Abstract:
A method and system for a light source detection system, comprising an aircraft carrying at least one camera. The system includes a database for storing information about the aircraft&#39;s motion, direction, and position and ground location information, such as the onboard navigation system database or an remotely accessible database. The system also includes a processor that accesses the database and is connected to the camera. The processor uses image analysis and processing techniques to determine the ground location corresponding to the light source from an image of that light captured by the camera. It determines the path traveled by that light and estimates its location as being a pre-selected distance vertically above the ground along the path traveled by the light to the aircraft when the image of the light was captured.

Description:
TECHNOLOGICAL FIELD 
       [0001]    The technology of the present disclosure relates generally to identifying and locating the position of objects in three dimensions. More specifically, the technological field relates to determining the location of a source of bright light from a moving aircraft. 
       BACKGROUND 
       [0002]    A growing problem for aircraft pilots is the use of lasers by those on the ground who direct laser light toward the aircraft. Laser light can blind or dazzle a pilot. More powerful lasers can damage the aircraft. Similarly, there are those who use firearms to shoot at aircraft. Whether out of mischief or the result of action by a hostile force, those who direct bright light sources at or who shoot at aircraft must be located. 
         [0003]    Military aircraft have equipment that targets and returns fire when they are engaged in combat. Civilian aircraft are not so equipped. A convenient onboard capability to detect the sources of laser light or shots from firearms directed against an aircraft would be an advantage in apprehending those responsible. 
       SUMMARY 
       [0004]    Herein is disclosed a bright light source detection and location system, comprising an aircraft carrying at least one camera and a processor with the capability to derive the location of the source from the images captured by the camera. A database for storing information about the aircraft&#39;s motion, direction, and position and ground location information, such as the aircraft&#39;s onboard computer is also used. The processor is configured to access the database and receive images captured by the camera and to determine a ground location corresponding to the location of a source of bright light detected in the captured images. 
         [0005]    An aspect of the disclosure is that the processor is configured to perform a spectral analysis in order to identify light that comes from a laser as opposed to light from a different type of source. The processor can also perform an analysis to identify a muzzle flash. The spectrum of a muzzle flash and its characteristics, such as being of short duration and having a pattern based on the type of firearm being used, can provide information confirming the source of the flash is from a firearm. 
         [0006]    Another feature of the disclosure is that the camera or the processor may have the capability to filter the light from the source. The camera may have a polarizing filter or a wavelength filter that removes part of the electromagnetic spectrum to eliminate portions of the image that are irrelevant or that obscure relevant portions. The processor may look for narrow spikes in the intensity of frequencies associated with lasers such as in the red to infrared region of the electromagnetic spectrum or may look for the spectral signature of gunpowder combustion products. 
         [0007]    Another aspect of the disclosure is that the processor may be located at an airport instead of being on board the aircraft. Since many incidents involving lasers and firearms take place when aircraft are landing or taking off, a processor located at the airport and which communicates with cameras on inbound and outbound aircraft would be able to provide the same information as an on-board processor without adding the weight of an extra processor to the aircraft. Ground-based processors stationed at an airport may also have more accurate topographic maps of the airport&#39;s surroundings that would improve accuracy in locating the source of the bright light. 
         [0008]    Yet another aspect of the disclosure is a communications link from the aircraft to local authorities who can receive the location of the source of laser light or the muzzle flash for them to begin their investigation much sooner after the image of the source was captured. 
         [0009]    Still another aspect of the disclosure is there may be at least two cameras taking images from different locations on the aircraft or taking images in different ways to increase the likelihood and accuracy of locating the source of light. For example, one camera may be a high resolution camera better able to detect a laser&#39;s light and another camera may be a high speed camera for recording images of muzzle flashes. There may also be an array of cameras. 
         [0010]    Still another aspect of the disclosure is that a multiplicity of cameras may be spread across more than one aircraft or platform, yet these data may be combined into a coherent information set just as if acquired by a single aircraft. 
         [0011]    An aspect of the disclosure is that the processor is configured to estimate a ground location of a source of a light when one camera records an image of a bright light that it determines is harmful and communicates the ground position of the source to authorities. 
         [0012]    Another aspect of the disclosure is the use of topographic information to more accurately determine ground location of the person who has the source of light. 
         [0013]    Still another aspect of the disclosure is a method of detecting the location of a light source directed at a moving aircraft, including the steps of capturing at least one image of light directed at the aircraft from a light source; determining the speed, direction and orientation of the aircraft at the time the image was captured, determining the direction running from the moving aircraft to the light source when the image was captured; and determining the ground location that corresponds to a preselected vertical distance between the line to the source and the ground. 
         [0014]    Yet another aspect of the disclosure is the processing of images captured by different cameras carried on an aircraft to determine if laser light or muzzle flashes are present in the images. 
         [0015]    These and other aspects of the disclosure will be apparent to those skilled in the art from a careful reading of the detailed description, accompanied by the following drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0016]    Having thus described variations of the disclosure in general terms, reference will now be made to the accompanying drawings, which are not necessarily drawn to scale, and wherein: 
           [0017]      FIG. 1  illustrates an example of the type of problem the present method and apparatus solves; 
           [0018]      FIG. 2  is an aircraft carrying three cameras, according to an aspect of the disclosure; 
           [0019]      FIG. 3  is a schematic diagram of an apparatus for detecting bright lights from a moving aircraft, according to an aspect of the disclosure. 
           [0020]      FIG. 4  is a schematic diagram of an alternative apparatus for detecting bright lights from a moving aircraft, according to an aspect of the disclosure; and 
           [0021]      FIGS. 5A, 5B, and 5C  are views of three different laser images captured by a camera, according to aspects of the disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0022]    Aspects of the present disclosure are directed to the detection and location of the source of bright light by moving aircraft. 
         [0023]    Laser light and muzzle flashes are both bright lights. The former is coherent and may subject to little scatter in the atmosphere; the latter are bright and brief, but while the flash itself does little harm at a distance, the bullet that is associated with the muzzle flash may cause harm. The images of both laser and muzzle flash, however, can be captured by a camera. 
         [0024]    The term camera is used herein generally to mean a device that is capable of capturing a two-dimensional image, and in particular a device capable of capturing a two-dimensional array of electromagnetic intensity data mapped to a view. The present camera may forward the image as a digital signal so that, for example, an analysis can be performed on the image using a suitably programmed special or general purpose computer. The analysis may include review of the image for any bright or intense lights captured. It may include a spectrum analysis to identify lights that, while they may not be the brightest in the image, are nonetheless red laser lights or have the characteristic spectrum of the combustion gases of gunpowder. The analysis may be a negative analysis, ruling out, for example, the brightest portion of sunlight by filtering the highest intensity light received from the sun, which peaks near 5500 angstroms so that other frequencies are more easily discernable. 
         [0025]    Analysis may have a time-dimension to it such as when there are several images of the same location received by the same high-speed camera that may show that a bright light source appeared suddenly at that location and then disappeared suddenly, which may indicate a muzzle flash from a firearm. Depending on the resolution of the camera, a flash of light may have a pattern that may not only serve to confirm that the light is a muzzle flash but also help to determine the type of firearm that was fired. 
         [0026]    The advantage of spectral analysis and filtration of the image is that, by ruling out irrelevant light information, the remaining light information can be more quickly identified for its relevance and source location analysis. 
         [0027]    The term source in this application refers to the device that generated the light, such as small arms or lasers. The location of the source is most likely located directly below the source such as the location where a person is standing who is holding a laser or small arm, which may be about five feet below the laser. That ground location can be determined by identifying the line to the source given the location and orientation of the camera at the instant the image was captured, and then looking for places along that line that are five feet above the ground. Without an accurate topographic map, the ground is assumed to be level, which assumption may be wrong. For example, if the person holding the source is standing on a ridge line, the source may be appear to be coming from a location far behind the ridgeline. 
         [0028]    In order to determine the location of a source, a line is generated from the camera where it was pointing when the image was captured to the location of the light in the captured image. The direction to the light can be determined from the location of the camera that is fixed to a specific part of an aircraft and pointed in a known direction, plus information about the speed, direction, and orientation of the aircraft, and map information regarding the land over which the aircraft was flying when the image was captured. Much of this information is routinely available from the onboard navigation system of an aircraft. The information about the camera, that is, where it is located with respect to the aircraft, can be obtained fairly precisely when the camera is installed. If the camera can be directed, the pan-tilt directions to the camera&#39;s servo motors can be ascertained. Installing more than one camera, indeed, installing an array of cameras, will reduce the uncertainty of the location analysis. Obtaining a series of images from the same camera can also reduce uncertainty as an additional image and an additional camera can contribute to the probability that the source location has been accurately determined. 
         [0029]    However, a single camera can estimate the location to the source. The quality of the location estimation is based in part on the nature and quality of the ground information. Accurate topographic maps provide better information than a map that assumes level ground. Additional sources of error are in knowing the speed of the aircraft with respect to ground, the direction, the altitude, and the orientation of the aircraft (roll, pitch, yaw). 
         [0030]    The weather is also a factor. For example, if the ground air is warm and the air near the aircraft is colder, the change in air temperature with elevation will act as a lens and bend the path of the light rays. This effect is greater if the aircraft is farther from the source or at a higher elevation. 
         [0031]    Increasing the number of cameras and having more than one image from each camera can reduce the uncertainty but also complicate processing and increase processing time. There is a tradeoff in accuracy and the number of cameras deployed. However, an array of cameras of very simple construction can be used, with, for example, a small aperture and no focusing lens, and an array of solid state devices for capturing the light. These cameras may include either of two types: semiconductor charge coupled devices (CCD) or N-type complementary metal oxide semiconductors (CMOS). CCD cameras are inexpensive but CMOS cameras are more effective in burst mode and more robust against bright lights. Further technological developments may provide other devices for use in capturing images inexpensively, accurately, effectively, and rapidly. 
         [0032]    When an image is captured by a camera and a light in that image is identified as a laser or muzzle flash, the next step is to determine the line between the light and the image of it in the camera when the camera captured it. While this calculation is complex, the analysis is known and can be programmed into a general or special purpose computer. See for example, Ashwini Amara, “Object Detection and Tracking Using Uncalibrated Cameras” (2010). New θ, Orleans Theses and Dissertations, Paper 1184, which is incorporated by reference herein in its entirety. 
         [0033]    Referring now to  FIGS. 1 and 2 , suppose a source  10  from a bright laser is detected by cameras on a flying aircraft  18  taking path  14 . Several images of source  10  are captured at different times as aircraft  18  flies along path  14 . The direction to source  10  for each one is provided as an angle between the direction to source  10  and another, well-established, reference direction provided by an on-board navigation system. This reference direction may simply be “North,” as indicated by the vertical arrows in  FIG. 1 , or it may be the direction to any known landmark. At each of the “i” locations of aircraft  18 , the angles θ 1 , θ 2 , and θ 3  between source  10  and North, and the distances R 1 , R 1n , and R n  to source  10  can be determined so that the location of source  10  at X 0 , Y 0  can be calculated. 
         [0034]      FIG. 2  shows aircraft  18  carrying three cameras or three arrays of cameras  22 ,  26 ,  30 . Whether one camera  30  captures three images of source  10  or all three cameras  22 ,  26 ,  30 , capture one image, three spaced-apart images are captured for analysis, one image for each of three platforms, which in this context means three locations in three-dimensional space. These three platforms cooperate to provide the solution to the location of source  10 . 
         [0035]    These cooperating platforms&#39; solution follows immediately. Suppose that one platform forms, from its series of angular detections, matrices P 1MA  and P 1MB  as above: 
         [0000]    
       
         
           
             
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         [0036]    Likewise a second platform forms its own similar matrices P 2MA  and P 2MB  as above. These matrices represent 16 numbers in the case of P 1MA , and 4 numbers in the case of P 1MB . If the first platform communicates these 20 numbers to the second platform, then it is well able to calculate an optimal solution to the location of source  10  as if it were in two places at once, namely, its own and its partners. The optimal solution is: 
         [0000]    
       
         
           
             
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         [0037]    The technique is perfectly extendable to an array of cameras approaching a source  10 , each communicating their two matrices of collected and processed detections, as above, in the form of their 20 numbers to a processor. Any one camera can then readily contribute to the solution, rather than capturing one image of the source  10  as in  FIG. 1 . The solution for some number ‘m’ of cameras is formed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             P 
                             
                               1 
                                
                               MB 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                                
                               MB 
                             
                           
                           + 
                           
                             P 
                             
                               3 
                                
                               MB 
                             
                           
                           + 
                           … 
                           + 
                           
                             P 
                             mMB 
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       
                         
                           
                             
                               
                                 y 
                                 . 
                               
                               0 
                             
                              
                             
                               t 
                               i 
                             
                           
                           - 
                           
                             y 
                             i 
                           
                         
                         = 
                         
                           tan 
                            
                           
                               
                           
                            
                           
                             
                               θ 
                               i 
                             
                              
                             
                               [ 
                               
                                 
                                   x 
                                   0 
                                 
                                 + 
                                 
                                   
                                     
                                       x 
                                       . 
                                     
                                     0 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                 
                                 - 
                                 
                                   x 
                                   i 
                                 
                               
                               ] 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0038]    Assume first that the source  10  (the pentagon in  FIG. 1 ) is not stationery but is moving at a constant velocity, with components x(dot) and y(dot). The appropriate regression function is of the form: 
         [0000]    
       
         
           
             
               
                 
                   g 
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               y 
                               0 
                             
                             + 
                             
                               
                                 
                                   y 
                                   . 
                                 
                                 0 
                               
                                
                               
                                 t 
                                 i 
                               
                             
                             - 
                             
                               y 
                               i 
                             
                             - 
                             
                               tan 
                                
                               
                                   
                               
                                
                               
                                 
                                   θ 
                                   i 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       x 
                                       0 
                                     
                                     + 
                                     
                                       
                                         
                                           x 
                                           . 
                                         
                                         0 
                                       
                                        
                                       
                                         t 
                                         i 
                                       
                                     
                                     - 
                                     
                                       x 
                                       i 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             σ 
                             i 
                           
                         
                         ] 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0039]    To be clear, the measurements here are any variable subscripted with an ‘i’. The solution is being solved for any variable that is subscripted with a ‘0’. 
         [0040]    A useful estimate of the standard deviation σ i  is 
         [0000]      σ i   2 ≈σ r   2   +r   i   2 σ φ   2  
 
         [0000]    where σ r  is the standard deviation on camera position itself when the angle measurement is made, r i  is a rough estimate of the range to the target (based upon brightness, or down-angle ranging, or other rough basis), and σ φ  is the standard deviation of the angle measurement itself. 
         [0041]    Now assume aircraft  18  is performing a series of detections by capturing a series of images. Consider again the scenario in  FIG. 1 . An image is made of a source  10 . Aircraft  18  may then maneuver. A subsequent series of images is captured. It is assumed that the platform possesses navigation features to give position, platform orientation and time at the moment for the images. To find the best solution to the position of source  10 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       g 
                     
                     
                       ∂ 
                       
                         x 
                         0 
                       
                     
                   
                   = 
                   
                     
                       0 
                       ⇒ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           
                             [ 
                             
                               
                                 
                                   y 
                                   0 
                                 
                                 + 
                                 
                                   
                                     
                                       y 
                                       . 
                                     
                                     0 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                 
                                 - 
                                 
                                   y 
                                   i 
                                 
                                 - 
                                 
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       θ 
                                       i 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           x 
                                           0 
                                         
                                         + 
                                         
                                           
                                             
                                               x 
                                               . 
                                             
                                             0 
                                           
                                            
                                           
                                             t 
                                             i 
                                           
                                         
                                         - 
                                         
                                           x 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                               
                                 σ 
                                 i 
                               
                             
                             ] 
                           
                            
                           
                             ( 
                             
                               
                                 - 
                                 tan 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       g 
                     
                     
                       ∂ 
                       
                         y 
                         0 
                       
                     
                   
                   = 
                   
                     
                       0 
                       ⇒ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 y 
                                 0 
                               
                               + 
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   0 
                                 
                                  
                                 
                                   t 
                                   i 
                                 
                               
                               - 
                               
                                 y 
                                 i 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     θ 
                                     i 
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         x 
                                         0 
                                       
                                       + 
                                       
                                         
                                           
                                             x 
                                             . 
                                           
                                           0 
                                         
                                          
                                         
                                           t 
                                           i 
                                         
                                       
                                       - 
                                       
                                         x 
                                         i 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       g 
                     
                     
                       ∂ 
                       
                         
                           x 
                           . 
                         
                         0 
                       
                     
                   
                   = 
                   
                     
                       0 
                       ⇒ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           
                             [ 
                             
                               
                                 
                                   y 
                                   0 
                                 
                                 + 
                                 
                                   
                                     
                                       y 
                                       . 
                                     
                                     0 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                 
                                 - 
                                 
                                   y 
                                   i 
                                 
                                 - 
                                 
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       θ 
                                       i 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           x 
                                           0 
                                         
                                         + 
                                         
                                           
                                             
                                               x 
                                               . 
                                             
                                             0 
                                           
                                            
                                           
                                             t 
                                             i 
                                           
                                         
                                         - 
                                         
                                           x 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                               
                                 σ 
                                 i 
                                 2 
                               
                             
                             ] 
                           
                            
                           
                             ( 
                             
                               
                                 - 
                                 tan 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             ) 
                           
                            
                           
                             t 
                             i 
                           
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       g 
                     
                     
                       ∂ 
                       
                         
                           y 
                           . 
                         
                         0 
                       
                     
                   
                   = 
                   
                     
                       0 
                       ⇒ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           
                             [ 
                             
                               
                                 
                                   y 
                                   0 
                                 
                                 + 
                                 
                                   
                                     
                                       y 
                                       . 
                                     
                                     0 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                 
                                 - 
                                 
                                   y 
                                   i 
                                 
                                 - 
                                 
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       θ 
                                       i 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           x 
                                           0 
                                         
                                         + 
                                         
                                           
                                             
                                               x 
                                               . 
                                             
                                             0 
                                           
                                            
                                           
                                             t 
                                             i 
                                           
                                         
                                         - 
                                         
                                           x 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                               
                                 σ 
                                 i 
                                 2 
                               
                             
                             ] 
                           
                            
                           
                             t 
                             i 
                           
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0042]    From equation (3), 
         [0000]    
       
         
           
             
               
                 
                   0 
                   = 
                   
                     
                       
                         - 
                         
                           y 
                           0 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         
                           y 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               y 
                               i 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         x 
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 tan 
                                 2 
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           x 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               
                                 tan 
                                 2 
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               x 
                               i 
                             
                              
                             
                               tan 
                               2 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0043]    From Equation (4), 
         [0000]    
       
         
           
             
               
                 
                   0 
                   = 
                   
                     
                       
                         - 
                         
                           y 
                           0 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             1 
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         
                           y 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               t 
                               i 
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             y 
                             i 
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         x 
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           x 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               x 
                               i 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0044]    From Equation (5), 
         [0000]    
       
         
           
             
               
                 
                   0 
                   = 
                   
                     
                       
                         - 
                         
                           y 
                           0 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         
                           y 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                                 2 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               y 
                               i 
                             
                              
                             
                               t 
                               i 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         x 
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               
                                 tan 
                                 2 
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           x 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                                 2 
                               
                                
                               
                                 tan 
                                 2 
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               x 
                               i 
                             
                              
                             
                               t 
                               i 
                             
                              
                             
                               tan 
                               2 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0045]    From Equation (6), 
         [0000]    
       
         
           
             
               
                 
                   0 
                   = 
                   
                     
                       
                         - 
                         
                           y 
                           0 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               t 
                               i 
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         
                           y 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               t 
                               i 
                               2 
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               y 
                               i 
                             
                              
                             
                               t 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         x 
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           x 
                           . 
                         
                         0 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                                 2 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               x 
                               i 
                             
                              
                             
                               t 
                               i 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0046]    Putting these together into a single matrix equation: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                           
                       
                        
                       
                         [ 
                         
                           
                             
                               tan 
                               2 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               t 
                               i 
                             
                              
                             
                               tan 
                               2 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             1 
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               t 
                               i 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               t 
                               i 
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               t 
                               i 
                             
                              
                             
                               tan 
                               2 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               t 
                               i 
                               2 
                             
                              
                             
                               tan 
                               2 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               
                                 t 
                                 i 
                                 2 
                               
                                
                               tan 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 i 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               t 
                               i 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               t 
                               i 
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         [ 
                         
                           
                             
                               t 
                               i 
                               2 
                             
                              
                             tan 
                              
                             
                                 
                             
                              
                             
                               θ 
                               i 
                             
                           
                           
                             σ 
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           [ 
                           
                             
                               t 
                               i 
                               2 
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               ] 
             
              
             
                 
               
                 
                   [ 
                   
                     
                       
                         
                           x 
                           0 
                         
                       
                     
                     
                       
                         
                           y 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             x 
                             . 
                           
                           0 
                         
                       
                     
                     
                       
                         
                           
                             y 
                             . 
                           
                           0 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     i 
                                   
                                    
                                   
                                     tan 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     y 
                                     i 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     i 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   y 
                                   i 
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     i 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                    
                                   
                                     tan 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     y 
                                     i 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     i 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     y 
                                     i 
                                   
                                    
                                   
                                     t 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0047]    The full optimal solution to the position of the source  10  and of its motion is had by the equation: 
         [0000]    
       
         
           
               
             
               
                 [ 
                 
                   
                     
                       
                         x 
                         0 
                       
                     
                   
                   
                     
                       
                         y 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           x 
                           . 
                         
                         0 
                       
                     
                   
                   
                     
                       
                         
                           y 
                           . 
                         
                         0 
                       
                     
                   
                 
                 ] 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     tan 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     t 
                                     i 
                                   
                                    
                                   
                                     tan 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       t 
                                       i 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   1 
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     t 
                                     i 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     t 
                                     i 
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     t 
                                     i 
                                   
                                    
                                   
                                     tan 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       t 
                                       i 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     t 
                                     i 
                                     2 
                                   
                                    
                                   
                                     tan 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       t 
                                       i 
                                       2 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     t 
                                     i 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     t 
                                     i 
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     t 
                                     i 
                                     2 
                                   
                                    
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   σ 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     t 
                                     i 
                                     2 
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   
                     - 
                     1 
                   
                 
                  
                 
                     
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       i 
                                     
                                      
                                     
                                       tan 
                                       2 
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       y 
                                       i 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       i 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     y 
                                     i 
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       i 
                                     
                                      
                                     
                                       t 
                                       i 
                                     
                                      
                                     
                                       tan 
                                       2 
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       y 
                                       i 
                                     
                                      
                                     
                                       t 
                                       i 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       i 
                                     
                                      
                                     
                                       t 
                                       i 
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       y 
                                       i 
                                     
                                      
                                     
                                       t 
                                       i 
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
         [0048]    At least four angle measurements of separate images—and more for greater accuracy—may be made to gauge an estimate of the position and motion of the source  10 . If the source  10  is assumed to be stationary, then the solution to the above equation is more conveniently expressed as: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       x 
                       0 
                     
                   
                 
                 
                   
                     
                       y 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         x 
                         . 
                       
                       0 
                     
                   
                 
                 
                   
                     
                       
                         y 
                         . 
                       
                       0 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   tan 
                                   2 
                                 
                                  
                                 
                                   θ 
                                   i 
                                 
                               
                               
                                 σ 
                                 i 
                                 2 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   tan 
                                    
                                   
                                       
                                   
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         [0049]    Now assume multiple cameras, such as cameras  22 ,  26 ,  30 , are all on aircraft  18  with captured images. 
         [0050]    The cooperating-camera solution follows immediately from above. Suppose that camera  22  forms, from its series of angular detections, the following matrices P 1MA  and P 1MB  as above: 
         [0000]    
       
         
           
             
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         [0051]    Likewise camera  26  forms its own similar matrices P 2MA  and P 2MB  as above. These matrices represent 16 numbers in the case of P 1MA , and 4 numbers in the case of P 1MB . If camera  22  communicates these 20 numbers to camera  26 , then camera  26  is well able to calculate an optimal solution to the location of the source  10  as if it were in two places at once, its own and that of camera  22 . The optimal solution is 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
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                       0 
                     
                   
                 
                 
                   
                     
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                       0 
                     
                   
                 
                 
                   
                     
                       
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                         MA 
                       
                     
                   
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         [0052]    The technique perfectly extends to an array of cameras approaching a source  10  communicating their two matrices of collected and processed images, as above, in the form of their 20 numbers to a processor. Any one camera can then readily form the best solution as if it were instantaneously at every camera point at once, rather than flying path  14  past source  10  as in  FIG. 1 . The solution for some number ‘m’ of air platform measurements is formed as: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       x 
                       0 
                     
                   
                 
                 
                   
                     
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                       0 
                     
                   
                 
                 
                   
                     
                       
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                         2 
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                     + 
                     
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                         3 
                          
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                     + 
                     … 
                     + 
                     
                       P 
                       mMA 
                     
                   
                   ) 
                 
                 
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                       0 
                     
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                       [ 
                       
                         
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                           0 
                         
                         + 
                         
                           
                             
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                               . 
                             
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                             t 
                             i 
                           
                         
                         - 
                         
                           x 
                           i 
                         
                       
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               . 
             
           
         
       
     
         [0053]    In  FIG. 1 , camera  22  is located on a different part of aircraft  18  than cameras  26  and  30 . Simultaneous images from cameras  22 ,  26 ,  30 , will have slightly different degrees of parallax because of their different locations on aircraft  18 . That different parallax can be used to reduce the uncertainty in the location of source  10 . In addition, cameras  22 ,  26 ,  30 , may be different types of cameras or may capture different images in different ways. For example, camera  30  may carry a filter  32  for visible light except red and infrared. Camera  26  may be a camera array. Camera  22  may be a high speed camera to better detect muzzle flashes. Cameras  22  and  26  are forward-looking; camera  30  is rearward-looking; other cameras may be side-looking. Cameras  22 ,  26 ,  30 , may capture images using an array of charge coupled devices or an array of complementary metal oxide semiconductors. 
         [0054]    The images are forwarded from cameras  22 ,  26 ,  30 , to a processor  34 , which has access to a database  38  of flight information carried by and in operational connection with said aircraft  18 . Processor  34  is in electronic communication with cameras  22 ,  26 ,  30  and with database  38  so that it receives captured images and has access to information about the motion of aircraft  18 . Database  38  stores flight information from the on-board navigation system  42 , including motion, direction, and position of aircraft  18  and ground location information, that is useful to process the data derived from the images and determine the ground location of source  10 . Specifically, database  38  contains altitude, roll, pitch, yaw, direction, air speed, ground speed, GPS (global positioning satellite) information, ground information such as topographic maps in particular, and weather information for the area, which may be useful to processor  34  for determining the location of the source  10  of a bright light. Processor  34  may be carried by aircraft  18 , or be remote from it, as illustrated in  FIG. 4 . Database  38  with stored navigation information data regarding aircraft  18  may additionally be located on the ground, such as at an airport, for example. If processor  34  and database  38  are on the ground, a transmitter  46  on aircraft  18  transmits data from cameras  22 ,  26 ,  30  to processor  34 . 
         [0055]    There are advantages to either location. Weight is a concern in aircraft  18 . Perhaps more importantly are the issues of battery power, heat generation, and the impact of vibration on electronic equipment. Because most instances of individuals shooting or shining lasers at aircraft  18  happen when aircraft  18  are taking off or landing, an airport may be a suitable place for processor  34 . Also, an airport may be more likely to have access to the topographic information its surroundings, and it can also be more recent information and well-verified information. 
         [0056]    Processor  34  is configured to perform a spectral analysis on the image. Spectral analysis of photographic image data is well-known, and is used here to determine which light source in an image is a threat, such as a bright light from a laser or a muzzle flash. Processor  34  may proceed by removing extraneous information from the captured image and then by identifying any bright light source  10  that remains. A filter  32  may help to remove extraneous information, such as that light having wavelengths of no interest. Filter  32  may also be used to identify light sources directly, such as those filters that remove light other than those in the red laser frequencies or those that remove light except those in the signature frequencies for gunpowder combustion products. Glare can mask light from sources so polarizing filters may be helpful especially if aircraft  18  is flying over water, snow or other reflective surfaces. 
         [0057]    When source  10  is located, its ground location can be forwarded via a communications link from processor  34  to authorities  50  such as a law enforcement agency. 
         [0058]    If source  10  is a laser, the image captured will contain a smaller dispersion pattern of laser light because laser light is coherent. That image may or may not contain internal structure. Internal structure refers to the portions of the dispersion pattern that are brighter than other parts and may contain a brightest point adjacent to a darker area.  FIGS. 5A-5C  illustrate dispersion patterns of detected laser light from images.  FIG. 5A  shows a pattern of detected laser light with internal structure;  FIG. 5B  shows a pattern of laser light with some internal structure; and  FIG. 5C  shows a pattern of laser light with no internal structure. Using edge detection software, the structure in a pattern such as those shown in  FIGS. 5A and 5B , can be analyzed. If structure is detected, the source is presumed to be coming from the sharpest corner of the detected structure. If no structure is detectable, the geometric center of the dispersion pattern is presumed to be the source. 
         [0059]    When introducing elements of the present disclosure or exemplary aspects thereof, the articles “a,” “an,” “the” and “said” are intended to mean that there are one or more of the elements. The terms “comprising,” “including” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. Although this disclosure has been described with respect to specific aspects, the details of these aspects are not to be construed as limitations.