Abstract:
A method for photogrammetric block adjustment of satellite imagery using a simplified adjustment model in-lieu of a physical camera model. A known relationship between image space line and sample coordinate and object space X, Y, Z coordinates is provided by a mathematical model. Observations comprise ground control points, tie points, or other observations for which approximate knowledge of object and image space coordinates is available. The photogrammetric block adjustment determines the parameters of the adjustment model to best fit the observations and measurements. Object coordinates can be calculated for features measured in the images after block adjustment. In one embodiment the method is utilized for photogrammetric block adjustment of satellite imagery described by a rational polynomial (RPC) camera model rather than by a physical camera model.

Description:
FIELD OF THE INVENTION 
     The present invention relates to photogrammetric block adjustments and more particularly to mapping image coordinates from a photographic picture or image to ground coordinates using a Rational Polynomial Camera block adjustment. 
     BACKGROUND OF THE INVENTION 
     Topographical maps, orthophoto maps, Digital Elevation Models, and many other metric products are generated from satellite and aerial images. In order to satisfy accuracy requirements for these metric products, a need exists for an accurate mathematical sensor model relating an image coordinate space to an object coordinate space. As shown in FIG. 1, an image coordinate space  110  is a 2-dimensional coordinate system defined only with respect to the image. An object coordinate space  120 , however, is a 3-dimensional coordinate system defined with respect to an object being imaged (the imaged object is not shown in FIG. 1) on the ground  130 . As will be explained further below, mathematical sensor models allow one to determine the object coordinates in object coordinate space  120  of the object being imaged from the image coordinates in image coordinate space  110 . However, these mathematical sensor models are complex. Therefore, it would be desirous to develop a simplified model. 
     SUMMARY OF THE INVENTION 
     The foregoing and other features, utilities and advantages of the invention will be apparent from the following more particular description of a preferred embodiment of the invention as illustrated in the accompanying drawings. It is an object of the present invention to describe a simplified method and apparatuses for photogrammetric block adjustment. The present invention uses simple offsets, scale factors, and polynomial adjustment terms in either image or object space to achieve effects analogous within a high degree of accuracy to the effects achieved by adjustment of physical model parameters. 
     It is a further object of the present invention to have a method of photogrammetric block adjustment that does not present the numerical ill-conditioning problems of classical techniques. The numerical conditioning is achieved by having only one adjustment parameter that represents multiple physical processes that have substantially the same effect. 
     Another object of the present invention is to simplify development of photogrammetric block adjustment software by utilizing a generic camera model describing object-to-image space relationships and a generic adjustment model for block-adjusting parameters of that relationship. Use of generic models reduces the effort associated with developing individual camera models. 
     Described is an algorithm for photogrammetric block adjustment of satellite images. The object-to-image relationship of each image in the block of images is described by a rational polynomial camera (RPC) mathematical model. To the basic object-image model of each image, an adjustment model is added. The adjustment model comprises simple offsets, scale-factors, and/or polynomial adjustment terms in either object or image space. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the present invention, and together with the description, serve to explain the principles thereof. Like items in the drawings are referred to using the same numerical reference. 
     FIG. 1 is a plan view of an image space coordinate system and an object space coordinate system; 
     FIG. 2 is a plan view of the image space coordinate system for a digital image; 
     FIG. 3 is a plan view of the image space coordinate system for an analog image; 
     FIG. 4 is a functional view of a frame image camera configuration having images of the ground; 
     FIG. 5 is a functional diagram of the image space coordinate system to the ground space coordinate system using a perspective projection model for the frame image camera configuration of FIG. 4; 
     FIG. 6 is a functional diagram of the perspective projection model for the pushbroom image camera configuration; 
     FIG. 7 is a functional diagram demonstrating a block adjustment of images; 
     FIG. 8 is a functional diagram showing the principle of RPC generation in accordance with the present invention; 
     FIG. 9 is a functional diagram showing application of one embodiment of the present invention to block adjust images; and 
     FIG. 10 is a flowchart illustrating a method of performing a block adjustment in accordance with one embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION 
     As shown in FIG. 1, the image space  110  is a 2-dimensional coordinate system defined with respect to the image. The object space  120  is a 3-dimensional coordinate system defined with respect to an object being imaged. As will be explained, conventional mathematical models relating the image space  110  to the object space  120  are complex. The present invention eliminates many of the complexities by using a parametric model, such as, for example, a rational polynomial camera model, instead of the conventional physical camera models. 
     Image Coordinate Systems 
     FIG. 2 shows an image coordinate system  200  for digital images. Image coordinate system  200  has a line axis  210  and a sample axis  220 . Image coordinate system  200  comprises a 2-dimensional matrix of a plurality of pixels  230 . The upper left corner or the center of the upper left pixel  240  defines the origin of the image coordinate system  200 . The line axis  210  points down along the first column of the matrix. The sample axis  220  points right along the first row of the matrix and completes the right-handed coordinate system. 
     FIG. 3 shows an image coordinate system  300  for analog (photographic) images. In image coordinate system  300 , fiducial marks A, B, C, and D define the image coordinate system. As one of skill in the art would recognize, the fiducial marks A, B, C, and D are typically mounted on the body of the camera and exposed onto the film at the time of imaging. In particular, a line  310  connecting fiducial marks A and C intersects a line  320  connecting fiducial marks B and D at intersection point  330 . The intersection point  330  defines the origin of image coordinate system  300 . A line  340  extending between the origin at intersection  330  and fiducial mark C define the x-axis (line  340  coincides with line  310 ). A line  350  extending upwards from intersection  330  at a right angle to the x-axis line  340  defines the y-axis and completes the right-handed coordinate system (line  350  does not necessarily coincide with line  320 ). 
     Camera Models 
     While many different sensor configurations can be used to image objects on the ground, two of the most popular sensor configurations for satellite and aerial imaging systems are the frame camera configuration and the pushbroom camera configuration. Typically, the frame camera configuration is used in aerial imaging systems. Conversely, satellite imaging systems typically use the pushbroom camera configuration. 
     Frame Camera Model 
     As shown in FIG. 4, the frame camera configuration images, or takes the picture of, an entire image (frame)  410  at one instance of time. Therefore, as the camera moves over the ground, the series of complete images  410  are taken in succession. A mathematical relationship exists that allows the each image  410  to be mapped to the ground  420 . 
     The relationship between the image coordinate system and the object coordinate system is shown in FIG.  5 . The relationship shown in FIG. 5 is known as the perspective projection model. In particular, FIG. 5 shows an image space  510  and a ground space  520  (or object space  520  ground space and object space are used interchangeably in this application). The camera taking the image is located at a perspective center (“PC”)  530 . Directly below the PC  530  in the image space  510  is the principal point  540  having image space coordinates x o  and y o . (As one of skill in the art would know, the image space is actually above the PC, but it is mathematically equivalent to model the system with the PC above the image space.) Image space  510  also has a point i  545  having image space coordinates x i  and y i . The PC  530  has ground space coordinates X pc , Y pc , and Z pc  (not shown in ground space  520 ). Furthermore, point i  545  corresponds to a point g  555  having ground space coordinates X g , Y g , and Z g . Finally, a line  560  can be drawing that connects PC  530 , point i  545 , and point g  555 . 
     The mathematical relationship of collinearity defines the line  560 . Under the principles of collinearity, the PC  530 , the image point i  545 , and the corresponding ground point g  555  all lie on the same line. For the frame camera model, the principle of collinearity can be expressed by the following set of equations:                  F     x   i       =         x   i     -     x   0     +     c              m   11          (       X   g     -     X   PC       )       +       m   12          (       Y   g     -     Y   PC       )       +       m   13          (       Z   g     -     Z   PC       )               m   31          (       X   g     -     X   PC       )       +       m   32          (       Y   g     -     Y   PC       )       +       m   33          (       Z   g     -     Z   PC       )               =   0            
            F     y   i       =         y   i     -     y   0     +     c              m   21          (       X   g     -     X   PC       )       +       m   22          (       Y   g     -     Y   PC       )       +       m   23          (       Z   g     -     Z   PC       )               m   31          (       X   g     -     X   PC       )       +       m   32          (       Y   g     -     Y   PC       )       +       m   33          (       Z   g     -     Z   PC       )               =   0            
            where   :     
          [           m   11           m   12           m   13               m   21           m   22           m   23               m   21           m   32           m   33           ]       =     [           cos                 φcos                 κ             cos                 ωsin                 κ     +     sin                 ωsin                 φcos                 κ               sin                 ωsin                 κ     -     cos                 ωsin                 φcos                 κ                   -   cos                   φsin                 κ             cos                 ωcos                 κ     -     sin                 ωsin                 φsin                 κ               sin                 ωcos                 κ     +     cos                 ωsin                 φsin                 κ                 sin                 φ             -   sin                   ωcos                 φ           cos                 ωcos                 φ                      ]               (     Eq   .              1     )                                
     is a rotation matrix; and 
     ω, φ and κ are the (pitch, roll, and yaw) attitude angles of the camera; 
     X g , Y g , Z g  are the object coordinates of point g  555 ; 
     X PC , Y PC , Z PC  are the object coordinates of point PC  530 ; 
     x i  and y i  are the image coordinates of point i  545 ; 
     x o  and y o  are the image coordinates of the principal point  540 ; and 
     c is the focal length of the camera. 
     The attitude angles of the camera (ω, φ and κ) and the position of the perspective center (X PC , Y PC , Z PC ) are the so-called exterior orientation parameters of the frame camera. The interior orientation parameters of the frame camera comprise the focal length (c), the principal point location (x 0  and y 0 ), and could optionally include lens distortion coefficients (which are not included in the above equations, but generally known in the art). Other parameters directly related to the physical design and associated mathematical model of the frame camera could also be included as is known in the art. 
     Pushbroom Camera Configuration 
     In the pushbroom camera configuration, a plurality of scan image lines  610  are taken at different instances of time, as shown in FIG.  6 . (Notice that FIG. 6 is shown generically without referencing specific points i and g.) A plurality of scan image lines  610  make up a complete image. In other words, a plurality of scan image lines  610  would comprise a single image  410  of the frame camera configuration described above, and each one of the plurality of scan images lines  610  would have its own perspective projection model, which is similar to the model described above for the frame camera configuration. In this model, the ground space to image space relationship for the pushbroom camera configuration can be expressed by a modified collinearity equation in which all exterior orientation parameters are defined as a function of time. The set of modified collinearity equations are:                  F     x   i       =         -     x   0       +     c                           m        (   t   )       11          ⌊       X   g     -       X        (   t   )       PC       ⌋       +         m        (   t   )       12          ⌊       Y   g     -       Y        (   t   )       PC       ⌋       +         m        (   t   )       13          ⌊       Z   g     -       Z        (   t   )       PC       ⌋                 m        (   t   )       31          [       X   g     -       X        (   t   )       PC       ]       +       m   32          [       Y   g     -       Y        (   t   )       PC       ]       +       m   33          [       Z   g     -       Z        (   t   )       PC       ]               =   0            
            F     x   i       =         y   i     -     y   0     +     c                           m        (   t   )       21          [       X   g     -       X        (   t   )       PC       ]       +         m        (   t   )       22          [       Y   g     -       Y        (   t   )       PC       ]       +         m        (   t   )       23          [       Z   g     -       Z        (   t   )       PC       ]                 m        (   t   )       31          [       X   g     -       X        (   t   )       PC       ]       +         m        (   t   )       32          [       Y   g     -       Y        (   t   )       PC       ]       +         m        (   t   )       33          [       Z   g     -       Z        (   t   )       PC       ]               =       0        
     [             m        (   t   )       11             m        (   t   )       12             m        (   t   )       13                 m        (   t   )       21             m        (   t   )       22             m        (   t   )       23                 m        (   t   )       21             m        (   t   )       32             m        (   t   )       33           ]     =     [           cos                 φ                   (   t   )                   cos                 κ                   (   t   )               cos                 ω                   (   t   )                   sin                 κ                   (   t   )       +     sin                 ω                   (   t   )                   sin                 φ                   (   t   )                   cos                 κ                   (   t   )                 sin                 ω                   (   t   )                   sin                 κ                   (   t   )       -     cos                 ω                   (   t   )                   sin                 φ                   (   t   )                   cos                 κ                   (   t   )                     -   cos                   φ                   (   t   )                   sin                 κ                   (   t   )               cos                 ω                   (   t   )                   cos                 κ                   (   t   )       -     sin                 ω                   (   t   )                   sin                 φ                   (   t   )                   sin                 κ                   (   t   )                 sin                 ω                   (   t   )                   cos                 κ                   (   t   )       +     cos                 ω                   (   t   )                   sin                 φ                   (   t   )                   sin                 κ                   (   t   )                   sin                 φ                   (   t   )               -   sin                   ω                   (   t   )                   cos                 φ                   (   t   )             cos                 ω                   (   t   )                   cos                 φ                   (   t   )             ]                   (Eq.  2)                                
     is a rotation matrix; and 
     ω, φ(t) and κ(t) are the (pitch, roll, and yaw) attitude angles of the camera; 
     X g , Y g , Z g  are the object coordinates of a point g located in the ground space imaged; 
     X(t) PC , Y(t) PC , Z(t) PC  are the object coordinates of the perspective center; 
     y 1  is the sample image coordinate of a point i located in the image space comprising a plurality of scan image lines; 
     x o  and y o  are the image (sample and line) coordinates of the principal point; 
     c is the focal length; and 
     t is time. 
     As one of skill in the art would recognize, the pushbroom camera model exterior orientation parameters are a function of time. In other words, the attitude angles (ω(t), φ(t) and κ(t)) and position of the perspective center (X(t) PC , Y(t) PC , Z(t) PC ) change from scan line to scan line. The interior orientation parameters, which comprise focal length (c), principal point location (x o  and y o ), and optionally lens distortion coefficients, and other parameters directly related to the physical design and associated mathematical model of the pushbroom camera, are the same for the entire image, which is a plurality of scan image lines  610 . 
     For both camera configurations, however, the exterior orientation parameters may be, and often are, completely unknown. In many instances some or all of these parameters are measured during an imaging event to one level of accuracy or another, but no matter how they are derived, the exterior orientation parameters are not perfectly known. Similarly, the interior orientation parameters, which are typically determined prior to taking an image, also are known to only a limited accuracy. These inaccuracies limit the accuracy with which the object coordinates of the point g  555  can be determined from image coordinates of the corresponding point i  545 , for example. Consequently, photogrammetrically derived metric products following only the above equations will have limited accuracy, also. 
     In order to improve accuracy, multiple overlapping images are customarily block adjusted together. Block adjustment of multiple overlapping images uses statistical estimation techniques, such as least-squares, maximum likelihood and others, to estimate unknown camera model parameters for each image. The images are tied together by tie points whose image coordinates are measured on multiple images, and tied to the ground by ground control points with known or approximately known object space coordinates and measured image positions. 
     Block Adjusting the Frame Camera Configuration 
     In the case of the frame camera model, which is expressed by the first set of collinearity equations (Eq. 1), the unknown model parameters for n overlapping images, with m ground control points and p tie points will be explained with reference to FIG.  7 . FIG. 7 shows 3 overlapping images  705 ,  710 , and  715  (i.e., n=3). In the simple case shown in FIG. 7, image  705  has image position tie points  721 ,  722 , and  723 ; image  710  has image position tie points  721 ,  722 ,  723 ,  724 ,  725 , and  726 ; and image  715  has image position tie points  724 ,  725 , and  726 , which means FIG. 7 has 6 tie points (i.e., p=6). Image  705  is “tied” to image  710  by tie points  721 ,  722 , and  723  and image  710  is “tied” to image  715  by tie points  724 ,  725 , and  726 . Furthermore, FIG. 7 has ground control points  732  and  734  (i.e., m=2). Notice that each image position tie point  721 - 726  has a corresponding ground position  741 - 746  and ground control points  732  and  734  have a corresponding image positions  752  and  754 . Finally, each image  705 ,  710 , and  715  has a PC  765 ,  770 , and  775 , respectively, which represents from where the camera took the image. Of course, each image ( 705 ,  710 , and  715 ) has six (6) associated exterior orientation parameters (which are attitude angles of the camera (ω, φ and κ) and position of the perspective center (X PC , Y PC , Z PC )). Further, each ground control point has three (3) object space coordinates (an X, Y, and Z coordinates). Moreover, each tie point has three (3) corresponding object space coordinates (X, Y, and Z coordinates). In summary, FIG. 7 shows: 
     (1) The model parameters to be estimated which are: 
     6×n exterior orientation parameters (18 parameters); 
     3×m object coordinates of ground control points (6 object coordinates of ground control points); 
     3×p object coordinates of tie points (18 object coordinates of tie points); and 
     optionally (an known in the art, but not represented for simplicity) 3 interior orientation parameters and other parameters directly related to the physical design and associated mathematical model of the camera, as required. 
     (2) Moreover, the observations used in this model comprise: 
     6×n exterior orientation parameters (18 parameters); 
     3×m object coordinates of ground control points (6 object coordinates of ground control points); 
     2×r (where r≧m) image coordinates of ground control points (4, i.e. 2 times 2 image coordinates of ground control points); 
     2×s (where s≧2p) image coordinates of tie points (24 image coordinates of tie points); 
     3 interior orientation parameters and other parameters directly related to the physical design and associated mathematical model of the camera, as required. 
     While generally understood in the art, r is the number of images of the ground control points. For example, FIG. 7 has two ground control points  732  and  734 . Point  732  is imaged by Point  752  and Point  734  is imaged by Point  754 , thus r is two. However, if ground control point  732  were also imaged in frame  710 , then r would equal three (i.e. one more image of a ground control point). Additionally, if Point  732  were also imaged in frame  715 , then r would equal four. Similarly understood in the art, s is the number of images of the tie points. For example, FIG. 7 has six tie points  721 ,  722 ,  723 ,  724 ,  725 , and  726 . As can be seen in FIG. 7, tie point  721  is imaged in Frame  705  and  710 . Similarly, each of the tie points is imaged in two frames, so s is twelve. If tie point  721  were also imaged in frame  715 , then s would be thirteen. If point  724  were also imaged in frame  705 , then s would be fourteen. Because a tie point must be in at least two frames, s must be greater than or equal to twice the number of tie points. 
     With the above information, application of the Taylor Series expansion to the set of collinearity equations represented as Eq. 1 results in the following linearized math model for the block adjustment with unknown exterior and interior orientation parameters, and object space coordinates of both the ground control and tie points:                      [           A   EO           A   S           A   IO             0       I       0           I       0       0           0       0       I         ]                [           δ   EO               δ   S               δ   IO           ]     +   v     =     [           w   P               w   S               w   EO               w   IO           ]                       
        or        
              A                 δ     +   v     =   w             (Eq.  3)                                
     with the a priori covariance matrix being:          C   w     =     [           C   P         0       0       0           0         C   S         0       0           0       0         C   EO         0           0       0       0         C   IO           ]                            
     The vector of corrections to the approximate values of the model parameters is given as:                         [           δ   EO               δ   S               δ   IO           ]     =     δ   =     β   -     β   0           ,                            
     β is a vector of unknown model parameters and β o  is a vector of approximate values of the unknown parameters, typically computed from approximate measurements. 
     δ EO =[dX o1  dY o1  dZ o1  dω 1  dφ 1  dκ 1  . . . dX on  dY on  dZ on  dω n  dφ n  dκ n ] T  are the unknown corrections to the approximate values of the exterior orientation parameters for n images, 
     δS=[dX 1  dY 1  dZ 1  . . . dX m+p  dY m+p  dZ m+p ] T  are the unknown corrections to the approximate values of the object space coordinates for m+p object points, 
     δ IO =[dc dx o  dy o ] T  are the unknown corrections to the approximate values of the interior orientation parameters. A EO  is the first order design matrix for the exterior orientation (EO) parameter,          A   EO     =       [           A     EO   1                 A     EO   2               ⋮         ]                   where               A     EO   i       =           ∂     F   i         ∂     β   EO   T                   β   o         =       [               ∂     F     x   i           ∂     X     o   1                             ∂     F     x   i           ∂     Y     o   1                             ∂     F     x   i           ∂     Z     o   1                             ∂     F     x   i           ∂     ω   1                           ∂     F     x   i           ∂     φ   1                           ∂     F     x   i           ∂     κ   1                       …               ∂     F     x   i           ∂     X   on                           ∂     F     x   i           ∂     Y   on                           ∂     F     x   i           ∂     Z   on                           ∂     F     x   i           ∂     ω   n                           ∂     F     x   i           ∂     φ   n                           ∂     F     x   i           ∂     κ   n                         ∂     F     y   i           ∂     X     o   1                             ∂     F     y   i           ∂     Y     o   1                             ∂     F     y   i           ∂     Z     o   1                             ∂     F     y   i           ∂     ω   1                           ∂     F     y   i           ∂     φ   1                           ∂     F     y   i           ∂     κ   1                       …               ∂     F     y   i           ∂     X   on                           ∂     F     y   i           ∂     Y   on                           ∂     F     y   i           ∂     Z   on                           ∂     F     y   i           ∂     ω   n                           ∂     F     y   i           ∂     φ   n                           ∂     F     y   i           ∂     κ   n                 ]                     |                                                                                                 β   o                                                  
     the first order design matrix for the EO parameters—for the i-th image point.          A   S     =       [           A       S   1                              A     S   2               ⋮         ]                   where               A     S   i       =           ∂     F   i         ∂     β   S   T                   β   o         =       [               ∂     F     x   i           ∂     X   1                           ∂     F     x   i           ∂     Y   1                           ∂     F     x   i           ∂     Z   1                 …                            ∂     F     x   i           ∂     X     m   +   p                             ∂     F     x   i           ∂     Y     m   +   p                             ∂     F     x   i           ∂     Z     m   +   p                                        ∂     F     y   i           ∂     X   1                           ∂     F     y   i           ∂     Y   1                           ∂     F     y   i           ∂     Z   1                 …                            ∂     F     y   i           ∂     X     m   +   p                             ∂     F     y   i           ∂     Y     m   +   p                             ∂     F     y   i           ∂     Z     m   +   p                                ]                     |                                                                                                 β   o                                                  
     The first order design matrix for the object space coordinates—for the i-th image point. 
     A IO  is the first order design matrix for the interior orientation (IO) parameters where          A   IO     =       [           A       IO   1                              A     IO   2               ⋮         ]                   where               A     IO   i       =           ∂     F   i         ∂     β   IO   T                   β   o         =       [               ∂     F     x   i           ∂   c                         ∂     F     x   i           ∂     x   o                           ∂     F     x   i           ∂     y   o                         ∂     F     y   i           ∂   c                         ∂     F     y   i           ∂     x   o                           ∂     F     y   i           ∂     y   o                 ]                     |                                                                                                 β   o                                                  
     is the first order design matrix for the IO parameters—for the i-th image point. 
     w p =F(β o ) is the vector of misclosures for the image space coordinates. 
     w s =β S observed −β S o  is the vector of misclosures for the object space coordinates. 
     w EO =β EO observed −β EO o  is the vector of misclosures for the exterior orientation parameters. 
     w IO =β IO observed −β IO o  is the vector of misclosures for the interior orientation parameters. 
     C w  is the a priori covariance matrix of the vector of observables (misclosures) w. 
     C p  is the a priori covariance matrix of image coordinates. 
     C S  is the a priori covariance matrix of the object space coordinates. 
     C EO  is the a priori covariance matrix of the exterior orientation parameters. 
     C IO  is the a priori covariance matrix of interior orientation parameters. 
     Because the math model is non-linear, the least-squares solution needs to be iterated until convergence is achieved. At each iteration step, application of the least-squares principle results in the following vector of estimated corrections to the approximate values of the model parameters: 
     
       
         {circumflex over (δ)}=( A   T   C   w   −1   A ) −1   A   T   C   w   −1   w   (Eq. 4)  
       
     
     At the subsequent iteration step, the vector of approximate values of the unknown parameters β 0  is replaced by the vector of estimated model parameters as shown by the following: 
     
       
         {circumflex over (β)}=β 0 +{circumflex over (δ)},  
       
     
     where {circumflex over (δ)} is the vector of corrections estimated in the previous iterative step and the math model is linearized again. The least-squares estimation is repeated until convergence is reached, i.e., when {circumflex over (δ)} is below some predetermined acceptable level. The covariance matrix of the estimated model parameters follows from application of the law of propagation of random errors. 
     
       
           C   {circumflex over (δ)} =( A   T   C   w   −1   A ) −1   (Eq. 5)  
       
     
     Block Adjustment of Pushbroom Camera Configuration 
     Similar to the frame camera adjustment above, a linearized block adjustment model can be implemented for the pushbroom camera model, as expressed by the set of collinearity equations represented by Eq. 2. Generally, the mathematical model described above for the frame camera can be used for pushbroom camera models because a plurality of scan image lines will comprise one frame. In other words, for the pushbroom camera model, the unknown model parameters for n overlapping images, each with j scan lines for each image, together with m ground control points and p tie points (FIG. 7 can be used to represent the Pushbroom model if one assumes each image  705 ,  710 , and  715  comprise a plurality of scan image lines or j image lines in this case) include the following: 
     6×n×j exterior orientation parameters 
     3×m object coordinates of ground control points, 
     3×p object coordinates of tie points, 
     3×n interior orientation parameters and other time or image dependent parameters directly related to the physical design and associated mathematical model of the pushbroom camera, as required. 
     Where the observations comprise: 
     6×n×j exterior orientation parameters 
     3×m object coordinates of ground control points, 
     2×r image coordinates of ground control points, 
     2×s image coordinates of tie points, 
     3×n interior orientation parameters and other time or image dependent parameters directly related to the physical design and associated mathematical model of the pushbroom camera, as required. 
     It should be noted that estimation of exterior orientation parameters for each image line is not practical, and it is a standard practice to use a model, such as a polynomial, to represent them in the adjustment model. 
     Otherwise, the mathematics are similar, and the modifications to the above equations are generally known in the art. 
     In general, satellite camera models are more complicated than frame camera models because of satellite dynamics and the dynamic nature of scanning, of push-broom, and other time dependent image acquisition systems commonly used for satellite image acquisition. Implementing such a complicated model is expensive, time consuming, and error prone. 
     As explained above, classical block adjustment models based on physical camera math models, whether pushbroom, frame or other camera configuration, are complicated and difficult to implement. This is, in part, due to the shear volume of interior and exterior parameters that need to be estimated and for which equations need to be solved. 
     Moreover, a multiplicity of parameters each having the same effect is another difficulty with classical block adjustment techniques using a physical camera model. For example, any combination of moving the exposure station to the right, rolling the camera to the right, or displacing the principal point of the camera to the left all have the same general effect of causing the imagery to be displaced to the left. Classical photogrammetry separately estimates exposure station, orientation, and principal point, leading to ill-conditioning of the solution. Only a priori weights control the adjustment parameters to reasonable values. Ill-conditioning can lead to unrealistic parameter values or even divergence of iteratively solved systems. 
     A further difficulty with classical techniques is that each camera design, be it a frame camera, pushbroom, or other, presents the software developer with the necessity of developing another adjustment model. 
     The Rational Polynomial Camera (“RPC”) block adjustment method of the present invention avoids all the aforementioned problems associated with the classical photogrammetric block adjustment approach. Instead of adjusting directly the physical camera model parameters, such as satellite ephemeris (position of the perspective center for each scan line), satellite attitude, focal length, principal point location, and distortion parameters, the method introduces an adjustment model that block adjusts images in either image space or object space. In other words, the RPC mathematical model describes the object-to-image relationship of each image in the block of images. To the basic object-image model of each image, an adjustment model is added. The adjustment model comprises simple offsets, scale-factors, and/or polynomial adjustment terms in either object or image space. 
     The main benefit of the RPC block adjustment model of the present invention is that it does not present the numerical ill-conditioning problems of classical techniques. This is achieved by having only one adjustment parameter to represent multiple physical processes that have substantially the same effect. Furthermore, using such a block adjustment model simplifies development of photogrammetric block adjustment software by using either an existing or a generic camera model describing object-to-image space relationships and a generic adjustment model for block-adjusting parameters of that relationship. Use of generic models reduces the effort associated with developing individual camera models. 
     While the methods of the present invention can be installed on almost any conventional personal computer, it is preferred that the apparatuses and methods of the present invention use a general purpose computer having at least 512 Mbytes of RAM and 10 gigabytes of storage. Also, it is preferable to use a computer with a stereo display capability. Moreover, one of ordinary skill in the art would understand that the methods, functions, and apparatuses of the present invention could be performed by software, hardware, or any combination thereof. 
     The RPC Function 
     a tie point image coordinate generation module configured to automatically generate the image coordinates of the at least one tie point and transmit the generated image coordinates of the at least one tie point to the tie point receiving module; and 
     a tie point ground coordinate generation module configured to automatically generate the ground coordinates of the at least one tie point and transmit the generated ground coordinates of the at least one tie point to the tie point receiving module. 
     The Rational Polynomial Camera (“RPC”) model is a generic mathematical model that relates object space coordinates (which comprise Latitude, Longitude, and Height) to image space coordinates (which comprise Line and Sample). The RPC functional model is of the form of a ratio of two cubic functions of object space coordinates. Separate rational functions are used to express the object space coordinates to the image line coordinate and the object space coordinates to the image sample coordinate. 
     While one of skill in the art will, on reading this disclose, recognize that the RPCs can be applied to numerous types of imaging devices, the description that follows will apply the apparatuses and methods of the present invention to the IKONOS® satellite imaging system by Space Imaging, LLC, which is located at 12076 Grant Street, Thornton, Colo. For the IKONOS® satellite, the RPC model is defined as follows: 
     Given the object space coordinates (Latitude, Longitude, Height) of a control point, where Latitude is geodetic latitude expressed in degrees in the product order datum, Longitude is geodetic longitude expressed in degrees in the product order datum, and Height is geodetic height expressed in meters height above the ellipsoid of the product order datum, then the calculation of image space coordinates begins by normalizing latitude, longitude, and height as follows, where the normalizing offsets and scale factors are estimated as shown below:              P   =       Latitude   -   LAT_OFF     LAT_SCALE             (Eq.  6)               L   =       Longitude   -   LONG_OFF     LONG_SCALE             (Eq.  7)               H   =       Height   -   HEIGHT_OFF     HEIGHT_SCALE             (Eq.  8)                                
     The normalized image space coordinates (X and Y, which are normalized sample and line image coordinates, respectively) are then calculated from the object space coordinates using their respective rational polynomial functions f(.) and g(.) as                Y   =       f        (     Latitude   ,   Longitude   ,   Height     )       =         Num   L          (     P   ,   L   ,   H     )           Den   L          (     P   ,   L   ,   H     )                  
        where           (Eq.  9)                           Num   L          (     P   ,   L   ,   H     )       =                  a   1     +       a   2     ·   L     +       a   3     ·   P     +       a   4     ·   H     +       a   5     ·   L   ·   P     +                                  a   6     ·   L   ·   H     +       a   7     ·   P   ·   H     +       a   8     ·     L   2       +       a   9     ·     P   2       +                                  a   10     ·     H   2       +       a   11          P   ·   L   ·   H       +       a   12     ·     L   3       +       a   13     ·                                  L   ·     P   2       +       a   14     ·   L   ·     H   2       +       a   15     ·     L   2     ·   P     +       a   16     ·                                  P   3     +       a   17     ·   P   ·     H   2       +       a   18     ·     L   2     ·   H     +       a   19     ·                                    P   2     ·   H     +       a   20     ·     H   3                      
        and           (Eq.  10)                           Den   L          (     P   ,   L   ,   H     )       =                  b   1     +       b   2     ·   L     +       b   3     ·   P     +       b   4     ·   H     +       b   5     ·   L   ·   P     +                                  b   6     ·   L   ·   H     +       b   7     ·   P   ·   H     +       b   8     ·     L   2       +       b   9     ·     P   2       +                                  b   10     ·     H   2       +       b   11          P   ·   L   ·   H       +       b   12     ·     L   3       +       b   13     ·                                  L   ·     P   2       +       b   14     ·   L   ·     H   2       +       b   15     ·     L   2     ·   P     +       b   16     ·                                  P   3     +       b   17     ·   P   ·     H   2       +       b   18     ·     L   2     ·   H     +       b   19     ·                                    P   2     ·   H     +       b   20     ·     H   3                      
        and           (Eq.  11)                 X   =       g        (     Latitude   ,   Longitude   ,   Height     )       =         Num   S          (     P   ,   L   ,   H     )           Den   S          (     P   .   L   .   H     )                  
        where           (Eq.  12)                           Num   S          (     P   ,   L   ,   H     )       =                  c   1     +       c   2     ·   L     +       c   3     ·   P     +       c   4     ·   H     +       c   5     ·   L   ·   P     +                                  c   6     ·   L   ·   H     +       c   7     ·   P   ·   H     +       c   8     ·     L   2       +       c   9     ·     P   2       +                                  c   10     ·     H   2       +       c   11          P   ·   L   ·   H       +       c   12     ·     L   3       +       c   13     ·                                  L   ·     P   2       +       c   14     ·   L   ·     H   2       +       c   15     ·     L   2     ·   P     +       c   16     ·                                  P   3     +       c   17     ·   P   ·     H   2       +       c   18     ·     L   2     ·   H     +       c   19     ·                                    P   2     ·   H     +       c   20     ·     H   3                      
        and           (Eq.  13)                         Den   S          (     P   ,   L   ,   H     )       =                  d   1     +       d   2     ·   L     +       d   3     ·   P     +       d   4     ·   H     +       d   5     ·   L   ·   P     +                                  d   6     ·   L   ·   H     +       d   7     ·   P   ·   H     +       d   8     ·     L   2       +       d   9     ·     P   2       +                                  d   10     ·     H   2       +       d   11          P   ·   L   ·   H       +       d   12     ·     L   3       +       d   13     ·                                  L   ·     P   2       +       d   14     ·   L   ·     H   2       +       d   15     ·     L   2     ·   P     +       d   16     ·                                  P   3     +       d   17     ·   P   ·     H   2       +       d   18     ·     L   2     ·   H     +       d   19     ·                                    P   2     ·   H     +       d   20     ·     H   3                       (Eq.  14)                                
     The normalized X and Y image space coordinates when de-normalized are the Line and Sample image space coordinates, where Line is image line number expressed in pixels with pixel zero as the center of the first line, and Sample is sample number expressed in pixels with pixel zero is the center of the left-most sample are finally computed as: 
     
       
         Line= Y·LINE   —   SCALE+LINE   —   OFF   (Eq. 15)  
       
     
     
       
         Sample= X·SAMP   —   SCALE+SAMP   —   OFF   (Eq. 16)  
       
     
     RPC Fitting 
     RPC fitting is described in conjunction with FIG.  8 . Preferably, a least-squares approach is used to estimate the RPC model coefficients a i , b i , c i , and d i  (where in the above example I=1, 2, . . . , 20) from a 3-dimensional grid of points generated using a physical camera model. The 3-dimensional grid of points, composed of object points  822 ,  824 ,  826 , etc. is generated by intersecting rays  820  with elevation planes  821 ,  830 , and  840 . For each point, one ray exists that connects the corresponding points in each elevation plane  821 ,  830 , etc. For example, point  812  in the image space, corresponds to point  822  in elevation plane  821  and ray  820  connects the two points, as well as the corresponding points in other elevation planes, not specifically labeled. The estimation process, which is substantially identical for the f(.) and g(.) rational polynomial functions and is, therefore, explained with respect to the line image coordinate only, is performed independently for each of the image space coordinates. 
     The offsets are computed as mean values for all grid points as:        LINE_OFF   =       1   n            ∑     i   =   1     n          Line   i                 SAMP_OFF   =       1   n            ∑     i   =   1     n          Sample   i                 LAT_OFF   =       1   n            ∑     i   =   1     n          Latitude   i                 LONG_OFF   =       1   n            ∑     i   =   1     n          Longitude   i                 HEIGHT_OFF   =       1   n            ∑     i   =   1     n          Height   i                                
     The scale factors are computed as: 
     
       
           LINE   —   SCALE =max(| Line   max   −LINE   —   OFF|,|Line   min   −LINE   —   OFF|)    
       
     
       SAMP   —   SCALE =max(| Sample   max   −SAMP   —   OFF|,|Sample   min   −SAMP   —   OFF|)    
     
       
           LAT   —   SCALE =max(| Latitude   max   −LAT   —   OFF|,|Latitude   min   −LAT   —   OFF|)    
       
     
     
       
           LONG   —   SCALE =max(| Longitude   max   −LONG   —   OFF|,|Longitude   min   −LONG   —   OFF|)    
       
     
     
       
           HEIGHT   —   SCALE =max(| Height   max   −HEIGHT   —   OFF|,|Height   min   −HEIGHT   —   OFF|)    
       
     
     The observation equation for ith observation (grid point) can be written as 
     
       
           N   i ( x   N )− D   i ( x   D )y i =0  (Eq. 17)  
       
     
     where for the line RPC 
     
       
           N   i ( x   N )= Num   L ( P   i   ,L   i   ,H   i )  (Eq. 18)  
       
     
     
       
           D   i ( x   D )= Den   L ( P   i   ,L   i ,H i )  (Eq. 19)  
       
     
     
       
         and  
       
     
     
       
           x   N =( a   0   ,a   1   ,a   2   ,a   3   ,a   4   ,a   5   ,a   6   ,a   7   ,a   8   ,a   9   ,a   10   ,a   11   ,a   12   ,a   13   ,a   14   ,a   15   ,a   16   ,a   17   ,a   18   ,a   19 ) T   (Eq. 20)  
       
     
     
       
           x   D =( b   0   ,b   1   ,b   2   ,b   3   ,b   4   ,b   5   ,b   6   ,b   7   ,b   8   ,b   9   ,b   10   ,b   11   ,b   12   ,b   13   ,b   14   ,b   15   ,b   16   ,b   17   ,b   18   ,b   19)   T   (Eq. 21)  
       
     
     y i  is the normalized line coordinate (Y) of the ith grid point. 
     For the sample RPC, the terms would be defined as: 
     
       
           N   i ( x   N )= Num   S ( P   i   ,L   i   ,H   i )  (Eq. 22)  
       
     
     
       
           D   i ( x   D )= Den   S ( P   i   ,L   i   ,H   i )  (Eq. 23)  
       
     
     
       
         and  
       
     
     
       
           x   N =( c   0   ,c   1   ,c   2   ,c   3   ,c   4   ,c   5   ,c   6   ,c   7   ,c   8   ,c   9   ,c   10   ,c   11   ,c   12   ,c   13   ,c   14   ,c   15   ,c   16   ,c   17   ,c   18   ,c   19 ) T   (Eq. 22)  
       
     
     
       
           x   D =( d   0   ,d   1   ,d   2   ,d   3   ,d   4   ,d   5   ,d   6   ,d   7   ,d   8   ,d   9   ,d   10   ,d   11   , d   12   ,d   13   ,d   14   ,d   15   ,d   16   ,d   17   ,d   18   ,d   19 ) T   (Eq. 23)  
       
     
     y i  is the normalized sample coordinate (X) of the ith grid point, of n grid points. 
     Then, the first order Taylor series expansion results in                      N   i          (     x   N0     )       -         D   i          (     x   D0     )            y   i       +       [         ∂     N   i         ∂     x   N   T                                          X   N0             ]                     dx   N       -       [         ∂     D   i         ∂     x   D   T                                          X   D0             ]                     y   i        d                   x   D         =   0          
        with           (Eq.  24)               A   =     [           A   1               A   2             ⋮             A   n           ]             (Eq.  25)                                
     being the first order design matrix and where                  A   i     =       [       [         -     ∂     N   i           ∂     x   N   T                                          X     N   0               ]     ,       [         ∂     D   i         ∂     x   D   T                                          X     D   0               ]          y   i         ]     =     [       A     N   i       ,     A     D   i         ]              
        and           (Eq.  25)                 A     N   i       =     [       -   1     ,     -     L   i       ,     -     P   i       ,     -     H   i       ,   …   ,     -     H   i   3         ]             (Eq.  26)                 A     D   i       =     [       y   i     ,       L   i          y   i       ,       P   i          y   i       ,       H   i          y   i       ,   …   ,       H   i   3          y   i         ]             (Eq.  27)                                
     also with              dx   =     [           dx   N               dx   D           ]             (Eq.  28)                                
     being the corrections to the vector of parameters; and with 
     
       
           B=diag└−D   i ( x   D0 )┘= diag ( B   i )  (Eq. 29)  
       
     
     being the second order design matrix and where 
     
       
           B   i =└−1,− L   i   ,−P   i   ,−H   i   , . . . ,−H   i   3 ┘×└1, x   D1   0   ,x   D2   0   ,x   D3   0   , . . . ,x   D19   0 ┘ T   (Eq. 30)  
       
     
     
       
         and  
       
     
     
       
           y=[y   1   ,y   2   , . . . ,y   n ] T   (Eq. 31)  
       
     
     being the vector of image (line or sample) coordinates, and              w   =     [             N   1          (     x   N0     )                   N   2          (     x   N0     )               ⋮               N   n          (     x   N0     )             ]             (Eq.  32)                                
     where the least-squares math model is: 
     
       
           By+w=Adx+v  with  C   y =σ 0   2   I   (Eq. 33)  
       
     
     where v is a vector of random unobservable errors. 
     Then, using the following transformation 
     
       
           z=By+w   (Eq. 34)  
       
     
     one finally gets 
     
       
           z=Adx+v  with  C   z   =BC   y   B   T =σ 0   2   BB   T   (Eq. 35)  
       
     
     The least-squares solution for the correction vector follows with 
     
       
           d{circumflex over (x)} =( A   T   C   z   −1   A ) −1   A   T   C   z   −1   z   (Eq. 36)  
       
     
     Because the original observation equations are non-linear, the least-squares estimation process outlined above needs to be repeated until convergence is achieved. 
     RPC Block Adjustment Math Models 
     One presently preferred embodiment of the present invention uses the RPC model for the object-to-image space relationship and the image space adjustment model. Each image has its own set of RPC coefficients to describe the geometry of that individual image. The image space RPC Block Adjustment model is defined as follows: 
     
       
         Line=Δ L+f (Latitude,Longitude,Height)+ε L   (Eq. 37)  
       
     
     
       
         Sample=Δ S+g (Latitude,Longitude,Height)+ε S   (Eq. 38)  
       
     
     where 
     Line, Sample are measured line and sample coordinates of a ground control or a tie point with object space coordinates (Latitude, Longitude, Height). The object space coordinates are known to a predetermined accuracy for the ground control points and estimated for the tie points. 
     ΔL, ΔS are the adjustment terms expressing the differences between the measured and the nominal line and sample coordinates of a ground control or tie point, which are initially estimated to be zero. 
     ε L  and ε S  are random unobservable errors, and 
     f and g are the given line and sample RPC models given by Eq. 9 and Eq. 12 (and the associated definitions), respectively. 
     Furthermore, this preferred embodiment of the present invention uses the image adjustment model defined on the domain of image coordinates with the following terms 
     
       
           ΔL=a   o   +a   S ·Sample+ a   L ·Line  (Eq. 39)  
       
     
     
       
           ΔS=b   o   +b   S ·Sample+ b   L ·Line  (Eq. 40)  
       
     
     where 
     a 0 , a S , a L , b 0 , b S , b L  are the unknown adjustment parameters for each image to be estimated, 
     Line and Sample are either measured line and sample coordinates, or nominal line and sample coordinates—given by the RPC function f(.) and g(.) (see Eqs. 6 to 16)—of a ground control or tie point 
     While some error is inevitable, it is preferred for the purpose of implementing the RPC block adjustment model to treat the Line and Sample coordinates (see Eqs. 39 and 40) as fixed and errorless, i.e., there should be no error propagation associated with the first order design matrix. As one of skill in the art of estimation systems will now recognize, extension of this method to include additional polynomial coefficients is straightforward. In the general case, the following model defined on the domain of image space coordinates represents the image space adjustment model: 
     
       
           ΔL=a   0   +a   S Sample+ a   L Line+ a   SL Sample Line+ a   L2 Line 2   +a   S2 Sample 2   +a   SL2 Sample Line 2   +a   S2L ·Sample 2 Line+ a   L3 Line 3   +a   S3 ·Sample 3 +  (Eq. 41)  
       
     
     
       
         and  
       
     
     
       
           ΔS=b   0   +b   S Sample+ b   L Line+ b   SL Sample Line+ b   L2 Line 2   +b   S2 Sample 2   +b   SL2 Sample Line 2   +b   S2L Sample 2 Line+ b   L3 ·Line 3   +b   S3 Sample 3 +  (Eq. 42)  
       
     
     While the methods and apparatus are being explained using a polynomial model, one of skill in the art would recognize that other parametric models could be used, such as orthogonal polynomial, Fourier series, wavelet model. The principle of the preferred embodiments of the proposed RPC Block Adjustment methodology is shown conceptually in the FIG.  9 . As can be seen, each image  905  and  910  (be it a single frame or a plurality of scan lines) has a plurality of ground control points  915  and a plurality of tie points  920 . These are used as observables to the least-squares adjustment that results in the estimated line and sample adjustment parameters a i  and b i , which define the adjustment terms ΔL and ΔS (see Eqs. 41 and 42). Application of the adjustment terms in conjunction with the original RPC equation  930  (Eqs. 6-16) results in the adjusted Line and Sample coordinates. 
     
       
         Line=Δ L+f (.)  
       
     
      Sample=Δ S+g (.) 
     Alternatively, the image space adjustment model can also be represented a polynomial model defined on the domain of object coordinates as: 
     
       
           ΔL=a   0   +a   P Latitude+ a   L Longitude+ a   H Height+ a   P2   
       
     
     
       
         Latitude 2   +a   L2 ·Longitude 2   +a   H2   
       
     
     
       
         ·Height 2   +a   PL ·Latitude·Longitude+ a   PH Latitude Height+ a   LH   
       
     
     
       
         ·Longitude Height+ a   P3 ·Latitude 3   +a   L3 Longitude 3   +a   H3   
       
     
     
       
         Height 3 +.  (Eq. 43) 
       
     
     and 
     
       
           ΔS=b   0   +b   P Latitude+b L Longitude+ b   H Height+ b   P2   
       
     
     
       
         Latitude 2   +b   L2 Longitude 2   +b   H2 Height 2   +b   PL   
       
     
     
       
         Latitude Longitude+ b   PH Latitude Height+ b   LH Longitude Height+ b   P3    
       
     
     
       
         Latitude 3   +b   L3 Longitude 3   +b   H3 Height 3 +  (Eq. 44)  
       
     
     As before, other parametric models can be used. 
     The adjustment model can also be formulated as the object space adjustment model with: 
     
       
         Line= f (Latitude+ΔLatitude,Longitude+ΔLongititde,Height+ΔHeight)+ε L   (Eq. 45)  
       
     
     
       
         and  
       
     
     
       
         Sample= g (Latitude+ΔLatitude,Longitude+ΔLongitude,Height+ΔHeight)+ε S   (Eq. 46)  
       
     
     where 
     Line, Sample are measured line and sample coordinates of a ground control or tie point with object space coordinates (Latitude, Longitude, Height). The object space coordinates are known to a predetermined accuracy for the ground control points and estimated for the tie points. 
     ΔLatitude, ΔLongitude, ΔHeight are the adjustment terms expressing the differences between the measured and the nominal object space coordinates of a ground control or tie point, which are initially estimated at zero. 
     ε l  and ε S  are random unobservable errors, 
     f and g are the given line and sample RPC models given by Eq. 9 and Eq. 12, respectively 
     Using this alternative embodiment, a polynomial model defined on the domain of the object coordinates represents the object space adjustment model as: 
     
       
         ΔLatitude= a   0   +a   P ·Latitude+ a   L ·Longitude+ a   H ·Height+ a   P2   
       
     
     
       
         ·Latitude 2   +a   L2 ·Longitude 2   +a   H2   
       
     
     
       
         ·Height 2   +a   PL ·Latitude Longitude+ a   PH   
       
     
     
       
         ·Latitude Height+ a   LH ·Longitude Height+ a   P3   
       
     
     
       
         ·Latitude 3   +a   L3 ·Longitude 3   +a   H3   
       
     
     
       
         ·Height 3 +  (Eq. 47)  
       
     
     
       
         ΔLongitude= b   0   +b   P ·Latitude+ b   L   
       
     
     
       
         ·Longitude+ b   H ·Height+ b   P2   
       
     
     
       
         ·Latitude 2   +b   L2 ·Longitude 2   +b   H2   
       
     
     
       
         ·Height 2   +b   PL ·Latitude Longitude+ b   PH   
       
     
     
       
         ·Latitude Height+ b   LH ·Longitude Height+b P3   
       
     
     
       
         ·Latitude 3   +b   L3 ·Longitude 3   +b   H3   
       
     
     
       
         ·Height 3 +  (Eq. 48) 
       
     
     and 
     
       
         ΔHeight= c   0   +c   P ·Latitude+ c   L   
       
     
     
       
         ·Longitude+ c   H ·Height+ c   P2   
       
     
     
       
         ·Latitude 2   +c   L2 ·Longitude 2   +c   H2   
       
     
     
       
         ·Height 2   +c   PL ·Latitude Longitude+ c   PH   
       
     
     
       
         ·Latitude Height+c LH ·Longitude Height+ c   P3   
       
     
     
       
         ·Latitude 3   +c   L3 ·Longitude 3   +c   H3   
       
     
     
       
         ·Height 3 +(Eq. 48)  
       
     
     Once again, other parametric models can be used. 
     RPC Block Adjustment Algorithm 
     The RPC adjustment models given above in Eqs. 37-49 allow block adjusting multiple overlapping images. The RPC block adjustment of multiple overlapping images uses statistical estimation techniques, such as least-squares, maximum likelihood and others, to estimate unknown camera model parameters for each image. Restating the above for simplicity, one preferred embodiment of the present invention uses the image space adjustment model 
     
       
         Line=Δ L+f (Latitude,Longitude,Height)+ε L   (Eq. 50)  
       
     
     
       
         Sample=Δ S+g (Latitude,Longitude,Height)+ε S   (Eq. 51)  
       
     
     where the adjustment terms (a O , a S , a L , b O , b S , b L ) are defined on the domain of image coordinates as: 
     
       
           ΔL=a   o   +a   S ·Sample+ a   L ·Line  (Eq. 52)  
       
     
     
       
           ΔS=b   o   +b   S ·Sample+ b   L ·Line  (Eq. 53)  
       
     
     or, if more degrees of freedom are required (in the presently preferred embodiment, sufficient accuracy is achieved using a first order equation), as: 
     
       
           ΔL=a   0   +a   S Sample+ a   L Line+ a   SL Sample Line+ a   L2 Line 2   +a   S2 Sample 2   +a   SL2 Sample Line 2   +a   S2L Sample 2 Line+ a   L3 Line 3   +a   S3 Sample 3 +  (Eq. 54)  
       
     
     
       
         and  
       
     
       ΔS=b   0   +b   S Sample+ b   L ·Line+ b   SL ·Sample Line+ b   L2 ·Line 2   +b   S2 Sample 2   +b   SL2 Sample Line 2   +b   S2L Sample 2 Line+ b   L3 Line 3   +b   S3 Sample 3 +  (Eq. 55) 
     Referring back to FIG. 9, the overlapping images  905  and  910  are tied together by tie points  920  whose image space coordinates are measured on those images  905  and  910 , and tied to the ground by ground control points  915  with known or approximately known object space coordinates and measured image positions. Thus, incorporating this information generically into the equations, then for the i-th image point on the j-th image, the image space adjustment model reads: 
     
       
         Line i   (j)   =ΔL   i   (j)   +f   (j) (Latitude i ,Longitude i ,Height i )+ε L   (j)   (Eq. 56)  
       
     
     
       
         and  
       
     
     
       
         Sample i   (j)   =ΔS   i   (j)   +g   (j) (Latitude i ,Longitude i ,Height i )+ε S   (j)   (Eq. 57)  
       
     
     
       
         with  
       
     
     
       
           ΔL   i   (j)   =a   0   (j)   +a   S   (j) ·Sample i   (j)   +a   L   (j) ·Line i   (j)   (Eq. 58)  
       
     
     
       
         and  
       
     
     
       
           ΔS   i   (j)   =b   0   (j)   +b   S   (j) ·Sample i   (j)   +b   L   (j) ·Line i   (j)   (Eq. 59)  
       
     
     As explained above, for the purpose of implementing the RPC block adjustment model of one presently preferred embodiment of the present invention, the Line i   (j)  and Sample i   (j)  coordinates in Eq. 56 and Eq. 57 should be treated as fixed and errorless, i.e., there should be no error propagation associated with the first order design matrix. 
     When the unknown math model parameters for Eq. 56 and Eq. 57 with the adjustment terms defined by Eq. 58 and Eq. 59 are applied for n overlapping images with m ground control points and p tie points there exist: 
     6×n adjustment parameters (X AD ), and 
     3×m object coordinates of ground control points, and 3×p object coordinates of tie points (X S ). 
     Observations comprise: 
     6×n a priori values of the adjustment parameters, which are estimate initially to be zero, 
     3×m object coordinates of ground control points, which are known or approximately known values, 
     2×r (r≧m) line and sample image coordinates of ground control points, which are measured from the image, and 
     2×s (s≧p) line and sample image coordinates of tie points, which are measured from the image. 
     With the above known and measured values application of the Taylor Series expansion to the RPC block adjustment equations 56-59 results in the following linearized model represented by Eq. 60 and 61 below:                        Line   i     (   j   )       =                  a   0     (   j   )       +       a   S     (   j   )       ·     Sample   i     (   j   )         +       a   L     (   j   )       ·     Line   i     (   j   )         +                                  f     (   j   )            (       Latitude     0      i       ,     Longitude     0      i       ,     Height     0      i         )       +                                  [         ∂     f     (   j   )           ∂   Latitude                       Latitude     0      i                 Longitude     0      i                 Height     0      i                 ]          dLatitude   i       +                                  [         ∂     f     (   j   )           ∂   Longitude                       Latitude     0      i                 Longitude     0      i                 Height     0      i                 ]          dLongitude   i       +                                  [         ∂     f     (   j   )           ∂   Height                       Latitude     0      i                 Longitude     0      i                 Height     0      i                 ]          dHeight   i       +     ɛ   L     (   j   )                      
        and           (Eq.  60)                       Sample   i     (   j   )       =                  b   0     (   j   )       +       b   S     (   j   )       ·     Sample   i     (   j   )         +       b   L     (   j   )       ·     Line   i     (   j   )         +                                  g     (   j   )            (       Latitude     0      i       ,     Longitude     0      i       ,     Height     0      i         )       +                                  [         ∂     g     (   j   )           ∂   Latitude                       Latitude     0      i                 Longitude     0      i                 Height     0      i                 ]          dLatitude   i       +                                  [         ∂     g     (   j   )           ∂   Longitude                       Latitude     0      i                 Longitude     0      i                 Height     0      i                 ]          dLongitude   i       +                                  [         ∂     g     (   j   )           ∂   Height                       Latitude     0      i                 Longitude     0      i                 Height     0      i                 ]          dHeight   i       +     ɛ   S     (   j   )                       (Eq.  61)                                
     Furthermore, this block adjustment model in matrix form reads:                      [           A   AD           A   S             I       0           0       I         ]                [           x   AD               dx   S           ]     +   v     =     [           w   P               w   AD               w   S           ]            
        or        
            Adx   +   v     =   w             (Eq.  62)                                
     with the a priori covariance matrix of the vector of observables (misclosures) w being:          C   n     =     [           C   p         0       0           0         C   AD         0           0       0         C   S           ]                            
     where: 
     dx is a vector of unknown model parameters to be estimated for the purpose of arriving at a solution such that:        dx   =       [           x   AD               dx   S           ]                   and                            
     x AD =[a 01  a S1  a L1  b 01  b S1  b L1  . . . a 0n  a Sn  a Ln  b 0n  b Sn  b Ln ] T;    
     which are the RPC adjustment model parameters for n images, 
     and 
     dx S =[dLatitude 1  dLongitude 1  dHeight 1  . . . dLatitude m+p  dLongitude m+p  dHeight m+p ] T;    
     which are the corrections to the approximate object space coordinates, x so , such that: 
     x S0 =[Latitude 01  Longitude 01  Height 01  dLatitude 0m+p  dLongitude 0m+p  dHeight 0m+p ] T    
     for the m ground control and the p tie points, 
     A is the first order design matrix,        A   =     [           A   AD           A   S             I       0           0       I         ]           where           A   AD     =     [           A     AD   1               ⋮             A     AD   i               ⋮         ]                            
     is the first order design matrix for the adjustment parameters and where          A     AD   i       =     [         0       …       1         Sample   i     (   j   )             Line   i     (   j   )           0       0       0       0       …       0           0       …       0       0       0       1         Sample   i     (   j   )             Line   i     (   j   )           0       …       0         ]                            
     is the first order design sub-matrix for the adjustment parameters, that is to say the design sub-matrix for the i-th image point on the j-th image, further          A   S     =     [           A     S   1               ⋮             A     S   i               ⋮         ]                            
     is the first order design matrix for the object space coordinates and          A     S   i       =     [         0       ⋯               ∂     f     (   j   )           ∂   Latitude                    Latitude     0      k                 Longitude     0      k                 Height     0      k                           ∂     f     (   j   )           ∂   Longitude                    Latitude     0      k                 Longitude     0      k                 Height     0      k                           ∂     f     (   j   )           ∂   Height                    Latitude     0      k                 Longitude     0      k                 Height     0      k                   0       ⋯       0           0       ⋯               ∂     g     (   j   )           ∂   Latitude                    Latitude     0      k                 Longitude     0      k                 Height     0      k                           ∂     g     (   j   )           ∂   Longitude                    Latitude     0      k                 Longitude     0      k                 Height     0      k                           ∂     g     (   j   )           ∂   Height                    Latitude     0      k                 Longitude     0      k                 Height     0      k                   0       ⋯       0         ]                            
     is the first order design sub-matrix for the object space coordinates of the k-th ground control or tie point—for the i-th image point on the j-th image, and (Latitude 0k , Longitude 0k , Height 0k ) are the approximate object space coordinates for point k. Then,          w   p     =     [           w     p   1               ⋮             w     p   i               ⋮         ]                            
     is the vector of misclosures for the image space coordinates, and          w     P   i       =     [             Line   i     (   j   )       -       f     (   j   )            (       Latitude     0      i       ,     Longitude     0      i       ,     Height     0      i         )                     Sample   i     (   j   )       -       g     (   j   )            (       Latitude     0      i       ,     Longitude     0      i       ,     Height     0      i         )               ]                            
     is the sub-vector of misclosures—for the i-th image point on the j-th image, and 
     
       
           w   AD =0  
       
     
     is the vector of misclosures for the adjustment parameters, and          w   S     =     [           w     S   1               ⋮             w     S   i               ⋮         ]                            
     is the vector of misclosures for the image space coordinates, and          w     S   i       =     [             Latitude   i     -     Latitude     0      i                     Longitude   i     -     Longitude     0      i                     Height   i     -     Height     0      i               ]                            
     is the sub-vector of misclosures—for the i-th image point, 
     C P  is the a priori covariance matrix of image coordinates, 
     C AD  is the a priori covariance matrix of the adjustment parameters, and 
     C S  is the a priori covariance matrix of the object space coordinates. 
     Following the methods described above, the math model in non-linear the least-squares solution needs to be iterated until convergence is achieved. At each iteration step, application of the least-squares principle results in the following vector of estimated corrections to the approximate values of the model parameters: 
     
       
           d{circumflex over (x)} =( A   T   C   w   −1   A ) −1   A   T   C   w   −1   w    
       
     
     At the subsequent iteration step the vector of approximate values of the object space coordinates x S0  is replaced by the estimated values {circumflex over (x)} S =x S0 +d{circumflex over (x)} S , 
     where d{circumflex over (x)} is the vector of corrections estimated in the previous iterative step and the math model is linearized again. The least-squares estimation is repeated until convergence is reached. 
     The covariance matrix of the estimated model parameters follows from application of the law of propagation of random errors with: 
     
       
           C   {circumflex over (x)} =( A   T   C   w   −1   A ) −1    
       
     
     While the preferred embodiment of the method has been described, those familiar with the art to which this invention relates will recognize various alternatives and embodiments for practicing the invention. 
     FIG. 10 is a flowchart  1000  illustrating a method of implementing the RPC model of the present invention. First, it is preferred that one or more images are acquired or input into the computer, Step  1002 . The images can be directly downloaded from the imaging device or input in another equivalent manner. Once the images are acquired, the tie points between any overlapping images are identified, and the image coordinates of the tie points are measured, Step  1004 . Next, any ground control points are identified and both the image and object coordinates of the ground control points are measured, Step  1006 . After any tie points and ground control points are determined, a RPC model for each image including adjustment parameters is established to model the nominal relationship between the image space and the ground space, initially, the adjustment parameters are set to predetermined a priori values (preferably zero), Step  1008 . Using the adjustment parameters, the observation equations are built, Step  1010 , and solved for parameter corrections, Step  1012 . The corrections are applied to the adjustment parameters to arrive at corrected adjustment parameters, Step  1014 . Next, it is determined whether the corrected adjustment parameters achieve convergence, Step  1016 . If convergence is achieved, the corrected adjustment parameters and covariance are same, Step  1018 . If convergence is not achieved, the adjustment parameters are replaced by the corrected adjustment parameters, Step  1020 , and Steps  1010  to  1016  are repeated. 
     While the invention has been particularly shown and described with reference to and in terms of the rational polynomial camera model, which is a presently preferred embodiment thereof, it will be understood by those skilled in the art that various other nominal relation between ground and image space with adjustment parameters or other changes in the form and details may be made without departing from the spirit and scope of the invention.