Navigation of an Autonomous Vehicle is based on its interaction with the environment, through information acquired by sensors. The perception of the environment is a major issue in autonomous and (semi)-autonomous systems. This work presents the embedded real-time visual perception problem applied to experimental platform. In this way, a robust horizon finding algorithm that finds the horizon line was proposed and applied to generate the navigable area. It permits to investigate dynamically only a small portion of the image (road) ahead of the vehicle. From a dynamic threshold search method based on Otsu segmentation and Hough transform, this system was robust to illumination changes and does not need any contrast adjustments. It can also be used with infrared camera.

I. INTRODUCTION

HE robot navigation is based on its interaction with the environment. In order to obtain information about it, sensors are needed, which in many cases may be limited in scope and subject to noise. However, when incorporating several types of sensors, there is an increase of autonomy and "intelligence" degrees, especially in relation to navigation in unknown environments. On the other hand, the type and quantity of sensors determine the volume of data for processing that requires a high computational cost.

Furthermore, the important factors are the variety and complexity of environments and situations. For military or civil purposes, some of these applications include: the Grand Challenge [START_REF] Thrun | Stanley, the robot that won the DARPA Grand Challenge[END_REF] and Urban Challenge [START_REF]Spirit of Berlin: An Autonomous Car for the DARPA Urban Challenge Hardware and Software Architecture[END_REF]; Advanced Driver Assistance Systems (ADAS) [START_REF] Gietelink | Development of advanced driver assistance systems with vehicle hardware-in-the-loop simulations[END_REF]; autonomous perception system [START_REF] Ulrich | Appearance-Based Obstacle Detection with Monocular Color Vision[END_REF], [START_REF] Bonin-Font | Visual Navigation for Mobile Robots: A Survey[END_REF], and aerial robots [START_REF] Kim | Swarming Unmanned Aerial Vehicles: Concept Development and Experimentation, A State of the Art Review on Flight and Mission Control[END_REF], [START_REF] Ettinger | Vision-guided flight stability and control for micro air vehicles[END_REF]. These applications have a common issue: providing to the robot/vehicle platform the capability of perceiving and interacting with its neighbour environment.

Although extremely complex and highly demanding, thanks to the great deal of information it can deliver, machine vision is a powerful means for sensing the environment and has been widely employed to deal with a large number of tasks in the automotive field [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF].

Due to the general applicability of it, the problem of mobile robots navigation is dealt with using more complex techniques [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF]. Besides their intrinsic higher computational complexity caused by a significant increment in the amount of data to be processed, the most common vision systems (such as the processing of non-monocular images) must also be robust enough to tolerate noise caused by platform movements and drifts in the calibration of the multiple cameras' setup.

For land vehicle navigation, the monocular vision systems have been designed to investigate the road information, and in order to decrease the volume of data for processing, some systems have been designed to investigate only a small portion of the road ahead of the vehicle where the absence of other vehicles has been assumed [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF]. Otherwise, the sky region is not a region of interest, and the horizon line threshold is applied to generate a road image [START_REF] Hann | Vision-based Lane-Vehicle Detection and Tracking[END_REF]. Stanford Racing Team [START_REF] Dahlkamp | Self-Supervised Monocular Road Detection in Desert Terrain[END_REF] implemented the horizon finding algorithm originally proposed by [START_REF] Ettinger | Vision-Guided Flight Stability and Control for Micro Air Vehicles[END_REF] to eliminate all pixels above that horizon.

Additionally, these techniques has been used for flight stability and control system to Micro Air Vehicles [START_REF] Ettinger | Vision-Guided Flight Stability and Control for Micro Air Vehicles[END_REF], for the control of the airship [START_REF] Rives | Linear Structures Following by an Airship using Vanishing Point and Horizon Line in Visual Servoing Scheme[END_REF] and landing aircraft control [START_REF] Entzinger | Modeling of the visual approach to landing using neural networks and fuzzy supervisory control[END_REF]. It also was employed as an absolute attitude sensor which is useful for low-level control of an unmanned aerial vehicle [START_REF] Hwangbo | Robust monocular vision-based navigation for a miniature fixed-wing aircraft[END_REF].

In this way, from a dynamic threshold search method based on Otsu segmentation [START_REF] Otsu | A threshold selection method from graylevel histogram[END_REF] and Hough transform [START_REF] Ballard | Generalized Hough transform to detect arbitrary shapes[END_REF], a robust horizon finding algorithm that finds the horizon line was proposed and applied to generate the navigable area. It permits to investigate dynamically only a small portion of the image (road) ahead of the vehicle. This algorithm is robust to illumination changes and does not need any contrast adjustments.

The Section II presents the sensor perception review and monocular vision contributions. From the Section III and IV the results are presented, and the conclusions are given in Section V.

II. SENSOR PERCEPTION

Several models of distributed vehicle navigation architecture are presented in the literature [START_REF] Thrun | Stanley, the robot that won the DARPA Grand Challenge[END_REF], [START_REF]Spirit of Berlin: An Autonomous Car for the DARPA Urban Challenge Hardware and Software Architecture[END_REF], [START_REF] Greco | Real-Time Forward Urban Environment Perception for an Autonomous Ground Vehicle Using Computer Vision And Lidar[END_REF]. The perception layer uses many types of sensors, including ultrasonic sensors, laser rangefinders, radar, cameras, etc.

However, these sensors are not perfect: ultrasonic sensors are cheap but suffer from specular reflections, and laser rangefinders and radar provide better resolution but are more complex and more expensive [START_REF] Ulrich | Appearance-Based Obstacle Detection with Monocular Color Vision[END_REF].

According to [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF], the vision-based sensors are defined as passive sensors, the image scanning is performed fast enough for Intelligent Transportation Systems, and it can be used for some specific applications: road marking

Robust Horizon Finding Algorithm for Real-Time Autonomous Navigation based on Monocular Vision

A. Miranda Neto, A. Corrêa Victorino, I. Fantoni and D. E. Zampieri T localization, traffic signs recognition, obstacle identification. However, vision sensors are less robust than millimeter-wave radars in foggy, night, or direct sun-shine conditions.

On the other hand, for range-based obstacle detection systems have difficulty for detecting small or flat objects on the ground, and range sensors are also unable to distinguish between different types of ground surfaces [START_REF] Ulrich | Appearance-Based Obstacle Detection with Monocular Color Vision[END_REF]. Notwithstanding, the main problem with the use of active sensors is represented by interference among sensors of the same type, hence, foreseeing a massive and widespread use of these sensing agents, the use of passive sensors obtains key advantages [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF].

Fig. 1 illustrates the monocular vision contribution to the DARPA Grand Challenge [START_REF] Dahlkamp | Self-Supervised Monocular Road Detection in Desert Terrain[END_REF]. This diagram shows that the reach of lasers was approximately 22 meters (left), whereas the monocular vision module often looks 70 meters ahead (right). It also shows the horizon detection for sky removal. On the safety front, the ADAS including front bumper mounted cameras. The progressive safety systems will be developed the manufacturing of an "intelligent bumper" (safety belt) peripheral to the vehicle in answering new features as: lane change assistant, blind spot detection, frontal and lateral pre-crash, etc. The objective in terms of cost to fill ADAS functions has to be very lower than the current Adaptive Cruise Control (500 Euros) [18].

Aware that in the majority of the navigation systems, the machine vision system is working together with other sensors, added to its low cost, this paper presents a monocular vision-based system that includes a robust horizon finding algorithm. Because it uses simple techniques and fast algorithms, the system is capable to achieve a good performance, where the compromise between processing time and images acquisition is fundamental.

III. ROBUST HORIZON FINDING ALGORITHM

In a general way, the navigation system must be capable to drive the vehicle (or to help the driver in its driver process), using the information received from all sensors. Processes conducted by the machine vision have as goal, in addition to the visual information, to process image data for machine perception. Its algorithms carry through operations on images, with the purpose to reduce noise and to segment them.

One way to perform segmentation of an image is to use thresholds. This type of segmentation technique, called thresholding, is very simple and computationally fast, however the identification of the ideal threshold can be sufficiently complicated. The best thing to do in this case is to use techniques and algorithms that search the thresholds automatically. Thresholding methods are divided in two groups: global and local. The global ones divide the image using only one threshold and the local ones are those that divide the image in sub-images and for each one of them a threshold is defined [START_REF] Neto | A Simple and Efficient Road Detection Algorithm for Real Time Autonomous Navigation based on Monocular Vision[END_REF].

What this work proposes is a global thresholding method, which seeks not the ideal threshold for the whole image, but an ideal threshold associated at each portion (local information), that contributes to the final decision. Details of this search are described below.

A. Image pre-processing

Most research groups face this problem using highly sophisticated image filtering algorithms [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF]. This work uses a color or gray-level image and smooth them using a Gaussian filter. The Gaussian smoothing operator is a 2-D convolution operator. It acts as low-pass frequency filters [START_REF] Gonzalez | Digital Image Processing[END_REF]. In order to reduce the number of data, it includes the resolution reduction of image (to 128x96).

If the image is colored, in order to utilize the most important information of the color image, the candidate color channel that was dominant in certain color space is selected to generate the histogram image [START_REF] Yanqing | Vision-based Road Detection by Monte Carlo Method[END_REF]. For us this space represents 60% of the image (Fig. 5). It is described by Eq. [START_REF] Thrun | Stanley, the robot that won the DARPA Grand Challenge[END_REF].

{ } ( ) B G R B G R N N N Cc , , max arg , , = (1)
where Cc means the color channel and R N means the number of the dominant color channel in certain referenced region.

B. Image segmentation and sky removal advantage

The purpose of segmentation here is the line horizon detection (horizon finding) for sky removal. Right after the images pre-processing, it can use the search for an ideal threshold based on Otsu [START_REF] Otsu | A threshold selection method from graylevel histogram[END_REF].

The Otsu method exhaustively searches for the threshold that minimizes the intra-class variance, defined as a weighted sum of variances of the two classes. The thresholding process is seen as the partitioning of pixels of an image in two classes (foreground and background): C1={0, 1,…, T} and C2={T+1, T+2,…, N-1}, where T is the chosen threshold and N the number of intensity levels of the image. It is described by: The class probabilities are estimated as:

( ) ( ) ( ) ( ) ( ) T T q T T q T W 2 2 2
∑ = = T i i H T q 1 1 ) ( ) ( and ∑ + = = N T i i H T q 1 2 ) ( ) ( (3)
The class means are given by:

∑ = = T i T q i iH T 1 1 1 ) ( ) ( ) ( µ and ∑ + = = N T i T q i iH T 1 2 2 ) ( ) ( ) ( µ (4)
Finally, the individual class variances are:

∑ = - = T i T q i H T i T 1 1 2 1 2 1 ) ( ) ( )] ( [ ) ( µ σ (5) ∑ + = - = N T i T q i H T i T 1 2 2 2 2 2 ) ( ) ( )] ( [ ) ( µ σ (6)
where H is the histogram of the selected channel in Eq. ( 1). One of the great advantages of this method is that it does not restrict itself to the type of histogram of the image, that is, it can be applied to unimodal, bimodal or multimodal histograms, but it presents better performance in images with bigger intensity variance. Its main disadvantage is its sensitivity to noise in the image, what can be reduced with the application of a filter.

Region recognition can be handled by popular thresholding algorithm such as Maximum Entropy, Invariant Moment and Otsu thresholding method (OTM). For road detection, because OTM supplies a more satisfactory performance in image segmentation, it was used to overcome the negative impacts caused by environmental variation [START_REF] Yanqing | Vision-based Road Detection by Monte Carlo Method[END_REF].

Furthermore, some authors consider the OTM as one of the best choices for real-time applications in machine vision [START_REF] Sahoo | A survey of thresholding techniques[END_REF], [START_REF] Lee | A Comparative Performance Study of Several Global Thresholding Techniques for Segmentation[END_REF]. It still remains one of the most referenced thresholding methods [START_REF] Sezgin | Survey over image thresholding techniques and quantitative performance evaluation[END_REF].

Although the Otsu method is an excellent method to choose an ideal threshold, it considers for all cases the information of the image as a whole (global information).

Let's see an example for land vehicle navigation, where the main task is road detection. Fig. 3 illustrates that probably for images that possess the horizon (sky) in its composition, the algorithm may not have a satisfactory result.

To deal with this problem, a robust horizon finding algorithm is proposed. After sky removal, it permits to investigate only a small portion of the image ahead of the vehicle, and the new result can be seen in Fig. 4.

C. Visual-based line horizon detection

According to [START_REF] Sezgin | Survey over image thresholding techniques and quantitative performance evaluation[END_REF], cluster-Otsu (clustering thresholding) minimizing the weighted sum of intra-class variance of the foreground and background pixels to establish an optimum threshold, that minimization of intra-class variance is tantamount to the maximization of between-class scatters. The Otsu method gives satisfactory results when the numbers of pixels in each class are close to each other. Using as reference Fig. 5, the horizon finding results can be seen in Table I. It presents an example of land vehicle navigation, where the range of horizon analysis was empirically limited to a maximum height of 60% of the image.

The horizon finding algorithm creates empirically cuts that divide image in ten parts (sub-images) of equal heights. All these sub-images are then submitted to the segmentation algorithm: Eq. ( 2), where the output is a vector with Otsu thresholds values of each sub-image. Then, a threshold is calculated at each sub-image (Fig. 6), which depends on local information and variance of the pixel neighborhood.

Subsequently, in order to find the optimal threshold that is capable to get a more efficient segmentation and to distinguish with bigger precision, for each found local Otsu threshold, all sub-images are again submitted to the segmentation algorithm. The purpose is to analyze how much each local sub-image contributes positively to increase or reduce horizon points.

Then, the algorithm initiates by analyzing the sub-image closest to the vehicle, that is, the first more inferior slice of the image, going from the bottom edge of the image (red line, Fig. 6) to the last more superior sub-image.

Using as reference Fig. 6, it obtains the local Otsu thresholds and its percentage of foreground points (horizon). They are represented in blue by the diagonal matrix in Table I, respectively, threshold (T) and percentage (P): 6 Another interaction is next necessary. For each local Otsu threshold found, all sub-images are again submitted to the segmentation algorithm.

Taking as reference the data in Table I, from an Otsu threshold equal to 135, the result of the first interactions can be seen in column (a). For example, it has in row (a)-10 its percentage of foreground points equal to 28 (blue) which refers to the sub-image closest to the vehicle. In the following, using the same Otsu threshold, in row (a)-9, the result equal to 14 refers to the second sub-image analyzed, and so on for others rows in column (a).

As a second example, in column (j) it has an Otsu threshold equal to 118. From this threshold (last more superior sub-image analyzed), in row (j)-10 the percentage of foreground points equal to 53 refers to the sub-image closest to the vehicle. In the following, in row (j)-9 the result equal to 23 refers to the second sub-image analyzed.

The columns (k) and (l) are, respectively, sum of rows and percentage difference between the value of a row and its previous one. Then, considering that the algorithm search the region of greatest difference in the image, the yellow filled row [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF] is the one where it finds the value 91.

The horizon line separates the image into two regions that have different appearance [START_REF] Ettinger | Vision-Guided Flight Stability and Control for Micro Air Vehicles[END_REF]. The choice of this method was based on [START_REF] Neto | A Simple and Efficient Road Detection Algorithm for Real Time Autonomous Navigation based on Monocular Vision[END_REF], because the use of a global segmentation method, not always the analysis of a bigger image portion can contribute for a better result in the most critical region (Fig. 3: in land navigation, the region closer to the vehicle), where obstacles should be detected and avoided as fast as possible. On the contrary, when separating the superior portion of the original image (sky removal), it is capable to get a more efficient segmentation and to distinguish with bigger precision the obstacles from the navigation area.

Once the Otsu horizon line is found, the Hough transform [START_REF] Ballard | Generalized Hough transform to detect arbitrary shapes[END_REF] is used, and a rapprochement between both results is triggered based on the weighted average. The Canny detector [START_REF] Canny | A computational approach to edge detection[END_REF] was employed as input of Hough transform due to its robust performance and accurate edge localization.

Finally, due to the fact that in small time intervals, there will be no abrupt changes in the positioning of new horizon line (NHL), it has decided to use a simple decaying filter followed by a Kalman filter (KF) [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]. It is presented in Fig. 7.

IV. IMPLEMENTATION AND RESULTS

A. Robust horizon finding algorithm (sky removal)

Fig. 8 to 11 present the performance of the horizon line detection process in unknown environments: desert road videos available by DARPA [START_REF]DARPA Grand Challenge Rulebook[END_REF]; urban environment data obtained by the intelligent vehicle platform presented in Fig. 14; and infrared road video available by FLIR [START_REF] Flir | Thermal imaging and the PathFindIR: An automotive night vision camera[END_REF].

For each image a real horizon line (black) was registered manually. In (a) the green line represents the Otsu horizon line detection; in (b) the magenta line represents the weighted average of the Hough transformation lines; in (c) the blue line represents the weighted average between the Otsu horizon line detection and the Hough transformation result; in (d), finally, the red line represents the robust horizon finding algorithm based on Otsu segmentation, Hough transformation and filtering (Fig. 7).

B. Real-time considerations

According to [START_REF] Laplant | Real-Time System Design and Analysis[END_REF], a real-time system must satisfy explicit response-time constraints, including failure. A real-time system is one whose logical correctness is based on both the correctness of the outputs and their timeliness.

For unstructured environments, the scenario for study is dynamic, with several elements in motion. As a consequence, running an autonomous system from a starting point to a destination, or assisting a driver with his task, involves carrying out complex, and non-deterministic operations and real-time constraints.

The main problem that has to be faced when real-time imaging is concerned and which is intrinsic to the processing of images is the large amount of data [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF]. Based on the idea to reduce information acquired in order to reduce processing time, and taking into account that it has been estimated that humans perceive visually about 90% of the environment information required for driving [START_REF] Bertozzi | Visionbased intelligent vehicles: state of the art and perspectives[END_REF], in [START_REF] Neto | Pearson's Correlation Coefficient for Discarding Redundant Information in Real Time Autonomous Navigation Systems[END_REF], [START_REF] Neto | Nondeterministic Criteria to Discard Redundant Information in Real Time Autonomous Navigation Systems based on Monocular Vision[END_REF] was proposed an automatic image discarding criteria method based on Pearson's Correlation Coefficient (PCC) [32], a low complexity and easy implemented solution. This method improves the performance of a realtime system by choosing, in an automatic way, which images to discard and which ones to process. The Pearson's method is widely used in statistical analysis, pattern recognition and image processing. It is described by Eq. ( 2) [32]:

∑ ∑ ∑ - - - - i m i i m i i m i m i ) y (y ) x (x ) y )(y x (x = r
Fig. 12 presents the computational mean time of this robust horizon finding algorithm in unknown environment data (Fig. 9 / desert road video 2) available by DARPA [START_REF]DARPA Grand Challenge Rulebook[END_REF]. In (b), from an empirical PCC threshold, it was 833 Hz (1.20 ms). In (a) it was 159 Hz (6.27 ms) without the discarding method. In (c), above the gray line (d), it has the discarded images.

V. CONCLUSION

The machine vision research area is still in evolution. This can be observed by the great number of research being published in the last few years. In this work, a realtime perception problem is applied to intelligent vehicles. It is a simple solution that solves the problem related to the visual perception of the environment during the navigation of intelligent vehicles and robots. From an image captured by a single camera, the purpose of it was to present a realtime algorithm capable of identifying the horizon area and apply it to generate the navigable area (region of interest). It is important to notice that our algorithm is not based on previous knowledge of the environment neither camera calibration.

Fig. 1 -

 1 Fig. 1 -Monocular vision: comparison of the laser-based (left) and the image-based (right) mapper. Circles are spaced around the vehicle at 10 meters distance. It also shows the horizon detection for sky removal [10].

2 σ

 2 is the intra-class variance,

Fig. 3 -

 3 Fig. 3 -Original image and Otsu segmentation.

Fig. 4 -

 4 Fig. 4 -Left: Otsu segmentation (road detection) after sky removal; Right: Global Otsu threshold (104); new Local Otsu threshold (32) after sky removal.

Fig. 5 -

 5 Fig. 5 -Sub-images and the horizon line detection.

Fig. 6 -

 6 Fig. 6 -The clusters-Otsu and its optimal local thresholds.

  (a) T = 135 / P = 28% | (f) T = 150 / P = 57% (b) T = 132 / P = 15% | (g) T = 143 / P = 48% (c) T = 123 / P = 7% | (h) T = 133 / P = 53% (d) T = 136 / P = 59% | (i) T = 125 / P = 53% (e) T = 154 / P = 80% | (j) T = 118 / P = 52%

Fig. 7 -

 7 Fig. 7 -Horizon Line (HL) and New Horizon Line (NHL).

Fig. 8 -

 8 Fig. 8 -Desert road video 1 [27].

Fig. 9 -

 9 Fig. 9 -Desert road video 2 [27].

Fig. 10 -

 10 Fig. 10 -Urban environment data obtained by the intelligent vehicle platform presented in Fig. 14

Fig. 11 -

 11 Fig.11-Infrared road video[START_REF] Flir | Thermal imaging and the PathFindIR: An automotive night vision camera[END_REF].

Fig. 12 .

 12 Fig. 12. The computational mean time in desert roads.

Fig. 13

 13 Fig.13presents the computational mean time of the robust horizon finding algorithm in unknown and urban environment data (Fig.10) obtained by the intelligent vehicle platform presented in Fig.14. In (b), from an empirical PCC threshold, it was 196 Hz (5.09 ms). In (b) it was 64 Hz (15.62 ms) without the discarding method. In (c), above the gray line (d), it has the discarded images.

Fig. 13 .

 13 Fig. 13. The computational mean time in urban environment.

Fig. 14 .

 14 Fig. 14. Carmen: The experimental vehicle with the monocular vision system.

  Manuscript received July 13, 2011. Arthur de Miranda Neto is Ph.D. Student in Mechanical Engineering at the State University of Campinas -Brazil and in Information Technology and Systems at the University of Technology of Compiègne -CNRS UMR 6599, Heudiasyc, France. Actually, he is researcher engineer at the CNRS. Alessandro Corrêa Victorino is associate professor at the University of Technology of Compiègne -CNRS UMR 6599, Heudiasyc, France.

Isabelle Fantoni is researcher at the University of Technology of Compiègne -CNRS UMR 6599, Heudiasyc, France. Douglas Eduardo Zampieri is associate professor of the Computational Mechanics Department, at the State University of Campinas, Brazil.

TABLE I LOCAL

 I OTSU THRESHOLDS VALUES OF THE FIG.

2[START_REF]Spirit of Berlin: An Autonomous Car for the DARPA Urban Challenge Hardware and Software Architecture[END_REF] where x i is the intensity of the i th pixel in image 1, y i is the intensity of the i th pixel in image 2, x m is the mean intensity of image 1, and y m is the mean intensity of image 2.

ACKNOWLEDGMENT

The authors wish to thank Mr. Gerald Dherbomez for his support in data acquisition. This work was supported in part by French National Research Agency / ANR PERCOIVE Project.


L Aldon

Leh Mathurin

Deli

P E Lippens

J Olivier-Fourcade

J C Jumas

Thermal stability of some glassy compositions of the Ge-As-Te ternary system

Keywords: chalcogenide glasses, X-ray diffraction, Thermal behavior, optical properties

Tellurium-based chalcogenide glasses have been studied extensively in the recent past not only because of their technological applications, but also due to their interesting physical properties. Tellurium is a good element for far infrared transmission but is a poor glass former. Glass forming domains are very narrow in binary Ge-Te or As-Te system. Better glass forming ability and thermal stability are observed in presence of arsenic and germanium atoms. These glasses possess a wide optical window that is maximal for the Ge 10 As 20 Te 70 glass (20µm). It made it possible to show that the substitution of germanium by selenium and antimony reduces the physical properties (, Eg, Tg, Tc 1 , Tc 2 , Tc 1 -Tg et T m ). Indeed germanium supports the formation of glasses, contrary to metal antimony.

Introduction

Vitreous materials have thermo-physical and mechanical properties that make them suitable for many applications in various devices using ligtht wave emission, transmission or detection. These device functions depend on optical and electronic characteristics, such as electrical conductivity [1,2], the gap width and its nature. Chalcogenide glasses offer a range of infrared transmitting materials that are transparent in the regions of 3-5µm and 8-14µm.

Thus, observed transmitting ranges in glasses are 0.6-15µm for sulfides, 1-18µm for selenides and 2-20µm for tellurides. Chalcogenide glasses have also a high refractive index, low optical losses and their possible applications as infrared optical materials have been discussed [3,4]. The Ge-As-Te ternary system shows the existence of the quasi-binary nGeTe-As 2 Te 3 crystalline phases (with n = 1, 3, 5) [12,13] separating two glass forming regions [14,15]. We have represented in Figure 1 the main crystalline phases, and some glassy compositions under investigation in the 'Te-rich' and 'As-rich' regions. Having in mind the fact that the area of glassy state of the Ge-Te and As-Te binary systems is relatively small, and that Tecontaining glasses exhibit an increased tendency towards crystallization [16], melt cooling was carried out by quenching in iced water. So, structural information on telluride glasses is needed in order to characterize the local order in these materials. Among the different experimental tools, differential scanning calorimetry and optical absorption spectroscopy provide valuable information for chalcogenide glasses. Structural information from EXAFS In the present paper, particular attention is focused on the compositions Ge 10 As 90-x Te x (with 35x80) and their preparation. We are interested in the substitution of As for Te in the widest tie line at 10% germanium.

Glasses of binaries As-Te

Experimental procedure

Glassy Ge 10 As 90-x Te x ingots (with 35x80) where prepared by melt-quenching method.

Chemical mixtures of Ge, As and Te (4N purity) were vacuum sealed (10 -5 torr) in silica ampoules (3mm id, 4mm od) and slowly heated in a furnace at 1000°C for 8 hours. The contents of the ampoule were about 1g. The ampoules were agitated continuously during the heating process and during several hours at the melting temperature in order to ensure good homogeneity of the glasses. The experimental set-up is shown in Figure 2. The melt was quenched in iced water at 273K to reach the glassy state of the samples. The quenching rate was estimated to be about 100K/s. Ingots thus obtained were annealed at about 50°C below the glass transition temperature for 8 hours. The same experimental process was used for compositions of the ternary system.

Characterization and quality of the samples were determined by X-ray diffraction (XRD) and differential scanning calorimetry (DSC). Infrared transmission windows were determined by optical absorption edge (Eg) measurements and far infrared transmission spectra (cut-off wavelength,  c ).

For XRD experiments, powdered samples were ground manually and placed in a special support. Experiments were carried out on an automated Phillips diffractometer in the bragg-Brentano (-2) geometry. The patterns were run with Cu as target using the Kα filtered radiation (Ni filter,  = 1.54Å) at 40keV and 20mA, and scintillation registration, and were obtained in the range 5-30° in the  scale within multiscan mode. Scanning rate was 0.04°/s, counting rate was 500ms and we used 30 scans.

Glass transition temperature (Tg) measurements and thermal behavior were investigated by using a SETARAM DSC 121 differential scanning calorimeter. The temperature calibration was performed using the well known melting temperature of high-purity indium supplied with the instrument. The crystallization thermograms were recorded as the temperature of the samples increased at uniform rate using a dynamic argon atmosphere. Typically, 50mg of sample in powdered form (particle size less than 20µm) were sealed in standard quartz tube and scanned over a temperature range from room temperature to about 873K at heating rate α = 10K/min. The values of the glass transition temperature (Tg), the crystallization peaks (Tc 1 , Tc 2 ) and the melting temperature (T m ) were determined with accuracy ±4K by using the microprocessor of the thermal analyser. Values are reported in Table 1. The densities were measured using the Archimedean method along with toluene as a reference medium.

Infrared transmission windows were determined by coupling optical absorption edges measurements (Eg given in Table 1) at low wavelengths (UV-visible and near infrared) and infrared transmission at medium and high wavelengths. First, we used a Beckman Acta M IV 5240 equipped with an integration sphere (0.5-3.5eV). Second, the experimental device was a FT-IR BOMEM DA 8 (2.5-50 µm). Samples were ground, mixed with powdered KBr, the pressed into pallets 13mm diameter and 0.4mm thick. As a example, the transmission window for the Ge 10 As 20 Te 70 glassy sample is given in Figure 3.

Results and discussion

3.1.

Glass forming ability

Numerous trials were carried out in order to optimize the quenching process (ampoule size, cooling media, melt temperature, batch weight). It was performed for Ge 20 Te 80 as can be seen in g.cm -3 ), glassy g-As 2 Te 3 (5.65 g.cm -3 ) and g-Ge 20 Te 80 (5.26 g.cm -3 ), and liquid Te at 725K

(5.77 g.cm -3 ) are given for comparison.

Thermal behavior and optical properties

Tg measured by DSC in plotted as function of the ratio [Te]/([Te]+ [As]) in Figure 7.

For the Ge 10 As 90-x Te x samples, Tg continuously decreases from 461K (x = 36) to 385K (x = 80) with the Te content. After DSC measurements above melting temperature, samples present pronounced changes in the XRD patterns. It is clear that crystallization took place, resulting in the formation of phases which may be identified as Te, As, As 2 Te 3 , As 2 GeTe 4 and As 2 Ge 3 Te 6 as can be seen in Figure 8. Variation of the absorption edges (Eg), represented in Figure 9, is very small due to similar electronic behavior for As and Te. On the contrary, glass transition temperature variations can be explained because of the substitution of As for Te in terms of connectivity.

Relationship between Tg and connectivity

For

Relationship between Tg and Eg

Values reported in

Tables Table 1 and Table 2 have been used to show (see Figure 10) the correlation between T g which reflects the covalency of the chemical bonding between glassy micro domains, and Eg which is the rigidity of the molecular units composing the micro domains. The linear correlation for the both groups ('Te-rich' and 'As-rich' glasses) dearly show that molecular units are the 2).

When one replaces germanium by selenium and antimony in the ternary system Ge-As-Te, one notes that the values of the physical properties evolve in the same direction when the ratio [Te]/([Te] + [As]) increase, but they are less low in the system Sb 2 Te 3 -As 2 Se 3 .

The aptitude to vitrify glasses of this system (Ge-As-Te) is more important, proving that they are more stable than those of the system Sb 2 Te 3 -As 2 Se 3 .

Crystallization process

In Figure 11 Concerning the thermal stability of the compositions under study, we can see in Figure 11 the 'As-rich' glasses present a more stable thermal domain than 'Te-rich' glasses. Hence, these compositions are more suitable for drawing glassy fibers devoted to IR sensors technology.

Conclusion

We investigated some ternary compositions in the Ge-As-Te system. Particular attention was made in the preparation of the glassy sample. We have optimized the quenching process in order to produce glasses avoiding crystallization. Then, optical properties of these glasses have been studied. Fundamental study of relationships between structure and properties are very important to decide if telluride glasses constitute good candidates for devices involving good optical transmission in the far infrared domain. We also studied their thermal stability in order to choose a glass presenting optimal performance for drawing fibers without crystalline domains. A comparison of the properties of glasses of the ternary system Ge-As-Te was done with those of glasses of the binary system Sb 2 Te 3 -As 2 Se 3 . It made it possible to show that the substitution of germanium by selenium and antimony reduces the physical properties (, Eg, Tg, Tc 1 , Tc 2 , Tc 1 -Tg et T m ). Indeed germanium supports the formation of glasses, contrary to metal antimony.

Tables

[ 5 ]

 5 and Ge-Te [6] have been extensively studied as regards to their optical properties because they are transparent to infrared radiation from 2 to 18µm [7,8]. These glasses are good candidates for exoplanet atmosphere detection [9]. Other studies have been carried out on their electrical properties, density, heat capacities, and electrical properties [10,11].

[ 17 ]

 17 , 125 Te Mössbauer spectroscopy [18], and also electronic properties have been earlier investigated [19,20].

Figure 4 .

 4 Trials at 870°C and 900°C (melt temperature before quenching) clearly show XRD peaks of the two crystalline c-GeTe and c-Te phases. At 960°C, broad bumps, typical for the glassy state, appear in the background. Some peaks still presents in the XRD pattern of the sample quenched at 1000°C in liquid nitrogen. For the same temperature, the melt quenched in iced water give a glassy sample without crystalline phases. This latter procedure was used for ternary compositions. The glassy nature of the synthesized materials has been confirmed by the absence of peaks in XRD patterns of annealed (50°C below Tg) samples as can be seen in Figure5. Broad bumps are typical for the glassy state. The position of observed bumps is slightly shifted towards small values of  with the tellurium content.Variations of the densities of Ge 10 As 90-x Te x glassy samples as a function of the ratio [Te]/([Te]+ [As]) are represented in Figure 6. As can be seen, in the glassy state, densities range from 5.26 to 5.63 g.cm -3 . Values of crystalline c-As 2 Te 3 and c-Te (6.23 g.cm -3 and 6.24

  each sample only one value of glass transition temperature (Tg) was measured by DSC. Decrease of Tg values with the Te content for Ge 10 As 90-x Te x glasses can be explained by a decrease of the network rigidity. Usual co-ordination numbers (8-N rule) for the Te, As and Ge atoms in chalcogenide glasses are 2, 3 and 4 respectively. The compositional variation of the investigated glasses is characterized by the Te-concentration (x at. %) and/or by the average co-ordination number Z [21,22]. For the Ge 10 As 90-x Te x glasses the average coordination number is Z = 3.1-0.01x. A model commonly used is the Sreeram's model [23,24] based on Gibbs and Di Marzio's work. In this model the authors assume that glasses are formed by chains of twofold co-ordinated atoms (S, Se, or Te in chalcogenide glasses). They make the assumption that T 0 equals to Tg for the ideal chalcogenide glasses (pure element) where Z equals 2. Considering thus the average co-ordination number Z, Tg is rules by the following law: T g = T 0 /[1-(Z-2)] where  is characteristic of the system. Hence drawing 1000/T g as a function of the crosslinking density (Z-2) gives an estimated value for 1000/T 0 . Values obtained in this work are T 0 = 340±4K and 1/ = 0.40±0.02 for the Ge 10 As 90-x Te x glasses. As a comparison we give here results obtained for binary T 0 = 343K in tellurium, T 0 = 316K in selenium, T 0 = 245K in sulfur based glasses [25,26].

3 . 5

 35 same and have similar structural behavior. These results are in a good agreement with those previously reported [16, 17]. Comparison with the values obtained in the binary system Sb 2 Te 3 -As 2 Se 3 The values of the ratio [Te]/([Te]+[As]) of the compositions Ge 10 As 54 Te 36 (0.400) and Ge 10 As 50 Te 40 (0.444) of the system Ge-As-Te itself close to those of the compositions Sb 12 As 28 Se 42 Te 18 (0.391) and Sb 14 As 26 Se 39 Te 21 (0.447) of the Sb 2 Te 3 -As 2 Se 3 system we can thus compare their physical properties (, Eg, Tg, Tc 1 , Tc 2 , Tc 1 -Tg and T m ) (Table

  , we have represented the thermal behavior (characteristic temperatures) as a function of the ratio [Te]/([Te]+ [As]). We proposed from XRD patterns and DSC measurements the following crystallization process. In the 'Te-rich' region, crystallization occurs at 440K for c-Te and at 500K for As 2 Ge 3 Te 6 . The crystallization process follows the reaction: g-Ge 10 As 90-x Te x (70≤x≤80c-Te + c-As 2 Ge 3 Te 6 In the 'As-rich' region, it occurs above 540K for c-As and As 2 GeTe 4 and depends on the composition. In this case, the crystallization process is: g-Ge 10 As 90-x Te x (40≤x≤60c-As + c-As 2 GeTe 4 For all compositions, melting point occurs in the range [630-650K].

Figure 1 .Figure 2 .

 12 Figure captions

Figure 3 .

 3 Figure 3. Transmission window of Ge 10 As 20 Te 70 glassy sample. Low wavelengths spectrum was obtained from optical absorption edge measurements.

Figure 4 .

 4 Figure 4. X-Ray diffraction patterns of as-quenched samples as a function of the melting temperature and the cooling media for the Ge 20 Te 80 composition. Observed diffraction peaks were attributed to the c-Te (■) and c-GeTe () crystalline phases.

Figure 5 .

 5 Figure 5. X-ray diffraction patterns of glassy samples under investigation. No diffraction peak was observed in these glasses.

Figure 6 .

 6 Figure 6. Evolution of the density as a function of the ratio [Te]/([Te]+[As]). Glassy g-As 2 Te 3 and g-Ge 20 Te 80, crystalline c-As 2 Te 3 and c-Te and liquid Te at 725K are drawn for comparison. Open circles (o) correspond to 'Te-rich' glasses. Solid squares (■) correspond to 'As-rich' glasses. The star labeled A corresponds to Sb 12 As 28 Se 42 Te 18 glassy composition.

Figure 7 .

 7 Figure 7. Evolution of the glass transition temperature (T g ) as a function of the ratio [Te]/([Te]+[As]). Open circles (o) correspond to 'Te-rich' glasses. Solid squares (■) correspond to 'As-rich' glasses. Stars labeled A and B correspond to Sb 12 As 28 Se 42 Te 18 and Sb 14 As 26 Se 39 Te 21 glassy compositions respectively.

Figure 8 .

 8 Figure 8. X-ray diffraction patterns of annealed samples after DSC measurements. Different crystalline phases have been attributed: c-Te and As 2 Ge 3 Te 6 in the 'Te-rich' region, and As and As 2 GeTe 4 in the 'As-rich' region.

Figure 9 .

 9 Figure 9. Evolution of the optical absorption edge (E g ) as a function of the ratio [Te]/([Te]+[As]). Open circles (o) correspond to 'Te-rich' glasses. Solid squares (■) correspond to 'As-rich' glasses. The star labeled A corresponds to Sb 12 As 28 Se 42 Te 18 glassy composition.

Figure 10 .

 10 Figure 10. Correlation between optical gap (E g ) and glass transition temperatures (T g ) for compositions described in this study. Open circles (o) correspond to 'Te-rich' glasses. Solid squares (■) correspond to 'As-rich' glasses. The star labeled A corresponds to Sb 12 As 28 Se 42 Te 18 glassy composition.

Figure 11 .

 11 Figure 11. Representation of the thermal behaviour of the glassy samples. Glass transition temperatures (■ ), crystallisation temperature (▲▼) and melting point (•) are draw as a function of the ratio [Te]/ ([Te]+[As]). Solid lines are drawn in order to give an idea of the thermal stability of the glassy compositions and represent (Tc 1 -Tg) quantity. (see text for details)

Table 1 .

 1 The ratio [Te]/([Te]+[As]) is taken in order to compare Ge 10 As 90-x Te x and Ge 20 Te 80 glasses. This ratio increases when Te content x increases. Densities are given in g.cm-3 . Optical absorption edges (E g ) are given in eV. Glass transition temperature (T g ), first (Tc 1 ) and eventually second (Tc 2 ) crystallization temperature, and melting temperature of glassy compositions are given in K (error bars ±4K).

	Composition	[Te]/	Density	Eg	Tg	Tc 1	Tc 2	Tc 1 -Tg	T m
		([Te]+[As])	(±0.18)	(±0.03)
	Ge 10 As 54 Te 36	0.400	5.26	-	461	580	-	119	639
	Ge 10 As 50 Te 40	0.444	5.54	0.92	449	549	571	100	649
	Ge 10 As 46 Te 44	0.488	5.34	0.89	439	539	550	100	651
	Ge 10 As 20 Te 70	0.777	5.63	0.83	403	442	497	39	628
	Ge 10 As 16 Te 74	0.822	5.32	0.81	397	433	506	36	648
	Ge 10 As 10 Te 80	0.888	-	0.79	385	449	504	64	645
	Ge 20 Te 80	1.000	5.26	0.72	420	443	503	23	640

Table 2 .

 2 Comparison of the values of the physical properties (, Eg, Tg, Tc 1 , Tc 2 , Tc 1 -Tg and T m ) of two compositions of the ternary system Ge-As-Te those their counterparts of the Sb 2 Te 3 -As 2 Se 3 binary system.


Arthur De

Miranda Neto

Alessandro Correa

Isabelle Fantoni

Douglas Eduardo Zampieri

Automatic

Automatic Regions-of-Interest Selection based on Pearson's Correlation Coefficient

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

ately, several applications for control of autonomous vehicles are being developed, and, in most cases, the machine vision is an important part of the set of sensors used for navigation. Some of these applications include: the aerial robots that offer great perspectives in many applications as search and rescue, real-time monitoring, high risk aerial missions, mapping, etc [START_REF] Bonin-Font | Visual Navigation for Mobile Robots: A Survey[END_REF], [START_REF] Kim | Swarming Unmanned Aerial Vehicles: Concept Development and Experimentation, A State of the Art Review on Flight and Mission Control[END_REF]. Similarly, the development of Unmanned Aerial Vehicles (UAVs) has been of interest for military applications, however, one limitation is their maximum flight time; therefore they cannot carry large fuel payloads [START_REF] Anton | Energy Harvesting for Unmanned Aerial Vehicles[END_REF]. Future exploration of Mars requires longendurance UAVs that use resources that are plentiful on Mars [START_REF] Klesh | Solar-Powered Aircraft: Energy-Optimal Path Planning and Perpetual Endurance[END_REF], [START_REF]The Vision for Space Exploration[END_REF], [START_REF]Energetically Autonomous Tactical Robot[END_REF], [START_REF] Finkelstein | Energetically Autonomous Tactical Robot and Associated Methodology of Operation[END_REF]. Finally, for military or civil purposes, vehicular applications [START_REF] Thrun | Stanley, the robot that won the DARPA Grand Challenge[END_REF], [START_REF]Spirit of Berlin: An Autonomous Car for the DARPA Urban Challenge Hardware and Software Architecture[END_REF], [START_REF] Gietelink | Development of advanced driver assistance systems with vehicle hardware-in-the-loop simulations[END_REF] have as objective the development of autonomous and semi-autonomous systems capable of traversing unrehearsed and off-road terrain, driving a car autonomously in an urban environment and also to help the driver in its driver task. The perception is a common task to all cases previously presented, and an important factor is the variety and complexity of environments and situations. These real-time intelligent platform developments have a common issue: providing to the platform the capability of perceiving and interacting with its neighbour environment, managing power consumption, CPU usage, etc. Then, taking into account the temporal coherence between consecutive frames, this work proposes an environment observer method based on Pearson's Correlation Coefficient that observes if there are no significant changes in the environment, permitting that regions-of-interest (ROI) are automatically selected in order to perform a task: road detection and obstacle avoidance.

The Section II presents a review of previous works. The Section III introduces the Pearson's Correlation Coefficient, followed by the Discarding Criteria method in Section IV. The Automatic Regions-of-Interest (ROI) Selection is presented in the Section V. The results and conclusions are given in Section VI and VII.

II. RELATED WORKS

A. Sensor Perception

Environment perception is a major issue in autonomous vehicles. It uses many types of sensors [START_REF] Thrun | Stanley, the robot that won the DARPA Grand Challenge[END_REF], [START_REF]Spirit of Berlin: An Autonomous Car for the DARPA Urban Challenge Hardware and Software Architecture[END_REF], including ultrasonic sensors, laser rangefinders, radar, cameras, etc. However, when incorporating several types of sensors, there is an increase of autonomy and "intelligence" degrees, especially in relation to navigation in unknown environments. In contrast, the type and quantity of sensors determine the volume of data for processing that requires, in most cases, a high computational cost. For unstructured environments, the scenario for study is dynamic, with several elements in motion. Thus, running a semi-or autonomous system involves carrying out complex, and non-deterministic operations in real time.

Moreover, a real-time system must satisfy explicit response-time constraints, including failure. This system is one whose logical correctness is based on both the correctness of the outputs and their timeliness [START_REF] Laplant | Real-Time System Design and Analysis[END_REF]. Furthermore, there is a considerable complexity in the sense that correctness not only depends on the logical ordering of events of the systems, but also on the relative timing between them [START_REF] Bouyer | Quantitative analysis of real-time systems[END_REF].

Aware that in the majority of the semi-and autonomous navigation systems, the machine-vision system is working

Automatic Regions-of-Interest Selection based on Pearson's Correlation Coefficient

A. Miranda Neto, A. Corrêa Victorino, I. Fantoni and D. E. Zampieri L together with other sensors, added to its low cost, this paper proposed a monocular vision-based sensor. Because it uses simple techniques and fast algorithms, the system is capable to achieve a good performance, where the compromise between processing time and images acquisition is fundamental.

Additionally, the vision-based sensors are defined as passive sensors and the image scanning is performed fast enough for Intelligent Transportation Systems [START_REF] Bertozzi | Vision-based intelligent vehicles: state of the art and perspectives[END_REF]. Furthermore, on the safety front, the progressive safety systems will be developed through the manufacturing of an "intelligent bumper" peripheral to the vehicle in answering new features as: blind spot detection, frontal and lateral precrash, etc. The objective in terms of cost to fill ADAS functions has to be very lower than the current Adaptive Cruise Control (500 Euros) [14].

B. Machine Vision

Autonomous robots can perform desired tasks in unstructured environments without continuous human guidance. An important variable is the state conditions in combination with environment events, because they may determine the system behavior.

In this way, Dynamic Power Management (DPM) is a design methodology for dynamically reconfiguring systems to provide the requested services and performance levels with a minimum number of active components or a minimum load on such components. For example, DPM and Real-Time Scheduling (RTS) techniques were presented in [START_REF] Mei | A case study of mobile robot's energy consumption and conservation techniques[END_REF]. They were applied to reduce the power consumption of mobile robots. At the same time that scheduling is a key concept in computer multitasking and real-time operating system, the DPM dynamically adjusts power states of components adaptive to the task's need, reducing the power consumption without compromising system performance.

The machine vision, part of the embedded computers, is an important component of the set of sensors. Although extremely complex and highly demanding, thanks to the great deal of information it can deliver, it is a powerful means for sensing the environment and it has been widely employed to deal with a large number of tasks in the automotive field [START_REF] Bertozzi | Vision-based intelligent vehicles: state of the art and perspectives[END_REF]. However, complex machine vision systems can lead to some losses due to the processing time.

A great amount of information would not necessarily lead to better decisions and could also harm the performance of the system, overloading it. Then, taking into account that it has been estimated that humans perceive visually about 90% of the environment information required for driving [START_REF] Bertozzi | Vision-based intelligent vehicles: state of the art and perspectives[END_REF], it is not a bad idea to reduce information acquired by a vision system, in order to reduce processing time.

The method proposed here is based in an automatic image discarding criteria [START_REF] Neto | Pearson's Correlation Coefficient for Discarding Redundant Information in Real Time Autonomous Navigation Systems[END_REF], [START_REF] Neto | Nondeterministic Criteria to Discard Redundant Information in Real Time Autonomous Navigation Systems based on Monocular Vision[END_REF], a low complexity and easy implemented solution. It improves the performance of a realtime system by choosing, in an automatic way, which images should be discarded and which ones should be treated at the visual perception system.

Furthermore, the fundamental premise for the applicability of DPM is that systems experience non-uniform workloads during operation time. A second assumption is that it is possible to predict, with a certain degree of confidence, the fluctuations of workload [START_REF] Benini | A Survey of Design Techniques for System-Level Dynamic Power Management[END_REF]. In this case, a simple DPM method shuts down a component when it is idle and it is essentially a prediction problem [START_REF] Mei | A case study of mobile robot's energy consumption and conservation techniques[END_REF]. Thus, according to [START_REF] Benini | A Survey of Design Techniques for System-Level Dynamic Power Management[END_REF], the rationale in all predictive techniques is that of exploiting the correlation between the past history of the workload and its near future in order to make reliable predictions about future events.

III. PEARSON'S CORRELATION COEFFICIENT (PCC)

According to [START_REF] Rodgers | Thirteen Ways to Look at the Correlation Coefficient[END_REF], an empirical and theoretical development that defined regression and correlation as statistical topics were presented by Sir Francis Galton in 1885. In 1895, Karl Pearson published the Pearson's Correlation Coefficient (PCC) [20]. The Pearson's method is widely used in statistical analysis, pattern recognition and image processing. Applications on the latter include comparing two images for image registration purposes, disparity measurement, etc [START_REF] Eugene | The Ineffectiveness of the Correlation Coefficient for Image Comparisons[END_REF]. It is described in [START_REF] Bonin-Font | Visual Navigation for Mobile Robots: A Survey[END_REF]:

∑ ∑ ∑ - - - - i m i i m i i m i m i ) y (y ) x (x ) y )(y x (x = r 2 2 1 (1)
where i x is the intensity of the th i pixel in image 1, i y is the intensity of the th i pixel in image 2, m x is the mean intensity of image 1, and m y is the mean intensity of image 2. The PCC threshold, 1 r , has value 1 if the two images are identical, 0 if they are completely uncorrelated, and -1 if they are completely anti-correlated, for example, if one image is the negative of the other.

IV. DISCARDING CRITERIA

The discarding criteria was presented in [START_REF] Neto | Pearson's Correlation Coefficient for Discarding Redundant Information in Real Time Autonomous Navigation Systems[END_REF] as a simple solution to improve the performance of a real-time navigation system by choosing, in an automatic way, which images should be discarded and which ones should be treated at the visual perception system. It was a new approach to the Pearson's Correlation Coefficient (PCC).

In Fig. 1, basically, if the PCC indicates that there is a high correlation between a reference frame and another new frame acquired, the new frame is discarded without being processed, Fig. 1 (c). In this case, some logical components may be shut down to save processor energy consumption, and/or to make the CPU available for running concurrent processes (the system can repeat a last valid command). Otherwise, the frame is processed and it is set as the new reference frame for the subsequent frame. For all cases presented in this paper a threshold is chosen in an empirical way, as explained next.

Fig. 1 -Discarding criteria [START_REF] Neto | Pearson's Correlation Coefficient for Discarding Redundant Information in Real Time Autonomous Navigation Systems[END_REF].

Whereas the main problem that has to be faced when realtime imaging is concerned and which is intrinsic to the processing of images is the large amount of data [START_REF] Bertozzi | Vision-based intelligent vehicles: state of the art and perspectives[END_REF], a logical dynamic optimization methodology based on Pearson's Correlation Coefficient (PCC) was introduced in [START_REF] Neto | Real-Time Dynamic Power Management based on Pearson's Correlation Coefficient[END_REF]. To better understand this proposal, the Fig. 2 presents the accumulated time of a hypothetical image processing time (15ms) versus the gain obtained by using the discarding criteria, which could allow significant savings in CPU power consumption. In desert context were discarded 470 of 530 frames, whilst in off-road context were discarded 5595 of 6740 frames. For these two cases, the discarding rate remains over 80%. However, this discarding rate is not dependent on the video sequence or image size, but on the obstacles / objects influence. As shown in Fig. 3, it is also important to notice that there is no diffeomorphism between the robot speed and the PCC variation, because if there are no changes between consecutive frames, the PCC threshold remains static. In this case, the isomorphism cannot be guaranteed and it ensures more efficiency for our proposal. In real, dynamic and unknown environments, this rate remains over 65%. For these examples, the PCC threshold was fixed at 0.85.

V. AUTOMATIC REGIONS-OF-INTEREST (ROI) SELECTION

According to the Pearson's correlation, in a certain analysis window (pair of frames), if the obstacle/object occupies a big portion of the scene, the PCC threshold tends to be low. Conversely, if obstacle/object occupies a small portion of the frame, it means that it is away from the vehicle and the system will have time enough to react. Nevertheless, where are these interest points/pixels? Or, which pixels (ROI) of the pair of images contributed most to the Pearson's coefficient computed? Which of them really need to be reprocessed (or resent to a server)?

Right after the Pearson's correlation in [START_REF] Bonin-Font | Visual Navigation for Mobile Robots: A Survey[END_REF], it has x m and y m , respectively: the mean intensities of images 1 and 2, i.e. From these two values, it begins again the process's correlation in [START_REF] Kim | Swarming Unmanned Aerial Vehicles: Concept Development and Experimentation, A State of the Art Review on Flight and Mission Control[END_REF], where for each pair of pixels analyzed, the only possible result is: [-1 or +1]. That is, all pixels with intensities below these means will be candidates for interest points. The Fig. 4 (c) and (f) present this process, where the red pixels (ROI) represent

1 2 - = r .      + - = - - - - ∑ ∑ ∑ 1 1 2 1 2 1 1 1 2 or ) r (y ) r (x ) r )(y r (x = r i Ym i i Xm i i Ym i Xm i (2)
where i x is the intensity of the th i pixel in image 1, i y is the intensity of the th i pixel in image 2, Xm r 1 and Ym r 1 were obtained in [START_REF] Bonin-Font | Visual Navigation for Mobile Robots: A Survey[END_REF]. A. Road detection Different techniques on automatic and semi-automatic road extraction methods are proposed in the literature. With respect to these specific tasks, a road detection method based on Otsu thresholding algorithm was proposed in [START_REF] Neto | A Simple and Efficient Road Detection Algorithm for Real Time Autonomous Navigation based on Monocular Vision[END_REF], [START_REF] Neto | Robust Horizon Finding Algorithm for Real-Time Autonomous Navigation based on Monocular Vision[END_REF].

As shown in Fig. 1, the road detection process (imaging processing) is performed only when the PCC indicates that there is a low correlation between the reference frame and the current frame. Otherwise, the current frame is discarded without being processed. When an image is discarded, the system keeps the previous segmentation result, which is linked to an Otsu threshold. This stored threshold represents the navigable area. Consequently, in order to further improve the navigable area detection, for each discarded image, in Fig. 1, it classifies only the interest pixels (ROI) obtained in (2) from the last stored Otsu threshold. As an example, Fig. 5 (d) presents the interest pixels from the correlation between the Fig. 5

B. Identifying the limits (boundaries) of the road

In order to identify the limits of the road (which includes the obstacles), many research works proposing methods to detect road boundary. As an example, the Canny edge detector [START_REF] Canny | A computational approach to edge detection[END_REF] can be employed as input of Hough transform [START_REF] Ballard | Generalized Hough transform to detect arbitrary shapes[END_REF] due to its robust performance and accurate edge localization. Nevertheless, to demonstrate another application for the proposed method here, Fig. 5 (f) presents the line detection using Hough transform from the Fig. 5 (d), where the white pixels represent 1 2 -= r .

C. Obstacle avoidance

According to [START_REF] Chatila | Dual research: Issues in robotics[END_REF], the robots enter in the military context especially when it is necessary to reduce human exposure to hazardous situations. Many of these robotic missions can be observed, among them, as shown in Fig. 6 (d), an Improvised Explosive Device (IED) detonation. In this context, obstacle avoidance is a robotic discipline that includes reactive control in real time, i.e. reactive obstacle avoidance.

In this way, as discussed earlier, in order to make reliable predictions about future events, a predictive technique explores the correlation between the past history of the workload and its near future [START_REF] Benini | A Survey of Design Techniques for System-Level Dynamic Power Management[END_REF]. On the other hand, the Pearson's method explores regression and correlation aspects. Then, in order to reduce the risk of collision, for the obstacle avoidance task, it uses a PCC threshold equal to 0.65, which put in evidence the past history properties. In a computational process running in parallel to what was discussed in section V-A, for each frame processed, evidencing the past history properties, another old Otsu threshold is also stored. Besides the results presented in the Fig. 6, this procedure also allows a greater level of security, especially when the camera does not "see" the navigation area (i.e. in front of a wall), as will be shown in next section.

Therefore, for each discarded image, in Fig. 1, it classifies only the interest pixels (ROI) obtained in (2) from the last Otsu threshold obtained by a road detection algorithm (reference frame). The horizontal red lines represent the horizon detection process [START_REF] Neto | Robust Horizon Finding Algorithm for Real-Time Autonomous Navigation based on Monocular Vision[END_REF].

VI. EXPERIMENTAL RESULTS

Besides the experimental DARPA test-banks [START_REF]DARPA Grand Challenge[END_REF], this section presents results on real, dynamic and unknown environments, and they were obtained using two experimental vehicles, from a moving vehicle with a Sony DFW-VL500 camera. In order to reduce the number of data, it includes a resolution reduction of image to 160x120 pixels.

From displacements on the outskirts of the Heudiasyc Laboratory in France, the test-bank presented in the Fig. 7, 9 and 10 contains images recorded using the Carmen vehicle shown in Fig. 11 (a). The data were collected while a driver was driving the vehicle. Results for two different types of image texture (road surfaces) were selected and its results are presented in the Fig. 7 (b) and (e).

The next cases present autonomous displacements at Renato Archer IT Center (CTI) in Brazil. This stage of testing evaluates the proposed algorithm at low speed on real-time conditions using the vehicle VERO shown in Fig. 11 (b). The VERO platform is equipped with SICK LMS and Hokuyo UTM30 outdoor laser scanners, GPS receiver, a monocular camera, and a CAN-bus interface which grants access to encoder data for the four wheels and steering, and allows commands to be sent to two independent motors driving the rear wheels and to control the steering angle. In the Fig. 8 (b) and (e), the trajectory control system had a single goal: to keep the robot in the center of the navigable path based on the mass of center. Fig. 8 (d) shows the task execution to go through a gate in off-road context. Fig. 8 (c) and (f) present the line detection using Hough transform from navigable area pixels. Two types of image texture (road surfaces) also were selected and its results are presented in Fig. 8 (b) and (e). As expected, in grass areas, on the parallelepiped streets or where an excessive noise is observed, the efficiency of the method decreases considerably, what can be improved with the application of a smoothing filter, and/or from regionmerging algorithm that mainly aims to represent homogeneous regions.

As shown in Fig. 9, it is not expected that a single camera provides all needed information to the safe navigation system to take decisions on routes. However, following what was presented in section VI-C: Obstacle avoidance, the new results in front of walls are presented in Fig. 10.

A video showing the application of this method is available in [29].

VII. CONCLUSION

This work proposes a real-time machine vision algorithm based on monocular vision. It is important to notice that this algorithm is not based on previous knowledge of the environment (lane shape, geometric inference, etc) neither camera calibration. A remarkable characteristic of methodology presented in this work is its independence of the image acquiring system and of the robot itself. The same implementation works in different mobile robots, with different embedded vision systems, without the need of adjusting parameters. Moreover, this visual-observer methodology may be extended to other sensors and components. Future work would be also focused to provide ground truth measurements from a front mounted radar and/or LIDAR system.

Fig. 2 -

 2 Fig. 2 -(a) Desert video [23]; (b) Off-road video [23]; In blue: the cumulative impact computations (ms) without the discarding criteria; In red: the cumulative computations (ms) by using the discarding criteria.

Fig. 3 -

 3 Fig. 3 -Real environment: Red line: discarding rate; Blue line: the vehicle speed; Green line: computational time (ms).

Xm r 1

 1 and Ym r 1 .

Fig. 4 -

 4 Fig. 4 -(a) and (b) are frames of the desert video [23]; (d) and (e) are frames of the off-road video [23]; (c) and (f) present the process's correlation in (2), where the red pixels (ROI) represent 1 2 -= r .

  (a) and (b), where the white pixels (ROI) . 5 (e), these interest pixels are classified as navigable area in blue.

Fig. 5 -

 5 Fig. 5 -(a) and (b) are frames of the off-road video [23]; (c) the road detection process result; (d) the interest pixels (white pixels); (e) the interest pixels are classified as navigable area in blue; (f) line detection using Hough transform from the interest pixels (ROI) in Fig. 5 (d). The horizontal red lines represent the horizon detection process [25].

  Fig. 6 (c) and (f) represent the interest pixels classified as obstacle, where the white pixels (ROI) represent 1 2 -= r . As shown in Fig. 6 (b) and (e), the interest pixels are classified as obstacle in yellow.

Fig. 6 -

 6 Fig. 6 -(a) is a frame of the off-road video [23]; (d) Improvised Explosive Device example; (b) and (e): the interest pixels are classified as navigable area in blue; (b) and (e): the interest pixels classified as obstacle in yellow.The horizontal red lines represent the horizon detection process[START_REF] Neto | Robust Horizon Finding Algorithm for Real-Time Autonomous Navigation based on Monocular Vision[END_REF].

Fig. 7 -

 7 Fig. 7 -(a) and (d) are frames in real environments; (b) and (e): the interest pixels are classified as navigable area in blue; (b) and (e): the interest pixels are classified as obstacle in yellow; (c) and (f): line detection using Hough transform from navigable area pixels. The horizontal red lines represent the horizon detection process [25].

Fig. 8 -

 8 Fig. 8 -(a) and (d) are frames in real environments; (b) and (e): the interest pixels are classified as navigable area in blue; (b) and (e): the interest pixels are classified as obstacle in yellow; (c) and (f): line detection using Hough transform from navigable area pixels. The horizontal red lines represent the horizon detection process [25].

Fig. 9 -

 9 Fig. 9 -(a) Original image (in front of a wall) and its road detection result in (c); (b) Canny edge detection result.

Fig. 10 -

 10 Fig. 10 -(a) and (d) are frames in real environments; (b) and (e): the interest pixels are classified as navigable area in blue; (b) and (e): the interest pixels are classified as obstacle in yellow; (f) and (i): line detection using Hough transform from navigable area pixels. The horizontal red lines represent the horizon detection process [25].

Fig. 11 -

 11 Fig. 11 -The experimental vehicles: (a) Carmen vehicle at Heudiasyc Laboratory in Compiegne, France; (b) Autonomous vehicle (VERO) at Renato Archer IT Center (CTI) in Campinas, Brazil.

  Manuscript received July 31, 2011. Arthur de Miranda Neto is Ph.D. Student in Mechanical Engineering at the State University of Campinas -Brazil and in Information Technology and Systems at the University of Technology of Compiègne, France. Actually, he is researcher engineer at the Heudiasyc Lab CNRS UMR 6599. Alessandro Corrêa Victorino is associate professor at the University of Technology of Compiègne -Heudiasyc Lab CNRS UMR 6599, France. Isabelle Fantoni is researcher at the University of Technology of Compiègne -Heudiasyc Lab CNRS UMR 6599, France. Douglas Eduardo Zampieri is associate professor of the Computational Mechanics Department, at the State University of Campinas, Brazil.

ACKNOWLEDGMENT

The authors wish to thank Dr. Luiz Gustavo Bizarro Mirisola, Dr. Samuel S. Bueno, Dr. Josué Jr. G. Ramos, Mr. Hélio Azevedo and Mr. Gerald Dherbomez for their kind attention to our work, for the useful discussions and support in data acquisition. This work was supported in part by French National Research Agency / ANR Percoive Project.


J Bertoin
email: jean.bertoin@math.uzh.ch

D Dufresne
email: dufresne@unimelb.edu.au

M Yor

Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion

Keywords: Brownian motion, exponential functional, Bougerol's identity, local time, Bessel processes

We present a two-dimensional extension of an identity in distribution due to Bougerol [Bou] that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend at the level of processes, we point at further striking relations in this direction.

1 Introduction

(1.1) To a linear Brownian motion (B s , s ≥ 0) starting from 0, we associate the exponential functional

A t = t 0 ds exp(2B s ) , t ≥ 0.
The distribution of A t is made accessible thanks to Bougerol's identity in law for fixed t , sinh(B t )

(law) = β(A t ), (1)
where (β(u), u ≥ 0) denotes a Brownian motion which is independent of (B s , s ≥ 0), hence of A t . Assuming (1), elementary computations yield the following characterization of the law of A t :

E 1 √ A t exp - x 2 2A t = a ′ (x) √ t exp - a 2 (x) 2t , x ∈ R , (2)
where a(x) = arg sinh(x) ≡ log(x + √ 1 + x 2 ) and a ′ (x) = 1 √ 1 + x 2 .

For further reference, we note some simple, but useful, consequences of (2), i.e.:

E 1 √ A t = 1 √ t , (3)
and, differentiating both sides with respect to t:

E exp(B t ) A 3/2 t ( * ) = E exp(2B t ) A 3/2 t = 1 t 3/2 , (4)
where ( * ) is obtained by time reversal of (B s , s ≤ t) from time t.

(1.2) It took some time, despite the original proof in [Bou], to understand simply and deeply why (1) holds. In [ADY], one finds the following arguments, among which the (essential) time reversal one:

for fixed t , β(A t ) is distributed as:

Now, it is easily shown, using Itô's formula, that the process in ( 5) is distributed as the process (sinh(B t ) , t ≥ 0).

(1.3) In the present paper, we obtain an extension of (1), by considering the two-dimensional vector (sinh(B t ), sinh(L t )), where (L t , t ≥ 0) denotes the local time at 0 of B. Our main result is:

Theorem 1. For fixed t, the three following two-dimensional random variables are identically distributed:

(sinh(B t ) , sinh(L t ))

(law) = (β(A t ) , exp(-B t )λ(A t )) (law) = (exp(-B t )β(A t ) , λ(A t )) , (6)
where (β(u), u ≥ 0) is a one-dimensional Brownian motion, with local time at 0, (λ(u), u ≥ 0), and β is independent from B.

It may be interesting to observe right now that Tanaka's formula shows that the local time at level 0 and time t of the process (sinh(B s ), s ≥ 0) is simply L t , whereas that of the process (exp(-B s )β(A s ), s ≥ 0) can be expressed as t 0 exp(-B s )dλ(A s ). Hence we have also the identity in distribution between two-dimensional processes

(sinh(B t ), L t ) t≥0 (law) = exp(-B t )β(A t ), t 0 exp(-B s )dλ(A s ) t≥0 . (7)
We stress that (6) cannot be extended to the level of processes; see the forthcoming Section 2.2. Hence the two identities in distribution ( 6) and ( 7) differ profoundly.

Theorem 1 is proved in Section 3. In Section 2, we discuss a number of consequences and equivalent statements to that of Theorem 1. For instance, the well-known equivalence in law, due to Paul Lévy, between the processes ((B t -B t , B t ) , t ≥ 0) and ((|B t | , L t ) , t ≥ 0) allows to present a version involving the supremum B t = sup s≤t B s instead of the local time version of Theorem 1.

Apart from this, Section 2 consists in the statements and discussions of four other theorems. Roughly speaking, these theorems were motivated by our desire to understand whether in (6), the two extreme identities hold for processes. This question has now been solved in the negative (see [BY]), but nonetheless there are some rather remarkable identities between jump intensity measures, which are described in Theorems 2-5, and which made us believe for some time in a 2-dimensional process identity extending (6). We let the reader discover the precise statements of these theorems in Section 2; their proofs are found in Section 4.

(1.4) Before we get into the precise arguments of the proofs of our various theorems, we should like to present an overall appreciation of the present work, by making a parallel with the way our understanding of Bougerol's identity (1) has improved: in [Y b ], a Mellin transform proof was given, based on the identity in law (23) below. Later, in [ADY], a time-reversal argument and stochastic calculus proof of (1) were found. We estimate that, at the moment, our understanding of Theorem 1 lies at the level of [Y b ], and that we should be able to develop some kind of understanding similar to [ADY]. However, such a proof eludes us for now; it is not clear that a time-reversal argument may be the missing stone. . . Nevertheless, we present some further extensions of Bougerol's identity different from the ones found in the volume [Y d ], which, hopefully should lead us in the future to a better understanding of Theorem 1.

2 Discussion of and some theorems closely related to Theorem 1

(2.1) Firstly we note that we may rewrite the identity in law (6) in a seemingly slightly weaker form

(sinh(|B t |) , sinh(L t )) (law) = (|β|(A t ) , exp(-B t )λ(A t )) (law) = (exp(-B t )|β|(A t ) , λ(A t )) . (8)
This induces no loss of generality, since the left-hand sides (without absolute values) of the expressions in (6) only differ from the ones with absolute values in (8) by multiplication by a symmetric Bernoulli variable, independent of the remaining quantities.

Secondly, it is well-known that the law of the two-dimensional vector (|B t |, L t ) is symmetric. More precisely, it is given by:

P(|B t | ∈ dx , L t ∈ dℓ) = 2(x + ℓ) √ 2πt 3 exp - (x + ℓ) 2 2t dx dℓ , x ≥ 0, ℓ ≥ 0 , (9)
as it can be checked from Lévy's identity (Theorem VI.2.3 in [RY], p.240) and the reflexion principle (Exercise 3.14 in [RY], p.110, or Proposition 2.8.1 in [KS], p. 95). Hence, the common law of ( 8) is also symmetric.

(2.2) We now consider the right-hand sides of the first and third vectors in (6) (or ( 8)), and we deduce therefrom

sinh(L t ) (law) = λ(A t ) . (10)
Now, unlike for Bougerol's identity (1), for which the possibility of an identity in law between processes is immediatley ruled out, since the left-hand side of (1) is not a martingale, whereas the right-hand side is, when one considers (10) it seems reasonable to wonder whether this identity might be valid at the level of the two increasing processes involved. However, the recent results in [BY] also rule out this possibility. In fact, it is this uncertainty which prevented us from publishing an earlier version of this paper, as in [DY].

We point out also that for each fixed t ≥ 0, we deduce from (7) the rather puzzling identity in law

λ(A t ) (law) = sinh t 0 exp(-B s )dλ(A s ) ,
which complements (10).

(2.3) Reformulation in terms of Brownian suprema. A celebrated identity in distribution due to Paul Lévy states that

((B t -B t , B t ) , t ≥ 0) (law) = ((|B t | , L t ) , t ≥ 0)
where B t = sup s≤t B s denotes the supremum of the Brownian trajectory up to time t. This enables us to reformulate (8) in the form

(sinh(B t -B t ) , sinh(B t )) (law) = ((β -β)(A t ) , exp(-B t )β(A t )) (law) = (exp(-B t )(β -β)(A t ) , β(A t )) ,
with β(t) = sup s≤t β(s). We leave to the interested reader further alternative reformulations of this identity in the same vein.

(2.4) A partial interpretation in terms of the Bessel clock.

We now discuss Bougerol's identity (1) in terms of a two-dimensional Bessel process. Specifically, let (R h , h ≥ 0) denote 2-dimensional Bessel process starting from 1, and let

H u = u 0 dh R 2 h , u ≥ 0 ,
the clock associated with R. The well-known skew-product decomposition of planar Brownian motion ([IM], p.270; [M1]; [M2]) shows that the clock H can be viewed as the inverse of the exponential Brownian functional A; consequently, considering the inverses of the increasing processes involved in (10), we obtain the following:

Corollary 1. Let (σ t , t ≥ 0) denote a stable (1/2) subordinator, precisely:

σ t := inf{u : λ u ≥ t} , t ≥ 0 , independent from (R h , h ≥ 0).
Then, for fixed s, one has:

H σs (law) = σ a(s) ( 11
)
where a(s) ≡ arg sinh(s).

It is interesting to compare Corollary 1 with the following consequence of (7):

Corollary 2. There is the identity in law between processes

(σ t ) t≥0 (law) = H σ η(t) t≥0
where σ is as in Corollary 1 and η : [0, ∞) → [0, ∞) is the inverse bijection of the continuous strictly increasing process

s → s 0 du R σu .
Proof: Indeed, (7) yields in terms of the Bessel clock

(L t ) t≥0 (law) = At 0 dλ(u) R u t≥0 .
Our statement now follows from the easy fact that the process σ η(t) , t ≥ 0 is the right-inverse of the continuous increasing process s →

s 0 dλ(u)
Ru . Again, one may wonder whether identity (11) holds at the level of increasing processes, however the results in [BY] rule out this possibility. It is then natural to ask for which functionals Φ :

C ↑ → R the identity E(Φ(H σ• )) = E(Φ(σ a(•) )) (12)
may hold, where C ↑ stands for the space of càdlàg increasing paths ω : R + → R + . An element of positive response is provided by the following result.

Theorem 2. Consider a measurable function Γ : R 3 + → R + with Γ(•, 0, •) = 0 and define

Φ(ω) = s≥0 Γ(ω s-, ∆ω s , s) , ω ∈ C ↑ ,
where ∆ω s = ω s -ω s-. Then (12) holds.

More precisely, if Γ(x, y, s) = f (x, s)g(y) for some measurable nonnegative functions f and g with g(0) = 0, then

E(Φ(H σ• )) = E(Φ(σ a(•) )) = C(f )D(g) , with C(f ) = ∞ 0 dλ √ 1 + λ 2 E[f (H σ λ , λ)] = ∞ 0 dλ √ 1 + λ 2 E[f (σ a(λ) , λ)] and D(g) = ∞ 0 dt √ 2πt 3 g(t) .
We observe that Corollary 2 implies that the range of subordinated clock H σ has the same distribution as the range of subordinator σ (and hence is a regenerative set). In particular we see that (12) holds whenever for a generic increasing path ω, Φ(ω) only depends on the range of ω. This provides a quick check of the identity

E(Φ(H σ• )) = E(Φ(σ a(•)
)) in the special case when the function Γ does not depend on the time parameter, i.e. Γ(x, y, s) = Γ(x, y).

(2.5) An amplification and a variant of Theorem 2.

Theorem 3. Let a, b ≥ 0 and Γ : R 2 + → R + a measurable function with Γ(•, 0) = 0. Introduce H a,b (Γ)(ℓ) = E λ≤ℓ (R σ λ-) a Γ(H σ λ-, H σ λ -H σ λ-) R b σ λ .
The following formula holds for Γ = f ⊗ g:

H a,b (f ⊗ g)(ℓ) = h - a-b (f, ℓ)h + b (g) ,
where:

h - c (f, ℓ) = ∞ 0 dt f (t)E e (c+1)Bt √ 2πA t (1 -e -ℓ 2 /2At ) h + b (g) = ∞ 0 dt g(t) √ 2π E e (2-b)Bt A 3/2 t .
It may be interesting to point out that these formulas become simpler in the special case when a = 0 and b = 1. Indeed, one gets using (4) that

h + 1 (g) = ∞ 0 dt √ 2πt 3 g(t)
and then, using (3) and ( 2) that

h - -1 (f, ℓ) = ∞ 0 dt f (t) √ 2πt 1 -a ′ (ℓ) exp - a 2 (ℓ) 2t
where a and a ′ are defined below (2).

We also stress that the quantities

m p,q (t) = E exp(pB t ) A q t (13)
arising in Theorem 3, have been studied in [CMY] and [Duf b , Duf c ].

(2.6) A variant of Theorem 2 which involves the windings of planar Brownian motion.

The following variant of Theorem 2 bears upon a relationship between the continuous winding process of planar Brownian motion, subordinated with (σ λ , λ ≥ 0), and the standard Cauchy process.

Theorem 4. Let Z u = |Z u | exp(iθ u ), u ≥ 0, denote complex valued Brownian motion, starting from: 1 + i0, with (θ u , u ≥ 0) its continuous winding process. Let (σ λ , λ ≥ 0) denote the inverse local time process of a linear Brownian motion, so σ is a stable(1/2) subordinator, which is assumed to be independent from (Z u , u ≥ 0). Finally, let (C α , α ≥ 0) be a standard Cauchy process.

For any measurable Γ : R × R → R + with Γ(x, 0) = 0, we have

E λ≤ℓ Γ(θ σ λ-, θ σ λ -θ σ λ -) = E λ≤ℓ Γ(C a(λ)-, C a(λ) -C a(λ)-)
for all ℓ's. In particular, for fixed ℓ ≥ 0, there is the equality in law:

θ σ ℓ (law) = C a(ℓ) . ( 14
)
The reader interested in some applications of these identities in law to functionals of the winding process (θ u , u ≥ 0) may refer to Vakeroudis [V]. In particular, the identity ( 14) allows to apply D. Williams' pinching method to yield yet another proof of Spitzer's celebrated theorem:

2 log t θ t (law) -→ t→∞ C 1 . (15)
(2.7) The joint Laplace-Mellin transform of (H σ λ , R σ λ ).

We now come back to Theorem 2, or rather we discuss part of its proof, as given in paragraph (3.2) below. A by product of Lemma 1 therein is:

E 1 R σ λ H σ λ = h = 1 √ 1 + λ 2 , (16)
an intriguing identity, which made us suspect for a moment that R σ λ and H σ λ might be independent. This is not the case, as we discovered by computing the joint Laplace-Mellin transform of (H σ λ , R σ λ ):

Theorem 5. The following formulae hold:

E 1 (R σ λ ) 2b exp(- µ 2 2 H σ λ ) = E (µ) 1 (R σ λ ) 2b+µ = C b,µ × F µ+1 2 -b, µ 2 + 1 -b, µ + 1; -1/λ 2 (1 + λ 2 ) 2b-1/2 (λ 2 ) µ+1 2 -b , ( 17
)
where F ≡ 2 F 1 denotes the classical hypergeometric family of functions with three parameters, and E (µ) refers to the expectation with respect to the probability measure P (µ) under which (R t , t ≥ 0) is a Bessel process with index µ (i.e. of dimension 2 + 2µ) and started from R 0 = 1, and

C b,µ = Γ(b + µ 2 + 1 2 )Γ(1 + µ 2 -b) Γ( 1 2 )Γ(1 + µ)
.

As a partial check for formula (17), we have made verifications with b = 1/2, b = -µ/2 (the result should be 1), and b = 0 (the result is exp(-µa(λ)) = (λ + √ 1 + λ 2 ) -µ ). Let us give some details for b = 1/2: We note that, for b = 1/2, (17) simplifies, as in the numerator

F µ 2 , µ 2 + 1 2 , µ + 1; -y = 2 1 + √ 1 + y µ ,
and in the denominator

(1 + λ 2 ) 1/2 (λ 2 ) µ/2 ≡ λ µ (1 + λ 2 ) 1/2 .
Hence, using the fact that C 1/2,µ = 2 -µ , formula (17) simplifies to:

2 -µ • 2 µ 1 + 1 + 1 λ 2 µ • 1 λ µ √ 1 + λ 2 ≡ 1 ( √ 1 + λ 2 + λ) µ √ 1 + λ 2 ,
which confirms identity ( 16), since the previous RHS expression equals

1 √ 1 + λ 2 E exp - µ 2 2 H σ λ .
3 Proof of Theorem 1

(3.1) We start by recalling some well-known facts about Brownian motion running up to an independent exponential time, which will be useful for the proof. In this subsection, S p denotes an exponential random variable with parameter p > 0, independent from the Brownian motion B. For t ≥ 0, denote g t = sup{u < t : B u = 0} the last zero of B before t. It is known that the processes (B u , u ≤ g Sp ) and (B g Sp +u , u ≤ S p -g Sp ) are independent. As a consequence, the variables L Sp (≡ L g Sp ) and B Sp are independent. Moreover, since L t and |B t | have the same law (see ( 9)), the same applies to L Sp and |B Sp |. Their common density is

2p exp(-2pu), u ≥ 0 (this is because P(L Sp ≥ ℓ) = P(S p ≥ τ ℓ ) = E[exp(-pτ ℓ )] = exp(-ℓ √ 2p), if τ ℓ is the time L • reaches ℓ
). An equivalent way to express this property is

√ 2e(|β(1)|, λ(1)) (law) = (e, e ′ ),
where on the left e

(law)
= S 1 is independent of β, and on the right the two variables are independent copies of S 1 .

(3.2) Recall the discussion in paragraph (2.1). Our main goal is to show that

(sinh(|B t |), sinh(L t )) (law) = (exp(-B t ) A t |β(1)|, A t λ(1)) . ( 18
)
This will be done by computing the joint Mellin transforms on either side, but before doing so we replace t with an exponential time S p and multiply both sides by √ 2e, assuming implicitly that S p , e, B and β are independent. What will be proved is:

√ 2e(sinh(|B Sp |), sinh(L Sp )) (law) = √ 2e(exp(-B Sp ) A Sp |β(1)|, A Sp λ(1)) . (19)
From the one-dimensional Bougerol identity (1), the left-hand side of ( 19) has the same distribution as

√ 2e(|N| A Sp , |N ′ | A ′ S ′ p ) , (20)
where N, N ′ are independent standard normals and A ′ a copy of A which is also independent of the other quantities. On the right-hand side of ( 19), use the facts in paragraph (3.1) to obtain

√ 2e(exp(-B Sp ) A Sp |β(1)|, A Sp λ(1)) (law) = (exp(-B Sp ) A Sp e , A Sp e ′ ) (law) = (exp(-B Sp ) A Sp |N| √ 2e , A Sp |N ′ | √ 2e ′ ) . (21)
Squaring, we are left with calculating the joint Mellin transforms of

e(A Sp , A ′ S ′ p ) and (exp(-2B Sp )A Sp e , A Sp e ′ ) , (22)
and verifying that they are equal (the N, N ′ on both sides of ( 20) and ( 21) can be cancelled).

The essential ingredient for these Mellin transforms is (see [Y a ], paper ♯6, p. 94)

A (ν) Sp (law) = β 1,a 2γ b , ( 23
)
where A (ν) denotes the exponential functional

A (ν) t = t 0 ds exp(2B (ν) s ) , t ≥ 0, of a Brownian motion with drift ν, B (ν) s
= B s + νs, β u,v is a beta variable with parameters (u, v), γ b is a gamma variable with parameter b, and

a = a(ν, p) = 1 2 (ν + 2p + ν 2 ) , b = b(ν, p) = 1 2 (-ν + 2p + ν 2 ) .

The Mellin transform of A (ν)

Sp is then

E(A (ν) Sp ) r = 2 -r Γ(1 + a)Γ(1 + r)Γ(b -r) Γ(1 + a + r)Γ(b) .
On the one hand, since a(0, p

) = b(0, p) = p 2 , E[(eA Sp ) c (eA ′ S ′ p ) d ] = 2 -c-d Γ(1 + c + d)Γ(1 + a)Γ(1 + c)Γ(b -c)Γ(1 + a)Γ(1 + d)Γ(b -d) Γ(1 + a + c)Γ(1 + a + d)Γ(b) 2 = 2 -c-d p 2 Γ(1 + c + d)Γ(1 + c)Γ( p 2 -c)Γ(1 + d)Γ( p 2 -d) Γ(1 + c + p 2 )Γ(1 + d + p 2 ) . ( 24
)
On the other hand, recalling the definition of A (ν) t and using the Girsanov-Cameron-Martin theorem, we obtain

E[(exp(-2B Sp )A Sp e) c (A Sp e ′ ) d ] = Γ(1 + c)Γ(1 + d)E[exp(-2cB Sp )(A Sp ) c+d ] = Γ(1 + c)Γ(1 + d)E[exp(2c S p )(A (-2c) Sp ) c+d ].
From the elementary identity

P(S p ∈ ds ; exp(ηS p )) = p p -η P(S p-η ∈ ds) , η < p , we find, letting q = p -2c 2 , E[(exp(-2B Sp )A Sp e) c (A Sp e ′ ) d ] = p q Γ(1 + c)Γ(1 + d)E[(A (-2c) Sq ) c+d ] = 2 -c-d p q Γ(1 + c)Γ(1 + d) Γ(1 + a(-2c, q))Γ(1 + c + d)Γ(b(-2c, q) -c -d) Γ(1 + a(-2c, q) + c + d)Γ(b(-2c, q)) . ( 25
)
Now, a(-2c, q) = -c + p 2 , b(-2c, q) = c + p 2 , and thus, comparing ( 24)-( 25),

p q Γ(1 + a(-2c, q))Γ(b(-2c, q) -c -d) Γ(1 + a(-2c, q) + c + d)Γ(b(-2c, q)) = p p -2c 2 Γ(1 -c + p 2 )Γ(-d + p 2 ) Γ(1 + d + p 2 )Γ(c + p 2 ) = p p -2c 2 ( p 2 -c)( p 2 + c) Γ( p 2 -c)Γ( p 2 -d) Γ(1 + c + p 2 )Γ(1 + d + p 2 ) = p 2 Γ( p 2 -c)Γ( p 2 -d) Γ(1 + c + p 2 )Γ(1 + d + p 2 )
.

Note that, along the way, it was necessary to assume q = p -2c 2 > 0, so that c needed to be taken small enough, and likewise for d, precisely: c, d < p 2 . But, even with these restrictions, we can conclude the proof of the identity in law of the two vectors in (22), thus ending the proof of (18).

(3.3) The second identity in Theorem 1 may be proved rather simply, by first noting that

(β(A t ), exp(-B t )λ(A t )) (law) = ( A t β(1), exp(-B t ) A t λ(1)) (26)
and then recalling (from the proof of Bougerol's identity in [ADY]) that time reversal

(B t -B (t-u) , 0 ≤ u ≤ t) (law) = (B u , 0 ≤ u ≤ t) implies (A t , exp(-2B t )A t ) (law) (exp(-2B t )A t , A t ) .
This completes the proof of Theorem 1.

4 Proofs of Theorems 2 to 5 (4.1) Proof of Theorem 2. (a) A key for the proof of Theorem 2 is the following interesting, and puzzling, identity, as discussed in Subsection 2.7.

Lemma 1. For any measurable function f : R + → R + , and any s ≥ 0, we have:

E 1 R σs f (H σs ) = 1 √ 1 + s 2 E[f (σ a(s) )].
Proof: For all q, t, ε > 0, we have from ( 11) that E(exp(-qσ a(t) ) -exp(-qσ a(t+ε) )) = E(exp(-qH σt ) -exp(-qH σ t+ε )) .

The LHS can be computed explicitly and we obtain

exp(-a(t) 2q) -exp(-a(t + ε) 2q)) ∼ ε √ 1 + t 2 2q exp(-a(t) 2q) , ε → 0 .
We next turn our attention to the RHS and apply the Markov property. In this direction, it is convenient to introduce a two-dimensional Bessel process R ′ which is independent of R and write H ′ for its clock. Likewise, σ ′ refers to an independent subordinator which has the same distribution as σ. For every r > 0, the notation P ′ r refers to the law under which R ′ 0 = r and E ′ r to the mathematical expectation under P ′ r . We point out that the scaling property implies the identities

E ′ r 1 -exp(-qH ′ σ ′ ε ) = E ′ 1 1 -exp(-qH ′ r -2 σ ′ ε ) = E ′ 1 1 -exp(-qH ′ σ ′ ε/r
) .

Of course we can also express the RHS as E 1 -exp(-qH σ ε/r ) . Writing σ t+ε = σ t + σ ′ ε , with σ ′ ε independent from σ t and R, we get from an application of the Markov property

E(exp(-qH σt ) -exp(-qH σ t+ε )) = E exp(-qH σt )E ′ Rσ t 1 -exp(-qH ′ σ ′ ε ) = E exp(-qH )E ′ 1 1 -exp(-qH ′ σ ′ ε/Rσ t ) = E exp(-qH σt ) 1 -exp{-a(ε/R σt ) 2q}
where the third equality stems from (11). Note that when ε → 0+, the preceding quantity is equivalent to

ε 2qE exp(-qH σt ) 1 R σt .
Putting the pieces together, we arrive at

E exp(-qH σt ) 1 R σt = 1 √ 1 + t 2 exp(-a(t) 2q) = 1 √ 1 + t 2 E(exp(-qσ a(t) ))
for all t, q > 0, which establishes Lemma 1.

(b) To end the proof of Theorem 2, we introduce the following notation concerning jump intensity measures:

H(Γ)(s) = E λ≤s Γ(H σ λ -, H σ λ -H σ λ -)
and ( 27)

K(Γ)(s) = E   α≤a(s) Γ(σ α -, σ α -σ α -)  
for given s, and Γ : R + × R + → R + , Borel, such that Γ(x, 0) = 0. Then, in order to finish the proof of Theorem 2, it suffices to take: Γ = f ⊗ g and to show the following:

H(f ⊗ g)(s) = h(f )(s) ∞ 0 dt √ 2πt 3 g(t) K(f ⊗ g)(s) = k(f )(s) ∞ 0 dt √ 2πt 3 g(t)
with, furthermore, the quantities h(f )(s) and k(f )(s) being equal, and equal to:

h(f )(s) = s 0 dλE 1 R σ λ f (H σ λ ) = s 0 dλ √ 1 + λ 2 E[f (H σ λ )] k(f )(s) = a(s) 0 dαE[f (σ α )] = s 0 dλ √ 1 + λ 2 E[f (σ a(λ) )] .
Now, concerning K(Γ)(s), since the Lévy measure of the subordinator

(σ α , α ≥ 0) is dt √ 2πt 3 ,
we have:

K(Γ)(s) = E a(s) 0 dαf (σ α-) ∞ 0 dt √ 2πt 3 g(t) = s 0 du √ 1 + u 2 E[f (σ a(u) )] ∞ 0 dt √ 2πt 3 g(t)
.

Concerning H(Γ)(s), starting again with the same argument (i.e: the knowledge of the Lévy measure of (σ u , u ≥ 0)), we obtain:

H(Γ)(s) = E s 0 dλ ∞ 0 dt √ 2πt 3 f (H σ λ )g(H σ λ +t -H σ λ ) = E s 0 dλ ∞ 0 dt √ 2πt 3 f (H σ λ )E ′ Rσ λ (g(H ′ t )) (28)
(from the Markov property for R). However, by scaling, we have:

E ′ ρ [g(H ′ t )] = E[g(H t/ρ 2 )] (29)
so that, plugging ( 29) in ( 28), we obtain:

∞ 0 dt √ 2πt 3 E ′ ρ [g(H ′ t )] = ∞ 0 dt √ 2πt 3 E[g(H t/ρ 2 )] = 1 ρ ∞ 0 du √ 2πu 3 E[g(H u )] . (30)
Note that, since the inverse of {u → H u } is: t → A t = t 0 dve 2Bv , we have:

∞ 0 du √ 2πu 3 E[g(H u )] = 1 √ 2π E ∞ 0 dtg(t) e 2Bt A 3 t = ∞ 0 dtg(t) 1 √ 2πt 3
by (4).

Going back to (28), we have obtained:

H(Γ)(s) = E s 0 dλ 1 R σ λ f (H σ λ ) ∞ 0 dt √ 2πt 3 g(t) (31)
b) Finally, to obtain the equality between H(Γ)(s) and K(Γ)(s), it remains to show, with the notation in the statement of Theorem 3, that:

h(f )(s) = k(f )(s) , for every f ≥ 0, Borel.
Again, it suffices to prove this for f λ (a) = e -λa , for any λ ≥ 0. Now we have:

E[exp(-νH σs )] = 1 + E θ≤s e -νHσ λ -e -νHσ λ - = 1 + E θ≤s e -νHσ λ -e -ν(Hσ λ -Hσ λ -) -1 = 1 + H(f ν ⊗ g ν )(s) (where : f ν (a) = exp(-νa); g ν (b) = (e -νb -1)) = 1 + h(f ν )(s) ∞ 0 dt √ 2πt 3 g ν (t) .
On the other hand:

E[exp(-νσ a(s) )] = 1 + a(s) 0 dαf ν (α) ∞ 0 dt √ 2πt 3 g ν (t) .
Thus, explicitly:

E[exp(-νH σs )] = 1 -(h(f ν )(s)) √ 2ν (32)
whereas:

E[exp(-νσ a(s) )] = 1 -k(f ν )(s) √ 2ν . (33)
Since the left hand sides of ( 32) and ( 33) are equal, so are the right hand sides, therefore:

∀f ≥ 0 , Borel , h(f )(s) = k(f )(s) . (34)
Hence, in complete generality:

H(Γ)(s) = K(Γ)(s) , (35)
which finishes the proof of Theorem 2.

(4.2) Proof of Theorem 3. Here are the main steps of this proof, which is quite similar to that of Theorem 2:

1) We first transform

H a,b (Γ)(ℓ) = E λ≤ℓ (R σ λ-) a f (H σ λ-) g(H σ λ -H σ λ-)1 (σ λ >σ λ -) (R σ λ ) b = E ℓ 0 dλ(R σ λ ) a f (H σ λ ) ∞ 0 dt √ 2πt 3 g(H σ λ +t -H σ λ ) (R σ λ +t ) b = E ℓ 0 dλ(R σ λ ) a f (H σ λ )E Rσ λ ∞ 0 dt √ 2πt 3 g(H t ) (R t ) b .
We begin by studying

h (+) (r, g) = E r ∞ 0 dt √ 2πt 3 g(H t ) R b t = E ∞ 0 dt √ 2πt 3 g(H t/r 2 ) (rR t/r 2 ) b (by scaling) = 1 r b E ∞ 0 dt √ 2πt 3 g(H t/r 2 ) (R t/r 2 ) b = 1 r b 1 r E ∞ 0 du √ 2πu 3 g(H u ) (R u ) b ≡ 1 r b+1 h (+) b (g) .
We then study: where the quantity m p,q (t) has been defined in (13).

h (+) b (g) = E ∞ 0 du √ 2πu 3 g(H u ) (R u ) b = E ∞ 0 dte 2Bt 2πA 3 t g(t) (exp(bB t )) = E ∞ 0 dte (2-b)Bt
3) Let us come back to

H a,b (Γ)(ℓ) = E ℓ 0 dλ(R σ λ ) a-b-1 f (H σ λ ) h (+) b (g) = h (-) a-b (f, ℓ)h (+) b (g), for F = f ⊗ g .
Thus, our next aim is to study:

h (-) c (f, ℓ) := E ℓ 0 dλ(R σ λ ) c-1 f (H σ λ ) .
We can re-express this quantity as

h (-) c (f, ℓ) = E τ ℓ 0 dL u (R u ) c-1 f (H u ) = ∞ 0 du √ 2πu P(L u < ℓ|B u = 0)E[(R u ) c-1 f (H u )] = ∞ 0 du √ 2πu P( √ u √ 2e < ℓ)E[(R u ) c-1 f (H u )] = ∞ 0 du √ 2πu P(e ≤ ℓ 2 /2u)E[(R u ) c-1 f (H u )] = ∞ 0 du √ 2πu (1 -exp(-ℓ 2 /2u))E[(R u ) c-1 f (H u )] = E ∞ 0 dA t √ 2πA t (1 -e -ℓ 2 /2At ) exp((c -1)B t )f (t) = E ∞ 0 dt e (c+1)Bt √ 2πA t (1 -e -ℓ 2 /2At )f (t) = ∞ 0 dtf (t)E e (c+1)Bt √ 2πA t
(1 -e -ℓ 2 /2At ) .

(4.3) Proof of Theorem 4. It is a simple consequence of Theorem 2, once one uses the wellknown skew product reresentation of θ t = γ Ht , where (γ u , u ≥ 0) is a real-valued Brownian motion independent from (H t , t ≥ 0) (we already gave some references before Corollary 1). Then all one needs to do is to "freeze" γ first, then apply Theorem 2, and finally use Spitzer's representation of the Cauchy process as

(C α , α ≥ 0) (law)
= (γ σα , α ≥ 0). Thus, it remains to prove the second equality. For this purpose, we use the same arguments as in the proof of ( 16) in [DY]; here are some details.

Let

J def = E (µ) 1 (R σ λ ) 2b+µ = 1 Γ(b + µ 2 ) ∞ 0 du u b+ µ 2 -1 E (µ) exp(-uR 2 σ λ ) .
There is a classical expression for E (µ) [exp(-uR 2 t )] (see, e.g. [RY] on page 441), which yields

E (µ) exp(-uR 2 σ λ ) = E (1 + 2uσ λ ) -1-µ exp - u 1 + 2uσ λ .
Using Fubini and making the change of variables v = 2uσ λ /(1+2uσ λ ), we obtain the following

  s ) dβ(s) , which, by time reversal (at time t) is also distributed as exp(B t ) t 0 exp(-B s ) dβ(s) .

(4. 4 )

 4 Proof of Theorem 5. The first equality follows from the (local) absolute continuity relationship between the laws of different Bessel processes, see, e.g., [Y a ].

expression for the integral

Hence, using Fubini again, we obtain:

To compute this last expectation, which we denote by K, we use

where γ 1/2 is a standard gamma(1/2)-variable. Then we obtain

Plugging this in (36), we obtain:

To derive the desired formula, we finally use a classical integral representation of 2 F 1 , together with

see formula (9.5.3) in Lebedev [Leb]. We leave the details to the reader.