Patent Publication Number: US-2021192778-A1

Title: Robust retroreflective photogrammetry markers

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present application is a nonprovisional application of, and claims priority to, U.S. Provisional Application Ser. No. 62/952,700 filed on Dec. 23, 2019, the contents of which are incorporated by reference herein. 
    
    
     FIELD OF THE INVENTION 
     The subject invention relates to photogrammetry markers and, in particular, to a retroreflective photogrammetry marker suitable for use in harsh environments and their method of use. 
     BACKGROUND 
     Photogrammetry is the art, science and technology of obtaining reliable information about physical objects and the environment through the process of recording, measuring and interpreting photographic images and patterns of electromagnetic radiant imagery and other phenomena. Accurate photogrammetry typically includes retroreflective markers placed within a scene that is to be imaged, thereby marking various locations or positions within the image. Retroreflective surfaces however can be imprecise and can be damaged by wear. Further, in some environments, such as a manufacturing environment where the retroreflective markers are used on robot grippers for example, abrasive cleaning agents such as dry ice blasting may be used on a frequent basis. The use of the abrasive cleaning agents can damage or remove the retroreflective markers. 
     Accordingly, while existing retroreflective markers are suitable for their intended purpose there is a need to provide a more robust retroreflective marker that is capable of operating in harsh environments. 
     SUMMARY OF THE INVENTION 
     In one exemplary embodiment of the invention, a method of determining an angular position of a retroreflective marker with respect to an imaging device is provided. The method includes: measuring a first angle at the imaging device of a light projected between the retroreflective marker and the imaging device along a deviated optical path, determining, at a processor, an angular deviation of the light from a straight-line path between the retroreflective marker and the imaging device, and determining, at the processor, a second angle at the imaging device, the second angle indicative of the straight-line path based on the first angle and the angular deviation. 
     In another exemplary embodiment of the invention, a retroreflective marker assembly is provided. The retroreflective marker assembly includes a retroreflective marker and a protective window, wherein light passes through the protective window to reflect from the retroreflective marker. 
     In yet another exemplary embodiment of the invention, a retroreflective marker assembly is provided. The retroreflective marker assembly includes a spherical base with a reflective coating and a mechanical element securing the reflective coating to the spherical base. 
     In yet another exemplary embodiment of the invention, a method of determining a parameter of object is provided. The method includes: placing a retroreflective marker assembly on a surface of the object, the retroreflective marker assembly comprising a support structure including one or more legs, each leg having a facet configured to polarize light upon reflection of the light from the facet, and a retroreflective marker located on a face of the support structure; obtaining a polarization angle of light reflected from a facet at an imaging device; determining an orientation of the retroreflective marker assembly with respect to the imaging device from the polarization of the light; and determining the parameter of the surface from the determined orientation of the retroreflective marker assembly. 
     In yet another exemplary embodiment of the invention, a retroreflective marker assembly is provided. The retroreflective marker assembly includes: a support structure including one or more legs, each leg having a facet configured to polarize light upon reflection of the light from the facet; and a retroreflective marker located on a face of the support structure. 
     The above features and advantages and other features and advantages of the invention are readily apparent from the following detailed description of the invention when taken in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other features, advantages and details appear, by way of example only, in the following detailed description of embodiments, the detailed description referring to the drawings in which: 
         FIG. 1A  shows a frontal view of an imaging device of an embodiment; 
         FIG. 1B  shows a top view of a first camera and a second camera of the imaging device of  FIG. 1A ; 
         FIG. 2  shows an imaging configuration in which the imaging device of  FIG. 1A  obtains information from an object; 
         FIG. 3  shows a marker assembly suitable for placing a retroreflective marker on a surface of an object for the purposes of photogrammetry; 
         FIG. 4  shows a diagram of the media of the marker assembly of  FIG. 3 ; 
         FIG. 5  shows the media of  FIG. 4 , illustrating an angular deviation of the position of retroreflective marker due to a presence of a protective window; 
         FIG. 6  shows a plot of angular deviation of light due to a presence of a protective window; 
         FIG. 7  shows a flowchart for determining a correction of a marker location based on a deviation of light occurring at the protective window; 
         FIG. 8  shows a marker assembly in an alternate embodiment. 
         FIG. 9  shows a cylindrical marker assembly in side view and plan view; 
         FIG. 10  shows a spherical marker assembly; 
         FIG. 11  shows a flowchart illustrating a method of manufacturing the spherical marker assembly of  FIG. 10 ; 
         FIG. 12  show a flowchart illustrating a method of photogrammetry to determine three-dimensional position and orientation of a marker assembly of  FIG. 3  using stereo images; 
         FIG. 13  shows a stereo image pair of a plurality of markers disposed within a scene; 
         FIG. 14  shows a processed image based on a left side image of  FIG. 13 ; 
         FIG. 15  shows a left side image and a right-side image showing three-dimensional coordinates calculated from two-dimensional positions determined in stereoscopic images; 
         FIG. 16  shows a stereoscopic imaging setup; 
         FIG. 17  shows a polarization enhanced photogrammetric marker assembly; 
         FIG. 18  shows the marker assembly of  FIG. 17  as viewed via several cameras; 
         FIG. 19  shows the polarization enhanced photogrammetric marker assembly; and, 
         FIG. 20  shows various alternative three-dimensional markers that can be used in various embodiments. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, its application or uses. It should be understood that throughout the drawings, corresponding reference numerals indicate like or corresponding parts and features. 
     Embodiments of the present invention provide advantages in use of robust retroreflective marker assemblies in photogrammetry. Methods are disclosed for correcting angular location of a retroreflective marker due to a presence of a protective window that deviates an optical path of light traveling from the retroreflective marker to an imaging device. In addition, novel structures for retroreflective marker assemblies are discussed as well as their use. In particular, a three-dimensional marker assembly is disclosed that employs facets which have an impact on the polarization of light reflected from them. The polarized light can be used to determine various parameters, such as orientation of the marker assembly with respect to an imaging device. 
       FIG. 1A  shows a frontal view of an imaging device  100  of the present invention in an embodiment, also referred to as a DMVS. In various embodiments, the imaging device  100  can be a triangulation scanner. The imaging device  100  includes a body  5 , a first camera  20  and a second camera  30 . The imaging device  100  can further include a projector  40  for projecting structured light patterns into a region. The first camera  20  and second camera  30  are separated by a baseline having a separation distance B 1 . A coordinate system  125  for the imaging device  100  is shown of illustrative purposes. The baseline is parallel to an x-axis of a coordinate system  125  of the imaging device  100 . 
       FIG. 1B  shows a top view of the first camera  20  and the second camera  30  of the imaging device  100  of  FIG. 1A . As shown in  FIG. 1B , a first-camera optical axis  22  of the first camera  20 , and a second-camera optical axis  32  of the second camera  30  lie on a common plane (i.e., plane x-z). The first-camera optical axis  22  and the second-camera optical axis  32  are generally unaligned with the z-axis of the imaging device  100 . In some embodiments, an optical axis  105  of the imaging device  100  passes through a center of symmetry of the imaging device  100 , for example. The first camera  20  includes a first-camera body  24  and a first-camera lens  26 . The first camera  20  further includes a photosensitive array and camera electronics (not shown). The second camera  30  includes a second-camera body  34  and a second-camera lens  36 . The second camera  30  further includes a photosensitive array, and camera electronics (not shown). 
     Referring back to  FIG. 1A , the imaging device  100  includes a control unit  50 . The control unit includes a processor  52  and a memory storage device  54  having programs  56  stored therein that when accessed by the processor  52  enable the processor to perform various operations for controlling operations of the imaging device  100  and its components as well as for image processing of various images captured at least one of the first camera  20  and the second camera  30 . Such image processing includes position and orientation determination for the various markers and marker assembly disclosed herein. The control unit  50  can further control operation of the projector  40  and can determine 3D coordinates of points projected onto an object by the projector  40 . The control unit  50  can be included inside the body  5  or may be external to the body  5 . In further embodiments, more than one processor can be used. In an embodiment, the imaging device  100  may determine the 3D coordinates in a similar manner to that described in commonly owned U.S. Patent Application 2017/0186183, which is incorporated by reference herein. 
     The imaging device  100  further includes a stereo illumination system that includes a first set  60  of light sources associated with the first camera  20  and a second set  70  of light sources associated with the second camera  30 . The light sources of the first set  60  and the light sources of the second set  70  can be light-emitting diodes in various embodiments. In various embodiments, the first set  60  of light sources is a single light source and the second set  70  of light sources is a single light source. In various embodiments, the first set  60  of light sources is a single ring of light, such as a ring of light-emitting diodes (LEDs), or LED right of light, and/or the second set  70  of light sources is a single LED ring of light, such as an LED ring of light. The first set  60  of light sources are located along a periphery or ring  62  that is concentric with a central or optical axis of the first camera  20 . In other words, each of the light sources in the first set  60  is located at the same position radially outward from the central axis of first camera  20 . Similarly, the second set  70  of light sources are located along a periphery or ring  72  concentric with a central axis of the second camera  30 . Each light source is oriented to project light rays along selected directions. In various embodiments, the orientation of the light rays are non-parallel to the z-axis of the coordinate system  125 . In one embodiment, the first set  60  of light sources includes four light sources (LED  1 , LED  2 , LED  3 , LED  4 ) and the second set  70  of light sources includes four light sources (LED  5 , LED  6 , LED  7 , LED  8 ). In alternate embodiments, each of the first set  60  and the second set  70  includes at least two light sources. 
     The light sources are coupled to the control unit  50 . The control unit  50  can control the times at which the light sources are turned on and off as well as the illumination levels of each light source. In various embodiments, the control unit operates the first set  60  of light sources separately or independently from the second set  70  of light sources, for example, to control stereo and /or mono lighting of a region. 
     As shown in  FIG. 1A , the first set  60  of light sources associated with the first camera  20  are labelled LED  1 , LED  2 , LED  3 , and LED  4 . LED  1  is located in a bottom right corner of the first camera  20  as viewed from the frontal view ( FIG. 1A ) of the imaging device  100 . Further in respect to the first camera  20 , LED  2  is located in a top right corner, LED  3  is located in a top left corner and LED  4  is located in a bottom left corner. 
     The light sources of the second camera  30  are labelled LED  5 , LED  6 , LED  7 , and LED  8 . LED  5  is located in a top right corner of the second camera  30  as viewed from the frontal view of the imaging device  100 . Further in respect to the second camera  30 , LED  6  is located in a top left corner, LED  7  is located in a bottom left corner and LED  8  is located in a bottom right corner. The particular method of numbering the light sources shown herein are for exemplary purposes and the claims should not be so limited. It is contemplated that in other embodiments, the positions of the light sources with respect to the cameras  20 ,  30  may be different. 
     In an embodiment, the arrangement of the first set  60  of light sources is a mirror image of the arrangement of the second set  70  of light sources about the optical axis  105  ( FIG. 1B ). The top LEDS (i.e., LED  2 , LED  3 , LED  5  and LED  6 ) are generally a same distance above the base line and the lower LEDS (i.e., LED  1 , LED  4 , LED  7  and LED  8 ) are generally as same distance below the baseline. In other words, the LED pairs are equidistant from the baseline. 
       FIG. 1B  shows the LEDs of the first camera  20  and second camera  30  from a top view. LED  2  and LED  3  are shown with respect to the first camera  20 , whereas LED  1  and LED  4  are behind LED  2  and LED  3 , respectively, and are therefore not visible in  FIG. 1B . Similarly, LED  5  and LED  6  are shown with respect to the second camera  30 , whereas LED  8  and LED  7  are behind LED  5  and LED  6 , respectively, and are therefore not visible in  FIG. 1B . 
     It should be appreciated that while embodiments herein describe the use of a retroreflective marker with the device  100 , this is for exemplary purposes and the claims should not be so limited. In other embodiments, the retroreflective marker that is described herein may be used with other photogrammetry systems, such as not limited to a single camera system, a stereoscopic camera system, or a system with a plurality of cameras. 
       FIG. 2  shows an imaging configuration  200  in which the imaging device  100  of  FIG. 1  obtains information from an object  202 . The object  202  includes a plurality of retroreflective marker  205  placed thereon. The imaging device  100  is located at a selected distance d from the object  202  and at an orientation with respect to the optical axis  105  of the imaging device  100 , as indicated by angle a between a light ray  210  and the optical axis  105 . 
     In accordance with an embodiment,  FIG. 3  shows a marker assembly  300  suitable for placing a retroreflective marker, such as the retroreflective marker  205  of  FIG. 2 , on a surface of the object  202  for the purposes of photogrammetry. The marker assembly  300  includes a base  302  having a top surface or viewing surface  308  on which a retroreflective marker  304  is disposed. In various embodiments, the retroreflective marker  304  is a circular film or material. The base  302  can have a rotationally symmetric mount that has a symmetry axis that is collinear or substantially collinear with the symmetry axis of the retroreflective marker  304 . A protective window  306  is placed over the retroreflective marker  304  in order to protect the retroreflective marker  304  from environmental elements, cleaning agents (e.g. dry ice blasting) and wear. The protective window  306  can be made of a suitable transparent material, such as glass. 
       FIG. 4  shows a schematic diagram of the media of the marker assembly  300  of  FIG. 3  that illustrates the effect the direction of light at the marker assembly  300 .  FIG. 4  illustrates a path taken by a beam of light as it passes through the protective window  306  of the marker assembly  300  of  FIG. 3 . The protective window  306  is shown on top of base  302  with the retroreflective marker  304  centrally located between of the protective window  306  and the base  302 . For purposes of illustration, the protective window  306  is made of glass having index of refraction n g  and having a thickness d. The region  406  above the protective window  306  is air, having index of refraction n=1. A selected light beam  402  projected from the imaging device at location A is incident on a glass-air interface  408  of the marker assembly  300  at a location B at orientation angle θ. Line  410  is normal to the glass-air interface  408  at point B. The glass-air interface  408  is parallel to the glass-base interface  412 . Therefore, the line  410  intersects the glass-base interface  412  at point E. 
     An undeviated extended path  420  has been illustrated in  FIG. 4  that continues the path of light beam  402  through the protective window  306  from point B to point C at the glass-base interface  412 . The undeviated extended path  420  travels horizontally by a distance k through the protective window  306  as it travels through the protective window  306 . Light beam  402  is however refracted at point B. The refracted light  404  at point B passes through the protective window  306  at a refractive angle θ g  and is incident on the retroreflective marker  304  at its center D. Therefore, the refracted light  404  travels a horizontal distance k-Δx as it passes through the protective window  306  to be incident at the center of the retroreflective marker  304  at point D. Since the camera sees the retroreflective marker  304  at point C, the presence of the protective window  306  in front of the retroreflective marker  304  produces a horizontal deviation Δx in the determined position of the center of the retroreflective marker  304 . This horizontal deviation Δx depends on the orientation angle θ of the retroreflective marker  304  with respect to a camera and a thickness d of the protective window  306 . 
     The horizontal deviation Δx can be determined using in terms of the orientation angle θ and various properties of the protective window  306  as discussed with respect to Eqs. (1)-(6). Triangle BDE, which includes the path of the refracted light  404 , yields the trigonometric relation of Eq. (1): 
     
       
         
           
             
               
                 
                   
                     
                       k 
                       - 
                       
                         Δ 
                          
                         x 
                       
                     
                     a 
                   
                   = 
                   
                     tan 
                      
                     
                       θ 
                       g 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
     Triangle BCE, which includes the undeviated extended path  420 , yields the trigonometric relation of Eq. (2): 
     
       
         
           
             
               
                 
                   
                     k 
                     d 
                   
                   = 
                   
                     tan 
                      
                     
                         
                     
                      
                     θ 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
           
         
       
     
     From Eq. (1) and Eq. (2), it follows that the horizontal deviation Δx is: 
       Δ x=d (tan θ−tan θ g )   Eq. (3)
 
     Snell&#39;s law states that: 
     
       
         
           
             
               
                 
                   
                     sin 
                      
                     
                       θ 
                       g 
                     
                   
                   = 
                   
                     
                       1 
                       
                         n 
                         g 
                       
                     
                      
                     sin 
                      
                     θ 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     Therefore, Eq, (3) can be rewritten using Eq. (4) and known trigonometric identities to obtain Eq. (5): 
     
       
         
           
             
               
                 
                   
                     Δ 
                      
                     x 
                   
                   = 
                   
                     
                       d 
                       ( 
                       
                         
                           tan 
                            
                           
                               
                           
                            
                           θ 
                         
                         - 
                         
                           
                             
                               1 
                               
                                 n 
                                 g 
                               
                             
                              
                             sin 
                              
                             θ 
                           
                           
                             
                               1 
                               - 
                               
                                 
                                   1 
                                   
                                     n 
                                     g 
                                     2 
                                   
                                 
                                  
                                 
                                   sin 
                                   2 
                                 
                                  
                                 θ 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       d 
                        
                       
                         ( 
                         
                           
                             tan 
                              
                             θ 
                           
                           - 
                           
                             
                               sin 
                                
                               θ 
                             
                             
                               
                                 
                                   n 
                                   g 
                                   2 
                                 
                                 - 
                                 
                                   
                                     sin 
                                     2 
                                   
                                    
                                   θ 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
     For small angles, Eq. (5) reduces to: 
     
       
         
           
             
               
                 
                   
                     Δ 
                      
                     x 
                   
                   ≈ 
                   
                     
                       d 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             1 
                             
                               n 
                               g 
                               2 
                             
                           
                         
                         ) 
                       
                     
                      
                     θ 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
     Thus, even for small angles, the horizontal deviation Δx is significant (e.g. ˜100 μm for a  1 mm glass plate with n g =1.5). The horizontal deviation Δx causes a deviation in a calculated angular location of the retroreflective marker, as discussed with respect to  FIG. 5 . 
       FIG. 5  shows the media of  FIG. 4 , illustrating an angular deviation of the position of retroreflective marker  304  that is due to the presence of the protective window  306  for the marker assembly  300  of  FIG. 3 . In various embodiments, a camera, such as the imaging device  100 , is used to obtain an image of the marker and determine the angular position of the retroreflective marker  304 . The imaging device  100  is oriented along its optical axis  105  and detects the retroreflective marker  304  along a projected light beam  504  that forms an angle α with optical axis  105 . The actual location of the retroreflective marker  304  with respect to the imaging device  100  is along straight-line path  506  between the imaging device  100  at point A and the center of the retroreflective marker  304  at point D. The straight-line path  506  has a length z and forms an angle ϕ with the optical axis  105 . The angular difference between the straight-line path  506  and the projected light beam  504  is therefore the deviation angle ϕ-α. 
     The deviation angle ϕ-α can be estimated based on the projected distance of the traveled by the projected ray and the horizontal deviation Δx: 
     
       
         
           
             
               
                 
                   
                     tan 
                      
                     
                       ( 
                       
                         φ 
                         - 
                         α 
                       
                       ) 
                     
                   
                   ≈ 
                   
                     
                       Δ 
                        
                       
                         x 
                         σ 
                       
                     
                     z 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
     where z is the absolute distance from the marker center D to the camera center A and 
       Δx σ =Δx sin σ  Eq. (8
 
     is the previously estimated horizontal deviation Δx projected onto the ray of observation and σ is the angle between the projected ray of the camera and the plane of the marker, hence we have the identity: 
     
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       π 
                       2 
                     
                     - 
                     θ 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
     with the approximation for Δx above, we get the following Eq. (10) for the orientation dependent shift in angle space: 
     
       
         
           
             
               
                 
                   
                     tan 
                      
                     
                       ( 
                       
                         φ 
                         - 
                         α 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       d 
                       z 
                     
                      
                     
                       ( 
                       
                         1 
                         - 
                         
                           1 
                           
                             n 
                             g 
                             2 
                           
                         
                       
                       ) 
                     
                      
                     θ 
                      
                     
                         
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     θ 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
       FIG. 6  shows a plot  600  of the right-hand side of Eq. (10) where the glass plate has a thickness d=1 mm and index of refraction n g= 1.5 and a distance between the camera and the marker is z=55 cm (i.e., a typical distance for the DMVS of  FIGS. 1A-B ). The curve shown in plot  600  can be used to correct for angular deviation of the image of the retroreflective marker due to the presence of the protective window  306 . From Eq. (10), it is clear the angular deviation between the projected light beam  504  and the straight-line path  506  can be determined used the thickness of the glass, refractive index of the glass and a perpendicular distance between the marker and the camera is known. The thickness and refractive index are generally known from specifications and the perpendicular distance can be determined. 
     The DMVS imaging device  100  generally has a pixel resolution of ˜1 mrad. The photogrammetric markers are typically located with an accuracy and precision significantly below 0.5 pixel. Hence, the deviation introduced by a glass plate, i.e., protective window  306 , can lead to a significant measurement error. 
       FIG. 7  shows a flowchart  700  for determining a correction of a marker location based on the deviation of light occurring at the protective window. Determining the correction includes knowing a geometrical relation between an imaging device and the marker location, such as a distance between the marker location and imaging device and a relative orientation between the marker and imaging device. This can be performed by observing the marker with from at least two locations. Either a single camera can be used to image the marker from two locations or a DMVS (or other suitable two-camera system) can be used to image the marker from a single position. In either scenario, the camera positions have a fixed or known geometric relation to each other. In box  702 , a geometric relation between the imaging device and the retroreflective marker assembly is obtained. In box  704 , a light beam is projected onto the retroreflective marker. In box  706 , a first angle α of light beam  504  at the imaging device (i.e., with respect to the optical axis of the imaging device) is determined. The projected light beam  504  travels a deviated optical path between the retroreflective marker and the imaging device. In box  708 , an angular deviation of the reflected light is determined (i.e., using Eq. (10)). In box  710 , a second angle (measured at the imaging device) indicative of a straight-line path  506  between the retroreflective marker and the imaging device angle is determined from the first angle and the angular deviation. 
       FIG. 8  shows a marker assembly  800  in accordance with an alternate embodiment. The alternate marker assembly  800  has the advantage that there is no deviation introduced to the position of the marker when it is observed under an angle. An aperture on the mask acts as the retroreflective marker. The marker assembly  800  includes a housing or base  802  including a recess  804 . In various embodiments, the base  802  is made of aluminum. A retroreflective film  806  is disposed on a bottom surface  808  of the recess  804  and a glass  810  material (or other transparent material) is secured in the recess  804  to sandwich the retroreflective film  806  between the bottom surface  808  and the glass  810 . The glass  810  can be secured in the recess by any suitable device including glue, double-sticky tape, etc. In an embodiment, the retroreflective film  806  includes a frame  812  that offsets the glass  810  from the bottom surface  806 . A mask  814  includes an aperture through which light can pass to travel through the glass  810  and reflect off of retroreflective film  806 . In one embodiment, the mask  814  on the glass  810  can be produced by masking a circular shape and coating the glass with a metal, such as by physical vapor deposition. In another embodiment, the mask  814  can be selectively deposited using a laser. In yet another embodiment, the metal can be deposited to coat the glass and an aperture can be selectively removed via a laser. 
       FIG. 9  shows a cylindrical retroreflective marker assembly  900  in side view  912  and plan view  914  in accordance with an embodiment. The cylindrical marker assembly  900  includes a cylindrical base  902  which can be made of aluminum, steel, or other suitable material. A reflective cylindrical film  904  made of reflective material surrounds the cylindrical base  902  and a glass cylindrical shell  906  surrounds the reflective cylindrical film  904 . Caps  908  are placed at opposite ends of the cylindrical marker assembly  900  and are secured to the cylindrical base  902  in order to encapsulate the reflective cylindrical film between the cylindrical base  902  and the glass cylindrical shell  906 . A sealant  910  can be used to seal any gaps between the glass cylindrical shell  906  and the caps  908 . 
       FIG. 10  shows a spherical marker assembly  1000 . The spherical marker assembly  1000  includes a spherical base  1002  with a reflective coating  1004  surrounding formed on an outer surface of the spherical base  1002 . An adhesive ring  1006  secures the reflective coating  1004  and the spherical base  1002  to a mounting pin  1008 . The number of possible angles at which the spherical marker assembly  1000  can be viewed is directly related to a ratio between the diameter of the sphere and the diameter the mounting pin  1008 . 
       FIG. 11  shows a flowchart  1100  illustrating a method of manufacturing the spherical marker assembly  1000  of  FIG. 10 . In box  1102 , a surface of the spherical base  1002  is pre-treated, for example, by sand blasting. In box  1104 , the surface of the pre-treated spherical base  1002  is coated with a powder. In box  1106 , the coating is heated to a liquidation temperature of the coating. In box  1108 , the sphere is coated with glass microspheres. In one embodiment, the glass microspheres are half-covered with a metal to improve reflectance. In another embodiment, the glass microspheres are half-covered with a chemical coating on one side to help orient the microspheres so that the reflective side is oriented toward the powder-coating. In box  1110 , the coating is further heated in order to complete a hardening process. In box  1112 , the coating is cooled. In boxes  1114 , a ring is attached to cover the edge of the coating e.g. to prevent corrosive effects on the edge. 
       FIG. 12  show a flowchart  1200  illustrating a method of photogrammetry to determine three-dimensional position and orientation of a marker assembly  300  of  FIG. 3  using stereo images such as by using the DMVS of  FIG. 1 . In box  1202 , stereo images are obtained. In box  1204 , two-dimensional positions of the marker are determined in each of the stereo images. This can include angular corrections based on Eq. (10) when a distance between the camera is known. In box  1206 , three-dimensional coordinates of the marker are determined using the two-dimensional positions of the marker within the images. In box  1208 , a direction or orientation of the marker is determined. The direction or orientation of the marker can be determined by measuring ellipses in the images, given that the retroreflective markers are circular in nature. In box  1210 , the three-dimensional coordinates of the marker are recalculated using the orientation direction of the marker determined in box  1208 . For a marker such as the marker of  FIG. 3 , this recalculation can be performed using the methods described herein with respect to Eqs. (1)-(10). (Recalcuation is not performed for the spherical marker of  FIG. 10 .) Recalculation uses the marker orientation, the distance between the camera and the marker and the observation angle in order to determine a deviation angle. Box  1212  is a decision box comparing a change in a computation of 3D coordinates in the previous iteration to a current computation of the 3D coordinates. If the change in the values of the 3D coordinates due to a single iteration is less than a selected threshold, then the calculations can come to an end at box  1214 . Otherwise, the computation process is iterated by returning to box  1208  with the updated vales of angle, distance, etc. 
     In one embodiment, a single iteration through the loop of boxes  1208  through  1212  is enough to provide accurate location and orientation values. For applications desiring higher levels of accuracy, a second calculation loop may be used. Alternatively, the calculation may iteratively update the 3D coordinates until the value of the change for the markers falls below the selected threshold. 
       FIG. 13  shows a stereo image pair  1300  of a plurality of markers disposed within a scene. The stereo image pair  1300  includes a left side image  1302  and a right-side image  1304 . The laser pattern  1306  projected from the DMVS is visible, as are several retroreflective markers  1308  and  1310 . The laser pattern  1306  can be ignored when processing the retroreflective markers  1308 ,  1310 . 
       FIG. 14  shows a processed image based on the left side image  1302  of  FIG. 13 . The markers  1402 ,  1404  are identified by circles or highlights indicating that the processor recognized them as markers. The two-dimensional position of each marker  1402 ,  1404  can then be extracted. For a single marker, the image processing step can extract a two-dimensional position of the center of mass of the marker, a size of the marker (based on a number of illuminated pixels that marker occupies in the image plane), and an ellipticity of the image of the marker (based on a ratio of long axis vs. short axis of the image of the marker). In addition, the membership of a marker within a group of markers can be determined from a spatially coded dot assembly. 
     The actual two-dimensional position of the marker can be extracted taking into account compensation for the angular deviation introduced by the glass plate, using, for example, Eq. (10). Using Eq. (10) assumes knowledge of the thickness and refractive index of the glass plate, the distance between the imaging device and the marker, and the orientation of the marker with respect to the imaging device. The thickness and index of refraction of glass plate are typically known from vendor specifications, but can be also determined using other methods. 
     The distance between the imaging device and the marker can be determined by triangulating a three-dimensional position of the marker. The first estimation of the three-dimensional position is based on the projected light beam  504  of  FIG. 5  and therefore provides an incorrect angle. Therefore, the original three-dimensional position will be off from its correct value based on this incorrect angle. Nonetheless, the difference/error is small compared to the overall distance z. Using an iterative process, the estimation of the three-dimensional distance can be improved in a stepwise fashion to within a selected criterion or resolution. 
       FIG. 15  shows a left side image  1502  and a right-side image  1504  showing three-dimensional coordinates calculated from the 2D positions determined in box  1204 . Two images are sufficient to determine the 3D coordinates. However, more than two images in various embodiments. For a multi-camera imaging device such as the DMVS ( FIG. 1 ), the relative camera positions are known and constant. This knowledge helps in estimating the 3D coordinates but is not necessary. Without this information, a specific scale is used to estimate the 3D coordinates. Additionally, with pre-known relative camera positions there is no need for a minimum number of markers or a specific marker alignment (i.e. at least 4 markers in non-collinear setup which ideally do not form a single plane are generally used). 
     The left side image  1502  and right-side image  1504  show two renderings of the 3D points. Each rendering is calculated from 2D position by triangulation. For a first iteration, the 3D coordinates can be based on 2D positions that have not had any corrections applied. Therefore, subsequent iterations can be used to correct the values of the 3D coordinates. 
       FIG. 16  shows a stereoscopic imaging setup. The imaging setup illustrates a process for determining an orientation of a marker M with respect to the cameras of the DMVS. The orientation of the marker M with respect to a selected camera (e.g., first camera P or second camera Q) can be determined based on the effects of angular orientation of the marker with respect to the imaging device on the marker&#39;s image. In various embodiments, the marker M is a circular marker. When observed at an angle, the circular marker M appears as an ellipse. A ratio of a short axis of the ellipse to a long axis of the ellipse, as well as orientation of these axes, can be used to calculate the basic orientation and an absolute value of the angle for a single marker M with respect to the selected camera. The calculations include the parameters of distance z and the angle θ. Given that the 3D coordinates are known, the distance z between the marker and the camera can be determined, because both the position of the camera and the position of the marker are known in a common coordinate system. 
     For a single image of a marker M, a ratio between the lengths of the long axis and short axis of a measured marker can be used to calculate the absolute value of θ, via the following equation: 
     
       
         
           
             
               
                 
                   
                     cos 
                      
                     θ 
                   
                   = 
                   
                     
                       〈 
                       
                         short 
                          
                         
                             
                         
                          
                         axis 
                       
                       〉 
                     
                     
                       〈 
                       
                         long 
                          
                         
                             
                         
                          
                         axis 
                       
                       〉 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
     where &lt;short axis&gt; is a length of the short axis of the ellipse and &lt;long axis&gt; is a length of the long axis of the ellipse. The angle θ is shown for each camera in  FIG. 12   
     The approximation of Eq. (11) is valid for small angular extensions of the marker and is valid for typical measurements, in which a marker is seen across ˜30×30 pixel (or less) in an image recorded with a normal lens system (i.e. no fisheye; less than ˜100 FoV). The direction of orientation can be determined along the short axis as well as an absolute value of θ. The sign of θ is however not determined. In order to obtain the sign, a second observation is taken from the second camera. 
     Still referring to  FIG. 16 , a normal vector {right arrow over (n )} to the marker M is shown. Observation vector v 1  points along a direction between the marker M and a first viewing point or first camera P. Observation vector v 2  points along a direction between the marker M and a second viewing point or second camera Q. The image of the circular marker M is shown to form an ellipse  1606  in the imaging plane  1602  as well as an ellipse  1608  in the imaging plane  1604 . 
     From the estimated value of θ, the dot product of the first viewing ray with the normal vector is given by: 
       {right arrow over ( v   1 )} ·{right arrow over (n)} =cos θ 1    Eq. (12)
 
     while the dot product of the second viewing ray with the normal vector is given by 
       {right arrow over ( v   2 )} ·{right arrow over (n)} =cos θ 2    Eq. (13)
 
     where all vectors v 1 , v 2  and n are normalized or unit vectors. 
     The direction or orientation of the marker is tilted with respect to each of the first camera P and the second camera Q. A direction perpendicular to the normal vector can be determined from the long axis of the measured ellipse and the observation vector. The long axis vector in the image plane is given by: 
     
       
         
           
             
               
                 
                   
                     a 
                     → 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             x 
                             L 
                           
                         
                       
                       
                         
                           
                             y 
                             L 
                           
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
     The long axis vector can be projected into the plane perpendicular to the viewing direction (e.g., imaging planes  1602 ,  1604 ) via the following equation: 
     
       
         
           
             
               
                 
                   
                     p 
                     → 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               v 
                               1 
                             
                             → 
                           
                           · 
                           
                               
                           
                            
                           
                             a 
                             → 
                           
                         
                         ) 
                       
                        
                       
                         
                           
                             v 
                             1 
                           
                           → 
                         
                         
                           
                              
                             
                               
                                 v 
                                 1 
                               
                               → 
                             
                              
                           
                           2 
                         
                       
                     
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                         a 
                         → 
                       
                       . 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
     The resulting dot product between the resulting vector {right arrow over (p)} and the normal vector n must be zero, i.e., 
         {right arrow over (p)}·{right arrow over (n)}= 0.   Eq. (16)
 
     The viewing directions {right arrow over (v 1 )} and {right arrow over (v 2 )} are known from the estimated or calibrated camera positions. The viewing directions are the direction vectors from their camera nodal point to the 3D position of the marker. Equations (14), (15) and (16) can therefore be determined for each camera. As a result, the system of equations for determining the normal vector of the marker is overdetermined. Therefore, for a DMVS system, the 3D information for the marker M can be determined with a single exposure. The normal vector can be determined with a least squares calculation. Once the normal vector is determined, the magnitude and sign of the angle θ can be determined for each camera of the DMVS system. 
     Other methods can be used in addition to the above described method for determining the orientation of a marker. In one embodiment, three or more markers can be placed on a common plane. From the distribution of the markers, it can be seen which markers belong to which assembly (e.g. all markers within a certain distance of each other will be grouped together. When the arrangement of all markers on each plane is known, the 6 degrees of freedom information of the plane can be calculated from a single image (including the distance from camera to marker). In another embodiment, the angle θ can be determined from multi-camera observations. Each observation confines the solution for the normal vector of the marker to two solutions. The estimate of angle θ is shared by at least two observations. 
       FIG. 17  shows a polarization enhanced photogrammetric marker assembly  1700 . The enhanced marker assembly  1700  has a three-dimensional structure, with a marker  1702  located at a central disk or central portion of a three-dimensional support  1710  having a top surface and a back surface. The central portion of the three-dimensional support  1710  is also referred to herein as a viewing face. The marker  1702  is located on the top surface. The material of the support structure can be a black anodized aluminum in an embodiment. The three-dimensional support  1710  includes a plurality of legs  1704   a,    1704   b,    1704   c,    1704   d,    1704   e,    1704   f,    1704   g,    1704   h  extending radially outward from a center D of the marker assembly  1700  by an equal distance and generally forming a rotationally symmetric structure. Each leg is bent or articulated at a selected radial distance from the center D by a selected angle to form a bent support section. Each support section includes a planar facet defining a normal vector (n1, . . . , n8). In the illustrative embodiment of  FIG. 17 , the enhanced photogrammetric marker assembly  1700  has eight legs and support sections. However, the enhanced photogrammetric marker assembly  1700  can have any number of legs and support sections in alternate embodiments. 
     Each facet of the support structure includes material for polarizing light upon its reflection from the facet. A polarization camera can be used to record the polarization enhanced photogrammetric marker assembly  1700 . Each pixel of a polarization camera records not only an intensity distribution of light but also a polarization state of the light. 
     Once the polarization state is recorded for each facet of the legs  1704   a - h,  the polarization state of each facet can be used to calculate the orientation and/or rotation of the marker assembly  1700  with respect to the camera. The additional information provided by the polarization states means that fewer markers per image are needed and less precision is need when setting up the markers in the scenery. 
       FIG. 18  shows the marker assembly  1700  of  FIG. 17  as viewed via several cameras. Image  1802  shows a image of the marker assembly  1700  from a standard camera. The illumination of the scenery does not allow for a meaningful interpretation of the shape of the object. Polarization image  1804  shows the marker assembly  1700  displaying an angle of polarization of the facets of the marker assembly  1700 . The polarization angle changes with the angular location of the facet along the marker assembly  1700 . Polarization image  1806  shows a degree of polarization of each facet, which is significantly different from zero. Using the polarization images ( 1804 ,  1806 ), the geometric property of the marker assembly can be determined. The measured state of polarization of a facet depends not only on the geometry of the marker assembly  1700  but also on the relative position of the camera with respect to the marker assembly  1700 . Thus, the polarization information can be used to obtain the relative rotation between the marker assembly  1700  and the camera. 
     A priori knowledge of the center D of the marker and of normal vectors of the facets is sufficient to constrain the position of the camera along a single line. The position of the camera along this line (i.e., the distance between the camera and the marker) can be estimated from the marker size as well as the sizes of other markers.\ 
     The normal vectors of each facet within the coordinate system of the marker assembly  1700  can be determined during a previous calibration step. Each facet can be described by a normal vector {right arrow over (n k )}, where k is an index of the facet. When the marker assembly  1700  is rotated by a rotation matrix  R  with respect to the camera, the normal vector is seen at the camera as: 
     
       
         
           
             
               
                 
                   
                     
                       
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                         k 
                       
                       → 
                     
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                   . 
                   
                       
                   
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                     ( 
                     12 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
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                          
                         
                             
                         
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                          
                         
                             
                         
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                         muth 
                       
                     
                     _ 
                   
                   = 
                   
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                             cos 
                              
                             
                                 
                             
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                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
     is the rotation matrix about the z-axis describing the azimuth angle which is measured by the camera along the viewing direction (which is set to be along the z-axis). Also 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       
                         az 
                          
                         
                             
                         
                          
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                          
                         
                             
                         
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                         muth 
                       
                     
                     _ 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
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                              
                             
                                 
                             
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                              
                             
                                 
                             
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                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
     is the rotation matrix about the y-axis describing the azimuth angle which is measured by the camera along the viewing direction (which is set to be along the z-axis). Thus, Eq. (12) can be rewritten as: 
     
       
         
           
             
               
                 
                   
                     
                       n 
                       k 
                     
                     → 
                   
                   = 
                   
                     
                       
                         R 
                         ¯ 
                       
                       
                         - 
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                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
     Since every Eqs. (13), (14) and (15) can be applied to each facet of the marker assembly  1700 , the resulting system of equations is overdetermined and can be used to solve for the rotation matrix R as well as the zenith angle of each facet  R azimuth,k   . 
       FIG. 19  shows the polarization enhanced photogrammetric marker assembly  1700  of  FIG. 17  with vectors illustrating how to resolve an ambiguity in the azimuth angle of a facet. It is noted that the azimuth angle of a facet determined using the Eqs. (13), (14) and (15) has an ambiguity of 180 degrees. In other words, the normal of the facet can be either pointing out of the facet or into the facet. This ambiguity can be resolved via the following method. 
     The two-dimensional center D of the marker assembly  1700  can be determined by locating the central disk of the marker is via known locations of the retroreflective markers on the marker assembly  1700 . A radial vector  1902  can then be drawn from the center of the marker to the center of facet  1904 . The facet  1904  includes a first normal vector  1906   a  that is pointing out of the facet  1904  and a second normal vector  1906   b  that is pointing into the facet  1904 . The correct normal vector is the normal vector for which the dot product between the normal vector and the radial vector  1902  is a positive value. 
       FIG. 20  shows various alternative three-dimensional markers that can be used in various embodiments. The alternative markers  2000  include polyhedrons with a first face  2002  that can include the various codes retroreflective surfaces disclosed herein and second faces  2004 ,  2006 ,  2008 ,  2020  that serve as the facets of the alternative markers  2000  that polarize light upon reflection. 
     As used herein, the term “module” or “unit” refers to an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), an electronic circuit, an electronic computer processor (shared, dedicated, or group) and memory that executes one or more software or firmware programs, a hardware microcontroller, a combinational logic circuit, and/or other suitable components that provide the described functionality. When implemented in software, a module can be embodied in memory as a non-transitory machine-readable storage medium readable by a processing circuit and storing instructions for execution by the processing circuit for performing a method. 
     While the invention has been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiments disclosed, but that the invention will include all embodiments falling within the scope of the application.