# Patent

## Title
Passive electro-optical tracker

## Abstract

A passive electro-optical tracker uses a two-band IR intensity ratio to discriminate high-speed projectiles and obtain a time-varying speed estimate from their time-varying temperature, as well as determining the trajectory back to the source of fire. In an omnidirectional system a hemispheric imager with an MWIR spectrum splitter forms two CCD images of the environment. Various methods are given to determine the azimuth and range of a projectile, both for clear atmospheric conditions and for nonhomogeneous atmospheric conditions. One approach uses the relative intensity of the image of the projectile on the pixels of a CCD camera to determine the azimuthal angle of trajectory with respect to the ground, and its range. A second uses a least squares optimization over multiple frames based on a triangle representation of the smeared image to yield a real-time trajectory estimate.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application is a continuation in part of U.S. patent application Ser. No. 12/709,780 titled “Passive Electro-Optical Tracker,” filed Feb. 22, 2010 by Agurok et al. This application claims benefit of commonly owned U.S. Provisional Patent Applications No. 61/208,700 titled “Multiple Object Omnidirectional Passive Electro-Optical Tracker” filed Feb. 25, 2009 by Agurok (through application Ser. No. 12/709,780) and No. 61/260,149 titled “Passive Electro-Optical Tracker” filed Nov. 11, 2009 by Agurok. All of those applications, which have at least one common inventor to the present application, are incorporated herein by reference in their entirety.


BACKGROUND OF THE INVENTION
Surveillance, detection, and tracking of multiple high-speed objects, particularly bullets, mortars, small rockets, and artillery shells, can help military forces immediately locate sources of enemy fire and trigger countermeasures, thereby reducing combat casualties.
The traditional techniques for detection and tracing of fast moving objects are different kinds of radar but unfortunately, traditional radar has inherently low resolution for such small objects. Worse yet, radar cannot scan 360° of azimuth with a high enough frequency (times per second) for timely bullet detection.
Fast moving projectiles rapidly heat up by several hundred degrees Kelvin, thereby radiating in the Mid-Infrared (MWIR) region, where electro-optical resolution is far superior, even to microwave radars. An MWIR system of the prior art was recently developed by Trex Enterprise Corp and is described in Ref [1]. This technique comprises a high-resolution, “roving fovea” MWIR telescope for projectile tracking and a coordinated pulsed-laser radar (lidar) for distance determination. The rapidly slewing “roving fovea” method is quite fast by mechanical standards, but it still takes 200 msec for one scan of a field of regard of only 36° square and, therefore, five seconds are required for a full hemisphere. This means that bullets will not be detected and tracked to any useful extent. Moreover, the use of lidar on a precise targeting platform results in a system that is too bulky and expensive to purchase and not useful for tracking anything smaller than rockets and artillery shells.
One object of the present invention is to make possible a compact, cost-effective passive electro-optical tracker of multiple high-speed objects in a combat environment. The Passive Electro-Optical Munitions Tracker (PET) described in this specification can assist in providing pinpoint 3D information in real time to back-track projectiles to their source of fire.
SUMMARY OF THE INVENTION
Fast moving projectiles rapidly heat up by several hundred degrees Kelvin, thereby radiating in the Mid-Infrared (MWIR) region, where electro-optical resolution is far superior even to microwave radars.
The temperature of fast moving projectiles depends directly on their speed. See Ref [1]. According to Wien's displacement law, the spectrum maximum of light emitted by a heated body shifts to shorter wavelengths as the temperature increases. See Ref [4]. The atmosphere has two high transmission windows in the MWIR region, at wavelengths from 3.0 to 4.2 μm and from 4.3 to 5.2 μm. The temperature of an object can be estimated by comparing the irradiance measured by the sensor for that object for these two sub-wavebands. Once this value is determined the speed of the projectile can then be calculated. The instantaneous speed data and array of azimuth and elevation obtained from the electro-optical sensor, together with the calibrated signal levels in each of its pixels, can be used to determine the ballistic trajectory by a proprietary application of the least-square method. This approach can determine the 3D trajectory of projectiles with very high degree of accuracy using passive electro-optical sensors without the need for scanning lidar.
The imaging system can be a fixed staring array that monitors the entire target region for every image cycle. This staring array may use a “fish-eye” or similar lens or mirror arrangement to view 360° of azimuth using a CCD or other sensor array on a flat image plane. The projectile tracking optics can be compact as 60 mm in diameter and no more than 100 mm in length, and can be mounted on an army vehicle or unmanned aerial vehicle (UAV) to support troops with tactical battlefield information. This passive electro-optical tracker can have a short enough reaction time not only to back-track projectiles and pinpoint the source of the fire nearly in real time, but also to trigger alarms and automatic countermeasures. The system can also be tied into a battlefield database to help distinguish friendly versus enemy fire, which can also be used to both save lives and quickly rule out projectiles which are not emanating from hostile positions. The latter can reduce the computing power needed to track hostile projectiles.
One objective of the present invention is to make possible a system reaction time short enough to backtrack projectiles and pinpoint the source of the fire to trigger automatic countermeasures, such as laser designation or even counter-sniper fire, before a second enemy shot can be fired. Also, an audible warning to troops in the target zone could allow a second or two of time to simply duck.
An embodiment of the invention provides a projectile tracking device, comprising detector apparatus for converting into electronic form images in at least two infrared wavebands, optics for projecting onto the detector apparatus an image of a scene in the at least two infrared wavebands, logic operative to obtain from the images in electronic form apparent brightnesses of the projectile at the optics in at least two infrared wavebands, logic operative to estimate the speed of the projectile varying over time from the varying ratio at successive times of the measured apparent brightnesses in said at least two infrared wavebands, logic operative to obtain from the images in electronic form an azimuth of the projectile from the optics at successive times; and logic operative to estimate the direction of the trajectory of the projectile from the measured azimuths in combination with the ratio between the measured apparent brightnesses.
Another embodiment of the invention provides a method of tracking a projectile in air, comprising measuring apparent brightnesses of the projectile at an observing location in at least two infrared wavebands at successive times, estimating the speed of the projectile as a function of time from the ratio of the apparent brightnesses measured in the at least two infrared wavebands at successive times, measuring an azimuth of the projectile from the observing location at successive times, and estimating the direction of the trajectory of the projectile from the measured azimuths in combination with the ratios between the measured apparent brightnesses.
In an embodiment, the logic is in the form of one or more programs running on a suitably programmed general purpose computer, and an embodiment of the invention also provides a suitable computer program or program to carry out the methods and to be embodied in the devices of the invention, and a non-transitory storage medium containing such a computer program or programs.
In an embodiment, the direction of the trajectory of the projectile may be estimated by estimating relative changes over time in the distance from the optics to the projectile from changes in at least one of the apparent brightnesses over time, including correcting the at least one measured apparent brightness for the change in absolute brightness as a function of temperature as a function of the estimated varying speed of the projectile.
In an embodiment, the acceleration may be related to speed using a lumped coefficient μ=−V′/V2, where μ is calculated from time dependent projectile velocity data and lumps together parameters that at least initially are not individually known.
In an embodiment, parameters of the projectile trajectory may be estimated using a method comprising measuring the azimuth of the projectile from the optics at more than two successive times, estimating the distances traveled by the projectile between the successive times using the speeds estimated at successive times, and estimating the direction of the trajectory of the projectile as a best fit to the successive distances and azimuths.
In an embodiment, a best fit may be estimated by defining imaginary triangles, each triangle formed by radii from the optics at two of the successive times, and a segment of trajectory of length calculated using the estimated speed or speeds of the projectile between the said two successive times bounding each triangle.
In an embodiment, the scene may be imaged at regular intervals of time, and the successive azimuths and speeds may then be derived from successive images.
In an embodiment, the speed and trajectory of the projectile may be extrapolated backwards in time to calculate a starting point at which the projectile would have had a speed corresponding to an initial speed of a weapon consistent with observed characteristics of the projectile.
In an embodiment, the speed and trajectory of the projectile may be extrapolated backwards in time to calculate a starting point at which the projectile would have been in a position providing a suitable station from which the projectile might have been fired.
In an embodiment, an origin of the projectile may be estimated by superimposing the calculated starting point with a local terrain map.
In an embodiment, parameters of the projectile trajectory may be estimated using a method comprising measuring the azimuth of the projectile from the optics at successive times, and estimating the direction of the trajectory of the projectile from the measured azimuths in combination with the ratios between the measured apparent brightnesses and/or the measured apparent brightnesses corrected for the change of absolute brightness with temperature.
In an embodiment, parameters of the projectile trajectory may be estimated by using a method comprising measuring the direction of the projectile from the optics at successive times, and estimating the trajectory of the projectile from the measured directions in combination with a sum or integral of the estimated speeds.
In an embodiment, parameters of the projectile trajectory may additionally be estimated by a method comprising estimating the normal from the optics to the projectile trajectory and hence the direction of the projectile trajectory by locating the point with zero value of the second derivative with respect to time of the direction to the projectile, further combining the estimated projectile speed with the projectile trajectory direction for calculating distances from the optics to the projectile.
Any one, two or more of the mentioned additional methods of estimating parameters may be combined in one device or method. In any embodiment in which parameters are measured or estimated by two or more methods, a final estimate may be provided using a comparison of estimates from at least two said methods of estimating parameters.
In any embodiment, a choice may be made between at least two methods of estimating parameters based on atmospheric conditions.
Embodiments of the invention provide devices and methods incorporating any combination of the above-mentioned features, including features operating optionally, as alternatives, or in combination.
One objective of the present invention is to make possible a passive electro optical system to provide accurate 3D tracking of the actual ballistic trajectories of projectiles, and determine the vital ballistic parameters, such as drag. Based on exterior ballistic laws, it is then possible to provide backward and forward extrapolation beyond the part of the ballistic trajectory directly visible to the tracking system, to locate the source of fire and/or predict the point of impact.



BRIEF DESCRIPTION OF THE DRAWINGS
The above and other aspects, features, and advantages of the present invention will be more apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein:
FIG. 1 is a graph of bullet speed as a function of temperature.
FIG. 2 is a graph of spectral radiant emittance of heated up gray bodies.
FIG. 3 is a graph of atmospheric spectral transmission.
FIG. 4 is a perspective view of an omnidirectional two-mirror optic.
FIG. 5 is an axial sectional view of another, similar two-mirror optic.
FIG. 6 shows an example of the annular format of an image from an omnidirectional optic similar to those of FIG. 4.
FIG. 7 is an axial sectional view of an omnidirectional fish-eye lens optic.
FIG. 8 is a diagram illustrating how range is calculated from successive observations of azimuth and speed.
FIGS. 9A and 9B are a flow chart of a combined projectile tracking algorithm.
FIG. 10 is a view similar to FIG. 5 of an omnidirectional optic with a beam splitter for directing dual subbands to separate detectors.
FIG. 11 is a graph showing optical transfer function as a function of spatial frequency.
FIG. 12 is a perspective view of an urban scene.
FIG. 13 is an image of the urban scene of FIG. 12 as projected onto a flat optical plane by an omnidirectional optic.
FIG. 14 is a diagram of spherical coordinates in the field of view.
FIG. 15 is a graph of projectile radiance spectrum at different temperatures.
FIG. 16 is a snapshot of the system output on a computer screen.
FIG. 17 is a schematic aerial view of a system operation scenario.
FIG. 18 is a three-dimensional representation of a projectile track in a further embodiment.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
A better understanding of various features and advantages of the present methods and devices may be obtained by reference to the following detailed description of the invention and accompanying drawings, which set forth illustrative embodiments. Although these drawings depict embodiments of the contemplated methods and devices, they should not be construed as foreclosing alternative or equivalent embodiments apparent to those of ordinary skill in the subject art.
1. Principles of Operation
1.1 Aerodynamics of Fast Moving Projectiles; Speed-Temperature Dependence.
The surface of fast-moving bullets and other projectiles is quickly heated by air friction to a speed-dependent temperature, as detailed in Ref [1]. FIG. 1 shows a graph 100 with an ordinate 101 of projectile speed in Mach numbers (relative to Mach 1=340.3 m/s) and an abscissa 102 of absolute temperature. High speed IR imaging of an example of a bullet from a rifle reveals that it is traveling with a speed of 840 m/s (Mach 2.5) at a distance of 1 meter from the rifle. Aerodynamic frictional heating of the bullet's nose at this distance reaches a temperature of 440° K [3]. By the time the bullet has reached 2 m into its flight, it will reach a temperature of 600° K, the maximum temperature associated with a bullet speed of Mach 2.5, as can be seen in FIG. 1. Therefore by 2 meters from the muzzle of the rifle the “signature” of such a rifle and bullet can be fully determined.
1.2 Infrared Signature of Hot Objects.
Heated up projectiles radiate light and have their spectral maxima in or near the MWIR region. A very useful fact is that projectiles heated up to different temperatures have a different ratio between the IR energies radiated in the 4.3 to 5.2 microns transmission band, referred to in this specification as wavelength Band1, and in the 3.0 to 4.2 microns transmission band, referred to in this specification as wavelength Band2. By finding the ratio between the energies in the two bands, Band1 and Band2, the absolute temperature can be determined with good accuracy. See Ref [2]. Once the absolute temperature is known, the speed can then be calculated with a high degree of confidence. Because only a ratio of two radiations is needed, the temperature, and therefore the speed, can be determined without needing to know the absolute radiation intensity, and therefore before determining the type, size, or distance of the target projectile.
The monochromatic intensity from a black body is given by the following equation from Planck's law [4]




  
    
      
        
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where u is thermal radiation intensity as a function of light frequency v and absolute temperature T. The physical constants in equation 1 are listed in Table 1. The total emissive power within a particular bandwidth is calculated by integrating equation 1 over the frequency interval of the bandwidth using the above equation from Planck or the modified Planck equation that gives the monochromatic intensity as a function of wavelength [4].








TABLE 1





Physical constants

















h
Planck's constant
6.626 0693(11) × 10−34 J s = 4.135




66743(35) × 10−15 eV s


b
Wien's displacement constant
2.897 7985(51) × 10−3 m K


kB
Boltzmann constant
1.380 3205(24) × 10−23 J K1 = 8.617




343(15) × 10−5 eV K−1


σ
Stefan-Boltzmann constant
5.670 400(40) × 10−8 W m−2 K−4


c
Speed of light
299,792,458 ms−1









Projectiles are typically made of metals. Such objects radiate as a gray body. See Ref [4]. A gray body's spectral radiation power is u′ (v,T)=εu(v,T), where u(v,T) is the spectral radiation of a black body (see Equation 1), and ε is the emissivity of the metal body (which is typically slowly varying over temperature but is constant over all wavelengths for a particular temperature). For a gray body ε≦1. Integrating u′ (v,T) over the two MWIR bands gives the ratios (as parameter C) shown in Table 2. FIG. 2 shows graphs 200 with functions u′ (v,T) for several temperatures: graph 201 for 500° K, graph 202 for 600° K and graph 203 for 700° K. Because ε is independent of wavelength, it will not affect the value of the parameter C, and will therefore not impair the estimation of temperature.







TABLE 2







Thermal Signature of Different Bullet Speeds












Speed, m/s
T, ° K
λpeak, μm
C = Band1/Band2
















700
511
5.67
1.68



800
570
5.08
1.4



900
628
4.61
1










FIG. 3 shows graph 300 of the infrared transmittance of air as a function of wavelength, with the two MWIR atmospheric transmission windows, Band1=4.3-5.3 μm, indicated at 301, and Band2=3.0-4.2 μm, indicated at 302, used in Table 2 calculations (for a 1800 m propagation distance at sea level).
1.3 Optical Tracker Architectures.
In an embodiment of the proposed system, a multilayer filter is used to split the two bands at 4.2 μm, so that light in one band is sent to one sensor and the other to a second sensor. An analysis shows that for this two band approach a 1% thermal and readout noise of the sensor results in 2% accuracy for the Band1/Band2 ratio, which in turn correlates to ±7° C. temperature accuracy.
Several optical architectures can be used for omni-directional optics of the system. FIG. 4 shows the 3D view of a prior art embodiment of an omnidirectional optical system 400 of Ref [5]. This sort of two mirror configuration is suitable in a ground mounted electro-optical tracker, where the blind spot directly upwards, caused by the secondary mirror, may be acceptable.
FIG. 5 shows a plan view of a further development using a two-mirror panoramic system 500 according to U.S. Pat. No. 6,611,282 (Ref. [6]).
FIG. 6 shows example of an annular field of view, 601, that would be seen by a sensor, which is 360° in azimuth and −20° to +80° degrees in elevation. This is typical of the image obtained using the optical systems of FIG. 4. The Azimuth is mapped directly onto the image, and the elevation is mapped directly to a radius within the annulus. FIG. 6 shows the image with the highest elevation at the largest radius. Depending on the optical geometry, the image may instead have the highest elevation at the smallest radius, as illustrated in FIG. 13 below. The skilled optical designer will also understand how to control the relationship between altitude in the field of view and radius in the image plane.
For UAV mounted systems the Fish-eye imager with hemispherical field of view, as shown FIG. 7 and indicated by the reference number 701, is preferable. As may be seen from FIG. 7, the fish-eye lens naturally produces an image with the center of the field of view at the center of the image, but the relationship between elevation (relative to the optical axis) and radius on the image is still to a significant extent controllable by the optical designer.
In the proposed system, the omni-directional optics, exemplified by FIG. 4, FIG. 5, or FIG. 7, is augmented with a spectrum splitting mirror (see FIG. 10) that enables two bore-sighted CCD images to be formed. A system controller collects and concurrently analyzes both streams of high-speed video data. By the next readout cycle there will be a temperature map highlighting objects of interest. Within a few more readout cycles, a complete 3D trajectory can be modeled, and the impact point and shooter position predicted. Also, once the range from the observing system to the projectile object is known, the absolute intensities of the original infrared images give an indication of the object's surface area, and hence bullet caliber, as well as providing a means of quickly calculating the planar trajectory angle that the bullet is traveling on (see angle β in FIG. 8 or FIG. 18).
In the case of multiple snipers firing in concert, there are the well-proven Kalman filtering algorithms utilized for strategic missile defense against even hundreds of projectiles. Kalman filtering is thus well suited for ‘connecting the dots’ for dozens of bullets at once, while simultaneously registering other items of military interest, particularly muzzle flashes, fires, detonations, and rocket exhaust, which could be tracked as well. Modern electronics have provided miniature field-deployable computers that can be adapted to such a video-computation load. Some of the early stages of pixel computation can be done integrally with the image readout.
FIG. 5 shows a possible type of hemispheric imaging with a two-mirror panoramic head. Primary mirror 501 is a convex hyperboloid and secondary mirror 502 is a concave ellipsoid. This pair transfers the omnidirectional input field into a flat internal ring shaped image 503 located between the mirrors. The beams are focused in this plane. A Double-Gauss projection lens 504 re-images this ring-shaped image onto receiver plane 505. It is necessary to perform this re-image, as the two-mirror optic produces a primary image in an intermediate plane located between the two mirrors. If the camera were positioned between the mirrors at the primary image plane location, the camera would obscure the incoming beams from the primary mirror to the secondary mirror. Given the practical constraints on the mirrors, it would be difficult to obtain a primary image behind the primary mirror. In addition, the mirrors 501, 502 typically produce a primary image so severely distorted that using a projection optic 504 to reduce the distortion is highly desirable.
The projection lens and CCD camera, which are typically the most expensive and most fragile parts of the tracker head, are also much less vulnerable when they are positioned behind the primary mirror, and can be placed in an armored enclosure for additional protection. The exposed parts of the optics are two mirrors, which can be reflective surfaces deposited on massively constructed, and therefore relatively robust, substrates. A polycrystalline alumina dome mounted on the primary mirror housing supports the secondary mirror and provides environmental protection for the whole optical assembly. Double Gauss lens 504 is a well known universal type of objective. It can compensate for a wide variety of aberrations—spherical, coma, astigmatism, and field curvature. In such an inherently distorted image, the optical system should not introduce any un-wanted aberrations since it could potentially destroy important information. Therefore, the omnidirectional optic should have controlled distortion so that, for example, the radial (elevations) distortion can be removed at the post processing.
The lens material for refractive optical elements must have a high transmission for the MWIR band. A suitable material can be germanium, silicon, or ZnSe, among others. The beamsplitter can be located in the object space so as to separate the two sub bands by reflecting one to a second camera. The ratio of image irradiance on the sensors in the two MWIR bands will give a temperature map, so that a small pixel-cluster can be tentatively identified as a projectile target and assigned an expected projectile speed. This also determines the radius within which the same projectile is expected to show up for the next readout cycle.
1.4 Tracking Algorithms.
A 2D example of the simulated trajectory of a bullet and its position over time is illustrated in FIG. 8. Table 3, which follows, provides the azimuth angle and range at several intervals of 1/30 sec. In this example the bullet tracking begins at point AO FIG. 8, which is 300 meters away from the omnidirectional lens (“omnilens”) at the origin O. For conciseness, it will be assumed that the tracker is a compact device at the origin O. However, the computer or other logic device that does the calculations could be separate from the optics. In that case, the origin O is the location of the imaging optics. The data calculated in Table 3, which assumes a constant bullet speed, are used to show the accuracy of the different tracking algorithms in the Passive Electro-Optical Munitions Tracker (PET) system for a variety of atmospheric conditions. One of the advantages of the proposed system is that several methods of analysis can be carried out independently of one another, thus providing checks and balances and a higher degree of confidence to the projected values.
In this calculated example, the bullet's trajectory, line 801, crosses the azimuth A0O, segment 802, at an angle β=40°. It is assumed that the tracker is taking new azimuth readings of the projectile every Δt= 1/30 sec, which results in array of azimuth angles αi 803. Every time segment Δt, the bullet travels a distance Δ=Δt*VB, shown as segments 810, where VB is a bullet speed (which in the PET system is calculated from bullet temperature). In this example, the trajectory was simulated for VB=853 m/s and A0O=300 m. In the following calculation, it is assumed that the azimuth angles αi are measured in the plane containing the bullet trajectory (approximated to a straight line) and the position O of the tracker. That usually gives more accurate results than measurements projected into the local absolute geodesic azimuth plane.
As long as the trajectory approximates to a straight line in the plane of computation, the choice of computational frame of reference does not typically significantly increase the computational burden because, as discussed elsewhere in this specification, the observed trajectory is in any case transposed from the (usually severely distorted) camera pixel array space to a real world space.
The first 10 sets of trajectory data for the array are shown in the Table 3, which follows:







TABLE 3







Trajectory data.












i
Time(s)
αi (degrees)
Ai O(meters)
















0
0.000
−30.000
300.000



1
0.033
−26.242
278.818



2
0.067
−21.888
259.030



3
0.100
−16.848
240.977



4
0.133
−11.046
225.079



5
0.167
−4.442
211.820



6
0.200
2.930
201.723



7
0.233
10.931
195.277



8
0.267
19.303
192.851



9
0.300
27.704
194.593



10
0.333
35.787
200.395










As was stated earlier, there is a slight delay for the bullet to heat up to a maximum temperature relating to its terminal velocity. In the simulations which follow it is assumed that by the first measurement the bullet has already heated up to its speed-associated temperature.
1.4.1 Algorithm I—Tracking the Bullet in Clear Atmosphere Using Effective Sensor Irradiance to Determine Apparent Brightness
Under clear and undisturbed atmospheric conditions (no smoke, fires, explosions, etc), the optical transmission during the tracking can be assumed to be the same in all directions from bullet to sensor and constant over time. For this condition the ratios between bullet image irradiance on the sensor (after sensor calibration—dark, light and bias frames) from one frame to the next directly relate to the direction of its trajectory angle 804 (β) of FIG. 8. In the case when the bullet is moving closer to the sensor, the irradiance ratios from one frame to the next always exceed a value of one (increasing over time, as the calibrated signal for each pixel or set of pixels is increasing). When the bullet is receding from the sensor these ratios are less than one (as the calibrated signal for each pixel or set of pixels is decreasing). This set of ratios, together with the measured azimuthal data from the bullet's trajectory (obtained by noting which pixels on the sensor are “activated” by the radiation from the bullet), and the speed calculation from two MWIR bands described above, provide the basis for the Apparent Brightness (AB) algorithm. The key principle of this algorithm is that the irradiance on the sensor for a particular frame is related to the square of the distance of the bullet from the sensor (correcting for dark noise and other factors, including the optical train, and the corrections for individual pixels associated with so-called “skew” bullet trajectory paths-see below for details). Using the “relative” distances for two frames and the change in azimuth angle 803 between these two frames, one can solve for the angle 804 (β) in FIG. 8. By using more sets of distances, a more exact value for β can be calculated through an application of the least square method. Once angle 804 (β) is calculated, the distance to the projectile can be determined by calculating the distance traveled by the bullet from one frame to the next, which is determined by its speed (derived from the ratio of the two MWIR bands). The law of sines can now be applied to find the actual distances. A more rigorous treatment of this algorithm follows.
The energy Ei coming to the sensor, which is sum of the energies in the two MWIR sub wavebands, is

Ei=EΩ/4π=Er2/4xi2  (2)

where:

E is the energy radiated by the bullet,

Ω is the solid angle of the light cone collected by the sensor,

r is the radius of the sensor's entrance pupil and

xi is the current distance.

So

Ei/E0=x02/xi2  (3)

From the triangle 805 (OA0Ai) in FIG. 8

xi/sin(β)=x0/sin(β+αi−α0)  (4)

Hence,

Ei/E0=sin2(β+αi−α0)/sin2(β)  (5)

So the direction of the bullet trajectory-angle 804 (β) can be found using least square optimization from criterion shown in Eq. (6)




  
    
      
        
          
            
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Linearization of Eq. (7) in regard of variable β gives, see Ref. [9]:





  
    
      
        
          
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The solution of Eq. (8) is





  
    
      
        
          
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                          ∂
                          
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                            i
                          
                        
                        /
                        
                          ∂
                          β
                        
                      
                      )
                    
                    2
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          
            ∂
            
              q
              i
            
          
          /
          
            ∂
            β
          
        
        =
        
          
            -
            2
          
          ⁢
          
            sin
            ⁡
            
              (
              
                β
                +
                
                  α
                  i
                
                -
                
                  α
                  0
                
              
              )
            
          
          ⁢
          
            
              sin
              ⁡
              
                (
                
                  
                    α
                    i
                  
                  -
                  
                    α
                    
                      0
                      ⁢
                      
                        
                      
                    
                  
                
                )
              
            
            /
            
              
                
                  sin
                  3
                
                ⁡
                
                  (
                  β
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




Angle β will be found after several optimization cycles solving Eq. (9). In each cycle, after Δβ is found, angle β is corrected using the expression β=β+Δβ. The cycle continues to be executed until the absolute value of the increment Δβ is sufficiently small to stop the process. Once β is determined then the law of sines can be used to calculate the distances of the bullet to the sensor at various points of its trajectory. The distance traveled by the bullet, A0Ai, provides the “missing” length for the application of the law of sines. The speed over a particular time period determines distance A0Ai. The approach works even if the speed is changing, as is described in Section 3.2 below. Further refinement is possible through the use of ballistic equations to determine the actual path of the bullet from its source. Application of the algorithm to 3D is simplified by the fact that bullets travel nearly in straight lines over the distances that prevail in the field. This allows the approach to assume that the path will be on a plane. This allows the same 2D algorithm to be used even when the bullet or other projectile is fired from an elevation position lower or higher than the intended target of the bullet.
A simulation was carried out using the above approach using the azimuthal trajectory data shown in Table 3, and taking into account the possible error in estimating the speed. The array of ratios Ei/E0 was calculated using Eq. (3) with 1% Gaussian noise. The results, described below, are very promising and indicate that the angle β can be estimated using the above approach to within 0.2° of the actual angle of 40°. This allows the distance from the tracker to the first frame where the bullet is detected (where it has heated up sufficiently to be detected) to be determined within 0.1 meters of the actual distance of 300 m. (The actual firing point may be slightly different as the heat up distance is on the order of 1 to 2 meters, even assuming the whole initial part of the bullet's trajectory is visible to the tracker). A detailed discussion follows which describes in more detail how the speed of the bullet is calculated with a representative CCD camera (such as those from FLIR, see Ref [7]) and how this information is used to arrive at the final determination of distances.
A representative sensor chip suitable for this application (FLIR makes several) has a square CCD sensor array 640 by 512 pixels. Therefore, the largest circle that can be imaged onto that chip has an outer perimeter of 1608 pixels, which for the omnidirectional system represents 0.22° of azimuth angle per pixel. The CCD camera records the bullet image during the entire 1/30 second integration time. Each pixel records the bullet image during as much of the 1/30 second integration time as the bullet is in the solid angle of the field viewed by that pixel. So for the first frame of bullet tracking, Δα1=3.758° (Table 3) and the bullet is viewed by 17 different pixels during the 1/30 second integration time. In the final image frame, the bullet image is a smear 17 pixels in width. To determine a defined instantaneous bullet position, the position at the middle of this smeared image for this frame will be chosen. Suppose that the bullet radiates a constant amount of IR energy into the camera during the time of integration for the frame. How much of the 1/30 of a second each pixel was irradiated for is then directly proportional to the integrated energy it has captured. It is obvious that the interior pixels of the smeared image all have the same integration time. However, the end pixels in the smeared image typically receive less energy than the interior ones, as the bullet position starts and ends in an interval less than 0.22° azimuth angle. The ratio of energy received by an end pixel to the energy received by an interior pixel provides the actual angular extent traveled by the bullet for the end pixel. So the bullet location in CCD pixel coordinates for each frame of the image can be found as

dk=[(k+1)+(h−1)]/2+[(Ih/Ih−1)/2−(Ik/Ik+1)/2] where  (11)

k is the pixel number at which the bullet entered the frame,

h is the pixel number at which the bullet exited the frame, and

Ik, Ih are the bullet signals of the ends of the smeared image.

The bullet may have a “skew” trajectory with a change in elevation (as well as azimuth) relative to the CCD pixel array, either because the bullet is actually changing elevation, or because of the distortion inherent in projecting a sphere onto a planar square grid. The smear line can then be found using Raster to Vector (R2V) algorithms, which are well developed and understood in the field of Computer Aided Design (CAD), see Ref [10], which is incorporated herein by reference. The computer software then creates a virtual pixel array, so that the image of the trajectory on the virtual pixel array resides on a single row or single column of the virtual pixels. This array has the same pixel size as an actual receiver (or could have a slightly different one if desired) but the pixel grid is rotated, positioning the first and last pixels of the smeared bullet image at the same elevation on the virtual array. The effective irradiance at each virtual pixel is the sum of recorded corrected values of the actual pixels covered by the virtual pixel weighted with coefficients equals to what portion of the actual pixel is covered by this virtual pixel [11]. Where the R2V calculation results in a smear image that is wider than one virtual pixel, the pixels in each column or row perpendicular to the length of the smear may be binned to give a single number for ease of calculation. The position of the bullet at the middle of the integration time will be calculated in the virtual array coordinates using the above equation and then the coordinate system is rotated back.
In the case of using an imager with precalculated distortion, like an omnidirectional lens, the smear curve is localized using the R2V technique. Following this, the local raster area is mapped from the image space to the object space. The distorted raster image in space is superimposed with a rotated raster array and the smear center will be found using aforementioned technique. Algorithms for this have been extensively developed for several applications including astrophotography, geographic information systems (GIS) and pattern recognition.
By knowing the distortion of the imaging lens, the coordinates of each pixel and its “partial” pixel size (end pixels) can be mapped to its respective azimuth position. Let us assume that this method yields an accuracy of the bullet spot location to 1⁄3 of a pixel (a reasonable assumption). Based on this assumption this will add a Gaussian noise with 0.075° standard deviation to azimuth array αi. To see the accuracy of this method the least squares estimate of trajectory angle 804 (β) began with an initial estimate of 20°. (However, a fairly accurate estimate of angle 804 (β) can be obtained by assuming that the square root of the integrated signal value for any two pixels, taking into account individual pixel calibrations, represents the “relative” distance of the bullet from those pixels.) As the azimuth angles associated with each pixel are known, angle 804 (β) can be estimated by application of AB least-square method shown above. The optimization cycles determine angles 804 (β): to be respectively 27.07°, 34.05°, 38.51°, 39.71°, 39.76° with the next step yielding a value for Δβ of 0.001°, which indicates convergence is achieved.
Now that β has been calculated (we will assume the calculated value is 39.76° or approximately 0.2° less than the actual value), we must include the error in measuring the bullet temperature. A bullet muzzle velocity of 853 m/s heats the bullet to about 601° K. Assuming there is a 7° K accuracy for the temperature estimation then in one worst case the measured temperature will be 594° K. This results in an estimated 840 m/s speed (FIG. 1), which is 15 m/s below the actual value. Using the predicted values for angle 804 (β) and the speed, we can determine the distance by application of the law of sines to triangle 805 (OA0A1) (FIG. 8) where angles 804 (β) and α1−α0 are known and segment A0A1=840 m/s*( 1/30 s) or 28 m. Solving for distance 802 OA0=x0 in the triangle yields an answer of 299.93 m, which is less than 0.1 m from the actual distance.
The tracking data results in an array of distances 802 to the successive positions of the projectile, and arrays of the projectile's azimuths and elevations measured in the local tracker coordinate system. If the tracker is expected to be mobile in use, the tracker may be equipped with a fiber gyroscope or similar device, by which the tracker's Tait-Bryan angles (yaw, pitch and roll) can be determined. With that information, the tracking data can be transformed to absolute local geodesic coordinates using the matrix formalism shown in ref. [12] or similar approaches known to those skilled in that art. If the tracker is expected to be stationary in use, the tracker may operate in its own coordinate space, or may be calibrated to an absolute coordinate space at startup.
This approach is sufficiently accurate for the case where there is homogeneous (clear) atmosphere and the optical transmission is the same for all azimuth angles during the bullet tracking. This is true for many practical cases. Even when the atmosphere over battlefield space is distorted by fire, explosions, smoke clouds and extensive arms fire, or other natural conditions, the above solution can be considered as the first approach to the next step in the process.
1.4.2 Tracking in Non-Homogeneous Atmosphere Conditions.
When dealing with non-homogenous conditions there are two possible approaches to determining the position of the bullet, the azimuth-geometrical method and the triangle fitting method. We will start with the geometrical method.
In this approach it is assumed that the bullet is not changing speed over a range of acquisition points. Further, this method only works well when the bullet path either ends near, or crosses, a perpendicular from the straight line path of the bullet to the position O of the tracking system. This perpendicular line is exemplified by dot-dash line 806 in FIG. 8. In this case, the system looks for two consecutive frames where triangles 807 (Ai−1AiO) and 808 (AiAi+1O) are close to being equal. This occurs when the absolute value of (Δαi−Δαi+1) is a minimum. Such points are exemplified by points i=8 and i=9 in Table 3, having respectively Δα8=8.372° and Δα9=8.401°. This approach clearly does not work for all bullet trajectories. The system can determine if the necessary condition has occurred by comparing the increase or decrease in Δα from one frame to the next. If there is a change in the sign of the second derivative from one frame to the next for these angles then the criteria has been met. In this case ΔΔα7=0.371°, ΔΔα8=0.029° and ΔΔα9=−0.318°, so the extremum criterion has been met. The linear interpolation of the second derivative of the angle a gives the location of its zero value at the position α8+0.0699*(α9−α8)=19.303°+0.0699*8.401°=19.89°. In this case β=[90°−(19.89°+30°)]=40.11°. The calculated azimuth value of 19.89° turns out to be close to the angle made by dot-dash line 806 of FIG. 8, which is the actual normal. This estimate has a confidence interval of (Δαi+Δαi+1)/2. This approach can be used in the case where the measured α-angles are not close to the angle made by the normal through the use of interpolation, although accuracy may decrease if the measured a angles are too far from the normal. Extrapolation can be used to predict the angular position of the actual normal even for the case where the trajectory of the bullet stops before reach the normal. This can be derived using the values for the first and second derivatives of the measured α-angles. Thus if we know the azimuths of the projectile up to point 8 in Table 3, we can extrapolate the position of zero value of ΔΔα, which is the estimated azimuth of the actual normal. With ΔΔα6=0.629° and ΔΔα7=0.371°, ΔΔα will be zero at the azimuth α=α7+(α7−α6)*[(0.371/0.629)/(1−0.371/0.629)]=21.9°. While this extrapolation provides a less accurate approach to determining the position of the actual normal than the earlier described method, with more data points some which cross the extremum, it is still close enough to be used to find the initial value of β for next step of optimization.
It is useful to compare the two estimates for the angle β obtained from the AB approach and the geometrical approach. If the Apparent Brightness method estimate is inside the confidence interval for the azimuth of the actual normal to the trajectory (the actual normal is somewhere within this interval) it should be chosen as the value used in the next optimization step. In this instance, the geometrical approach corroborates that the AB approach is working well. The clear atmosphere approach is then more reliable than the geometric one and β=39.76° is chosen. If the discrepancy between the two approaches is greater than the confidence interval, then the average of the values of angle β obtained from the AB and geometrical estimation approaches is used in final method of triangle fitting.
The final method of triangle fitting relies on the fact that the ratio of the powers in the two key MWIR wavebands will not be significantly affected by nonhomogeneous atmospheric conditions. The absolute transmission may be affected, but the transmission in each MWIR sub waveband will be affected nearly the same way along any line of sight even under these conditions. So, the projectile temperature and hence its speed, as well as the array of azimuth angles, will be known. In this optimization, triangles with known base lengths (product of bullet speed and frame time) and known angles at the apex (which are derived from the azimuth angle) are used. A series of initial distances 802 (A0O) are tested. For each distance 802 (A0O) estimate, the angle β is found which optimally fits the array of azimuth-speed data. The pair {A0O, β} which best fits the data with the minimum deviation from the so-called triangle requirement, is the so-called triangle fitting (TF) solution.
It should be obvious that the TF algorithm can be sped up by using angle 804 (β) and distance 802 (A0O) found by the first two algorithms. However, the TF algorithm is quite robust on its own. A more detailed description of the algorithm follows.
For every triangle A0AiO the law of sines yields:

i*Δ/sin(αi−α0)=OA0/sin(αi−α0+β)  (12)

Hence we wish to minimize





  
    
      
        R
        =
        
          
            ∑
            
              i
              =
              1
            
            N
          
          ⁢
          
            
              [
              
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        α
                        i
                      
                      -
                      
                        α
                        0
                      
                    
                    )
                  
                
                -
                
                  
                    (
                    
                      
                        i
                        *
                      
                      ⁢
                      
                        Δ
                        *
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              α
                              i
                            
                            -
                            
                              α
                              0
                            
                            +
                            β
                          
                          )
                        
                      
                    
                    )
                  
                  /
                  
                    OA
                    0
                  
                
              
              ]
            
            2
          
        
      
    
    
      
        (
        13
        )
      
    
  




where distance 802 (OA0) and angle 804 (β) are unknown variables to be estimated.

Equation 13 is the criterion for nonlinear least-square optimization. The optimization goes in cycles with a number of estimates of distance 802 (OA0) and then an optimization of angle 804 (β). The (OA0, β) pair delivering the minimum R is chosen as the solution. In the following example a bullet has a muzzle velocity of 853 m/s which heats the bullet to about 601° K. As done previously it is assumed that temperature estimation has an accuracy of ±7° K which using the lower error figure, yields a temperature of 594° K, and an estimated speed of 840 m/s. An initial distance 802 (OA0) is 290 meters and distances up to 310 meters are tried. That range of initial distances represents a confidence interval of ±10 meters around a distance of 299.93 meter, which is the AB solution for the distance, and covers the possible impact of non-homogeneous atmospheric conditions on the accuracy of the AB solution. Mismatches R in Equation 13 and least square optimized values of angle 804 (β) were tabulated at 0.2 meter increments for distance 802 (OA0). Several solutions with close to minimum values of R are shown in Table 4.







TABLE 4







Least Squares optimizations of angle 804(β)









A0 O
β
R













290
m
40.23°
0.0052


295.2
m
40.02°
0.0010


295.4
m
40.016°
0.00097


295.6
m
40.0086°
0.00093


295.8
m
40.005°
0.00093


296
m
39.99°
0.00097


296.2
m
39.98°
0.001


296.4
m
39.97°
0.001


296.6
m
39.96°
0.0011


296.8
m
39.96°
0.0012


310
m
39.42°
0.0277









The best estimate of distance 802 (A0O) in Table 4 is 295.8 m and β=40.005° with an “R” number of 0.00093. Some reduction in accuracy is expected for a turbid atmosphere. Nevertheless the results are sufficiently accurate to back-track to the shooter. To achieve the highest potential tracking accuracy, cooled MWIR CCD's should be used. Such CCDs have much less inherent dark noise (and thus increased signal to noise ratio) and less sensitivity variation from one pixel to the next compared to non-cooled sensors. In any case the highest accuracy will be achieved when a proper calibration is carried out on the system prior to its being deployed.
1.4.3 Tracking with a Conventional Staring Camera.
The tracking optics can alternatively comprise a conventional imaging lens with a 40° field of view. The beamsplitter can be mounted in image space to separate the two MWIR sub wavebands, or two mutually aligned cameras with band filters can be used. Such a two-camera optical arrangement restricts the field of regard for sniper location, but will be advantageous for light gathering power and consequently reduce the error in bullet temperature-speed estimates. A 40° field camera with a VGA CCD array will give 0.08° azimuth accuracy for each pixel. Assuming 1⁄3 of a pixel tracking accuracy and 4° K bullet temperature estimation accuracy, the bullet speed in the tracking example above can be estimated as 846 m/s instead of the actual value of 853 m/s. A trajectory optimization was carried out using 8 points of trajectory (809 in FIG. 8), spanning azimuths from −30° to 10° relative to the normal 806. The optimization estimated distance 802 (A0O) as 297.8 m and angle 804 (β) is 40.12°. In this example the tracking accuracy is close to the accuracy of tracking with an omnidirectional lens. The improvement of the signal to noise ratio did not result in better tracking accuracy because the number of tracked points is small. The major advantage of the 40° field of view system is that its greater light-gathering power enables it to resolve a bullet further away than the omnidirectional one. For existing VGA size sensors and a typical bullet, the omnidirectional system has a range of 0.6 km, whereas the narrow field system can resolve out to 2 km. This makes it ideal for applications on UAV or helicopter.
1.5 Self Protection
Real time MWIR imagery analysis can warn against incoming rounds. A projectile's track with unchanging azimuth but increasing MWIR signature is a collision indicator. Depending upon the vehicle there may be time for evasive maneuvers, at least to alter the impact point to a less vulnerable part of the vehicle, or onto reactive armor.
1.6 PET Operation Flow Chart.
A flow chart incorporating all three of the projectile tracking algorithms is shown in FIG. 9A and FIG. 9B, collectively referred to as FIG. 9. The “Apparent Brightness” solution referred to in FIG. 9A may be either the AB method of Section 1.4.1 above or the CABM method of Section 3.2 below. The Triangle Fitting solution of FIG. 9B may be either the TF solution of the second half of Section 1.4.2 above or the CTFM method of Section 3.5 below. Where the Corrected methods of Section 3 are combined with the methods of Section 1.4, it should be borne in mind that the Corrected methods will usually be much more accurate. Therefore, when CABM and CTFM are available, it may be preferred to omit the Geometrical Algorithm. Alternatively, the uncorrected methods may be retained as a backup or double-check. For example, if CABM is available but CTFM is not, then the TF or Geometrical method may be provided as a fallback for use when atmospheric conditions are too non-homogeneous for the Apparent Brightness method to be reliable.
Alternatively, the Azimuth-Geometrical method may be used for verification after the trajectory has been determined by one of the Corrected methods. The Azimuth-Geometrical method described in Section 1.4.2 above relies on the position of the minimum of the difference in angles (Δαi−Δαi+1), whereas the speed corrections described in Section 3 below apply to the line segments ΔiΔi+1. As a result, when the speed is changing the azimuth line that is orthogonal to the trajectory will be offset from the point of minimum difference in angle per frame. The correction will depend on the distance from the detector head to the projectile, and cannot be accurately applied unless that distance is already known. If an Azimuth-Geometrical method is attempted, the residual error will be least when the rate of deceleration of the projectile by drag is small (for example, in the case of howitzer shells and other high-altitude projectiles near the middle of their trajectory.
However, it is anticipated that for most situations, the AB approach described above, or the Corrected AB approach described below with reference to FIG. 18, will work just fine on its own.
The practical limit on how quickly a trajectory can be determined is likely to be the time in which a sufficient number (typically at least 10) of distinguishable projectile positions over the trajectory can be recorded.
1.7 Preliminary Design and System Functionality Considerations.
1.7.1 PET Omnidirectional Optics.
To estimate the system power budget, possible operational distances and other characteristics, a preliminary optical design of the dual-band omnidirectional lens has been developed with a two-mirror panoramic head, initially only for ground based projectile tracking. The proposed design uses a cooled FUR camera, the Photon HRC [7], with an InSb 640×512 pixels CCD with a 15 μm pitch. The layout of the lens 1000 is shown in FIG. 10. The lens has a ring-shaped image 1001 with 3.85 mm radius for an input field from 45° above horizon to 15° below. The optical transfer function 1100 is shown in FIG. 11. While a CCD with 15 μm pitch would require 33 line-pairs/mm resolution, the designed lens attains 60 lp/mm. The surplus resolution is due to attempts to increase the light gathering power of the lens to obtain the maximum operational distance. In the preliminary design image space the F number (F#) was 1.5, with entrance pupil diameter of 1.6 mm. While this is a good enough F#, further improvements are believed to be possible to achieve F# of 1 and entrance pupil diameter of 2 mm.
1.7.2 Object to Image Space Transformation in the Omnidirectional Lens.
The omnidirectional lens 1000 shown in FIG. 10 images any objects located at lower elevation angles to the outer circle of the CCD image and objects located at high elevation angles to the inner circle. FIG. 12 shows a 3D suburban scene 1200 and FIG. 13 its annular image 1300. Those objects with an elevation of negative 15° are imaged at a radius 3.85 mm at the image plane of the lens shown in FIG. 10. Objects with elevation 0° are imaged at a radius 3.8 mm and objects with elevation +45° are imaged at radius 1.86 mm. So the ratio of length of image circle of input field with elevation 45° to length of image circle of input field with elevation 0° is one half. The undistorted imaging in spherical coordinates 1400, which is shown in FIG. 14, requires that the ratio of length of circle 1401 (R45) to length of circle 1402 (R0) is 0.707. So the omnidirectional lens produces some distortion of the output field relative to the input field in spherical coordinates, with a 30% reduction of resolution for high elevation angles. This distortion will not dramatically reduce the tracking accuracy. The tracking accuracy for undistorted atmospheric conditions for bullets moving at 0° elevation is 0.1 m in distance and 0.2° in azimuth. This degrades to 0.14 m and 0.28° for bullets moving at 45° elevation, which is still good enough to locate the source of fire and undertake timely countermeasures.
1.7.3 Power Budget.
The main characteristic of sensitivity of IR receivers is Specific Detectivity D*

D*=√{square root over (Af)}/NEP, where  (14)

A is pixel area,

f is the acquisition frequency,

NEP is noise equivalent power.

The Specific Detectivity D* of a cooled InSb CCD is 1012 (cm √{square root over (Hz)}/W), see Ref. [1]. So for a 15 μm pixel and 30 Hz acquisition rate, the NEP is 8*10−15 W. The M16 round has a muzzle velocity of 930 m/s (Mach 2.7). Extrapolation of FIG. 1 gives a temperature of 650° K. The projectile blackbody radiance Q in the MWIR waveband is shown in FIG. 15.
As discussed above, the bullet emits IR radiation as a gray body. Ref. [1] suggests that it has an emissivity ε of 0.8. Measurements of the emissivity of an unused bullet with a brass jacket using a calibrated camera and heater give a value of 0.3, see Ref. [3]. High-velocity bullets typically have steel jackets. For dull steel, ε=0.44, see Ref. [8], and for steel alloys it can reach a value of 0.95. This can explain the difference in emissivity data shown in Refs. [1] and [3]. Assuming an emissivity 0.5 for a bullet therefore should be a reasonably conservative value.
The M16 bullet is 5.56 mm diameter and 23 mm in length. Treating its radiating area as a cone with a 23 mm height and 5.56 mm diameter at the base, the radiating area S is 2.26 cm2. The projectile blackbody radiance 1500 (U) according to Ref. [1] is shown in FIG. 15. In each 1 μm sub-waveband (3.0-4.2 μm and 4.2-5.3 μm) the bullet radiates energy

Q=U*S*εW/Sr·=0.1*2.26*0.5W/Sr=0.12W/Sr·  (15)

Given that the preliminary designed omnidirectional optics achieves a 2 mm diameter entrance pupil then the solid angle ω from the bullet to the entrance pupil at distance D can be calculated as follows:

ω=π(d/2)2/D2  (16)

and at a distance of 500 m

ω=3.1415*10−6/25*104=0.12*10−10Sr  (17)

Therefore the energy, E, which reaches the receiver is,

E=Q*ω=0.12(W/Sr)*0.12*10−10Sr=144*10−14W  (18)

These calculations do not take into account any atmospheric transmission losses, which are usually minimal. FIG. 3 shows there is a 95% transmission at a distance of 1800 m. According to the Bouguer law, at 500 m the transmission will be 98.5%. The pixel NEP is 8*10−15 W.
If the bullet is moving directly towards the sensor and the distance is 500 m the signal to noise ratio (SNR)

SNR=144*10−14/8*10−15=180  (19)

and at 300 m distance the SNR will be 500.

This is better than a 2% noise ratio used in the projectile tracking simulation in previous sections. The power budget results have therefore done better than the earlier conservative estimates.
If the bullet is moving tangentially to the sensor direction there will be a smear across multiple pixels during the 1/30 sec frame recording time. For 1/30 sec the bullet will travel 31 m. At a distance of 500 m from the sensor this amounts to 3.54° in angular coordinates. It will give the maximum smear and lower signal to noise ratio at low elevation angles, which are imaged at the receiver's external circle. The external circle length for a 640*512 pixels CCD is 1608 pixels, or 0.22° per pixel in azimuth direction. During one frame there will be a smear across 16 pixels. Each pixel will get 1/16 of the bullet's energy, and the SNR is reduced to only 11. Since the acceptable SNR for projectile detection and tracking is 5, see Refs. [1], [3], this will suffice for bullet tracking in angular coordinates, but far from the 2%, or SNR=50, needed for bullet temperature determination. Fortunately the bullet temperature does not change during several consecutive frames. Denote the bullet energy received by a pixel in Band1 (Table 1) as A and in Band2 as B. So the temperature indicating ratio C is

C=(A+V)/(B+V)  (20)

where V is the pixel noise level.

As is typical for MWIR image analysis, background subtraction mode will remove the dark current so that V will have zero mean and

C=(A/B+V/B)/(1+V/B)=(C0+v)/(1+v)  (21)

where C0 is the true intensity ratio between the sub-wavebands.

The variable v has a Gaussian distribution with zero mean value and a standard deviation of half of 1/SNR, which at SNR=11 will be 0.045. The important property of C0 is that it depends only on projectile temperature. As the projectile image moves from pixel to pixel the atmospheric conditions at the line of sight could change, but in equal proportions in both MWIR wavebands. Thus the temperature-dependent ratio C can be averaged over number of pixels to reduce its standard deviation. Expanding this in Taylor form gives

C=(C0+v)(1−v . . . )=C0+v−vC0−v2=C0+(1−C0)v−v2  (22)

The averaging of C over N pixels (one frame will be 16 pixels, but it is better to use several sequential frames) will give C1 approaching C0. Because the mean value of (1−C0)v is zero the mean value of the C1 is [9]

 C1=C0−σ2  (23)

where σ2=(1⁄2*[1/SNR])2 is variance of the v. In this case σ2=(0.045)2=0.003.

The variance δ2 of C1 is,

δ2=(1−C0)σ2/N+3σ4  (24)

where N is number of pixels being averaged.

For an M16 round heated to 650° K, C0 is close to 1, and the standard deviation of C1 is δ=√{square root over (3σ4)}, or 0.005 (0.5%). To get C0 we need to average C and increase it by σ2. The estimation of accuracy, C0=A/B, at 500 m distance is 0.5%, which is better than the 2% previously used. The computer simulation for C0=1, σ=0.045 and number of averaged pixels 32 (two frames), with a different set of noise component v, gives results shown in Table 5.







TABLE 5







Simulation for C0 = 1, σ = 0.045












Experiment number
1
2
3
4
5





Estimation of C0
0.9922
1.0028
1.002
1.0086
0.9987









The standard deviation of C0, δ, in Table 5 is 0.0054 (0.54%) which is in good agreement with the theoretical estimates. These calculations did not include any background radiation.
The normal background radiation in the MWIR waveband is relatively low, 0.12 mW/cm2/Sr, see Ref [1], while a heated bullet has radiance three orders of magnitude higher, at 0.1W/cm2/Sr. A normal landscape is conveniently dark in the MWIR bands of interest, because objects on the ground radiate primarily in longer infrared wavelengths, while the sun radiates primarily in shorter wavelengths, mostly visible and ultraviolet. Operating in background-subtraction mode, the informative image will be the difference between the frame with a bullet and the previous frame without, tacitly assuming that the background did not change in 1/30 of second. This operation will suppress the influence of the MWIR background radiation, which is already very small in comparison. This will also be helpful in more stressing situations of percussive background signals and other heat sources, as well as multiple rounds at once.
1.8 The Tactical Scenario of the Use of Passive Electro-Optical Tracker (PET)
The PET system will continuously track multiple projectiles in the omnidirectional field of view and calculate their trajectories. In addition, it will be able to record all muzzle flashes which occur in the imager's direct sight. The PET system will calculate the complete trajectory for fired projectiles, even if only part of the trajectory was available due to the obscurations in the battlefield environment. PET will determine the source of fire and the hit points, and will be able to determine the type of projectiles and firing weapons. PET can use this projectile data and GPS information to render the local battlefield showing trajectories, source of fire and hit locations, and the time of fire. Using this data and display, field personnel may be able locate the sniper after the first shot, separate friendly fire from hostile fire, deliver automatic warnings, and prepare countermeasure actions. An example of the system screen snapshot 1600 is shown in FIG. 16. An example of the system operation scenario 1700 is shown in FIG. 17. In FIG. 17 is shown the projectile tracker 1701 with a double waveband omnidirectional imager 1702, controller 1703 and display 1704. The tracker begins to track the bullet at the point 1705 and the bullet is exiting the tracker imager field at the point 1706. After calculating the distance to the initial point of tracking 1705 and the azimuth 1707 (β) of the bullet's trajectory, the tracker extrapolates the trajectory and superimposes it with the local 3D map to locate the source of fire 1708 and the point of hit 1709.
2.0 Projected Performance and Specification of the PET System
A performance specification can be projected for the PET system based on commercially available cameras (non-military) at the time of this application and the type of optical system employed (omnidirectional vs. 40°). The resolution and range for the two optical approaches for current high-velocity bullets are as follows. Also, dimensional and other functional parameters are included.













Accuracy
AB solution (good atmosphere conditions):



Distance: - 0.2 m or better



Azimuth: - 0.2° or better



TF solution (intensive firing conditions):



Distance: - 4 m or better



Azimuth: - 0.2° or better


Range
2 km for the system with 40° field of view



0.6 km for the system with omnidirectional lens


Weight
1400 grams


Dimensions
60 mm diameter, 180 mm length


Power consumption
20 W


Reaction time
Projectile flight time plus 1 sec.


Platforms
Ground vehicle, UAV









3. Method of Tracking Projectiles with More Realistic Ballistic Trajectories.

The preceding embodiments and examples assume that the projectile travels in a straight line with fairly constant speed. This simplified assumption works very well for many instances and types of projectiles. However, in some cases a more robust tracking algorithm needs to be employed, one that can track velocity (as opposed to speed), where a projectile has variable velocity (variable direction and/or speed). The following embodiment provides a method for determining the position and direction of a projectile over time that includes one of the most important parameters causing a projectile's trajectory to depart from uniform motion in a straight line, drag. This method is extended to include a second parameter that is important in many cases, gravity. In addition, a method is described that can accurately predict the firing position of a projectile even if the projectile is not visible the full length of its flight. These approaches do not require information as to the mass, size or coefficient of drag of the munition or projectile.
3.1 Ballistic Trajectory Simulation that Includes the Effects of Drag and Gravity.
In order to track projectiles traveling over an actual ballistic trajectory, both drag and gravity must be considered. Equation (25) below shows the classical equations assuming the projectile travels in the x-z plane with the x-coordinate axis horizontal and the z-coordinate axis vertical. Thus, movement along the x-coordinate axis is a projectile's movement in the horizontal direction (affected by drag) and the z-coordinate axis represents its movement in the vertical direction (affected by gravity and drag). In this case the components of acceleration are:

x′′=−(0.5pACd/M)Vx′=−μVx′

z′′=−(0.5pACd/M)Vz′−g=−μVz′−g  (25)

where:

p is density of the air (typically 1.29 kg/m3);

A is the cross-sectional area of the projectile;

M is the mass of the projectile;

g is gravitational constant (9.807 m/s2);

V is the speed of the projectile; and

Cd is the projectile's drag coefficient, see Ref. [14].

The μ term on the right lumps together the parameters of air density p, and the cross-sectional area A, drag coefficient Cd, and mass M of the projectile. This lumped value will be shown to be very useful in tracking of a projectile when details about the projectile (especially A, Cd, and M) are not initially known or may never be known. In order to verify the new method several ballistic curves were simulated. The simulations were carried out by an iterative step by step approach using the following set of equations from Ref. [15].

Δx=x′Δt+(1⁄2)x′′Δt2+(1⁄6)x′′′Δt3,

Δx′=x′′Δt 

Δz=z′Δt+(1⁄2)z′′Δt2+(1⁄6)z′′′Δt3,

Δz′=z′′Δt  (26)

and

x′′′=−μ(V′x′+Vx′′),

z′′′=−μ(V′z′+Vz′′),

V=(x′x′′+z′z′′)/V 

V={(x′)2+(z′)2}1/2  (27)

Such an approach, proposed in Ref. [15], allows one to use the maximum time step Δt over the simulated trajectory, while providing high accuracy for the determination of the projectile position at each time step. An example of one of these simulations follows and will be used to exemplify the novel method of tracking in this invention.
In this simulation it is assumed that the projectile has an initial speed of 965 m/s at the initial tracking point. The above set of equations provides 0.1 m/s velocity accuracy at the end of 1 sec flight simulation time using a moderate Δt increment of 0.0005 sec.
A three-dimensional representation of the projectile track is shown in FIG. 18.
In FIG. 18, perspective view 1800 of a projectile trajectory is shown in an XYZ Cartesian coordinate system. The detector head of the tracking device is at the origin O of the coordinate system. The track begins at a point A. Also there is trajectory curve 1801 (also represented by curve L) in plane 1802 (also represented by P). Plane P corresponds to the x-z plane of Equations 25 through 27. However, in a real situation, plane P is skew to the XYZ coordinate system of the tracking device, with its angle and offset from the origin O initially unknown. The elevation and azimuth angles in the device XYZ coordinate system of the points on the ballistic curve are respectively angle 1803 (E) and angle 1804 (Az). Angle 1805 (δ) is the azimuth in the plane XOY of the projectile trajectory plane P at a given point relative to the radius from the detector head O to that point. In FIG. 18, the azimuth angle δ is shown for point A′. Angle 1806 (γ) is elevation angle of the projectile velocity at the initial tracking point A. The tracking simulation was carried out for a bullet with a diameter d=5.6896 mm, mass M=3.24 grams, and drag coefficient Cd=0.295. Those values are typical for an M16 rifle bullet, see Ref. [14]. The speed at the initial tracking point A was set at 965 m/s. Distance 1807 (D) from the center of the Cartesian coordinate system O to point A was set at 300 m. Finally, it was assumed that azimuth Az of point A is minus 30° and its elevation, E, is 5° with angle γ=0° (i.e. horizontal firing) and angle δ=40°.
As was stated already, the bullet trajectory was calculated with Δt intervals of 0.0005 sec. Following this, data points for the trajectory were generated at intervals of 1/30 sec to emulate the speed of a typical camera frame rate. It was determined that interpolation was required to achieve the necessary accuracy, as 1/30 is an irrational number. In FIG. 18, points 1808 (Ai) are the borders of the trajectory segments captured in successive camera frames, whereas points 1809 (Fi) represent points in the middle of the frames. That is to say, assuming for simplicity that each frame records the field of view over a fixed period ( 1/30 sec), and that the interval between the end of one frame and the beginning of the next is negligible, the ith frame shows the bullet as a smear extending from Ai−1 to Ai, with the middle of the smear at Fi. Coordinates of the point Fi were calculated as the mean value of the coordinates of the border points Ai−1 and Ai. A procedure for obtaining the point Fi from the distorted image on the actual camera CCD array is described above with reference to FIG. 8. The bullet velocity at points Fi was calculated as the mean integral value of the velocity over the frames. The bullet temperature as a function of air speed was calculated using Eq. (28) below, from Ref [1]:

T=300·[1+0.5(1.4−1.)0.82 M2]  (28)

where M is the instantaneous bullet Mach number.

The data for the first ten 1/30 sec frames are shown in Table 6.












TABLE 6








Distance
Bullet
Bullet


N frame
Fi elev. °
Fi Az °
OFi meter
Speed m/s
Temp. K




















1
5.20
−28.035
288.47
943.072
672.3


2
5.62
−28.64
266.72
901.162
639.9


3
6.05
−18.73
247.64
862.819
611.6


4
6.47
−13.29
231.31
827.608
586.7


5
6.86
−7.36
217.86
795.160
564.6


6
7.20
−0.98
207.37
765.161
545.1


7
7.45
5.73
199.91
737.344
527.5


8
7.61
12.59
195.44
711.480
511.8


9
7.65
19.44
193.84
687.369
497.7


10
7.58
26.07
194.89
664.840
484.9









3.2 Corrected Apparent Brightness Method (CABM).

The Apparent Brightness (AB) method was described above in Section 1.4.1. It will now be shown how this method can be modified to handle the case of a projectile with a relatively fast-changing velocity (which is a consequence of drag) that does not travel in a straight line.
The AB approach can be applied to the ten frames of data of Table 6 by assuming that segments of trajectory curve are always located very close to the velocity vector at the initial point Ai−1 of each frame. The trajectory is then approximated to a series of straight lines, though with varying speed. The series of straight lines can then be approximated to a single straight line, taking an average direction, as described below. This assumption is valid because the drop in elevation as a consequence of gravity is relatively small compared to the length of travel of the projectile for any given time interval. For example, in the case of the simulation represented by Table 6, from the point F1 at the center of frame 1 to the point F10 at the center of frame 10 the bullet travels a distance 261 meters, while the sag of the curve is only 0.43 m. It will be shown that this assumption simplifies the calculation without negatively affecting the accuracy of predicting the range and azimuth at various points on trajectory. As explained above with reference to FIG. 8, that simplification enables the computation to be carried out in the plane defined by the velocity vector and the origin O, avoiding the need to transpose the observed trajectory into a different frame of reference. However, in the following description it is preferred to carry out the computation in a frame of reference with the Z axis vertical. Such a vertical frame of reference makes it easier to relate the computed trajectory back to a map of the terrain on which the start and end points of the trajectory can be located. Such a vertical frame of reference also makes it easier to reintroduce the vertical acceleration due to gravity, which then affects only the Z component of the calculation.
A projectile following a ballistic trajectory undergoes a continuous monotonic reduction in its speed. For a typical bullet trajectory, the negative acceleration (rate of reduction of speed) can be adequately approximated to be proportional to a constant times the speed squared (See Eq. 25). As the temperature of a projectile is directly related to its speed (Eq. (28)), its absolute brightness decreases along with its temperature. Therefore, the apparent brightness will change as a function of the distance the object has traveled along its trajectory, as well as of its distance from the observer. Supposing that the temperature can be established using the dual-waveband method as described in Section 1.2 above, then one can correct for the change in temperature based on the Stefan-Boltzman law (leaving the apparent brightness dependent only on the dynamically changing distance from the detector head at O). This is what we now call the Corrected Apparent Brightness Method, which is stated mathematically below in the equations that follow.
The original criterion for the Apparent Brightness Method was revealed in Equation (6) above. The modified criterion for new method to find the trajectory azimuth angle 1812 (β) of FIG. 18 is




  
    
      
        
          
            ∑
            
              i
              =
              1
            
            10
          
          ⁢
          
            
              [
              
                
                  
                    (
                    
                      
                        E
                        i
                      
                      ⁢
                      
                        T
                        1
                        4
                      
                    
                    )
                  
                  /
                  
                    (
                    
                      
                        E
                        1
                      
                      ⁢
                      
                        T
                        i
                        4
                      
                    
                    )
                  
                
                -
                
                  
                    
                      sin
                      2
                    
                    ⁡
                    
                      (
                      
                        β
                        +
                        
                          α
                          i
                        
                      
                      )
                    
                  
                  /
                  
                    
                      sin
                      2
                    
                    ⁡
                    
                      (
                      β
                      )
                    
                  
                
              
              ]
            
            2
          
        
        =
        min
      
    
    
      
        (
        29
        )
      
    
  




where αi are the angles 1813 between the radius from point O to the center F1 of the first trajectory segment and the successive radii from point O to the centers Fi 1809 of the trajectory segments in the subsequent frames. Expressed differently, angle αi 1813 is equal to the arccosine of the scalar product of the direction cosines of segments OF1 and OF; 1810. Angle β, in this example, is the average angle at the vertex F1 for a series of triangles OF1F2 thru OF1F10. This approximation is valid because triangles OF1F2 thru OF1F10 have a common side OF1 and there is only a small angle, approximately 0.09°, between line segments F1F2 and F1F10.

The solution of Eq. (29) is found iteratively. The mathematical formalism for the solution of Eq. (29) was shown above in Equations (7 to 10). After angle 112 (β) is determined, the next step is to find the path of the bullet for each of the 10 frames (from point F1 to the point F10). It can be found by integrating over time using the known array of velocities (using the dual waveband method described above to determine temperature and solving for velocity using Eq. (28)). Distance OF1, which is the distance from origin O to the center of the first frame, can be found by applying the Law of Sines. In the example related to Table 6, using the first 10 frames, the triangle OF1F10 can be solved using the Law of Sines, and we have:

OF1=F1F10 sin(β+α10)/sin(α10)  (30)

where segment F1F10 is the integral of the projectile velocity over the time span between the centers of the trajectory segments of the first and tenth frames.

In an actual system there will be some system noise. A simulation using a Gaussian noise with a standard deviation of 2 m/s was applied to the bullet velocities array to determine how accurate the CABM was under more realistic conditions. A Gaussian noise of 0.07 degrees, equivalent to a 1⁄3 pixel measurement accuracy, was also added to azimuth angles 1813 (αi) of FIG. 18. The results of the optimization, for the first three sequences of 10 frames each, is shown in Table 7.











TABLE 7








Distance to
Distance to


Sequence

β predicted
center of 1st
center of 1st


Number
β actual °
using CABM °
frame - actual
frame - CABM



















1
42.225
42.201
288.47
288.76


2
102.201
102.267
198.30
198.64


3
139.402
139.558
298.03
298.46









Even after three sequences, the accuracy of the predicted values is very similar to the results obtained above for the uncorrected Apparent Brightness Method, thereby confirming the validity of the new approach.
In order to retrieve the complete ballistic trajectory, including the coordinates of the point of impact and the source of fire, additional coordinates of the trajectory need to be determined. In order to accomplish this, the coefficient μ defined in Eq. (25) must first be determined. The coefficient μ can be found from the array of projectile time-speed data by solving equations given in Ref. [16]. Equation (25), from Ref. [16], can be rewritten as

V′=−μV2  (31)

The solution of Eq. (31) is given in Ref [16]:

V(t)=V0/[1+μV0(t−t0)]  (32)

where V0 is the projectile speed at the initial point of tracking. Coefficient μ can be found from the criterion:





  
    
      
        
          
            ∑
            
              i
              =
              0
            
            N
          
          ⁢
          
            
              {
              
                
                  V
                  i
                
                -
                
                  
                    V
                    0
                  
                  /
                  
                    [
                    
                      1
                      +
                      
                        μ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            V
                            0
                          
                          ⁡
                          
                            (
                            
                              
                                t
                                i
                              
                              -
                              
                                t
                                0
                              
                            
                            )
                          
                        
                      
                    
                    ]
                  
                
              
              }
            
            2
          
        
        =
        min
      
    
    
      
        (
        33
        )
      
    
  




where Vi is an array of projectile speeds obtained from the tracking data. The speed data is obtained by first determining the temperatures using the dual-waveband method and then calculating the speeds through the use of Eq. (28).

The linearization of criterion (33) gives




  
    
      
        
          
        
        ⁢
        
          
            ∑
            
              i
              =
              0
            
            N
          
          ⁢
          
            (
            
              
                
                  V
                  i
                
                -
                
                  
                    V
                    0
                  
                  /
                  
                    [
                    
                      1
                      +
                      
                        μ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            V
                            0
                          
                          ⁡
                          
                            (
                            
                              
                                t
                                i
                              
                              -
                              
                                t
                                0
                              
                            
                            )
                          
                        
                      
                    
                    ]
                  
                
                +
                
                  
                     
                    
                      
                        {
                        
                          
                            
                              V
                              0
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  t
                                  i
                                
                                -
                                
                                  t
                                  0
                                
                              
                              )
                            
                          
                          /
                          
                            
                              [
                              
                                1
                                +
                                
                                  μ
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    
                                      V
                                      0
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          t
                                          i
                                        
                                        -
                                        
                                          t
                                          0
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              ]
                            
                            2
                          
                        
                        }
                      
                      ⁢
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      μ
                    
                    )
                  
                  2
                
              
              =
              
                
                  min
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                  
                  ⁢
                  or
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      N
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              V
                              i
                            
                            -
                            
                              
                                V
                                0
                              
                              /
                              
                                [
                                
                                  1
                                  +
                                  
                                    μ
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        V
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            t
                                            i
                                          
                                          -
                                          
                                            t
                                            0
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                                ]
                              
                            
                            +
                            
                              
                                {
                                
                                  
                                    
                                      V
                                      0
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          t
                                          i
                                        
                                        -
                                        
                                          t
                                          0
                                        
                                      
                                      )
                                    
                                  
                                  /
                                  
                                    
                                      [
                                      
                                        1
                                        +
                                        
                                          μ
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            
                                              V
                                              0
                                            
                                            ⁡
                                            
                                              (
                                              
                                                
                                                  t
                                                  i
                                                
                                                -
                                                
                                                  t
                                                  0
                                                
                                              
                                              )
                                            
                                          
                                        
                                      
                                      ]
                                    
                                    2
                                  
                                
                                }
                              
                              ⁢
                              Δ
                              ⁢
                              
                                
                              
                              ⁢
                              μ
                            
                          
                          )
                        
                        *
                      
                      ⁢
                      
                        {
                        
                          
                            V
                            0
                            2
                          
                          ⁢
                          
                            
                              (
                              
                                
                                  t
                                  i
                                
                                -
                                
                                  t
                                  0
                                
                              
                              )
                            
                            /
                            
                              
                                [
                                
                                  1
                                  +
                                  
                                    μ
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        V
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            t
                                            i
                                          
                                          -
                                          
                                            t
                                            0
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                                ]
                              
                              2
                            
                          
                        
                        }
                      
                    
                  
                
                =
                0.
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




Equation (34) can be solved iteratively assuming at each step a fixed value for μ, calculating Δμ, correcting μ, and repeating the first step (plugging the new value for μ into Eq. (34)) until convergence is achieved. The optimization in the example simulation, assuming a starting value of μ=0.002, for the first three iterations, μ, yields the following values: 0.001430, 0.001498 and 0.001499. By the third iteration the value of μ is nearly indistinguishable from the actual value of the coefficient for the tracked bullet 0.001493.
For nearly all cases the above approach is sufficiently accurate. If a more accurate value for μ is required that includes the effect of gravity (a secondary effect that has an effect on the accuracy of parameter μ in the range of 0.1%), a more complex solution for determining this parameter can easily carried out. In this case the system of differential equations (25) is solved using the known data (scalar velocities and accelerations in two dimensions and the gravity constant). Several approaches to solving this set of equations are known to those skilled in the field of mathematics. A suitable approach is the method of least squares, where μ is fit to the system of equations (25).
The above procedure assumes that the drag coefficient is nearly constant over the lifetime of the track. That is true for many projectiles. For example, for most bullets the drag coefficient changes very little at speeds above Mach 1.5. Below these speeds the temperature of the bullet becomes too low to be detected with good accuracy by the dual-waveband method, based on the sensitivity associated with present day IR camera/optics. Even if there is some change in the drag coefficient value it is typically so small at the supersonic speeds of modern day projectiles that it introduces only a small error in the prediction of projectile range and azimuth.
3.3 Back Tracking Procedure.
The following section delineates the procedure for tracking the projectile back to the source of fire. The general principle is to start at a known or measured position along the trajectory and to iteratively trace backwards using negative time steps (negative gravity, drag, μ etc.) until a particular criterion is reached. For example, if a bullet type can be determined, the trajectory can be traced backward in time to the point at which a preselected maximum speed, corresponding to the muzzle velocity of the sort of gun that fires that sort of bullet, is reached. The actual firing position will be somewhere along the trajectory which falls within a particular range of speeds. Where the projectile is other than a bullet, an analogous initial speed may be identified. Another approach is to compare the predicted trajectory with a three-dimensional map of the battlefield looking either for an intersection with a solid object or for a likely point where a shooter might be stationed (window in building, point on hill, etc.). A more detailed description of an embodiment of the procedure will now be described.
The first step is to determine the 3D coordinates for point F1 (xf1, yf1, zf1) from the distance OF1 and azimuth (AzF1) and elevation (EF1) data already obtained for the center of the trajectory segment of the first frame. Distance OF1 was found using CABM described above. The elevation and azimuth information is obtained from the central position of the smeared image on the sensor of each trajectory segment. The method of calculating this from the smeared-image central-point position is described in Section 1.4.1 above, mathematically expressed in Eq. (11). The three-dimensional coordinates are then calculated as follows:

xf1=OF1 cos(EF1)sin(AzF1)

yf1=OF1 cos(EF1)cos(AzF1)

zf1=OF1 sin(EF1)  (35)

In addition to the coordinates of the point F1, in order to perform the back tracking algorithm one needs to know the velocity vector at this point. To find the direction cosines of the velocity vector one must calculate a second coordinate on the trajectory. Ideally, the second coordinate should be calculated using the least square optimization algorithm described above, which reduces the impact of noise. In the above example such a point would be F11. This is the center of frame number eleven (xf11, yf11, zf11), which was the first frame of the second sequence of 10 frames used in CABM of Table 7. The coordinates of the point F11 (xf11, yf11, zf11) are calculated in the same way as the coordinates for point F1. However, if fewer frames are available then one can choose the positional data for a second frame that is not determined by least square optimization, albeit with a slightly reduced accuracy.
Plane 1802 (plane P), in which the trajectory and its velocity vectors reside, is assumed to be vertical (orthogonal to the plane X,Y of FIG. 18), see Ref. [14]. So the two exemplary points F1 and F11 unambiguously define plane 1802. This condition yields the direction cosines (Nx, Ny, Nz) of vector 1814 (vector N in FIG. 18). Vector N is orthogonal to the plane P. For the vector N, we have Nz=0, because plane P is orthogonal to the plane X,Y [14]. Vector N is also orthogonal to the segment F1F11. The direction cosines FFx and FFy of the segment F1F11 can then be determined as follows:

S{(Xf11−xf1)2+(yf11−yf1)2+(Zf11−zf1)2}1/2 

FFx=(xf11−xf1)/S 

FFy=(yf11−yf1)/S 

As vector N is orthogonal to the segment F1F11 then the following is also true:

Nx*FFx+Ny*FFy=0.

Nx=sqrt(1·−Ny2)  (36)

or

Ny*FFy+sqrt(1−Ny2)*FFx=0  (37)

Eq. (37) gives us the value of the Ny and substituting Ny into Eq. (36) yields Nx. This procedure completes the definition of vector.
The velocity vector of the projectile velocity at point F1 with direction cosines (vx, vy, vz) is located in the plane P and the CABM determines angle β for the segment OF1. With this information the following relationships can be derived:

−(xf1*vx+yf1*vy+zf1*vz)/(xf12+yf12+zf12)1/2=cos(β)  (38)

Nx*vx+Ny*vy=0  (39)

vx2+vy2Vz2=1  (40)

and

vy=−Nx*vx/Ny  (41)

The direction cosine vy can be substituted into equations (38) and (40). This substitution into Eq. (38) yields a solution for vz:

vz={−cos(β)*sqrt(xf12+yf12+zf12)−vx*xf1+yf1*Nx*vx/Ny}/zf1  (42)

By substituting vz from Eq. (42) and vy from Eq. (41) into Eq. (40), it is possible to solve for vx. Then, by substituting vx into Eq. (41) it is possible to solve for vy, and by substituting vy and vx into Eq. (38) it is possible to solve for vz.
By knowing the velocity vector of the projectile velocity, the coordinates of the center of the first tracked frame and the coefficient μ for Eq. (31) and Eq. (35), the projectile can be back tracked using Eq. (25) with negative time steps Δt. The velocity-drag data suggest in the above example that the projectile is an M16 bullet. This bullet has a muzzle velocity 975 m/s. To reach this muzzle velocity using the above back tracking algorithm, it took 44 steps, each with a Δt of −0.0005 sec.
The coordinates of the center F1 of the first tracked trajectory segment using the present algorithms are xf1=−135.19 m, yf1=253.896 m, zf1=26.177 m with an estimated projectile speed v=945.38 m/s. The coordinates of the trajectory point at the 44th back step with a projectile velocity v=975.1 m/s were found to be: x=−154.59m, y=260.93m, 26.28m. The simulated actual point for a velocity of 975 m/s is displaced from the projected point by only 0.383 m. This means, in this case, if the source of fire was located in a building one could track the shooter with an accuracy sufficient to locate a specific window or other known feature.
An alternative to assuming a muzzle velocity for the projectile is to continue the back tracking of projectile's trajectory until it intersects a likely point on the battle landscape such as a hillside, building, or trench. In that case the source of fire can be identified and the muzzle velocity of the source of fire can be accurately estimated, confirming or correcting any initial guess at the type of weapon. If there are several suspicious points, for example, because the likely part of the trajectory is parallel to the ground and/or passes through open cover, a best guess can be determined by comparing the muzzle velocities at these suspicious points with known weapon specifications. It should be apparent that once the muzzle velocity and point of fire are determined, together with the other trajectory data (array of velocities), the coefficient μ will unambiguously suggest the type of firing weapon.
3.4 Forward Tracking.
The forward tracking procedure is similar to the back tracking but needs to take into consideration that with reduction of the projectile speed down to 1.5 Mach number, the drag coefficient increases. By knowing the drag coefficient of a projectile moving with high supersonic speed, the drag coefficient at lower speeds can be predicted with high accuracy using correction coefficient K=1/√{square root over (M)}, as shown in Ref. [14] Ch. 5.7. When a projectile's flight-time is dozens of frames, timely notification of the target would enable terminal defenses to engage it or deploy defensive countermeasures against it.
3.5 Tracking in Non-Homogeneous Atmosphere Conditions Including Drag and Gravity Effects.
The CABM is sufficiently accurate for the case when there is a homogeneous (clear) atmosphere and the optical transmission is the same for all azimuth angles during projectile tracking. This is true for many practical cases. Even when the atmosphere over the battlefield space is distorted by fire, explosions, smoke clouds and extensive arms fire, the above CABM solution can be considered as the first approach to the Corrected Triangles Fitting Method (CTFM). The CTFM relies on the fact that the ratio of the powers in the two key MWIR sub wavebands will not be significantly affected by nonhomogeneous atmospheric conditions. The absolute transmission may be affected, but the transmission in each MWIR sub waveband will be affected nearly the same way along any line of sight even under these conditions. Therefore, the projectile temperature and hence its variable speed, as well as the array of azimuth-elevation angles, will be known.
In this optimization the triangles with known base lengths (integrals of the projectile speed over acquisition time) and known angles 1813 (angles αi) at the vertex, which are derived using the azimuth-elevation angles of segments 1810 (segments OFi in FIG. 18) are used. One main difference from the original triangles fitting method is that CTFM calculates the triangle bases using the time integral of the variable projectile speed as opposed to the product of a constant speed times the frame time. An additional difference is that optimization is carried out over a restricted number of frames. In the following examples the optimization is carried out over ten successive frames, to reduce the effect of the curved trajectory. A series of initial distances 1810 (distance OF) as shown in FIG. 18 for the first set of ten frames, and corresponding distances OF11, OF21, etc. for each successive set of 10 frames) are tested. For each distance OF1, OF11, OF21 . . . , the angle 1812 (β) is found which optimally fits the data. The pair {OFj1, β} which best fit the data with the minimum deviation from so-called triangle requirement, is what is called the corrected triangle fitting solution.
It should be obvious that the CTFM algorithm can be sped up by using angle and distance 1810, if those angles have already been found from the CABM. However, the CTFM is quite robust on its own. A more detailed description of the algorithm follows.
For every triangle OF1Fi (FIG. 18) the law of sines yields:

Si/sin(αi)=OF1/sin(αi+β)  (43)

where





  
    
      S
      i
    
    =
    
      
        ∫
        
          t
          0
        
        
          t
          i
        
      
      ⁢
      
        
          V
          ⁢
          
            
          
          (
          t
          )
        
        ⁢
        
          d
          t
        
      
    
  
  ,
  
    
      or
      ⁢
      
        
      
      ⁢
      
        S
        i
      
    
    =
    
      
        ∑
        
          k
          =
          0
        
        
          k
          =
          i
        
      
      ⁢
      
        
          V
          k
        
        ⁢
        Δ
        ⁢
        
          
        
        ⁢
        t
      
    
  




(where Δt is the duration of the frame).

Hence in the example with ten frames, the value R to be minimized can be defined as follows:




  
    
      
        R
        =
        
          
            ∑
            
              k
              =
              1
            
            10
          
          ⁢
          
            
              [
              
                
                  sin
                  ⁡
                  
                    (
                    
                      α
                      k
                    
                    )
                  
                
                -
                
                  
                    S
                    k
                  
                  ⁢
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            α
                            k
                          
                          +
                          β
                        
                        )
                      
                    
                    /
                    
                      OF
                      1
                    
                  
                
              
              ]
            
            2
          
        
      
    
    
      
        (
        44
        )
      
    
  




where the distance OF1 and angle β are unknown variables that must be first estimated.

Equation (44) is the criterion for nonlinear least-square optimization. The linearization of criterion R can be stated as follows:




  
    
      
        R
        =
        
          
            
              ∑
              
                k
                =
                1
              
              N
            
            ⁢
            
              
                [
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        k
                      
                      )
                    
                  
                  -
                  
                    
                      S
                      k
                    
                    ⁢
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              α
                              k
                            
                            +
                            β
                          
                          )
                        
                      
                      /
                      
                        OF
                        1
                      
                    
                  
                  -
                  
                    
                      {
                      
                        
                          S
                          k
                        
                        ⁢
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  α
                                  k
                                
                                +
                                β
                              
                              )
                            
                          
                          /
                          
                            OF
                            1
                          
                        
                      
                      }
                    
                    ⁢
                    Δβ
                  
                
                ]
              
              2
            
          
          =
          min
        
      
    
    
      
        (
        45
        )
      
    
  




The optimization of angle β is carried out in a series of iterative cycles. In each cycle a correction Δβ to angle β is calculated as:




  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            β
          
          =
          
            
              {
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  10
                
                ⁢
                
                  
                    [
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            k
                          
                          )
                        
                      
                      -
                      
                        
                          S
                          k
                        
                        ⁢
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  α
                                  k
                                
                                +
                                β
                              
                              )
                            
                          
                          /
                          
                            OF
                            1
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        S
                        k
                      
                      ⁢
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                α
                                k
                              
                              +
                              β
                            
                            )
                          
                        
                        /
                        
                          OF
                          1
                        
                      
                    
                    ]
                  
                
              
              }
            
            /
            Q
          
        
        ,
        
          

        
        ⁢
        
          
        
        ⁢
        
          Q
          =
          
            
              ∑
              
                k
                =
                1
              
              10
            
            ⁢
            
              
                [
                
                  
                    S
                    k
                  
                  ⁢
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            α
                            k
                          
                          +
                          β
                        
                        )
                      
                    
                    /
                    
                      OF
                      1
                    
                  
                
                ]
              
              2
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




Then the new β=β+Δβ is calculated and the cycle is continued until there is a convergence of the absolute value of Δβ that is less than a required value (one that is as close to zero as needed). All the resultant values for segment OF1 are tested and the (OF1, β) pair that achieves the minimum R is chosen as the solution.
The optimization for the next ten frames repeats Equations (43)-(46), using distance OF11 instead of distance OF1 and angles (αk−α11) instead of angles αk. Subsequent blocks of ten frames are optimized similarly.
Once the values of the pair (OF1, β) have been optimized as described above using the data for frames 1 to 10, the value of distance OF11 can be recalculated using the trajectory optimized over frames 1 to 10. That recalculated value may be used as the initial value for the optimization of frames 11 to 20, and so on. Because the distance OF1, OF11, etc. at the beginning of each block of 10 frames is an optimizable variable, the optimization may result in a series of 10-frame segments of trajectory that do not quite join up end to end, unless a more complicated recursive optimization, such as a second-level least-squares optimization over the once-optimized OF1, OF11, etc., is used. Alternatively, the optimization may be performed for the last block of 10 frames independently (in which case, optimization of the intermediate blocks may not be necessary). As was discussed in Sections 3.3 and 3.4 above, only the first and last blocks of 10 frames are really needed to extrapolate the trajectory backward to the source of fire and forward to the point of impact, and the CTFM process as described above is believed to be capable of providing sufficiently accurate trajectories for each of those blocks without a second-level optimization.
In the example the data for the projectile trajectory of Table 6 was used. A Gaussian noise with a standard deviation of 2 m/s was applied to the bullet velocities array to determine how accurate the CTFM was under more realistic conditions. A Gaussian noise of 0.07 degrees, equivalent to 1⁄3 pixel measurement accuracy, was also added to azimuth angles 1813 (αi) of FIG. 18. Seven potential solutions using this method for the first ten frames together with their respective values of R are shown in Table 8.









TABLE 8





OF1
β



meters
degree
R

















281.76
38.65
0.000990


288.26
41.81
0.000011


288.51
41.91
0.0000058


288.76
42.01
0.0000037


289.01
42.11
0.0000043


289.26
42.22
0.0000074


292.76
43.36
0.000335









The best estimate for distance OF1 in Table 8 is 288.76 m which has β=42.01° and R equal to 0.0000037. These results show that the CTFM has a similar accuracy to the CABM (The results of CABM were shown in the Table 7) for the first set of ten frames. For the case of frame eleven to twenty, the optimized distance OF11 is 198.39 m and angle β=102.51°, which shows a similar degree of accuracy. However, the results of CTFM optimization for frames twenty one to thirty show a slight reduction of accuracy compared with the CABM. The CTFM optimization predicts OF21 is 300.71 m, whereas the actual distance is 298.03 m (Table 7). For angle β CTFM predict 139.22° with the actual angle of 139.402°, still an excellent result. The reason for this reduction of accuracy in the later frames is that the azimuth angular noise level does not change over the trajectory, whereas the angles αk in Equations (43) to (46) in the least-square optimization are getting smaller as the projectile recedes from the tracker. (As listed in Table 6, the closest approach is just after F9.) So in this case the noise is going up relative to the absolute size of the angle to which that noise is applied. The differences in azimuth angles are (α10−α1)=53.9° and (α30−α21)=12.1°. This changes the signal to noise ratio in the input data for the least-square method and results in the reduction in accuracy. Some reduction in accuracy is expected for a turbid atmosphere. Nevertheless the results are sufficiently accurate to back-track to the shooter and still much better than any known technology that is field deployable (one or more orders of magnitude better with regard to accuracy to determine range and azimuth).
CONCLUSION
The mathematical approach to solving the trajectory of bullets described herein can be extended to handle larger, heavier projectiles, where the projectile travels a much longer distance than a bullet. In the embodiments of FIGS. 8 and 9, the trajectory was assumed to be at constant speed in a straight line. In the Corrected Apparent Brightness Method of FIG. 18, the trajectory was assumed to be a ballistic curve at a speed that varies with acceleration along the line of flight. The trajectory can be accurately predicted and estimated by the application of well known ballistic laws (exemplified in Refs. [13], [14]) and properly taking into account the prevailing conditions (wind direction, location on the Earth, etc.) on the battlefield. Some of the secondary effects that could be considered in this analysis include: the Coriolis and Magnus effects; atmospheric conditions such as air density, temperature, pressure and humidity; lift and sideways aerodynamics; and wind. However, one skilled in the field of the mathematics of ballistics or projectiles would be able to extend the method revealed herein once the principles taught are fully understood, and such extensions are within the scope of the present invention.
Such a skilled person would know how to balance the complexity of the algorithms and model chosen with the required or desired tracking accuracy of the system, the nature of the projectiles to be tracked, and the limitations of the available hardware and software (computer overhead, resolution of camera, etc.). For example, the solution is simplified, and the calculation overhead is reduced, if the trajectory is assumed to follow a known curve in a single plane. Applying the methodology of FIG. 18 with a horizontal backward acceleration due to drag and a vertical downward acceleration due to gravity opposed by drag, using both parts of Eq. (25), may then be a better trade-off between accuracy and simplicity. As may be seen from Eq. (25), the X and Z components both include the overall speed V. For maximum precision, it would therefore be necessary to compute the X and Z components in parallel, updating the value of V at each stage from Equations (26) and (27). Even if the real curve cannot be handled analytically, it may be possible to match an observed trajectory with sufficient accuracy to a stored numerical model of a typical real trajectory.
When discussing the case of a bullet, it was assumed in the methods of section 1.4 that the trajectory could be approximated to a straight line, and the most convenient plane for the computation illustrated in FIG. 8 was then usually the plane containing the tracking device O and the trajectory line. However, if the actual trajectory is a curve (as was the case for the methods of section 3) in a plane (for example, artillery and mortar shells are often fired in a high, arching trajectory with only a much smaller sideways deflection) then projecting the trajectory into this plane may be preferred. However, in matching the speed to the incremental travel distance Δ, allowance must then of course be made for the component of velocity perpendicular to the plane of computation. If the arching trajectory is viewed from the side, its shape can be extracted before projecting into a plane.
At various places in the description, an integral or sum of a time varying quantity is used. As is pointed out, for example, in the alternative definitions of Si immediately following Equation (43), these are conceptually alternative. A true integral would be more precise, but a sum over steps determined by the frame rate of the detector camera is more practical, and is adequate for the purposes of the present methods and apparatus. Accordingly, references to “integral” or “sum” should in general be understood as encompassing both a sum and a true integral.
FIG. 16 shows the methods of the present embodiments being carried out on a conventional laptop computer. The described methods may be carried out, and the described calculating apparatus may be embodied, in any suitable form of logic, including a general purpose computer suitably programmed, application specific circuitry, or a combination. General purpose computers are cheap and easy to obtain, which may be a significant advantage when they are being used in a location under enemy fire, and are therefore likely to be damaged. However, an application specific integrated circuit (ASIC) or the like may give better performance in certain of the computationally intensive embodiments. A general purpose personal computer with an ASIC accelerator card may be attractive in some configurations. Provision for “tamper resistance,” to destroy or disable key parts of the device if it falls into the possession of an unauthorized user may be desired and, depending on the strength of the protection desired, may be easier to include in specially constructed hardware.
Although distinct embodiments have been described and shown in the several drawings, features from the different embodiments may be combined in a single embodiment.
Accordingly, reference should be made to the appended claims, rather than to the foregoing specification, as indicating the scope of the invention.
REFERENCES


    1. D. Leslie, H. Hyman, F. Moore, M. Squire, Surveillance, detection and 3D infrared tracking of bullets, rockets, mortars, and artillery, SPIE Proc. Vol. 4393, pp. 193-201, 2001.
    2. G. Williams, A. Barter “Dual band MWIR/LWIR radiometer for absolute Temperature measurement” SPIE Proc. Vol. 6205-23, 2006.
    3. A. Richards “Application for high-speed infrared imaging” SPIE Proc. Vol. 5580, pp. 137-145, 2005.
    4. W. Rohsenow, J. Hartnet, “Handbook of Heat Transfer”, McGraw-Hill, 1973, Chapter 15.
    5. A Takeya, Tl Kuroda, K. Nishiguchi and A Ichikawa, Omnidirectional vision system Using Two mirrors, SPIE Proceedings, Volume 3430, Pages 50-60 (1998)
    6. U.S. Pat. No. 6,611,282 B1 to Trubko, et al.
    7. www.FLIR.COM/US.
    8. R. Siegel, J. Howell, “Thermal radiation heat transfer”, McGraw-Hill, 1980.
    9. D. J. Hudson, “Statistics, Lectures on Elementary Statistics and Probability”, Geneva, 1964.
    10. T. Wu, “R2V Conversion: Why and How ?”, GeoInformatics, 6, Vol. 3, 2000, pp. 28-31.
    11. D. Hearn, M. Baker, “Computer Graphics”, Sec. 5-8, Prentice Hall, 1997.
    12. K. Symon, “Mechanics”, Adison-Wesley, 1971.
    13. C. L. Farrar, D. W. Leeming, “Military Ballistics”, Elsevier Science Ltd; 1983.
    14. R. McCoy, “Modern External Ballistic”, Schiffer Publ. Co., 1999.
    15. C. Frohlich, “Aerodynamic drag crisis and its possible effect on the flight of baseballs”, Am. J. of Phys., Vol. 52, No. 4, 1984.
    16. L. Ng, T. Karr, “Performance Analysis of Bullet Trajectory estimation. Approach, Simulation, and Experiments”, LLNL, November, 1994.

## Claims

1. A projectile tracking device comprising:
detector apparatus for converting into electronic form images in at least two infrared wavebands;
optics for projecting onto the detector apparatus an image of a scene in the at least two infrared wavebands;
logic operative to obtain from the images in electronic form apparent brightnesses of the projectile at the optics in at least two infrared wavebands;
logic operative to estimate the speed of the projectile varying over time from the varying ratio at successive times of the measured apparent brightnesses in said at least two infrared wavebands;
logic operative to obtain from the images in electronic form an azimuth of the projectile from the optics at successive times; and
logic operative to estimate the direction of the trajectory of the projectile from the measured azimuths in combination with the ratio between the measured apparent brightnesses;
wherein the logic operative to estimate the direction of the trajectory of the projectile further comprises logic operative to estimate relative changes over time in the distance from the optics to the projectile from changes in at least one of the apparent brightnesses over time, including logic operative to correct the at least one measured apparent brightness for the change in absolute brightness as a function of temperature as a function of the estimated varying speed of the projectile.

2. The tracking device of claim 1, wherein the logic is further operative to estimate parameters of the projectile trajectory using a method comprising measuring the azimuth of the projectile from the optics at successive times, and estimating the direction of the trajectory of the projectile from the measured azimuths in combination with the ratios between the corrected measured apparent brightnesses.

3. The tracking device of claim 2, wherein the logic is further operative to estimate parameters of the projectile trajectory using at least a method comprising measuring the direction of the projectile from the optics at successive times, and estimating the trajectory of the projectile from the measured directions in combination with the time-varying estimated speed; and
wherein the logic is operative to choose based on atmospheric conditions between projectile trajectory parameters calculated with at least said two methods of estimating parameters.

4. The projectile tracking device of claim 1, wherein the logic is operative to relate acceleration to speed using a lumped coefficient μ=−V′/V2, where μ is calculated from time dependent projectile velocity data and lumps together parameters that at least initially are not individually known.

5. The tracking device of claim 1, wherein the logic is further operative to estimate parameters of the projectile trajectory using a method comprising measuring the azimuth of the projectile from the optics at more than two successive times, estimating the distances traveled by the projectile between the successive times using the speeds estimated at successive times, and estimating the direction of the trajectory of the projectile as a best fit to the successive distances and azimuths.

6. The tracking device of claim 5, wherein the logic is operative to estimate the best fit by defining imaginary triangles, each triangle formed by radii from the optics to the projectile at two of the successive times, and a segment of trajectory of length calculated using the estimated speed or speeds of the projectile between the said two successive times bounding each triangle.

7. The tracking device of claim 6, wherein the logic is further operative to estimate parameters of the projectile trajectory using a method further comprising measuring the direction of the projectile from the optics at successive times, and estimating the trajectory of the projectile from the measured directions in combination with a sum or integral of the estimated speeds;
and to provide a final estimate using a comparison of estimates from at least two said methods of estimating parameters.

8. The tracking device of claim 1, wherein the detector apparatus images the scene at regular intervals of time, and the successive azimuths and speeds are derived from successive images.

9. The tracking device of claim 1, further comprising logic operative to extrapolate the speed and trajectory of the projectile backwards in time to calculate a starting point at which the projectile would have had a speed corresponding to an initial speed of a weapon consistent with observed characteristics of the projectile.

10. The tracking device of claim 1, wherein the operative logic comprises a general purpose computer suitably programmed to carry out the operations of the logic.

11. A nontransitory computer readable storage medium carrying a computer program that will cause a suitable general purpose computer to carry out a method comprising:
measuring apparent brightnesses of a projectile at an observing location in at least two infrared wavebands at successive times;
estimating the speed of the projectile as a function of time from the ratio of the apparent brightnesses measured in at least two infrared wavebands at successive times;
measuring an azimuth of the projectile from the observing location at successive times;
estimating the direction of the trajectory of the projectile from the measured azimuths in combination with the ratios at different times between the measured apparent brightnesses; and estimating relative changes in the distance from the optics to the projectile over time from changes in at least one of the apparent brightnesses over time, and correcting the measured apparent brightnesses for the change in absolute brightness as a function of temperature as a function of the estimated varying speed of the projectile.

12. A method of tracking a projectile in air, comprising:
measuring apparent brightnesses of the projectile at an observing location in at least two infrared wavebands at successive times;
estimating the speed of the projectile as a function of time from the ratio of the apparent brightnesses measured in the at least two infrared wavebands at successive times;
measuring an azimuth of the projectile from the observing location at successive times;
estimating the direction of the trajectory of the projectile from the measured azimuths in combination with the ratios between the measured apparent brightnesses; and
estimating relative changes in the distance from the optics to the projectile over time from changes in at least one of the apparent brightnesses over time, and correcting the measured apparent brightnesses for the change in absolute brightness as a function of temperature as a function of the estimated varying speed of the projectile.

13. The method of claim 12, further comprising combining the estimated projectile speeds with the estimated projectile trajectory direction for calculating distances from the sensor to the projectile.

14. The method of claim 12, comprising estimating parameters of the projectile trajectory using a method comprising measuring the azimuth of the projectile from the optics at more than two successive times, estimating the distances traveled by the projectile between the successive times using the speeds estimated at successive times, and estimating the direction of the trajectory of the projectile as a best fit to the successive distances and azimuths.

15. The method of claim 12, wherein estimating the best fit comprises defining imaginary triangles, each triangle formed by radii from the optics at two of the successive times, and a segment of trajectory of length calculated using the estimated speed or speeds of the projectile between the said two successive times.

16. The method of claim 12, comprising calculating projectile trajectory parameters by a method comprising measuring the azimuth of the projectile from the observing location at successive times, and estimating the direction of the trajectory of the projectile from the measured azimuths in combination with the ratios between the corrected measured apparent brightnesses.

17. The method of claim 16, further comprising calculating projectile trajectory parameters by at least a method further comprising measuring the direction of the projectile from the observing location at successive times, and estimating the trajectory of the projectile from the measured directions in combination with the estimated speed; and
choosing based on atmospheric conditions between projectile trajectory parameters calculated with at least two methods of calculating projectile trajectory parameters.

18. The method of claim 12, further comprising extrapolating the speed and trajectory of the projectile backwards in time to calculate a starting point at which the projectile would have had a speed corresponding to an initial speed of a weapon consistent with observed characteristics of the projectile.