Patent Publication Number: US-2023136084-A1

Title: Method for determining calibration parameters for an imaging device, and associated computer program product

Description:
The present invention relates to imaging-based surveillance, and relates more specifically to a method for automatically determining calibration parameters for an imaging device, in particular in order to estimate real distances based on images from an imaging device. 
     The prior art discloses detection algorithms, for example implemented using neural networks, in particular convolutional neural networks or CNN, in order to identify elements of interest in images. 
     For example, it is known to detect people or faces in security camera images. 
     Proceeding from this, if the parameters of the camera (called calibration parameters), such as the height at which it is positioned, its angle of inclination and its focal length, are known, a few simple geometric considerations and projections make it possible to estimate the quantities of the detected elements of interest, and generally to measure any “real” distance in the observed scene. 
     The difficulty is linked essentially to easily determining the calibration parameters. Indeed, thousands of cameras are already installed without their parameters being known. Of course, it would be possible to send people to take measurements on all of these cameras, but this would be a time-consuming, expensive and constrictive process. 
     There is also the need to estimate real distances based on images from an imaging device of which no characteristics are known, for example based on a video posted on the Internet or else based on a video taken using a mobile device the height and inclination of which vary little over time. 
     One automatic calibration method has been proposed, this being described in the document “A simplified nonlinear regression method for human height estimation in video surveillance”, Li, S., Nguyen, V. H., Ma, M. et al. 
     This method ends up estimating the height of people based on a surveillance camera video. In this method, a person of known size walking in front of the camera is chosen, so as to acquire multiple images in which the person is at various positions in space. Next, it is possible to determine the calibration parameters through regression on the positions of said person&#39;s head and feet. 
     This method is based on the assumption that a person has an essentially vertical representation in an image and that therefore the component along a vertical ordinate axis of a foot-head vector provides more information than the component along a horizontal abscissa axis of said foot-head vector, said component along the ordinate axis being able to be associated with the height of the person via the calibration parameters. The calibration parameters may thus be determined easily, using simplified mathematical equations neglecting the terms along the abscissa axis. 
     This method has the drawback of being suitable for limited use of the actual estimation of the height of people in images taken by a camera, and of requiring a specific prior calibration scenario during which a person of known size moves in front of said camera. 
     Indeed, this method does not give satisfactory calibration parameters for computing any real distances based on the images. When the representations of people in the images are inclined by a perspective effect, the estimated heights are inaccurate. 
     As an alternative, the document “Surveillance Camera Autocalibration based on Pedestrian Height Distributions”, Jingchen Liu, Robert T. Collins, Yanxi Li has proposed to use the estimation of vanishing points to evaluate the calibration parameters for the camera. It is thus possible to use any images from the camera as long as they show a large number of people assumed to be vertical (without specifically having to ask a subject to walk). 
     However, it is observed that estimating vanishing points remains a complex task in itself, which is sometimes difficult to apply generally and constitutes an excessively expensive aim, meaning that this method does not provide satisfaction. 
     The invention aims to solve the abovementioned problems of the prior art by proposing a method for determining calibration parameters for an imaging device that is completely automatic, simple, robust and universal. 
     The invention relates to a method for determining calibration parameters for an imaging device by way of a computer, based on at least one image acquired by the imaging device and comprising multiple people or based on a plurality of images (i, i 1 -i N ) acquired by the imaging device ( 10 ) at different instants in time and comprising at least one person, the method for determining calibration parameters comprising the following steps:
         a step of selecting, by way of the computer, a plurality of pairs of points in the at least one image, each pair of points comprising a head point associated with the head of a person and a foot point associated with the feet of said person, each point of a pair of points being referenced in the at least one image by image coordinates comprising an image abscissa and an image ordinate, each pair of points defining a foot-head segment in the three-dimensional real world of the environment of the imaging device,   a step of determining calibration parameters by way of the computer, wherein the computer comprises an estimator able to determine an estimated image ordinate of a head point based on the image ordinate of the foot point of a person and on the calibration parameters through an approximation that, in the three-dimensional real world of the environment of the imaging device, all foot-head segments have the same predetermined height and are vertical,       

     the estimator being two-dimensional and furthermore being able to determine an estimated image abscissa of a head point based on the image coordinates of the foot point, determining the calibration parameters comprising minimizing an error between said estimated image coordinates of the head point estimated by the two-dimensional estimator and said image coordinates of the corresponding head point. 
     According to one aspect of the invention, the selection step comprises determining the foot point as a midpoint between a right foot point and a left foot point. 
     According to one aspect of the invention, the calibration parameters comprise:
         a transverse angle of inclination θ of the imaging device,   a focal length f of the imaging device,   a height c at which the imaging device is positioned,       

     the estimator determining the estimated image coordinates (x′t, y′t) based on the image coordinates of the foot point (xp, yp) using the following equations: 
     
       
         
           
             
               
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     According to one aspect of the invention, the error minimization comprises using a regression algorithm that minimizes a mean squared error function between said estimated image coordinates of the head point estimated by the estimator and said image coordinates of the corresponding head point. 
     According to one aspect of the invention, the error minimization comprises an exhaustive search for the calibration parameters (f, θ, c) over a predetermined discretized domain. 
     According to one aspect of the invention, the error minimization comprises using a gradient descent method. 
     According to one aspect of the invention, the error minimization comprises two steps, a first step comprising an exhaustive search for intermediate calibration parameters over a predetermined discretized domain, and a second step comprising using a gradient descent method initialized on said intermediate calibration parameters resulting from the first step. 
     According to one aspect of the invention, the selection step furthermore comprises a filtering step in which the computer filters pairs of points of the plurality of pairs of points based on a visibility criterion of said points, in order to select the plurality of pairs of points. 
     According to one aspect of the invention, the at least one image acquired by the imaging device is divided into a plurality of areas, the plurality of pairs of points being selected by the computer in the selection step such that each area contains a number of pairs of points less than or equal to a predetermined threshold. 
     According to one aspect of the invention, the plurality of pairs of points results from a plurality of images acquired by the imaging device, the number of images corresponding to a predefined time period in which images are recorded by the imaging device. 
     According to one aspect of the invention, each image of the plurality of images results from a selection of images from the imaging device by the computer corresponding to a period greater than 100 ms. 
     The invention also relates to a computer program product comprising program instructions implementing the steps of the method for determining calibration parameters when the program instructions are executed by a computer. 
     Other advantages and features of the invention will become apparent upon reading the description and the drawings. 
    
    
     
         FIG.  1    shows a geometric model of an image reference frame and a real environment. 
         FIG.  2    shows a skeleton-point model of a person in an image. 
         FIG.  3    shows one example of a system for implementing the method for determining calibration parameters according to the invention. 
         FIG.  4   a    shows the steps of the method for determining calibration parameters according to the invention. 
         FIG.  4   b    shows one particular embodiment of a minimization step of the method for determining calibration parameters according to the invention. 
         FIG.  5   a    illustrates a superposition of multiple images divided into a plurality of areas, before points are selected by a computer of the invention. 
         FIG.  5   b    illustrates the same superposition of multiple images from  FIG.  5   a   , after points are selected by the computer of the invention. 
         FIG.  6    shows a superposition of five images in which three people are moving. 
         FIG.  7    shows an image comprising a multitude of people. 
     
    
    
     With reference to  FIG.  1   , calibration parameters for the imaging device are understood to mean parameters for matching the spatial coordinates of a point in the field of the imaging device, referred to as “real” coordinates as they are expressed in a terrestrial reference frame, with the planar coordinates of the representation of this point in the image acquired by the imaging device, referred to as “image” coordinates, that is to say the projection thereof. 
     The calibration parameters are shown in  FIG.  1    and are:
         a transverse angle of inclination θ of the imaging device  10 , that is to say the pitch angle defined by the angle between the main axis of the camera A and a horizontal direction,   a focal length f of the imaging device  10 ,   a height cat which the imaging device  10  is positioned.       

     The method of the invention comprises determining each of these three calibration parameters. It should be noted that it is still entirely possible for one or the other of these three parameters to be known, directly or indirectly, for example an imaging device may have a calibrated focal length f, or else the height c may have been expressly chosen when installing the camera. 
       FIG.  1    shows an imaging device  10  positioned at a height above the ground. The environment of the imaging device is referenced in the three-dimensional real world by a terrestrial reference frame the origin of which is a point on the ground vertical to the imaging device  10 . The axes of the terrestrial reference frame comprise an axis Y oriented upwards and passing through the imaging device  10 , and two axes X, Z located in the plane of the ground above which the imaging device  10  is positioned. The imaging device has the coordinates (X,Y,Z)=(0, c, 0) in the terrestrial reference system. 
     A standing person is located in the field of view of the imaging device  10 . In the terrestrial reference frame, the person is referenced by a head point Pt with coordinates (Xt, Yt, Zt), associated with the head of the person, and a foot point Pp with coordinates (Xp, Yp, Zp), associated with the feet of the person. The head point Pt and the foot point Pp define a foot-head segment PT in the three-dimensional real world of the environment of the imaging device  10 . 
     A two-dimensional image reference system is defined in an image i acquired by the imaging device  10 . The image reference system has the centre of the image i as origin and comprises two axes, a horizontal abscissa axis x and a vertical ordinate axis y. In the image reference frame, the head point pt of a person has the coordinates (xt, yt), and the foot point pp of a person has the coordinates (xp, yp). 
     The imaging device  10  is preferably fixed, at least with regard to the calibration parameters. It could for example be installed on a rail that allows horizontal translational movements. However, the imaging device could be fully mobile provided that its movements are known, and a person skilled in the art will specifically know how to apply the corresponding equations for changing reference system. 
     Some assumptions are furthermore advantageously made regarding the context:
         the roll angle of the imaging device  10  is assumed to be negligible,   the yaw angle of the imaging device  10  is assumed to be negligible,   the distortion in an image i acquired by the imaging device  10  is assumed to be negligible,   the optical centre of the imaging device  10  corresponds to the centre of the image i,   the ground of the environment of the imaging device  10 , in the field of view of the imaging device  10 , is flat.       

       FIG.  2    illustrates an image i in which a person is modelled in the image i by a fifteen-point skeleton model comprising a head point pt, a left foot point ppg and a right foot point ppd. 
     Advantageously, the foot point pp of the invention is the midpoint between the left foot point ppg and the right foot point ppd. 
       FIG.  3    illustrates a system comprising an imaging device  10 , a people detector  11  and a computer  20 . 
     The imaging device  10  is able to acquire images i 1 , i 2 , i 3  of a scene of its environment. The imaging device  10  is preferably a video camera, but may be a photographic camera able to take for example images of a scene in quick succession. 
     The people detector  11  is configured to detect people in images i 1 , i 2 , i 3  acquired by the imaging device  10 , and to determine key points of a person, for example so as to generate a fifteen-point skeleton as illustrated in  FIG.  2   . Advantageously, the key points comprise a head point pt corresponding to the top of the person&#39;s head, a left foot point ppg corresponding to the person&#39;s left heel, and a right foot point ppd corresponding to the person&#39;s right heel. 
     The people detector  11  may be split into two separate sub-devices able to communicate with one another, a first device being able to detect people in the image i and a second device being able to generate skeleton points corresponding to the person detected by the first device, for example through regression. 
     Each skeleton point ppg, ppd, pt is associated with two-dimensional coordinates in the image i 1 , i 2 , i 3 . 
     The two-dimensional coordinates of the skeleton points are able to be used by the computer  20  to execute the method of the invention, directly or after a potential change of reference system if the two-dimensional coordinates are not referenced in the image reference system as described and illustrated in  FIG.  1   . 
     Advantageously, each skeleton point ppg, ppd, pt is furthermore associated with a visibility score regarding visibility in the image i 1 , i 2 , i 3 . The visibility score makes it possible to assess the visibility of said skeleton point in the image, and thus therefore represents a reliability indicator for the two-dimensional coordinates of the skeleton point. 
     The computer  20  comprises an extractor  24  able to extract, from among skeleton points provided by the people detector  11 , a head point pt and a foot point pp, for each person detected by the people detector  11 . 
     The computer  20  comprises a selector  23  able to select, from among these head and foot points pt and pp, a plurality of pairs of points (pt, pp) comprising a head point pt and a foot point pp associated with the same person, based on multiple criteria taken on their own or in combination:
         a visibility score associated with the head point pt and/or with the foot point pp, said visibility scores being able to be deduced from the visibility scores associated with the skeleton points provided by the people detector  11 ,   a distribution, a location and/or a number of head points pt and foot points pp in the image i 1 , i 2 , i 3 ,   a temporal or geographic proximity of the head and/or foot points pt and/or pp to one another.       

     The computer  20  comprises a two-dimensional estimator  21  able to determine estimated image coordinates of a head point pt based on the image coordinates (xp, yp) of said foot point pp and on the calibration parameters f, θ, c. 
     The computer  20  comprises a minimizer  22  able to minimize an error between the image coordinates of the head point pt that are estimated by the estimator  21  and the image coordinates (xt, yt) of the head point pt that originate from the people detector  11 . 
     The extractor  24 , the selector  23 , the estimator  21  and the minimizer  22  are advantageously software modules of the computer  20 . 
       FIG.  4   a    and  FIG.  4   b    illustrate the steps of the method according to the invention, which comprise:
         a selection step  102 , in which a computer  20  selects a plurality of pairs of points (pt, pp), each pair of points (pt, pp) being associated with a person in an image i, i 1 , i 2 , i 3  taken by an imaging device  10 ,   a step  103  of determining calibration parameters for the imaging device  10  based on the selected plurality of pairs of points (pt, pp).       
     To be able to select a plurality of pairs of points (pt, pp), the computer  20  generally needs a plurality of images i 1 , i 2 , i 3  from the imaging device  10  that are acquired at different instants in time, for example multiple images from a video acquired by the imaging device  10 . Multiple images i 1 , i 2 , i 3  allow the computer  20  to select a plurality of pairs of points (pt, pp), even if for example a single person is moving in the scene. 
     According to one example,  FIG.  6    shows a superposition of five images i 1 -i 5  comprising three people, the images i 1 -i 5  being taken at various instants in time by the imaging device  10 . This superposition of images i 1 -i 5  comprises fifteen pairs of points (pt, pp) since the three people are moving in the field of the imaging device  10 . 
     In the case for example of a crowd of people in the field of view of the imaging device  10 , a single image i acquired by the imaging device  10  allows the computer  20  to select a plurality of pairs of points (pt, pp). Such an image i is for example shown in  FIG.  7   . 
     The computer  20  therefore needs pairs of points (pt, pp) originating from at least one image i, i 1 , i 2 , i 3 , i 4 , i 5  acquired by the imaging device  10  and comprising at least one person. These pairs of points (pt, pp) are provided for example by a people detector  11 , as described above. 
     In the selection step  102 , the computer  20  selects a plurality of pairs of points (pt, pp) in the at least one image i, i 1 , i 2 , i 3 , i 4 , i 5 , each pair of points (pt, pp) comprising a head point pt associated with the head of a person and a foot point pp associated with the feet of said person. 
     The computer  20  may select pairs of points (pt, pp) as images are acquired by the imaging device, in particular as points are provided by a people detector  11 . 
     As an alternative, the computer  20  may select the plurality of pairs of points (pt, pp) after multiple image acquisitions, for example after a predefined acquisition time period in which images are acquired by the imaging device  10 , in particular after a people detector  11  has provided points corresponding to multiple images. 
     Each point pt, pp of a pair of points (pt, pp) is referenced in the at least one image i, i 1 , i 2 , i 3 , i 4 , i 5  by image coordinates (xt, yt), (xp, yp) comprising an image abscissa xt, xp and an image ordinate yt, yp. 
     The coordinates of a head point pt in an image i are (xt, yt). 
     The coordinates of a foot point pp in an image i are (xp, yp). 
     To determine the calibration parameters (f, θ, c) in a more reliable manner, it is preferable for the plurality of pairs of points (pt, pp) to correspond as far as possible to a homogeneous distribution in the one or more images i, i 1 , i 2 , i 3 , that is to say to a relatively homogeneous distribution of people in the field of view of the imaging device  10 . 
     Thus, as illustrated in  FIGS.  5   a  and  5   b   , according to one particular embodiment, the at least one image i, i 1 , i 2 , i 3 , i 4 , i s  acquired by the imaging device  10  is divided into a plurality of areas Z 1 -Z 9 , for example nine areas as illustrated in the figures. 
     To avoid an excessively great imbalance between the various areas Z 1 -Z 9 , in particular an area Z 8 , Z 9  populated excessively with pairs of points (pt, pp) in comparison with another area Z 1 -Z 7 , the computer  20  selects the plurality of pairs of points (pt, pp) such that each area Z 1 -Z 9  contains a number of pairs of points (pt, pp) less than or equal to a predetermined threshold. The difference in the number of pairs of points between two areas thus cannot exceed this predetermined threshold. 
     According to this selection mode, the computer  20  retains all pairs of points of an area Z 1 -Z 9  in which the number of pairs of points (pt, pp) is less than or equal to the predetermined threshold. 
     According to this selection mode, the computer  20  reduces the number of pairs of points (pt, pp) in an area Z 1 -Z 9  in which the number of pairs of points (pt, pp) is greater than the predetermined threshold until reaching said predetermined threshold for the number of pairs of points (pt, pp) in said area Z 1 -Z 9 . 
     As an alternative, the computer  20  selects the number of pairs of points (pt, pp) in each area Z 1 -Z 9  until reaching said predetermined threshold for the number of pairs of points (pt, pp) in said area Z 1 -Z 9 , and stops selecting pairs of points for example after a predefined duration has elapsed. Some areas will therefore be populated to the maximum extent with pairs of points (pt, pp), corresponding to the predetermined threshold, and other areas will be less populated with pairs of points (pt, pp), for example because the imaging device captured fewer people in these other areas during the predefined duration. 
     In  FIG.  5   a   , the illustrated pairs of points (pt, pp) originate from a plurality of N images i 1 -i N  acquired by the imaging device  10  for example over 10 seconds, the points pt, pp shown corresponding to people detected by a people detector  11 . 
     In  FIG.  5   b   , the illustrated pairs of points (pt, pp) represent the plurality of pairs of points (pt, pp) selected by the computer  20 , such that each area Z 1 -Z 9  contains a number of pairs of points (pt, pp) less than or equal to the predetermined threshold, in this case equal to two. 
     For example, the selection of pairs of points (pt, pp) to limit the number of pairs of points (pt, pp) in an area Z 1 -Z 9  to a number less than or equal to the predetermined threshold is random or pseudorandom. 
     For example, the selection of pairs of points (pt, pp) to limit the number of pairs of points (pt, pp) to a number less than or equal to the predetermined threshold is such that preference is given to a distribution of pairs of points that is as uniform as possible in the area. 
     For example, the selection of pairs of points (pt, pp) to limit the number of pairs of points (pt, pp) to a number less than or equal to the predetermined threshold is such that, for two pairs of points (pt, pp) having image coordinates that are geographically close in the image, if the distance between two head points pt and/or two foot points pp is less than a predetermined distance, one of the pairs of points (pt, pp) may be removed. 
     To avoid redundancy or quasi-redundancy of pairs of points (pt, pp) that are excessively geographically close to one another in an image, the computer may select pairs of points (pt, pp) corresponding to images from the imaging device by way of the computer that are spaced in time by a predetermined period, for example a period of between 100 ms and 500 ms. 
     The selection step  102  comprises an optional filtering step  101 , in which the computer  20  filters pairs of points (pt, pp) based on a visibility criterion of said points in order to select the plurality of pairs of points (pt, pp). 
     For example, the visibility criterion is based on a visibility score associated with the head and foot points pt and pp and provided by the people detector  11 . According to one exemplary embodiment, below a threshold for a visibility score associated with a head point pt or with a foot point pp of a person, the corresponding pair of points (pt, pp) is not retained in the plurality of pairs of points. 
     In a determination step  103 , a two-dimensional estimator  21  of the computer  20  determines estimated image coordinates (x′t, y′t) of a head point pt based on the image coordinates (xp, yp) of the foot point pp of a person and on the calibration parameters (f, θ, c). 
     To determine the estimated image coordinates (x′t, y′t) of a head point pt based on the image coordinates (xp, yp) of the foot point pp, the estimator  21  makes the following approximations:
         in the three-dimensional real world of the environment of the imaging device, people are all of the same size, that is to say that all foot-head segments PT have the same predetermined height H.   in the three-dimensional real world of the environment of the imaging device, people are all standing and upright, that is to say that all foot-head segments PT are vertical.       

     The projection matrix P of the imaging device  10  is defined based on the calibration parameters (f, θ, c) as follows: 
     
       
         
           
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               ] 
             
           
         
       
     
     The terrestrial reference frame and the image reference system are as shown in  FIG.  1   . 
     An image point with coordinates (x,y) in the image reference system corresponds to a real point with coordinates (X,Y,Z) in the terrestrial reference frame, via the calibration parameters (f, θ, c). 
     More specifically, it is possible to obtain a homogeneous representation (xh,yh,wh) of an image point through multiplication by the projection matrix P of the homogeneous representation (Xh,Yh,Zh, 1) of a corresponding real point, using the following relationship: 
     
       
         
           
             
               [ 
               
                 
                   
                     xh 
                   
                 
                 
                   
                     yh 
                   
                 
                 
                   
                     wh 
                   
                 
               
               ] 
             
             = 
             
               
                 P 
                 · 
                 
                   [ 
                   
                     
                       
                         Xh 
                       
                     
                     
                       
                         Yh 
                       
                     
                     
                       
                         Zh 
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         f 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           f 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         
                           
                             - 
                             f 
                           
                           · 
                           
                             ( 
                             
                               sin 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             f 
                           
                           · 
                           c 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           sin 
                           ⁡ 
                           ( 
                           θ 
                           ) 
                         
                       
                       
                         
                           cos 
                           ⁡ 
                           ( 
                           θ 
                           ) 
                         
                       
                       
                         
                           
                             - 
                             c 
                           
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         Xh 
                       
                     
                     
                       
                         Yh 
                       
                     
                     
                       
                         Zh 
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             	 
             
               
                 [ 
                 
                   
                     
                       xh 
                     
                   
                   
                     
                       yh 
                     
                   
                   
                     
                       wh 
                     
                   
                 
                 ] 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         f 
                         · 
                         Xh 
                       
                     
                   
                   
                     
                       
                         
                           f 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           Yh 
                         
                         - 
                         
                           f 
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           Zh 
                         
                         - 
                         
                           f 
                           · 
                           c 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           Yh 
                         
                         + 
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           Zh 
                         
                         - 
                         
                           c 
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Switching back to Cartesian coordinates gives the following relationship between an image point with coordinates (x,y) in the image reference system and a real point with coordinates (X,Y,Z) in the terrestrial reference frame: 
     
       
         
           
             
               [ 
               
                 
                   
                     x 
                   
                 
                 
                   
                     y 
                   
                 
               
               ] 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         f 
                         · 
                         X 
                       
                       / 
                       
                         ( 
                         
                           
                             
                               sin 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                             · 
                             Y 
                           
                           + 
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                             · 
                             Z 
                           
                           - 
                           
                             c 
                             · 
                             
                               sin 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         ( 
                         
                           
                             f 
                             · 
                             
                               cos 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                             · 
                             Y 
                           
                           - 
                           
                             f 
                             · 
                             
                               sin 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                             · 
                             Z 
                           
                           - 
                           
                             f 
                             · 
                             c 
                             · 
                             
                               cos 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             
                               sin 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                             · 
                             Y 
                           
                           + 
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                             · 
                             Z 
                           
                           - 
                           
                             c 
                             · 
                             
                               sin 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     Using the approximations made by the estimator  21  of the computer  20 , any person in the field of view of the device  10  has a vertical foot-head segment PT of height H. 
     Thus, it is possible to write that, for a person, in the terrestrial reference frame: 
         Xp=Xt,Zp=Zt,Yp= 0and  Yt=H.    
     The image coordinates (xt, yt) of a head point (pt) and the image coordinates (xp, yp) of a foot point (pp) may thus be expressed based on their real coordinates (Xt, Yt, Zt), (Xp, Yp, Zp) and on the calibration parameters (f, θ, c) as follows: 
     
       
         
           
             	 
             
               
                 [ 
                 
                   
                     
                       xp 
                     
                   
                   
                     
                       yp 
                     
                   
                 
                 ] 
               
               = 
               
                 
                   f 
                   
                     
                       
                         cos 
                         ⁡ 
                         ( 
                         θ 
                         ) 
                       
                       · 
                       Zp 
                     
                     - 
                     
                       c 
                       · 
                       
                         sin 
                         ⁡ 
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
                 · 
                 
                   [ 
                   
                     
                       
                         Xp 
                       
                     
                     
                       
                         
                           
                             
                               - 
                               
                                 sin 
                                 ⁡ 
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             · 
                             Zp 
                           
                           - 
                           
                             c 
                             · 
                             
                               cos 
                               ⁡ 
                               ( 
                               θ 
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               [ 
               
                 
                   
                     xt 
                   
                 
                 
                   
                     yt 
                   
                 
               
               ] 
             
             = 
             
               
                 f 
                 
                   
                     H 
                     · 
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                   + 
                   
                     
                       cos 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                     · 
                     Zp 
                   
                   - 
                   
                     c 
                     · 
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                 
               
               · 
               
                 [ 
                 
                   
                     
                       Xp 
                     
                   
                   
                     
                       
                         
                           Yp 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                         - 
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           Zp 
                         
                         - 
                         
                           c 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     On the basis of these expressions and after simplification, the estimator  21  of the computer  20  determines estimated image coordinates (x′t, y′t) of a head point pt of a person in an image i using the following equations: 
     
       
         
           
             
               
                 x 
                 ′ 
               
               ⁢ 
               t 
             
             = 
             
               
                 
                   
                     
                       cos 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                     · 
                     c 
                     · 
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         - 
                         
                           f 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         + 
                         
                           f 
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   - 
                   
                     c 
                     · 
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                 
                 
                   
                     H 
                     · 
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                   + 
                   
                     
                       cos 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                     · 
                     c 
                     · 
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         - 
                         
                           f 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         + 
                         
                           f 
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   - 
                   
                     c 
                     · 
                     
                       sin 
                       ( 
                       θ 
                     
                   
                 
               
               · 
               xp 
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 y 
                 ′ 
               
               ⁢ 
               t 
             
             = 
             
               
                 
                   
                     H 
                     · 
                     
                       cos 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                   - 
                   
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                     · 
                     c 
                     · 
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         - 
                         
                           f 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         + 
                         
                           f 
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   - 
                   
                     c 
                     · 
                     
                       cos 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                 
                 
                   
                     H 
                     · 
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                   + 
                   
                     
                       cos 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                     · 
                     c 
                     · 
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         - 
                         
                           f 
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                           · 
                           yp 
                         
                         + 
                         
                           f 
                           · 
                           
                             sin 
                             ⁡ 
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   - 
                   
                     c 
                     · 
                     
                       sin 
                       ⁡ 
                       ( 
                       θ 
                       ) 
                     
                   
                 
               
               · 
               f 
             
           
         
       
     
     The units in relation to these equations are as follows:
         the average height H of a person is in centimetres,   the focal length f is in pixels,   the angle θ is in radians,   the height c is in centimetres.       

     The height H is a parameter to be adjusted. It represents an average height of the people. 
     Next, a minimizer  22  of the computer  20  determines the calibration parameters (f, θ, c) by performing an error minimization between said estimated image coordinates (x′t, y′t) of the head point (pt) estimated by the two-dimensional estimator  21  and said image coordinates (xt, yt) of the corresponding head point (pt). 
     For example, the height H is fixed at a predetermined value corresponding to an average size of a person, 160 cm. 
     The error minimization comprises for example using a regression algorithm that minimizes a mean squared error function between said estimated image coordinates (x′t, y′t) of the head point pt estimated by the estimator  21 , and said image coordinates (xt, yt) of the corresponding head point pt. 
     For a plurality of pairs of points (pt, pp) comprising M pairs of points (pt, pp), the aim is then to find the calibration parameters (f, θ, c) that minimize the following error function: 
     
       
         
           
             
               ( 
               
                 f 
                 , 
                 θ 
                 , 
                 c 
               
               ) 
             
             = 
             
               
                 argmin 
                 
                   ( 
                   
                     f 
                     , 
                     θ 
                     , 
                     c 
                   
                   ) 
                 
               
               ⁢ 
               
                 1 
                 M 
               
               ⁢ 
               
                 
                   ∑ 
                   1 
                   M 
                 
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           xt 
                           - 
                           
                             
                               x 
                               ′ 
                             
                             ⁢ 
                             t 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           yt 
                           - 
                           
                             
                               y 
                               ′ 
                             
                             ⁢ 
                             t 
                           
                         
                         ) 
                       
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Advantageously, the error minimization comprises an exhaustive search for the calibration parameters (f, θ, c) over a predetermined discretized domain. 
     According to one exemplary embodiment, the domain is defined by the following three ranges f ∈ [0.1·hi, 6·hi] 
     
       
         
           
             
               θ 
               ∈ 
               
                 [ 
                 
                   
                     - 
                     
                       π 
                       4 
                     
                   
                   , 
                   
                     π 
                     2 
                   
                 
                 ] 
               
             
             , 
           
         
       
     
     c ∈ [50, 4500], where hi is the height of the image expressed in pixels, each range being discretized into fifty values that are distributed uniformly over each of the ranges. 
     Thus, 125 000 values of triplets of calibration parameters (f, θ, c) are verified, the calibration parameters (f, θ, c) minimizing the error between the estimated image coordinates (x′t, y′t) of the head point pt estimated by the two-dimensional estimator and the image coordinates (xt, yt) of the head point pt being those retained by the minimizer  22  of the computer  20 . 
     In the case of using a regression algorithm that minimizes a mean squared error function, the calibration parameters (f, θ, c) minimizing the mean squared error function are those retained by the minimizer  22  of the computer  20 . 
     The error minimization comprises for example using a gradient descent method. 
     In particular, the gradient descent method is based on the L-BFGS-B algorithm. 
     Advantageously, the error minimization combines two examples proposed above. The error minimization thus comprises two steps, a first step  103   a  comprising an exhaustive search for intermediate calibration parameters (fi, θi, ci) over a predetermined discretized domain, and a second step  103   b  comprising using a gradient descent method initialized on said intermediate calibration parameters (fi, θi, ci) resulting from the first step. 
     In the case of using a regression algorithm that minimizes a mean squared error function, the gradient descent method is initialized on the intermediate calibration parameters (fi, θi, ci) minimizing the following error function: 
     
       
         
           
             
               ( 
               
                 fi 
                 , 
                 
                   θ 
                   ⁢ 
                   i 
                 
                 , 
                 ci 
               
               ) 
             
             = 
             
               
                 argmin 
                 
                   ( 
                   
                     f 
                     , 
                     θ 
                     , 
                     c 
                   
                   ) 
                 
               
               ⁢ 
               
                 1 
                 M 
               
               ⁢ 
               
                 
                   ∑ 
                   1 
                   M 
                 
                 
                   
                     ( 
                     
                       
                         
                           ( 
                           
                             xt 
                             - 
                             
                               
                                 x 
                                 ′ 
                               
                               ⁢ 
                               t 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             yt 
                             - 
                             
                               
                                 y 
                                 ′ 
                               
                               ⁢ 
                               t 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
     The number of images needed for the method of the invention depends on the computer selection method. 
     According to the method of the invention and according to a first method, the computer  20  selects the plurality of pairs of points (pt, pp) from among points relating to a number of images i 1 -i N  corresponding to a predefined acquisition time period in which images are acquired by the imaging device  10 . 
     According to a second method in which the pairs of points (pt, pp) are selected by the computer  20  as images are acquired by the imaging device  10 , the number of images i 1 -i N  corresponds to a number of pairs of selected points (pt, pp) that is reached in relation to a predetermined value, globally or in relation to areas Z 1 -Z 9  in the images.