neoliberal times of securitisation, the inhabitants fear and insecurities often become the legitimation for public policies [START_REF] Shirlow | The geographies and politics of fear[END_REF] in order to repress, displace, and erase non-conforming marginalised drug users from particular neighbourhoods. We refrain from such an approach. Instead, we prefer a more diverse approach of positive as well as negative emotions -because drug use is about both pleasure and risk -and we priority question the emotions of drug users -as showed by the panel of interviewees in this paper.

Qualitative mapping with social geography and critical cartography

On the contrary, of a positivist understanding of maps representing an objective space mostly with quantitative data, our approach of mapping is based on social geography and critical cartography.

In our understanding, space is not a mere objective physical dimension, but produced [START_REF] Lefebvre | La production de l'espace[END_REF], throughout the time and histories of actors, throughout the scales and interdependencies, materialising power relationships, structuring everyday lives and destinies, yet inhabited and transformed by subjectivities. Spaces are complex social constructions and often involved in conflicts between social groups and different uses. That is why the presence of drug users in public spaces, their access to places and resources, as well as, the autonomy or control of every person in public are central questions for public debate and research. These questions are fundamental to understand the where and why of drug use, and we will address them through the lenses of individual lives of marginalised drug users. In order not to objectify their lives in the city, we refer to the distinction made by social geography between living and lived spaces2 : The living space is the area of our everyday lives, a mere list of places (such as home, friends' homes, frequently visited public places, routes). The lived space, a core concept of the French speaking social geography, designates the way we inhabit these places and we interact with others, the relationships we establish with them, the representations we have from them. With the concept of lived spaces, the research focus on how the ways in which our participants inhabit spaces, the everyday practices they use and the emotions they associate with these spaces.

How to map lived spaces -subjective spaces? We follow the critical cartography perspective [START_REF] Harley | Deconstructing the map[END_REF][START_REF] Crampton | An Introduction to Critical Cartography[END_REF][START_REF] Kindynis | Ripping up the Map: Criminology and Cartography Reconsidered[END_REF][START_REF] Belina | Kriminalitätskartierung als Methode der Kritischen Kriminologie? [Crime mapping as a method of critical criminology?[END_REF]) -which does not only criticize the abstractions and objectifications of positivists maps, but also proposes methods to create, produce and work with maps, in a qualitative way, enhancing the complexities of the production of urban spaces.

Qualitative drawn maps have been for a long time developed as a research method in different disciplines such as urban planning, psychology, geography or urban theory [START_REF] Lynch | The Image of the City[END_REF][START_REF] Downs | Maps in minds: reflections on cognitive mapping[END_REF][START_REF] Ziervogel | Mental-Map-Method in research of neighbourhood managment[END_REF]. We enhanced the process of "mental maps", a semi-directed interview in the course of which interviewees draw the places of their everyday lives and their representation of space. In doing so we develop our own method of emotional mapping, one contribution to the lasting interest of geographers into mapping emotions [START_REF] Nold | Emotional Cartography: Technologies of the Self[END_REF][START_REF] Olmedo | Conceptuality and sensibility in the sensible cartography] Implications philosophiques[END_REF][START_REF] Mekdjian | Médier les récits de vie. Expérimentations de cartographies narratives et sensibles[END_REF][START_REF] Rekacewicz | Cartographier les émotions[END_REF][START_REF] Muis | Psychogéographie et carte des émotions, un apport à l'analyse du territoire ? [Psychogeography and emotion maps, a contribution to the analysis of territories?[END_REF][START_REF] Dernat | Représentations socio-cognitives de l'espace géographique[END_REF])

Mapping emotions

In order to obtain maps and narratives about lived spaces (and not only living spaces), and in order not to fall back into the traps of "fear" maps (s. above), we decided to include emotions as a major part of the mapping process. Recently there is a growing interest in the role of emotions, especially the role of pleasure within the context of drug use (Ducan et al., 2017). Emotions are increasingly recognized as an important factor that influences how and why people do drugs. We consider emotions as constructed and individual lived expressions of social conditions. For the mapping interviews, our first aim was to integrate a diverse but uniform spectrum of emotions, which enables an understanding of them in drug places of different cities and the comparability of drawn maps and transcribed texts. We developed a colour wheel of chosen emotions with a range of positive feelings such as relaxation (green), happiness and pleasure (yellow) or bliss and desire (orange) and a range of negative emotions such as blue (worry and insecurity), violet (disgust) and red (hostility and aggression)3 . As in figure 1 4 , the colours are arranged on a continuous wheel for hues gradations, between positive and negative as well as, more and less intense emotions.

(Insert Fig. 1 about here).

Methodology of "emotional mapping"

We conduct qualitative semi-directive interview and first ask our interviewees about their living spaces -from housing to visiting friends, sometimes going back into their past. We explain the process of mapping, and step by step they draw each space and place in black on a white big sheet of paper. There is no need for accurate localisations.

Later, the interviewees talk about their everyday life experiences in relationship with drug and drug places: they chose colours out of the colour wheel and circle or colour particular places with different emotions. (Insert Fig. 2

about here).

The drawn maps are representations of individual geographies and enable us to gain a deeper understanding of specific drug places or drug user's geographies (Behnken & Zinnecker, 2013, p.547). The entanglement of visual and textual material, the emotional map and the interview transcript represent the lived space and is therefore a lens into the way drug users experience urban space, producing further knowledge about pleasures and insecurities of a potentially vulnerable population group, also about cities as complex landscapes of drug consumption.

Experiences and practices from the field of research

After 26 interviews conducted by the authors of this article, our experiences of emotional mapping was enriching. First, we noted that paper, colour pens, and the map are mediating objects between interviewee and interviewer, helping to structure the dialogue. The people we interviewed were part of a heterogeneous group. The diversity showed in age, gender, housing conditions drug use and duration of drug use.

The interviewees appreciated the recognition of the importance of their knowledge. Some noted enhancement of feelings for being heard, the possibility of expressing themselves via pictorial and creative way. Thus, the mapping enhances the quality of the interview itself as the map mitigates the "typical" researcher-subject situation and enables the interviewees to be an expert of their own maps / their lived space.

Nevertheless, there are huge variations in the mapping results. Despite our efforts of standardising the interview process with common guideline, common emotion legend, each of the interviewee approaches the method in their own way, also depending on their fitness. The individual representation of space sets limits in the drawing of the maps, some maps do not show more than a few signs, which mark the everyday places of the person. Furthermore, likewise to other social research methods, the results depends on the willingness of the interviewees, especially if they are asked to draw criminalised cats like drug use (insert Fig. 3 about here).

In this article, we use two different ways of quotes: statements of the interviewees and maps. In order to anonymise the data, only some details of the maps are quoted, or the map was re-drawn be the authors, without any personal data from the original drawing, and everything was translated into English. All names of the interviewees quoted here are pseudonyms.

Urban lived spaces of drug users

Marginalized drug users in urban drug scenes face many challenges in their everyday life, as their living space is highly structured by daily routines. Mapping interviews with drug users from open drug scenes in Munich, Nuremberg, and Berlin give a unique view of the drug users' perspectives and experiences of these cities. From housing to encounters with (repressive) institutions, these urban experiences and mobilities are associated mainly with negative feelings. The visual and text material collected during the interviews confirms how emotions are the results of the encounter between subjectivity and social structures: an encounter taking place and drawing a social geography of the city.

The anxieties of precarious housing situations

The interviewees start with their current sleeping accommodation. Some have a home, sometimes shared with a non-consuming family. The living space "home" may lead to Some lost their former home due to separations with their partners or families or were evicted for lack of payment. Others cannot go back to their home because of being searched by a warrant to arrest. Some arrived in a new city in the last few months or years as existing drug consumers and had never had home, work or family in that time.

In the maps, the participants sketch sleeping accommodations often decentralized and at the edges of the paper, which indicates a peripheral location in the city, but also a minor role in the lived spaces (Insert Fig. 5 & 6 about here). They use these places mainly to sleep and are not willing or allowed to stay there during daytime. The housing situation is very stressful for most of the drug users and influences the drug use habits.

The interviewees describe an increase of drug consumption with each step towards precarity. Martin, a 36-year-old drug user from Munich, outlines the situation in the homeless shelter where he stays as unbearable and links this situation to his (increase of) drug use: "So it's… you won't make it without being high. I don't know anyone who doesn't drink or blaze or take other stuff." (Martin,30,Munich) In his experience, the mere presence in the shelters leads to an increased drug use.

Nevertheless, the pressure to consume drugs at the sleeping accommodations must be understood as socially constructed. Consuming in safe spaces such as home or shelter, is generating mixed feelings due to the transgression and the perceived risk of eviction.

Ambivalent joys and worries of urban routines

Our interviewees spend most of their time outside, going through their major routine: making money, buying, and using drugs. Thus, public space plays a major role as a lived space and particularly as a social meeting space. The term open drug scene is often negatively connoted and used as a catchphrase by politicians, journalist, and other actors. Open drug scenes are heterogeneous group of drug users, depending on earnings, housing, etc. Beside drug related issues, the social meeting places have an important function for coping in everyday-life (Weber, 2011, p. 109 f).

The people who meet at these places can be friends, acquaintances or strangers, while many highlights that their "street buddies" are in no way "real friends". Social meeting places can be in the public spaces such as the edge of a square, on the bench of a train station or alleys or other interstices between buildings, which allow staying there.

These places are often located next to dealing places, substitution doctors or harm reduction facility. Alternatively, the facilities of drug services also have an important function as social meeting places. The wide range of emotions visualises the important role of social meeting places in the maps. In some maps these places appear as "rainbow", covered by most of the colours in the emotion wheel. (Insert Fig. 7 about here).

The emotions associated with the everyday routine places are often described as inconsistent or conflicting. On the one hand, these are the places where one can meet one's friends and experience a social life or where one gets what one needs. On the other hand, the mere presence at these places is often connoted with drug use and the continuation of daily habits. Therefore, many of them want to change the everyday routine of drug consumption and subsequently, the marginalisation by changing the spatial environment: Miriam sees a significance in changing her lived space as the only possibility for changing her drug habits. Nevertheless, she struggles with doing so.

Avoiding the risks of repression and conflict

Within the autonomous routine, encounters with particular institutions -such as the police, traffic controllers, emergency services, and hospital personal -make up the lived spaces of marginalised drug users. Places, which are known to be hotspots of police surveillance and repression (because of drug dealing and use), are avoided as much as possible. As Alf (27, Berlin) puts it: "From my point of view, they [the police] bother... Then comes the whole confiscation stuff... that's, off again, collecting bottles, working... Evictions... when they see you again in the next 48 hours, they take you with them... From my point of view it is harassment".

Here again, as an answer to the question "Are there places where you never go?" "Mainly where there is the police! I don't want to have to do anything with them... There are places where they always go through and drive through, such as Kotti, Hermannplatz, Ku'damm, etcetera..."

Our interviewees report experiencing hostility from institutions, not only because they would "do their job", but mainly because of repeated experiences of humiliation and violence. Users also experience hostility (as threats and aggressions) from inhabitants while trying to hide in staircases, cellars and attics for consumption. The avoidance of such public and private places influences their mobilities in the one's very own neighbourhood and shapes their representation of the city that is made of hostile public or private places.

Police controls significantly structure movements and mobility of drug users in the lived space. With "junky jogging", drug users describe the daily routine of getting evicted from one public place and moving collectively to another, until control and banishment start over again. It reinforces the experience of being not a proper part of the "civil society", as several interviewees from Munich pointed out.

The emotional geographies of drug use

The lived spaces of marginalised drug users are characterised by a high spatial precariousness and vulnerability. Their emotional geography stresses unrest, hostility and aggression. Drug use is the main activity directing their routine. Therefore, we ask -what are the emotional geographies of their consumption places? Does pleasure have a role? Does the availability of DCRs make a difference?

For the few interviewees who have a home, it would be their preferred place for drug use. In case of cohabitation with sober partners or parents, consuming home often would be a transgression -so it happens only in secret. Homes, shelters and assisted housing are most of the times peripheral and far away from the city centre, where the routine and harm reduction take place. Therefore, drug consumption in a personal or private space happen only in the evening, for the last shot, and once the day is over.

Thus, it can be said that drug use is driving consumers throughout the city.

Drug consumption rooms as anchors of everyday lives

In Berlin, all interviewees were clients of the same DCR. Some come almost everyday there, others occasionally; most of them do not go to any of the other three DCR in Berlin.

Nonetheless, all emotional maps highlighted the DCR with positive emotions of safety, quietness and happiness: it was the place with the most positive and safe feelings, even before the home shared with a partner or parents -undermined by conflicts around addiction with the dearest ones. The possibility of drug consumption in a safe context responds to a need of quietness, hygiene, and overall safety; facilities such as sterile equipment, meal, shower, and laundry are a significant relief for most of the homeless clients; the availability of information and support from the health and social workers make DCRs a helpful resource. Most of them were regular users reporting how important this place became for their everyday lives, particularly because of feeling accepted and welcomed, not having to face the moral prompting to quit drugs and experiencing a relief from the discrimination experienced with police and health workers. In this place, marginalised drug users find acceptance and relief. For many of them, it is the only room or closed space, available for them. This one place is an anchor of their precarious routines throughout adverse and dangerous public spaces.

(insert Fig. 8 about here)

The DCRs are essential for drug users, because drug consumption is possible there and support is available. These are the places where one can autonomously conform to addiction. Nevertheless, most of the time, our interviewees consume outside of the DCR. There are different reasons for this. Neither the schedule of DCRs, nor their restricted number in each city could respond to the needs of the interviewees. The opening times, limited to the afternoons of opening days, leave longer closing hours: the whole weekend, as well as evening and morning of everyday. Even within opening hours, the distance between the place of retrieving drugs and the DCR for the person experiencing withdrawal symptoms is significantly long. In such circumstances, only a small part of consumption practices might take place in a DCR. Nevertheless, DCRs still play a crucial role for drug users.

Fragile existences of drug use in public

In cities without DCRs (and even in the cities with DCRs), marginalised drug users, homeless or not, are banned from finding places for drug use within public spaces and their interstices. Places of drug use are determined by the routine of making money and buying drugs, which is happening in the centre of the city. Regularly, withdrawals symptoms lead interviewees to use the drug immediately after buying it, in the very first interstice, half-hidden as one can find. Drug users have to find places for drug injecting and smoking, as these must be places within their reach, among the spaces one is allowed to be present and have privacy for that. Indeed, the places of drug use are not an adequate drug consumption places and the injecting process at these places is characterized by hurry and worries for the user. Charlie, 28 years old, from Munich describes this process below: "In underground car parks as well as on the stairs. Just between the cars, kneel down, boil it up, get ready." (Charlie, Munich)

The hurry, however, originates from the risks of "getting caught", as well as, the craving to consume drugs as soon as she gets it:

Geographical explanations between emotional appropriation and sociospatial exclusion

Social drug research showed how the question "why people do drugs" has to take into account many factors such as biographic or medical ones, as an act of reward-seeking, as a strategy to avoid distress, as a cultural practice of late capitalism. All these factors are materialised in urban spaces. The provisory results of emotional mapping showed howplace plays a huge role in explaining drug consumption. The obvious answer to our question "why do marginalised drug users do drugs in the public places of the city?" is: because there is no other place left where they could do so. However, another answer could be: it is exactly because they are excluded from private spaces and institutional support and evicted from public spaces that they do more drugs than they would. This plays at two scales: At an (inter-)individual scale, the emotional appropriation of spaces explains drug use in the moment. At a broader urban and social scale, socio-spatial exclusion processes explain how marginalised drug users are easily trapped in a day-to-day drug consumption routine.

Emotional appropriation of spaces

Beside urban (drug) policies, which affect the lived spaces of the drug users -in creating, suppressing or criminalising drug places -the users themselves constructed their lived space and influences of the cities geographies, as well as, their internalised drug habits. Due to what we call emotional appropriation, drug users experience different places within the city and overlay them with subjective emotions and spatial knowledge. The emotional mapping interviews illustrate that drug places are highly intertwined with specific, inconsistent, and overlaying emotions. Experiences (related to emotions) stabilise in the all-day places, where they are experienced. During this process, drug users appropriate specific places emotionally. The emotional appropriation of drug places becomes the individual explanation of behaviour such as drug use. Indeed, the desire for drugs can be understood as an emotion regarding drug use. As a result, drug user themselves find answers of the question "why using drugs" at the spatial setting. Through the emotional appropriation of a specific place, this space becomes the psychological cause for drug use.

From our perspective, these drug places are constructed not only through the lived experiences and appropriation of drug users, but also through urban (drug) policies, which influence the geographies of drug consumption within the urban space. The way drug users experience public drug places is determined through political (il)legalisations, harm reduction policies (e.g. establishment of DCR's) as well as through measures of repression (e.g. police controls) and displacement effects. The reason to use drugs in public places is rare and are rather individual preferences, even if mostly due to a lack of alternative possibilities. Drug users without a permanent home and, in cities without a DCR (or during the closing times in cities with DCR) remain public places in order to consume drugs out of necessity and to meet other people or undertake various activities.

Socio-spatial exclusion and day-to-day drug use routines

From our perspective illegalisation, tabooing, and moralising of some substances and their consumptions foster social exclusion of some drug user groups. The individualisation of personal vulnerabilities -such a social isolation or psychological issues -fails to point the production of both isolation and interpersonal violence in neoliberal times and to record the failure of the social and medical system to address this issues. In this theoretical perspective, drug use leads to social exclusion [START_REF] Becker | Outsiders: Studies in the Sociology of Deviance[END_REF] -exclusion from social relationships, from the work market, from sufficient healthcare, and stable housing. Exclusion expresses spatially, as more places become inaccessible, because of a lack of financial resources, economic capital, body and physical appearances, willingness to conform as soon as one does not fit any more to the roles of each space. The almost impossible access to private or semi-private spaces, the permanent eviction back to public spaces are also examples of sociospatial exclusion.

"I didn't really inject for three years, and I do it again since six weeks. Probably because all the things that come together" refers Evelyn (28, Berlin) to her eviction from accompanied housing, living now in a shelter.

In its turn, spatial exclusion deprives marginalised drug users from access to determinant resources. The public spaces appropriated by marginalised drug users are the last one in which they can stay alive. The interviews showed clearly how each step toward more socio-spatial exclusion was a step towards more consumption.

Exclusion fosters subjective and objective vulnerability, and thus, explains consumption.

Conclusion

Within a perspective rooted in critical cartography and social geography, we proposed emotional mapping as a method of describing and conceiving urban lived spaces of drug users in European cities. This method is not only helpful for understanding marginalised urban drug users, but also it is suitable to understand the lived spaces of other social groups -whether marginalised or not, occasional or regular users, or nonconsumers frequenting public places used for drug consumption and dealing.

With our method of emotional mapping, we highlighted in which manner places play an important role in explaining drug consumption. At a macro scale, structural phenomenon of socio-spatial exclusion in European metropolis explain how neoliberal, social, economic, and health policies produce massive anomie. Spatial eviction can lead to a feeling of helplessness, abandonment from the institution and social loneliness, and thus fosters drug use. At a micro scale, personal emotional appropriation explains individual patterns of drug use. Emotions are intertwined in personal experiences as well as in structural power relationship and their spatialisation reflect both.

Emotional Mapping is still in the explorative phase within the broader DRUSEC-Project. Our aim is to extent emotional mapping by persons from different social groups within the city, as a way to understand better urban space and the conflictual geographies of the city as a complex production. We will apply emotional mapping to cities in Germany and France and suggest that this method is suitable for European cities, which present similar mixed inner-city neighbourhoods known as drug places, similar intricate living spaces from many different groups and similar patterns of urban policies.

Figures:

  very positive or negative emotions -and is such, is lived in different ways, if the person is autonomous or lives with family or partner. (Insert fig. 4 about here). But the majority of our interviewees do not have a permanent home. Most of them live in uncertain housing conditions for years; stay at homeless shelters, friends' places, squat cellars and attics, use the sleeping accommodations of drug services or sleep on the street. The reasons of homelessness in the open drug scenes are diverse.

"

  I just want to leave. I have to change my social contacts somehow. (…) I just need a new and stable environment around me. (…) Again, and again, you go back to the places you know. And again, and again, you meet the people you know. That's very difficult." (Miriam, 39, Nuremberg)

"

  If you don't feel well, you have the seal (consumption unit of heroin) in your mouth, you buy yourself a syringe and then you want to absorb the stuff as soon as possible. And then you just go in the next best backyard." Most of the drug users would prefer a DRC instead of the "the next best backyard". Most of the interviewees reported aggressive encounters while consuming. Public consumption places are almost exclusively connoted with negative emotions such as insecurity, worry or disgust (insert Fig. 9 about here). A part of the respondents named non-specific places for drug use or refused to draw consumption places. This leads us to the interpretation that these places of drug use are rather an unavoidable situation the people have to confront in their daily life, than a choice of the lived space. Public drug consumption places are whether stable nor securing, but situations of frailty that change due to factors as (police) controls, public construction work or other measurements that influences the urban environment for drug user. As a consequence, the emotional geography of drug use itself is polarised between DCRs and public spaces -in the cities where DCRs exist.

Figure 1 :Figure 4 :

 14 Figure 1: DRUSEC Colour-Wheel of emotions

Figure 8 :

 8 Figure 8: Zet's Map: Quietness and joy at the DCR

Figure 1 :

 1 Figure 1: DRUSEC Colour-Wheel of emotions

Figure 5 :

 5 Figure 5: Martin's Map: decentralized sleeping accommodation

Figure 7 :

 7 Figure 7: L.Schweino's Map: Social contact and the rainbow of emotions

[START_REF] Downs | Maps in minds: reflections on cognitive mapping[END_REF]

These terms are derived from the French social geographical concepts "espace de vie" und "espace vécu".

We didn't choose the colors of emotions according to one or another theory, since we don't build this research on an underlying theory of emotions. The relationship between color and signification (emotion) is the one of a legend: a contingent code for a specific purpose.

All illustrations of this article can be found in color on following research blog: we will be able to provide the link to the blog on the 14 th of may and in all cases before proof reading