Patent Publication Number: US-11640680-B2

Title: Imaging system and a method of calibrating an image system

Description:
TECHNICAL FIELD 
     Embodiments relate to calibration of an imaging system. Especially, embodiments described herein relate to the calibration of an imaging system comprising at least three cameras having a common field of view. 
     BACKGROUND 
     Traffic surveillance and analysis within cities is an important and growing technology field. Traditional monocular and stereo vision cameras are commonly used and provide a large amount of data about the traffic flow. The more views of a scene that are available, the more geometric data can be extracted which allows for more detailed analysis and estimations. 
     One approach for imaging traffic that combines practical aspects and data quality may be a trinocular camera array, a camera rig with three cameras, providing more data than a traditional stereo camera rig, while being just as easy to mount. Compared to a stereo camera, the trinocular camera array generates more images with different perspectives, and therefore can provide more geometric data. A calibration process is necessary to set up a trinocular camera array, that finds both the rectification and the orientation of the camera rig, as well as internal orientation, such as the accurate distance between the cameras. Two commonly used approaches are those by Tsai (Tsai, R. Y. (1987). A versatile camera calibration technique for high-accuracy 3d machine vision metrology using off-the-shelf tv cameras and lenses. IEEE J. Robotics and Automation, 3:323-344) and Zhang (Zhang, Z. (2000). A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22:1330-1334. MSR-TR-98-71, Updated Mar. 25, 1999), each relying on a single type of external data source: known 3D points in an imaged scene (and their corresponding 2D positions in the images) in the former and controlled checkerboard patterns in the latter. A 2D point is a point in a two-dimensional space and its position is given by two Cartesian coordinates. Correspondingly, a 3D point is a point in a three-dimensional space and its position is given by three Cartesian coordinates. A common strategy used by traffic researchers is to use a variant of Tsai&#39;s calibration method, which calibrates the cameras using only measured 3D points in an imaged scene. This approach does not take into account the particular geometry of the trinocular camera array and instead calibrates each camera individually. Because all the 3D point data correspond to positions far from the camera, there is a risk that an incorrect calibration is found. Another commonly used approach is to calibrate using only checkerboards. This approach may result in calibration data that perform badly at distances too far from the cameras, where checkerboards are impractical to place or are of too low resolution to correctly resolve. 
     Therefore, what is needed is a calibration technique for a camera array with three or more cameras that avoids pitfalls of standard calibration techniques. 
     SUMMARY OF THE INVENTION 
     A first aspect of the disclosure is a method of calibrating an imaging system, the imaging system comprising at least three cameras having a common field of view, wherein each of the at least three cameras has a camera centre, and wherein the camera centres are positioned along a line, the method comprising: imaging a calibration object with the at least three cameras to generate a set of calibration object images, wherein the calibration object is located nearby the at least three cameras and in the common field of view, generating a set of first parameters by processing the set of calibration object images, imaging a set of distant scene position points with the at least three cameras to generate a set of position point images, generating a set of second parameters by processing the set of position point images, and generating a set of calibration parameters in dependence on the set of first parameters and the set of second parameters. 
     An advantage with embodiments disclosed herein is a simplified and improved calibration procedure. 
     Another advantage with embodiments disclosed herein is that thanks to the calibration procedure, depth measurement may be performed with increased precision in images acquired by the imaging system. 
     Optionally, the set of calibration parameters is generated in dependence on the set of first parameters, the set of second parameters, and an axiom that each camera centre is positioned along the line being a straight axis. The axiom that the camera centres are exactly on a straight line gives a virtual camera setup exactly on the straight line, which has the advantage of allowing rectification and efficient depth estimations of imaged points when the real camera set up is on the straight line. Having all the cameras on a straight line is a requirement for joint rectification of more than two cameras. When all cameras are rectified jointly, depth estimations of imaged points become more efficient, as the disparities between points in the different cameras can be efficiently computed in a search. The search is performed along the x-axis and therefore utilizes computer cache memory and computational resources more efficiently. 
     The set of calibration parameters may be generated in dependence on the set of first parameters, the set of second parameters, and a minimization of an estimated distance of each camera centre from the straight axis. This has the advantage of finding calibration parameters that are optimized for the real camera set up, wherein the cameras are arranged with the camera centres almost along a straight line, i.e. wherein the cameras are arranged with the camera centres with small deviations from the straight axis. 
     Optionally, a spacing between adjacent pairs of the camera centres varies. This has the advantage of enabling precise depth measurements at both short distances and long distances since the two adjacent cameras arranged close to each other provide accurate depth measurements at short distances and the two adjacent cameras arranged at a larger spacing provide accurate depth measurements at larger distances. In some examples, depth measurements may be made using a non-adjacent pair of cameras. 
     Optionally, the set of calibration parameters comprises at least one of: a set of distances between the camera centres, a position of the imaging system, a rotation of the imaging system, a position of the calibration object corresponding to at least one of the calibration object images, and a rotation of the calibration object corresponding to at least one of the calibration object images. 
     The step of generating a set of first parameters may comprise determining a set of radial distortion parameters and/or tangential distortion parameters from the calibration object images. The step of generating the set of first parameters may further comprise determining a set of intrinsic parameters for cameras from the calibration object images. The step of generating the set of first parameters may further comprise determining a set of rectification parameters for cameras from the calibration object images. Thanks to the determined set of parameters artefacts in images taken by the cameras may be corrected. 
     By the term ‘distortion’ when used in this description is meant deviation from a rectilinear projection, i.e. a deviation from a projection in which straight lines in a scene remain straight in an image. It is a form of optical aberration. 
     By the term ‘intrinsic parameters’ when used in this description is meant parameters such as focal length, image sensor format, and principal point. 
     By the term ‘rectification’ when used in this description is meant rectification of an image. The rectification is a transformation process used to transform the images to share all horizontal axes, or, more technically, to have horizontal epipolar lines. It may be used in computer stereo vision to find matching points between images. 
     Optionally, the set of second parameters may be generated in dependence on the set of position point images and the set of first parameters. A set of initial positions estimates for the at least three cameras may be determined. A set of optimized positions estimates for the at least three cameras may then be determined. This provides the advantage of reducing the risk that joint optimizations (described below) get stuck in local minima or suffer from slow convergence. 
     The step of generating the set of calibration parameters may comprise determining a set of jointly optimized positions comprising a position and a rotation for each of the at least three cameras, wherein the jointly optimized positions are determined according to a regression analysis in dependence on the set of first parameters and the set of second parameters. This has the advantage of providing a solution that is as consistent as possible with all available calibration data, both near and far from the cameras. 
     Another aspect of the disclosure is an imaging system comprising: at least three cameras having a common field of view, wherein each of the at least three cameras having a camera centre, and wherein the camera centres are positioned along a line, the at least three cameras being configured to: image a calibration object to generate a set of calibration object images, wherein the calibration object is located nearby the at least three cameras and in the common field of view, and image a scene comprising a set of distant scene position points to generate a set of position point images, and a processing unit configured to generate a set of calibration parameters in dependence on the set of calibration object images and the set of position point images. This has the advantage of providing an improved calibration procedure that enables that depth measurement may be performed with increased precision in images acquired by the imaging system. 
     The imaging system may further comprise a housing wherein the at least three cameras are located in the housing. Alternatively, at least one of the at least three cameras may be configured to be located separately from the remaining cameras. 
     Another aspect of the disclosure is a method of installing an imaging system, the imaging system having at least three cameras having a common field of view, wherein each of the at least three cameras having a camera centre, and wherein the camera centres are positioned along a line, comprising: installing the imaging system at a fixed location, controlling the imaging system to capture one or more images of a calibration object, wherein the calibration object is located nearby the at least three cameras and in the common field of view, controlling the imaging system to capture one or more images of a scene comprising a set of distant scene position points, providing the imaging system with a set of position data corresponding to the set of scene position points, controlling the imaging system to generate a set of calibration parameters in dependence on the images of the calibration object, images of the scene, and the set of position data. This has the advantage of providing a faster and more convenient installation procedure with an improved calibration precision. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other features and advantages of the present invention will become apparent from the following detailed description of an example, with reference to the accompanying drawings, in which: 
         FIG.  1    is diagram of a trinocular imaging apparatus according to an aspect of the description, 
         FIG.  2    is a flowchart of the imaging processing method sequence according to one aspect of the description, 
         FIG.  3    is a flowchart of the imaging processing method according to an aspect of the description, 
         FIG.  4    is diagram of an imaging apparatus and a calibration object within a field of view according to an aspect of the description, 
         FIG.  5    is diagram of an imaging apparatus and a scene within a field of view of the imaging apparatus according to an aspect of the description, 
         FIG.  6    is diagram of an imaging apparatus and a scene and an object of interest within a field of view of the imaging apparatus according to an aspect of the description, 
         FIG.  7    is a flowchart of the imaging processing method sequence according to another aspect of the description, and 
         FIG.  8    is a flowchart of the installation and calibration method for an imaging system of an aspect of the description. 
     
    
    
     DESCRIPTION 
     The present description relates to apparatuses and techniques for providing improved object detection using camera arrays. Throughout the description, the same reference numerals are used to identify corresponding elements. 
       FIG.  1    is a diagram of an imaging system  10  according to an aspect of the description. The imaging system  10  comprises at least three cameras. In the example of  FIG.  1   , the three cameras are labelled  30   a ,  30   b , and  30   c . The examples provided in the present description predominantly focus on three camera configurations. However, the apparatus and methods disclosed herein may be scaled to imaging systems comprising four, five, six, or more cameras without any known limitation. 
     Each of the cameras  30   a - c  comprises a lens configuration with an ‘optical centre’ or camera centre  31 . Each of the cameras  30   a - c  is arranged so that they have a substantially common field of view  32 . i.e. so that an object or scene can be within the field of view of each camera  30   a - c  simultaneously. By the expression substantially common field of view is meant that the cameras  30   a - c  are arranged so that they have substantially overlapping fields of view. Furthermore, the cameras  30   a - c  are arranged such that the camera centre  31  of each camera  30   a - c  is positioned along a straight axis  35 . However, it should be understood that the camera centres  31 , due to e.g. small deviations when mounted as a camera array, may not be exactly positioning along the straight axis  35 , even if it assumed that they are positioned along the straight axis  35 . As will be described in more detail below, a calibration process is necessary to set up a trinocular camera array, that finds both the rectification and the orientation of the imaging system  10 , as well as internal orientation such as accurate distances between the individual cameras  30   a - c . Further and as described in this disclosure, the small deviations of the camera centres  31  from the straight axis  35  may be determined in the calibration process. To perform this calibration process, having three cameras  30   a - c  on a straight line is an advantage over a stereo camera, because it provides an additional constraint (as compared to the stereo camera, which is always on some straight line). 
     In one example, a spacing between adjacent pairs of the camera centres  31  varies, i.e. a first camera spacing  33  between a first camera  30   a  and second camera  30   b  is smaller or larger than a second camera spacing  34  between a second camera  30   b  and third camera  30   c . For example, the first camera spacing  33  may be in the range of 10 cm to 30 cm and the second camera spacing  34  may be in the range of 5 cm to 15 cm, just to give some examples. 
     In one example, all of the cameras  30   a - c  are co-located within a housing  20 . Locating all the cameras  30   a - c  in a single housing  20  enables a simplified installation, where only one housing  20  needs to be installed at a site. In another example, at least one of the cameras  30   a - c  is arranged to be located separately from the remaining cameras  30   a - c , i.e. at a different location while still maintaining a substantially common field of view  32  with the remaining cameras  30 . This configuration provides better geometric data, but the mounting and calibration process may be more time consuming and complex. 
     In an example, the imaging system  10  further comprises a processing unit  15  configured to receive imaging data generated by the cameras  30   a - c  and perform image analysis and parameter generation according to the following description. The processing unit  15  may be implemented by special-purpose software (or firmware) run on one or more general-purpose or special-purpose computing devices. In this context, it is to be understood that each “element” or “means” of such a computing device refers to a conceptual equivalent of a method step; there is not always a one-to-one correspondence between elements/means and particular pieces of hardware or software routines. One piece of hardware sometimes comprises different means/elements. For example, the processing unit  15  may serve as one element/means when executing one instruction but serve as another element/means when executing another instruction. In addition, one element/means may be implemented by one instruction in some cases, but by a plurality of instructions in some other cases. Naturally, it is conceivable that one or more elements (means) are implemented entirely by analogue hardware components. 
     The processing unit  15  may include one or more processing units, e.g. a CPU (“Central Processing Unit”), a DSP (“Digital Signal Processor), an ASIC (“Application-Specific Integrated Circuit), discrete analog and/or digital components, or some other programmable logical device, such as an FPGA (“Field Programmable Gate Array”). The device may further include a system memory and a system bus that couples various system components including the system memory to the processing unit. The system bus may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The system memory may include computer storage media in the form of volatile and/or non-volatile memory such as read only memory (ROM), random access memory (RAM) and flash memory. The special-purpose software, the exclusion data, the reference values, and any other data needed during execution, may be stored in the system memory, or on other removable/non-removable volatile/non-volatile computer storage media which is included in or accessible to the computing device, such as magnetic media, optical media, flash memory cards, digital tape, solid state RAM, solid state ROM, etc. The processing unit  15  may include one or more communication interfaces, such as a serial interface, a parallel interface, a USB interface, a wireless interface, a network adapter, etc., as well as one or more data acquisition devices, such as an ND converter. The special-purpose software may be provided to the processing unit  15  on any suitable computer-readable medium, including a record medium, and a read-only memory. 
       FIG.  2    is a flowchart of the calibration sequence according to one example of the description.  FIG.  3    provides a more detailed example of the calibration sequence including the order of data flow within the calibration sequence.  FIG.  4    is a diagram of imaging system  10  and a calibration object  60  within the common field of view  32  of the cameras  30 . 
     In step  210 , the cameras  30  of the imaging system  10  are imaging a calibration object  60  to generate a set of images (i.e. calibration object images  110 ) each showing the calibration object  60 . Imaging system  10  may be installed in a fixed position, e.g. on a post or building and directed towards a scene  71 . In one example, imaging system  10  is installed in an elevated position, between 5 and 10 metres above the ground. In this example, scene  71  may comprise a set of objects (e.g. a road and a sidewalk) arranged at ground level. In one example, the distance of a set of objects in a scene  71  (and corresponding scene position points  70 , cf.  FIG.  5   ) may be at least 2 metres from the imaging system  10  and are defined as being ‘distant’ from the imaging system  10  in the description below. Preferably, the objects of scene  71  are at least 10 metres from imaging system  10 . 
     The calibration object  60  may be a checkerboard or similar high contrast object with a known set of dimensions and a set of positions (e.g. corners) on the calibration object  60  that are easy to locate in a 2D image of the calibration object  60  via known automatic image processing. Positioning the calibration object  60  in front of the imaging system  10  for step  210  may be done in a temporary manner and may be as simple as holding it up by hand. Step  210  may be repeated a number of times while varying the position of the calibration object  60  with respect to the imaging system  10  in order to generate a larger set of calibration object images  110  useful for the processing steps described below. For example, the calibration object  60  may be placed in front of the imaging system  10  in several different rotations and positions. The calibration object  60  will be positioned closer to the imaging system  10  during calibration step  210  than at least some of the objects of scene  71 . The calibration object  60  may be positioned within 10 metres from imaging system  10  during step  210 , wherein it is described as being ‘nearby’ in the description below. Preferably, calibration object  60  is positioned within 2 metres from imaging system  10  during step  210 . 
     In step  220 , a set of first parameters  121 ,  122 ,  123  is generated by processing the set of calibration object images  110 . The processing unit  15  determines the set of first parameters  121 ,  122 ,  123 . 
     First, a set of distortion parameters are generated from the calibration object images  110  to determine a set of radial distortion parameters  121  and/or tangential distortion parameters  121  for cameras  30 . In one example, the following technique is used to determine a set of radial distortion parameters  121  and tangential distortion parameters  121  from calibration object images  110 . Where the calibration object  60  is a checkerboard, a corner detector is applied to the calibration object images  110  of the checkerboard in all three cameras  30 . Points belonging to what are supposed to be straight lines are identified using some standard method, for example using x-corner detection followed by checkerboard structure recovery. The parameters α 1 , α 2 , β 2 , x c  and y c  are optimized for the distortion model 
     
       
         
           
             
               
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     where (x u , y u ) are the undistorted coordinates, (x c , y c ) are the centre of distortion, (α 1 , α 2 ) are radial distortion parameters  121   a  and (β 1 , β 2 ) are tangential distortion parameters  121   b , while the original, distorted, pixel coordinates (x d , y d ) are given by x r =x d −x c  and y r =y d −y c , and r=√{square root over (x r   2 +y r   2 )}. The optimization tries to find parameters that straighten the lines via the Nelder-Mead algorithm. The loss for straightening the lines is a total least squares line fitting per line. Once the optimal parameters are found, finding (x d , y d ) given (x u , y u ) can be done by solving a depressed quartic as a starting point, and then optimizing with fixed point iterations. 
     Second, a set of intrinsic parameters  122  are generated from the calibration object images  110  for the cameras  30 . In one example, the following technique is used to determine a set of intrinsic parameters  122  from the calibration object images  110 . Using the method presented by Zhang the intrinsic parameters  122  are found individually for each camera  30 . The points on the checkerboard are converted to undistorted coordinates, using the above radial distortion parameters  121  and tangential distortion parameters  121 , prior to the computation. The non-rectified intrinsic camera matrices K i   0  (where i is the index for each camera) were parametrized as 
     
       
         
           
             
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     where f i  is the focal length, γ i  is the aspect ratio, S i  is the skew and (c i,x ,c i,y ) is the principal point. 
     Third, a set of rectification parameters  123  are generated from the calibration object images  110  for cameras  30 . In one example, the following technique is used to determine a set of rectification parameters  123  from calibration object images  110 . The distortion-free images are rectified as follows: 
     An optimization is performed to find rectification parameters  123  (or homographies H i ), which comprise the rotations of the cameras  30 , as well as focal length adjustment parameters α i , such that the y coordinates of all corresponding 2D points are substantially matching after applying the homographies H i , defined as
 
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     where R i  is a rotation matrix. The homographies H i  are then used to create a set of rectified images, where the distortion has been removed using the set of radial distortion parameters  121  and tangential distortion parameters  121  and the same intrinsic matrix K applies to the cameras in all images. The found intrinsic matrix has the form 
     
       
         
           
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     where the common α is 1. α may be optimized as described below. 
     The set of first parameters  121 ,  122 ,  123  may comprise at least one of the set of radial distortion parameters  121 , the tangential distortion parameters  121 , the set of intrinsic parameters  122 , and the set of rectification parameters  123 . 
     In step  230 , the cameras  30  of the imaging system  10  are imaging a set of distant scene position points  70  to generate a set of images (i.e. position point images  130 ) each depicting the scene  71 .  FIG.  5    is diagram of the imaging system  10  and the scene  71  within the common field of view  32  of the cameras  30 . As described above, the scene  71  may comprise a set of objects (e.g. a road and a sidewalk). Step  230  may be repeated a number of times in order to generate a larger set of position point images  130  useful for the processing steps described below. 
     As part of step  230 , position data  72  are provided to imaging system  10 . The position data comprise a set of scene position points  70  describing a 3D position of a point in the scene  71  (e.g. a position on one or more of the set of objects in scene  71 ) as well as a corresponding 2D position within each of the position point images  130 . This may be carried out using traditional techniques, such as surveying measurements carried out using electronic/optical instruments, e.g. total stations. 
     In step  240 , a set of second parameters  130 ,  141 ,  142  are generated by processing the set of position point images  130 . The processing unit  15  determines a set of second parameters  130 ,  141 ,  142 . 
     First, a set of initial extrinsic parameters is generated from the position point images  130  and from the set of first parameters  121 ,  122 ,  123  to determine a set of initial positions estimates  141  for each of the cameras  30  (i.e. initial estimates P i  and T j ). In one example, the following technique is used to determine the initial positions estimates  141 . Using the measured 3D position of scene position points  70  as well as the new intrinsics K, a rotation R and a translation t 2  for the camera  30   b  (i.e. the middle one) is found via Direct Linear Transformation (DLT) such that the camera equation
 
λ x=K [ Rt   2 ] X  
 
substantially holds for each 2D point x and the corresponding 3D point X in homogeneous coordinates, for some constant λ. Afterwards, the two outer cameras (e.g.  30   a  and  30   c ) are found by identifying the horizontal axis as the first row of R and moving in that direction by the approximately known distances to the outer cameras, by
 
 C   2   =−R   T   ×t   2  
 
 P   1   =K [ R,−R ×( C   2   +d   2,1   R   1 )]
 
 P   2   =K [ R,−R×C   2 ]
 
 P   3   =K [ R,−R ×( C   2   −d   2,3   R   1 )]
 
where d 2,1  and d 2,3  are the approximate distances between cameras  30   b  and  30   a , and  30   b  and  30   c , respectively. C 2  is the camera centre  31  of the camera  30   b  and R 1  is the first row of the rotation matrix R.
 
     Then, the 2D corner points of the checkerboards u are triangulated into 3D points U jk ′ using the new cameras P i =K[R t i ] (where the index i denotes which camera, j denotes which checkerboard orientation and k denotes which corner on the checkerboard). This triangulation is performed using DLT. The 3D points of the checkerboards in the coordinate system of the plane itself, U k , are known. In this plane, z=0 for all the corner points, while the x and y positions are given by the square grid of known distances. A 4×4 transform 
               T   j     =     [           R   j           t   j             0       1         ]           
is computed so that, approximately, U j,k ′=T j U k , for each plane position j.
 
     Second, a set of preliminarily fine-tuned extrinsic parameters is generated from the initial positions estimates  141  to determine a set of optimized positions estimates  142  for each of the cameras  30 . In one example, the following technique is used to determine the optimized positions estimates  142 . Using the initial positions estimates  141  (i.e. P i  found above) as an initial point, Levenberg-Marquardt optimization is performed using the measured 3D points V 1  and their corresponding 2D points v il , where the index l denotes which point. Assuming the end residuals will belong to a Gaussian distribution, the maximum likelihood estimate is computed as the sum of squares of residuals of the form
 
res x =( P   i   V   l ) x /( P   i   V   l ) z −( v   il ) x /( v   il ) z  
 
res y =( P   i   V   l ) y /( P   i   V   l ) z −( v   il ) y /( v   il ) z  
 
and the optimization is performed over the parameters α, R (in Euler angles) and t 2 .
 
     The set of second parameters  130 ,  141 ,  142  comprises at least one of the set of position point images  130 , the set of initial positions estimates  141 , and the set of optimized positions estimates  142 . 
     In step  250 , a set of calibration parameters  150 , comprising jointly optimized final extrinsic parameters, is generated in dependence on the set of first parameters  121 ,  122 ,  123  and the set of second parameters  130 ,  141 ,  142 . Further, the set of calibration parameters  150  may be generated in dependence on an axiom that each camera centre is positioned along the line being a straight axis  35 . The processing unit  15  determines a set of calibration parameters  150  for cameras  30 . 
     First, the processing unit  15  determines a set of jointly optimized positions  143  from the optimized position estimates  142 , i.e. from the second set of parameters  130 ,  141 ,  142 , and the first rectification parameters  123 , i.e. from the first set of parameters  121 ,  122 ,  123 . In one example, a regression analysis is used to determine the jointly optimized positions  143  in dependence on the optimized position estimates  142 , first rectification parameters  123 , and an axiom that each camera centre  31  of the cameras  30  is positioned along the line being a straight axis  35 . 
     An example of a suitable regression analysis now follows. The parametrization is updated to include parameters ε 2,1  and ε 2,3  to allow for fine adjustments of the distances between the cameras, as:
 
 C   2   =−R   T   ×t   2  
 
 P   1   =K [ R,−R ×( C   2 +( d   2,1 +ε 2,1 ) R   1 )]
 
 P   2   =K [ R,−R×C   2 ]
 
 P   3   =K [ R,−R ×( C   2 −( d   2,3 +ε 2,3 ) R   1 )]
 
     while at the same time adding the checkerboards into the same optimization. The following residuals are optimized over, as a sum of squares:
 
res 1x =( P   i   V   l ) x /( P   i   V   l ) z −( v   il ) x /( v   il ) z  
 
res 1y =( P   i   V   l ) y /( P   i   V   l ) z −( v   il ) y /( v   il ) z  
 
res 2x =( P   i   T   j   U   k ) x /( P   i   T   j   U   k ) z −( u   ijk ) x /( u   ijk ) z  
 
res 2y =( P   i   T   j   U   k ) y /( P   i   T   j   U   k ) z −( u   ijk ) y /( u   ijk ) z  
 
     The optimization is done over the parameters α, R (in Euler angles), t 2 , ε 2,1 , ε 2,3 , R j  (by Euler angles) and t j . e.g. a total of 69 parameters if there are 10 checkerboard orientations j. This final optimization jointly finds a solution where an axiom holds that the camera centre  31  of all three cameras  30   a ,  30   b , and  30   c  all lie on the straight axis  35 , and both the measured 3D points and the checkerboards are projected to the points where they appear in the de-warped and rectified images. Alternatively, the final optimization may jointly find a solution comprising a minimization of an estimated distance of the camera centre  31  of all three cameras  30   a ,  30   b , and  30   c  from the straight axis, and both the measured 3D points and the checkerboards are projected to the points where they appear in the de-warped and rectified images. 
     Second, the processing unit  15  determines a set of calibration parameters  150  in dependence on jointly optimized positions  143 . The set of calibration parameters  150  can comprise a set of distances between the camera centres  31 , a position of the imaging system  10 , a rotation of the imaging system  10 , a position of the calibration object  60  corresponding to at least one of the calibration object images  110 , and a rotation of the calibration object  60  corresponding to at least one of the calibration object images  110 . 
     The step of determining the jointly optimized positions  143  in dependence on the optimized position estimates  142 , first rectification parameters  123 , and an axiom that each camera centre  31  of the cameras  30  is positioned along the line being a straight axis  35  provides several advantages. Both the calibration object images  110  and the position point images  130  contain important and partially complementary information, so a calibration technique that uses both is particularly advantageous. Data from nearby points (i.e. data from calibration object images  110 ) are particularly useful for finding intrinsic parameters such as the focal length, principal point, and precise relative positions of the cameras, while data from points further away (i.e. data from position point images  130 ) are more useful for extrinsic parameters like the rotation and translation of the entire imaging system  10 . Adding an explicit parametrization that each camera centre  31  of the cameras  30  is positioned along the line being a straight axis  35 , advantageously prevents the method from finding solutions where the cameras are not on a straight line. 
       FIG.  6    is a diagram of the imaging apparatus  10  viewing scene  71  with an object of interest  75  within a field of view of the imaging apparatus  10 . Using the set of calibration parameters  150 , the imaging system  10  can precisely determine the 3D position of the object of interest  75  within scene  71 , i.e. to determine its position in the three Cartesian coordinates x, y, and z. 
     While  FIG.  2    shows a calibration sequence in which step  210  is performed before step  230 , it is understood that the calibration sequence may be performed in alternative orders. E.g. Step  230  may be performed before step  210 .  FIG.  7    shows a calibration sequence according to another aspect of the description where step  230  is performed at a time independent of step  210 . 
       FIG.  8    is a flowchart of an installation and calibration method for an imaging system  10  according to an example. 
     In step  310 , the imaging system  10  is installed at a fixed location in a position such that the cameras  30  of imaging system  10  share a common field of view  32  of a scene  71 . This step may comprise installation of imaging system  10  by fixing a housing  20  of imaging system  10  to a post, building, support frame, or stationary vehicle. This step may be carried out by hand or in an automated manner, e.g. using a robot. For example, the position may be an elevated position at which the imaging system  10  is configured to monitor a large area. This may be the case in traffic surveillance and/or city surveillance, just to give some examples. 
     In step  320 , the imaging system  10  is controlled by a user or an automated system to capture one or more images of a calibration object  60 , wherein the calibration object  60  is located nearby the at least three cameras  30  and in the common field of view  32 . Imaging system  10  may be configured to automatically capture one or more images of a calibration object  60  upon detecting that the calibration object  60  is within the common field of view  32 . The calibration object  60  may be positioned within the common field of view  32  by hand or in an automated manner, e.g. using a robot. 
     In step  330 , the imaging system  10  is controlled by a user or an automated system to capture one or more images of scene  71 . Imaging system  10  may be configured to capture one or more images of a scene  71  upon detecting that the calibration object  60  is no longer within the common field of view  32  or at any time that the calibration object  60  is not within the common field of view  32 . As described above, the scene  71  comprises a set of objects at a distance of typically at least 2 metres from imaging system  10 . Step  330  may occur after step  320 , before step  320 , or where the calibration object  60  does not obscure scene  71 , in parallel with step  320 . 
     In step  340 , the imaging system  10  is provided with a set of position data  72  corresponding to the set of scene position points  70  in the scene  71 . The scene position points  70  comprise 3D positions measured using an electronic/optical instrument used for surveying and building construction, such as a total station. The position data  72  may be uploaded to imaging system  10  via a direct interface, a wired or wireless network, or any other means to transmit electronic data. 
     In step  350 , imaging system  10  generates a set of calibration parameters  150  in dependence on the images of the calibration object  60 , images of the scene  71 , and the set of position data  72 . 
     As the calibration object  60  may be relatively small, e.g. small enough to be hand held, this method advantageously allows the smaller calibration object  60  to be used for close calibration. Where the camera is arranged up high, only the small calibration object  60  needs to be lifted up high to be brought close to the imaging system  10 . In the meantime, the more sensitive position data  72  can be taken at stable positions where a measurement device can be held still for several seconds, like on the ground or on a wall. 
     In some embodiments, the imaging system  10  is configured to be moveable in one or more directions, e.g. in a pan direction and/or a tilt direction. In such embodiments, the described calibration process may be performed for each one out of a plurality of pan and/or tilt positions of the imaging system  10 . The calibration parameters  150  obtained may be stored and associated with the pan and/or tilt position of the imaging system  10 . When the imaging system  10  is in operation, e.g. in a traffic or city surveillance operation, the calibration parameters  150  associated with the pan and/or tilt positions being the same as or most closely being the same as the operational pan and/or tilt positions of the imaging system  10  will be used in e.g. depth estimations. 
     It should be understood that one or more of the steps performed using the calibration object  60 , e.g. the checkerboard, may be performed in a laboratory or at a manufacturing site when assembling the imaging system  10 , while one or more of the steps using the distant set of objects of the scene  71  may be performed at the installation site, i.e. the operation site, of the imaging system  10 . Thus, the steps of the described calibration process do not have to be performed all at the same location.