摄影测量学中, 将同名的物点与像点满足的函数关系描述为共线方程。在不考虑影像畸变情况下, 共线方程常表达为欧式坐标下的解析形式^[1]:

$ \left\{ \begin{array}{l} x = - f\frac{{{a_1}\left( {X - {X_{\rm{s}}}} \right) + {b_1}\left( {Y - {Y_{\rm{s}}}} \right) + {c_1}\left( {Z - {Z_{\rm{s}}}} \right)}}{{{a_3}\left( {X - {X_{\rm{s}}}} \right) + {b_3}\left( {Y - {Y_{\rm{s}}}} \right) + {c_3}\left( {Z - {Z_{\rm{s}}}} \right)}},\\ y = - f\frac{{{a_2}\left( {X - {X_{\rm{s}}}} \right) + {b_2}\left( {Y - {Y_{\rm{s}}}} \right) + {c_2}\left( {Z - {Z_{\rm{s}}}} \right)}}{{{a_3}\left( {X - {X_{\rm{s}}}} \right) + {b_3}\left( {Y - {Y_{\rm{s}}}} \right) + {c_3}\left( {Z - {Z_{\rm{s}}}} \right)}}, \end{array} \right. $

(1)

式中, [x, y]^T为像点坐标, f为主距, [X, Y, Z]^T为物方点坐标, [X_S, Y_S, Z_S]^T为摄站坐标, a₁~c₃为由外方位角元素构成的旋转矩阵元素。

共线方程贯穿于摄影测量学整个学科体系, 数据处理时一般先对其进行线性化处理。由于解析形式下的共线方程式为非线性函数, 且旋转矩阵元素与外方位角元素之间存在非常复杂的三角函数关系, 所以在求导过程中即使利用多次变换技巧, 像点坐标对外方位角元素求导后的解析结果也极其复杂。在计算机视觉领域, 同名的物点与像点的函数关系称为投影方程, 投影方程与共线方程仅个别参数定义有所差异^[2]。投影方程一般描述为矩阵形式, 通过矩阵分析方法获得相应问题结论, 方程形式非常简洁。因此, 可将共线方程也表达为矩阵形式, 利用矩阵分析理论获得其线性化解。

1 齐次坐标描述的共线方程

在计算机视觉领域, 习惯用齐次坐标下的矩阵形式表示投影方程^[3-6]:

$ \lambda \mathit{\boldsymbol{\tilde x}}{\rm{ = }}\mathit{\boldsymbol{M\tilde X}}, $

(2)

或

$ \lambda \left[ {\begin{array}{*{20}{c}} x\\ y\\ 1 \end{array}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{m_{11}}}\\ {{m_{21}}}\\ {{m_{31}}} \end{array}}&{\begin{array}{*{20}{c}} {{m_{12}}}\\ {{m_{22}}}\\ {{m_{32}}} \end{array}}&{\begin{array}{*{20}{c}} {{m_{13}}}\\ {{m_{23}}}\\ {{m_{33}}} \end{array}}&{\begin{array}{*{20}{c}} {{m_{14}}}\\ {{m_{24}}}\\ {{m_{34}}} \end{array}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} X\\ Y\\ Z\\ 1 \end{array}} \right], $

(3)

式中, M为投影矩阵; λ为尺度因子; $\mathit{\boldsymbol{\tilde x}}{\rm{ = }}{\left[ {{\mathit{\boldsymbol{x}}^T}{\rm{,}}\;\;1} \right]^T}{\rm{ = }}{\left[ {\begin{array}{*{20}{c}} x&y&1 \end{array}} \right]^T}$, ${\mathit{\boldsymbol{\tilde x}}}$为欧氏坐标下像点的物理坐标(mm); $\mathit{\boldsymbol{\tilde X}}{\rm{ = }}{\left[ {{\mathit{\boldsymbol{X}}^T}{\rm{,}}\;1} \right]^T}{\rm{ = }}{\left[ {X\;\;Y\;\;Z\;\;1} \right]^T}$, ${\mathit{\boldsymbol{\tilde X}}}$为欧氏坐标形式下同名物方点坐标。对比式(1)和(3), 显然, 投影矩阵M包含主距f、外方位元素线元素X_s及由外方位角元素构建的旋转矩阵R, 因此, 投影矩阵可构造为

$ \begin{array}{l} \mathit{\boldsymbol{M}} = \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{R}}^{\rm{T}}}\left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_{\rm{s}}}} \right]\\ = \left[ {\begin{array}{*{20}{c}} { - f}&{}&{}\\ {}&{ - f}&{}\\ {}&{}&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1\\ {}\\ {} \end{array}}&{\begin{array}{*{20}{c}} {}\\ 1\\ {} \end{array}}&{\begin{array}{*{20}{c}} {}\\ {}\\ 1 \end{array}}&{\begin{array}{*{20}{c}} { - {X_{\rm{s}}}}\\ { - {Y_{\rm{s}}}}\\ { - {Z_{\rm{s}}}} \end{array}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - f{a_1}}\\ { - f{a_2}}\\ {{a_3}} \end{array}}&{\begin{array}{*{20}{c}} { - f{b_1}}\\ { - f{b_2}}\\ {{b_3}} \end{array}}&{\begin{array}{*{20}{c}} { - f{c_1}}\\ { - f{c_2}}\\ {{c_3}} \end{array}}&{\begin{array}{*{20}{c}} {f{a_1}{X_{\rm{s}}} + f{b_1}{Y_{\rm{s}}} + f{c_1}{Z_{\rm{s}}}}\\ {f{a_2}{X_{\rm{s}}} + f{b_2}{Y_{\rm{s}}} + f{c_2}{Z_s}}\\ { - \left( {{a_3}{X_{\rm{s}}} + {b_3}{Y_{\rm{s}}} + {c_3}{Z_{\rm{s}}}} \right)} \end{array}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{m_{11}}}\\ {{m_{21}}}\\ {{m_{31}}} \end{array}}&{\begin{array}{*{20}{c}} {{m_{12}}}\\ {{m_{22}}}\\ {{m_{32}}} \end{array}}&{\begin{array}{*{20}{c}} {{m_{13}}}\\ {{m_{23}}}\\ {{m_{33}}} \end{array}}&{\begin{array}{*{20}{c}} {{m_{14}}}\\ {{m_{24}}}\\ {{m_{34}}} \end{array}} \end{array}} \right]。 \end{array} $

(4)

式中, K为由主距构成的对角矩阵; R为像空间坐标系到像空间辅助坐标系间的旋转矩阵; X_s为图像投影中心在像空间辅助坐标系中的欧氏坐标。

结合式(1)和(3)可以得到

$ \begin{array}{l} \left\{ \begin{array}{l} x = \frac{{{m_{11}}X + {m_{12}}Y + {m_{13}}Z + {m_{14}}}}{{{m_{31}}X + {m_{32}}Y + {m_{33}}Z + {m_{34}}}} = \frac{U}{\lambda },\\ y = \frac{{{m_{21}}X + {m_{22}}Y + {m_{23}}Z + {m_{24}}}}{{{m_{31}}X + {m_{32}}Y + {m_{33}}Z + {m_{34}}}} = \frac{V}{\lambda }, \end{array} \right.\\ \left\{ \begin{array}{l} U = {m_{11}}X + {m_{12}}Y + {m_{13}}Z + {m_{14}},\\ V = {m_{21}}X + {m_{22}}Y + {m_{23}}Z + {m_{24}},\\ \lambda = {m_{31}}X + {m_{32}}Y + {m_{33}}Z + {m_{34\;}}。 \end{array} \right. \end{array} $

(5)

因此, 式(1)~(3)和(5)在表达共线方程式时是一致的。

2 共线条件方程线性化的矩阵模型

以欧拉角描述的影像外方位元素(X_s, Y_s, Z_s, j, w, k)为例, 将式(1)线性化处理, 得到像点坐标误差方程式:

$ \left\{ \begin{array}{l} {v_x} = {a_{11}}\Delta {X_{\rm{s}}} + {a_{12}}\Delta {Y_{\rm{s}}} + {a_{13}}\Delta {Z_{\rm{s}}} + {a_{14}}\Delta \varphi + {a_{15}}\Delta \omega + \\ \;\;\;\;\;\;\;{a_{16}}\Delta \kappa - {a_{11}}\Delta X - {a_{12}}\Delta Y - {a_{13}}\Delta Z - {l_x},\\ {v_y} = {a_{21}}\Delta {X_{\rm{s}}} + {a_{22}}\Delta {Y_{\rm{s}}} + {a_{23}}\Delta {Z_{\rm{s}}} + {a_{24}}\Delta \varphi + {a_{25}}\Delta \omega + \\ \;\;\;\;\;\;\;{a_{26}}\Delta \kappa - {a_{21}}\Delta X - {a_{22}}\Delta Y - {a_{23}}\Delta Z - {l_y}。 \end{array} \right. $

(6)

式(6)可用矩阵表达为

$ {\mathit{\boldsymbol{V}}_{\rm{p}}} = \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{X}}_{{\rm{EoP}}}} + \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{X}}_{{\rm{Tie}}}} - {\mathit{\boldsymbol{L}}_{\rm{p}}} $

(7)

式中, ${\mathit{\boldsymbol{V}}_{\rm{p}}} = {\left[ {\begin{array}{*{20}{c}} {{v_x}}&{{v_y}} \end{array}} \right]^{\rm{T}}}$, ${\mathit{\boldsymbol{L}}_{\rm{p}}} = {\left[ {\begin{array}{*{20}{c}} {{l_x}}&{{l_y}} \end{array}} \right]^{\rm{T}}}$, $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \end{array}}&{\begin{array}{*{20}{c}} {{a_{14}}}&{{a_{15}}}&{{a_{16}}} \end{array}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \end{array}}&{\begin{array}{*{20}{c}} {{a_{24}}}&{{a_{25}}}&{{a_{26}}} \end{array}} \end{array}} \end{array}} \right]$, $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - {a_{11}}}&{ - {a_{12}}}&{ - {a_{13}}} \end{array}}\\ {\begin{array}{*{20}{c}} { - {a_{21}}}&{ - {a_{22}}}&{ - {a_{23}}} \end{array}} \end{array}} \right]$, ${\mathit{\boldsymbol{X}}_{{\rm{EoP}}}} = {\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\Delta {X_{\rm{s}}}}&{\Delta {Y_{\rm{s}}}}&{\Delta {Z_{\rm{s}}}} \end{array}}&{\begin{array}{*{20}{c}} {\Delta \varphi }&{\Delta \omega }&{\Delta \kappa } \end{array}} \end{array}} \right]^{\rm{T}}}$, ${\mathit{\boldsymbol{X}}_{{\rm{Tie}}}} = {\left[ {\begin{array}{*{20}{c}} {\Delta X}&{\Delta Y}&{\Delta Z} \end{array}} \right]^{\rm{T}}}$, a₁₁~a₂₆表示共线函数分别对各参数的偏导数。

这里, 采用复合函数求导方法(链式法则)推导像点坐标对外方位元素的导数, 即

$ \begin{array}{l} \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \end{array}}&{\begin{array}{*{20}{c}} {{a_{14}}}&{{a_{15}}}&{{a_{16}}} \end{array}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \end{array}}&{\begin{array}{*{20}{c}} {{a_{24}}}&{{a_{25}}}&{{a_{26}}} \end{array}} \end{array}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial x}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial x}}{{\partial {Z_{\rm{s}}}}}} \end{array}}&{\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial \varphi }}}&{\frac{{\partial x}}{{\partial \omega }}}&{\frac{{\partial x}}{{\partial \kappa }}} \end{array}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial y}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial y}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial y}}{{\partial {Z_{\rm{s}}}}}} \end{array}}&{\begin{array}{*{20}{c}} {\frac{{\partial y}}{{\partial \varphi }}}&{\frac{{\partial y}}{{\partial \omega }}}&{\frac{{\partial y}}{{\partial \kappa }}} \end{array}} \end{array}} \end{array}} \right]\\ = {\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial {m_{11}}}}}&{\frac{{\partial x}}{{\partial {m_{12}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{13}}}}}&{\frac{{\partial x}}{{\partial {m_{14}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{21}}}}}&{\frac{{\partial x}}{{\partial {m_{22}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{23}}}}}&{\frac{{\partial x}}{{\partial {m_{24}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{31}}}}}&{\frac{{\partial x}}{{\partial {m_{32}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{33}}}}}&{\frac{{\partial x}}{{\partial {m_{34}}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial y}}{{\partial {m_{11}}}}}&{\frac{{\partial y}}{{\partial {m_{12}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{13}}}}}&{\frac{{\partial y}}{{\partial {m_{14}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{21}}}}}&{\frac{{\partial y}}{{\partial {m_{22}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{23}}}}}&{\frac{{\partial y}}{{\partial {m_{24}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{31}}}}}&{\frac{{\partial y}}{{\partial {m_{32}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{33}}}}}&{\frac{{\partial y}}{{\partial {m_{34}}}}} \end{array}} \end{array}} \right]_{2 \times 12}} \end{array} $

$ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{12}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{13}}}}{{\partial {X_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{14}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{21}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{22}}}}{{\partial {X_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{23}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{24}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{31}}}}{{\partial {X_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{32}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{33}}}}{{\partial {X_{\rm{s}}}}}}&{\frac{{\partial {m_{34}}}}{{\partial {X_{\rm{s}}}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{12}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{13}}}}{{\partial {Y_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{14}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{21}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{22}}}}{{\partial {Y_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{23}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{24}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{31}}}}{{\partial {Y_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{32}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{33}}}}{{\partial {Y_{\rm{s}}}}}}&{\frac{{\partial {m_{34}}}}{{\partial {Y_{\rm{s}}}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{12}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{13}}}}{{\partial {Z_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{14}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{21}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{22}}}}{{\partial {Z_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{23}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{24}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{31}}}}{{\partial {Z_{\rm{s}}}}}} \end{array}}&{\frac{{\partial {m_{32}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{33}}}}{{\partial {Z_{\rm{s}}}}}}&{\frac{{\partial {m_{34}}}}{{\partial {Z_{\rm{s}}}}}} \end{array}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{12}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{13}}}}{{\partial \varphi }}} \end{array}}&{\frac{{\partial {m_{14}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{21}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{22}}}}{{\partial \varphi }}} \end{array}}&{\frac{{\partial {m_{23}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{24}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{31}}}}{{\partial \varphi }}} \end{array}}&{\frac{{\partial {m_{32}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{33}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{34}}}}{{\partial \varphi }}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial \omega }}}&{\frac{{\partial {m_{12}}}}{{\partial \omega }}}&{\frac{{\partial {m_{13}}}}{{\partial \omega }}} \end{array}}&{\frac{{\partial {m_{14}}}}{{\partial \omega }}}&{\frac{{\partial {m_{21}}}}{{\partial \omega }}}&{\frac{{\partial {m_{22}}}}{{\partial \omega }}} \end{array}}&{\frac{{\partial {m_{23}}}}{{\partial \omega }}}&{\frac{{\partial {m_{24}}}}{{\partial \omega }}}&{\frac{{\partial {m_{31}}}}{{\partial \omega }}} \end{array}}&{\frac{{\partial {m_{32}}}}{{\partial \omega }}}&{\frac{{\partial {m_{33}}}}{{\partial \omega }}}&{\frac{{\partial {m_{34}}}}{{\partial \omega }}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{12}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{13}}}}{{\partial \kappa }}} \end{array}}&{\frac{{\partial {m_{14}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{21}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{22}}}}{{\partial \kappa }}} \end{array}}&{\frac{{\partial {m_{23}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{24}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{31}}}}{{\partial \kappa }}} \end{array}}&{\frac{{\partial {m_{32}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{33}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{34}}}}{{\partial \kappa }}} \end{array}} \end{array}} \right]_{6 \times 12}^{\rm{T}}\\ = {\mathit{\boldsymbol{J}}_M} \cdot {\mathit{\boldsymbol{J}}_e}。 \end{array} $

(8)

事实上, 在数值解算中, 只要弄清楚J_M和J_e, 就可以得到系数矩阵A, 这也是矩阵方法的另一个优势。

由式(5)易得

$ \begin{array}{l} {\mathit{\boldsymbol{J}}_M} = \\ \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial {m_{11}}}}}&{\frac{{\partial x}}{{\partial {m_{12}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{13}}}}}&{\frac{{\partial x}}{{\partial {m_{14}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{21}}}}}&{\frac{{\partial x}}{{\partial {m_{22}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{23}}}}}&{\frac{{\partial x}}{{\partial {m_{24}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{31}}}}}&{\frac{{\partial x}}{{\partial {m_{32}}}}} \end{array}}&{\frac{{\partial x}}{{\partial {m_{33}}}}}&{\frac{{\partial x}}{{\partial {m_{34}}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial y}}{{\partial {m_{11}}}}}&{\frac{{\partial y}}{{\partial {m_{12}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{13}}}}}&{\frac{{\partial y}}{{\partial {m_{14}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{21}}}}}&{\frac{{\partial y}}{{\partial {m_{22}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{23}}}}}&{\frac{{\partial y}}{{\partial {m_{24}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{31}}}}}&{\frac{{\partial y}}{{\partial {m_{32}}}}} \end{array}}&{\frac{{\partial y}}{{\partial {m_{33}}}}}&{\frac{{\partial y}}{{\partial {m_{34}}}}} \end{array}} \end{array}} \right]\\ = \left[ \begin{array}{l} \frac{X}{\lambda }\;\;\frac{Y}{\lambda }\;\;\frac{Z}{\lambda }\;\;\frac{1}{\lambda }\;\;\;0\;\;\;0\;\;\;0\;\;\;0\;\; - \frac{{xX}}{\lambda }\;\; - \frac{{xY}}{\lambda }\;\; - \frac{{xZ}}{\lambda }\;\; - \frac{x}{\lambda }\\ \;0\;\;\;0\;\;\;0\;\;\;0\;\;\frac{X}{\lambda }\;\;\frac{Y}{\lambda }\;\;\frac{Z}{\lambda }\;\;\frac{1}{\lambda }\;\; - \frac{{yX}}{\lambda }\;\; - \frac{{yY}}{\lambda }\;\; - \frac{{yZ}}{\lambda }\;\; - \frac{y}{\lambda } \end{array} \right]\\ = \frac{1}{\lambda }\left[ {\begin{array}{*{20}{c}} X\\ 0 \end{array}\;\;\begin{array}{*{20}{c}} Y\\ 0 \end{array}\;\;\begin{array}{*{20}{c}} Z\\ 0 \end{array}\;\;\begin{array}{*{20}{c}} 1\\ 0 \end{array}\;\;\begin{array}{*{20}{c}} 0\\ X \end{array}\;\;\begin{array}{*{20}{c}} 0\\ Y \end{array}\;\;\begin{array}{*{20}{c}} 0\\ Z \end{array}\;\;\begin{array}{*{20}{c}} 0\\ 1 \end{array}\;\;\begin{array}{*{20}{c}} { - xX}\\ { - yX} \end{array}\;\;\begin{array}{*{20}{c}} { - xY}\\ { - yY} \end{array}\;\;\begin{array}{*{20}{c}} { - xZ}\\ { - yZ} \end{array}\;\;\begin{array}{*{20}{c}} { - x}\\ { - y} \end{array}} \right]. \end{array} $

(9)

为了求解J_e, 对投影矩阵$\mathit{\boldsymbol{M}}{\rm{ = }}\mathit{\boldsymbol{K}}{\mathit{\boldsymbol{R}}^{\rm{T}}}\left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_{\rm{s}}}} \right]$两边对外方位元素求导可得

$ \begin{array}{l} \mathit{\boldsymbol{M}}{\rm{ = }}\mathit{\boldsymbol{K}}{\mathit{\boldsymbol{R}}^{\rm{T}}}\left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_{\rm{s}}}} \right]\\ \frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial ({X_{\rm{s}}},\;{Y_{\rm{s}}},\;{Z_{\rm{s}}},\;\varphi ,\;\omega {\rm{,}}\;\kappa )}}{\rm{ = }}\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{R}}^T}\left[ {\mathit{\boldsymbol{I}},\; - {\mathit{\boldsymbol{X}}_{\rm{s}}}} \right]}}{{\partial ({X_{\rm{s}}},\;{Y_{\rm{s}}},\;{Z_{\rm{s}}},\;\varphi ,\;\omega {\rm{,}}\;\kappa )}}。 \end{array} $

$ \begin{array}{l} \frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial {X_{\rm{s}}}}}{\rm{ = }}\mathit{\boldsymbol{K}}{\mathit{\boldsymbol{R}}^T}\frac{{\partial \left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_{\rm{s}}}} \right]}}{{\partial {X_{\rm{s}}}}}\\ \;\;\;\;\;\; = \left[ {\begin{array}{*{20}{c}} { - f{a_1}}&{ - f{b_1}}&{ - f{c_1}}\\ { - f{a_2}}&{ - f{b_2}}&{ - f{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&0&0&{ - 1} \end{array}}\\ {\begin{array}{*{20}{c}} 0&0&0&0 \end{array}}\\ {\begin{array}{*{20}{c}} 0&0&0&0 \end{array}} \end{array}} \right]\\ \;\;\;\;\;\; = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} {f{a_1}}\\ {f{a_2}}\\ { - {a_3}} \end{array}} \end{array}} \right]。 \end{array} $

(10)

其他变量求导依此类推,

$ \frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial {Z_{\rm{s}}}}}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} {f{c_1}}\\ {f{c_2}}\\ { - {c_3}} \end{array}} \end{array}} \right],\frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial {Y_{\rm{s}}}}}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} {f{b_1}}\\ {f{b_2}}\\ { - {b_3}} \end{array}} \end{array}} \right]。 $

对于φ-ω-κ转角系统, 旋转矩阵

$ \begin{array}{l} {\mathit{\boldsymbol{R}}^T} = \left[ {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {\cos \varphi \cos \kappa - \sin \varphi \sin \omega \sin \kappa }&{\cos \omega \sin \kappa }&{\sin \varphi \cos \kappa + \cos \varphi \sin \omega \sin \kappa }\\ { - \cos \varphi \sin \kappa - \sin \varphi \sin \omega \cos \kappa }&{\cos \omega \cos \kappa }&{ - \sin \varphi \sin \kappa + \cos \varphi \sin \omega \cos \kappa }\\ { - \sin \varphi \cos \omega }&{ - \sin \omega }&{\cos \varphi \cos \omega } \end{array}} \right]。 \end{array} $

则

$ \begin{array}{l} \frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial \varphi }}{\rm{ = }}\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{R}}^T}}}{{\partial \varphi }}\left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_{\rm{S}}}} \right] = \mathit{\boldsymbol{KR}}_\varphi ^T\left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_{\rm{S}}}} \right]\\ = \left[ {\begin{array}{*{20}{c}} { - f}&{}&{}\\ {}&{ - f}&{}\\ {}&{}&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} { - {c_1}}&0&{{a_1}}\\ { - {c_2}}&0&{{a_2}}\\ { - {c_3}}&0&{{a_3}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&{}&{}&{ - {X_{\rm{s}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {}&1&{}&{ - {Y_{\rm{s}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {}&{}&1&{ - {Z_{\rm{s}}}} \end{array}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {f{c_1}}\\ {f{c_2}}\\ { - {c_3}} \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} { - f{a_1}}\\ { - f{a_2}}\\ {{a_3}} \end{array}}&{\begin{array}{*{20}{c}} { - f{c_1}{X_{\rm{s}}} + f{a_1}{Z_{\rm{s}}}}\\ { - f{c_2}{X_{\rm{s}}} - f{a_2}{Z_{\rm{s}}}}\\ {{c_3}{X_{\rm{s}}} - {a_3}{Z_{\rm{s}}}} \end{array}} \end{array}} \right]。 \end{array} $

(11)

其他变量求导依此类推, 可得

$ \begin{array}{l} \frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial \omega }} = \mathit{\boldsymbol{KR}}_\omega ^T\left[ {\mathit{\boldsymbol{I}},\; - {\mathit{\boldsymbol{X}}_{\rm{s}}}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - f{a_3}\sin \kappa }\\ { - f{a_3}cos\kappa }\\ {\sin \varphi \sin \omega } \end{array}}&{\begin{array}{*{20}{c}} { - f{b_3}\sin \kappa }\\ { - f{b_3}\cos \kappa }\\ { - \cos \omega } \end{array}}&{\begin{array}{*{20}{c}} { - f{c_3}\sin \kappa }\\ { - f{c_3}\cos \kappa }\\ { - \cos \varphi \sin \omega } \end{array}}&{\begin{array}{*{20}{c}} {f\sin \kappa ({a_3}{X_{\rm{s}}} + {b_3}{Y_{\rm{s}}} + {c_3}{Z_{\rm{s}}})}\\ {f\cos \kappa ({a_3}{X_{\rm{s}}} + {b_3}{Y_{\rm{s}}} + {c_3}{Z_{\rm{s}}})}\\ { - (\sin \varphi \sin \omega {X_{\rm{s}}} - \cos \omega {Y_{\rm{s}}} - \cos \varphi \sin \omega {Z_{\rm{s}}})} \end{array}} \end{array}} \right]。 \end{array} $

(12)

$ \begin{array}{l} \frac{{\partial \mathit{\boldsymbol{M}}}}{{\partial \kappa }} = \mathit{\boldsymbol{KR}}_\kappa ^T\left[ {\mathit{\boldsymbol{I}}, - {\mathit{\boldsymbol{X}}_s}} \right]\\ = \left[ {\begin{array}{*{20}{c}} { - f}&{}&{}\\ {}&{ - f}&{}\\ {}&{}&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{a_2}}&{{b_2}}&{{c_2}}\\ { - {a_1}}&{ - {b_1}}&{ - {c_1}}\\ 0&0&0 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&{}&{}&{ - {X_{\rm{s}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {}&1&{}&{ - {Y_{\rm{s}}}} \end{array}}\\ {\begin{array}{*{20}{c}} {}&{}&1&{ - {Z_{\rm{s}}}} \end{array}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - f{a_2}}\\ {f{a_1}}\\ 0 \end{array}}&{\begin{array}{*{20}{c}} { - f{b_2}}\\ {f{b_1}}\\ 0 \end{array}}&{\begin{array}{*{20}{c}} { - f{c_2}}\\ {f{c_1}}\\ 0 \end{array}}&{\begin{array}{*{20}{c}} {f{a_2}{X_{\rm{s}}} + f{b_2}{Y_{\rm{s}}} + f{c_2}{Z_{\rm{s}}}}\\ { - (f{a_1}{X_{\rm{s}}} + f{b_1}{Y_{\rm{s}}} + f{c_1}{Z_{\rm{s}}})}\\ 0 \end{array}} \end{array}} \right]。 \end{array} $

(13)

通过以上式子可以得到$\frac{{\partial ({m_{11}},\;...,\;{m_{34}})}}{{\partial (\varphi ,\omega ,\kappa )}}$, 从而得到J_e。

由式(5)同样易得

$ \begin{array}{l} {\mathit{\boldsymbol{J}}_X} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial X}}}&{\frac{{\partial x}}{{\partial Y}}}&{\frac{{\partial x}}{{\partial Z}}} \end{array}}\\ {\begin{array}{*{20}{c}} {\frac{{\partial y}}{{\partial X}}}&{\frac{{\partial y}}{{\partial Y}}}&{\frac{{\partial y}}{{\partial Z}}} \end{array}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{{{m_{11}} - x{m_{31}}}}{\lambda }}&{\frac{{{m_{12}} - x{m_{32}}}}{\lambda }}&{\frac{{{m_{13}} - x{m_{33}}}}{\lambda }} \end{array}}\\ {\begin{array}{*{20}{c}} {\frac{{{m_{21}} - y{m_{31}}}}{\lambda }}&{\frac{{{m_{22}} - y{m_{32}}}}{\lambda }}&{\frac{{{m_{33}} - y{m_{33}}}}{\lambda }} \end{array}} \end{array}} \right]\\ = \frac{1}{\lambda }\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{m_{11}} - x{m_{31}}}&{{m_{12}} - x{m_{32}}}&{{m_{13}} - x{m_{33}}} \end{array}}\\ {\begin{array}{*{20}{c}} {{m_{21}} - y{m_{31}}}&{{m_{22}} - y{m_{32}}}&{{m_{33}} - y{m_{33}}} \end{array}} \end{array}} \right]\\ = \frac{1}{\lambda }\left\{ {\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{m_{11}}}&{{m_{12}}}&{{m_{13}}} \end{array}}\\ {\begin{array}{*{20}{c}} {{m_{21}}}&{{m_{22}}}&{{m_{33}}} \end{array}} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {x{m_{31}}}&{x{m_{32}}}&{x{m_{33}}} \end{array}}\\ {\begin{array}{*{20}{c}} {y{m_{31}}}&{y{m_{32}}}&{y{m_{33}}} \end{array}} \end{array}} \right]} \right\}。 \end{array} $

(14)

至此, 基于矩阵分析方法的共线方程线性化系数推导完毕。需要指出的是, 该方法对由不同姿态元素构造的旋转矩阵都有很好的适用性, 如反对称矩阵法、四元数及轴角^[7]等, 差异之处仅在于投影矩阵对姿态元素变量的求导上, 即式(11)~(13)。

由式(14)易知, 对于物方点及外方位线元素的求导结果与解析方法一致, 而对于外方位角元素, 因篇幅有限, 本文仅证明$\frac{{\partial x}}{{\partial \varphi }}$, $\frac{{\partial x}}{{\partial \omega }}$, $\frac{{\partial x}}{{\partial \kappa }}$与解析法^[1]的一致性。

3 矩阵分析法结论证明

1) $\frac{{\partial x}}{{\partial \varphi }} = {\mathit{\boldsymbol{J}}_M} \cdot {\mathit{\boldsymbol{J}}_\varphi }$

$ = \left[{\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial {m_{11}}}}}&{\frac{{\partial x}}{{\partial {m_{12}}}}} {\frac{{\partial x}}{{\partial {m_{13}}}}}&{\frac{{\partial x}}{{\partial {m_{14}}}}} {\frac{{\partial x}}{{\partial {m_{21}}}}}&{\frac{{\partial x}}{{\partial {m_{22}}}}} {\frac{{\partial x}}{{\partial {m_{23}}}}}&{\frac{{\partial x}}{{\partial {m_{24}}}}} {\frac{{\partial x}}{{\partial {m_{31}}}}}&{\frac{{\partial x}}{{\partial {m_{32}}}}} {\frac{{\partial x}}{{\partial {m_{33}}}}}&{\frac{{\partial x}}{{\partial {m_{34}}}}} \end{array}} \right] \cdot $

${\left[{\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{12}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{13}}}}{{\partial \varphi }}} {\frac{{\partial {m_{14}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{21}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{22}}}}{{\partial \varphi }}} {\frac{{\partial {m_{23}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{24}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{31}}}}{{\partial \varphi }}} {\frac{{\partial {m_{32}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{33}}}}{{\partial \varphi }}}&{\frac{{\partial {m_{34}}}}{{\partial \varphi }}} \end{array}} \right]^T}$

$ = \frac{1}{\lambda }\left[ {\begin{array}{*{20}{c}} X&Y&Z 1&0&0 0&0&{ - xX} { - xY}&{ - xZ}&{ - x} \end{array}} \right] \cdot $

${\left[{\begin{array}{*{20}{c}} {f{c_1}}&0&{ - f{a_1}} { - f{c_1}{X_{\rm{s}}} + f{a_1}{Z_{\rm{s}}}}&{f{c_2}}&0 { - f{a_2}}&{ - f{c_2}{X_{\rm{s}}} + - f{a_2}{Z_{\rm{s}}}}&{ - {c_3}} 0&{{a_3}}&{{c_3}{X_{\rm{s}}} - {a_3}{Z_{\rm{s}}}} \end{array}} \right]^T}$

$ = \frac{1}{\lambda }\left[ {f\left( {X - {X_{\rm{s}}}} \right){c_1} + x\left( {X - {X_{\rm{s}}}} \right) \cdot {c_3} - f\left( {Z - {Z_{\rm{s}}}} \right) \cdot {a_1} - x\left( {Z - {Z_{\rm{s}}}} \right){a_3}} \right]$。

考虑到$\left[ {\begin{array}{*{20}{c}} {X - {X_{\rm{s}}}}\\ {Y - {Y_{\rm{s}}}}\\ {Z - {Z_{\rm{s}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}\\ {{b_1}}&{{b_2}}&{{b_3}}\\ {{c_1}}&{{c_2}}&{{c_3}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\bar X}\\ {\bar Y}\\ {\bar Z} \end{array}} \right]$及$\lambda = \bar Z$, 则

$ \begin{array}{l} \frac{{\partial x}}{{\partial \varphi }} = \frac{1}{{\bar Z}}\left[ {f\left( {{a_1}\bar X + {a_2}\bar Y + {a_3}\bar Z} \right){c_1} + x\left( {{a_1}\bar X + {a_2}\bar Y + {a_3}\bar Z} \right) \cdot {c_3}} \right.\\ \left. {\;\;\;\;\;\;\;\;\; - f\left( {{c_1}\bar X + {c_2}\bar Y + {c_3}\bar Z} \right) \cdot {a_1} - x\left( {{c_1}\bar X + {c_2}\bar Y + {c_3}\bar Z} \right){a_3}} \right]\\ \;\;\;\;\;\; = \frac{1}{{\bar Z}}\left[ {f\left( {{a_2}\bar Y + {a_3}\bar Z} \right){c_1} - f\left( {{c_2}\bar Y + {c_3}\bar Z} \right) \cdot {a_1} + x\left( {{a_1}\bar X + {a_2}\bar Y} \right) \cdot {c_3} - x\left( {{c_1}\bar X + {c_2}\bar Y} \right){a_3}} \right]\\ \;\;\;\;\;\; = \frac{1}{{\bar Z}}\left[ {f\left( {{a_2}{c_1} - {c_2}{a_1}} \right)\bar Y + f\left( {{a_3}{c_1} - {c_3}{a_1}} \right)\bar Z + x\left( {{a_1}{c_3} - {c_1}{a_3}} \right)\bar X + x\left( {{a_2}{c_3} - {c_2}{a_3}} \right)\bar Y} \right]\\ \;\;\;\;\;\; = \frac{1}{{\bar Z}}\left( {f{b_3}\bar Y - f{b_2}\bar Z + x{b_2}\bar X - x{b_1}\bar Y} \right) = - f{b_3}y - f{b_2} - \frac{{{x^2}}}{f}{b_2} + \frac{{xy}}{f}{b_1}。 \end{array} $

2) $\frac{{\partial x}}{{\partial \omega }} = {\mathit{\boldsymbol{J}}_M} \cdot {\mathit{\boldsymbol{J}}_\omega }$

$ = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial {m_{11}}}}}&{\frac{{\partial x}}{{\partial {m_{12}}}}} {\frac{{\partial x}}{{\partial {m_{13}}}}}&{\frac{{\partial x}}{{\partial {m_{14}}}}} {\frac{{\partial x}}{{\partial {m_{21}}}}}&{\frac{{\partial x}}{{\partial {m_{22}}}}} {\frac{{\partial x}}{{\partial {m_{23}}}}}&{\frac{{\partial x}}{{\partial {m_{24}}}}} {\frac{{\partial x}}{{\partial {m_{31}}}}}&{\frac{{\partial x}}{{\partial {m_{32}}}}} {\frac{{\partial x}}{{\partial {m_{33}}}}}&{\frac{{\partial x}}{{\partial {m_{34}}}}} \end{array}} \right] \cdot $

${\left[ {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial \omega }}}&{\frac{{\partial {m_{12}}}}{{\partial \omega }}}&{\frac{{\partial {m_{13}}}}{{\partial \omega }}} {\frac{{\partial {m_{14}}}}{{\partial \omega }}}&{\frac{{\partial {m_{21}}}}{{\partial \omega }}}&{\frac{{\partial {m_{22}}}}{{\partial \omega }}} {\frac{{\partial {m_{23}}}}{{\partial \omega }}}&{\frac{{\partial {m_{24}}}}{{\partial \omega }}}&{\frac{{\partial {m_{31}}}}{{\partial \omega }}} {\frac{{\partial {m_{32}}}}{{\partial \omega }}}&{\frac{{\partial {m_{33}}}}{{\partial \omega }}}&{\frac{{\partial {m_{34}}}}{{\partial \omega }}} \end{array}} \right]^T}$

$ = \frac{1}{\lambda }\left[ {\begin{array}{*{20}{c}} X&Y&Z 1&0&0 0&0&{ - xX} { - xY}&{ - xZ}&{ - x} \end{array}} \right]$

$\left[ {\begin{array}{*{20}{c}} { - f{a_3}\sin \kappa } { - f{b_3}\sin \kappa }&{ - f{c_3}\sin \kappa }&{f\sin \kappa ({a_3}{X_{\rm{s}}} + {b_3}{Y_{\rm{s}}} + {c_3}{Z_{\rm{s}}})} \end{array}} \right.$

$\;\;\begin{array}{*{20}{c}} { - f{a_3}cos\kappa }&{ - f{b_3}\cos \kappa }&{ - f{c_3}\cos \kappa }&{f\cos \kappa ({a_3}{X_{\rm{s}}} + {b_3}{Y_{\rm{s}}} + {c_3}{Z_{\rm{s}}})} \end{array}$

${\left. {\begin{array}{*{20}{c}} {\;\;\sin \varphi \sin \omega }&{ - \cos \omega }&{ - \cos \varphi \sin \omega }&{ - (\sin \varphi \sin \omega {X_{\rm{s}}} - \cos \omega {Y_{\rm{s}}} - \cos \varphi \sin \omega {Z_{\rm{s}}})} \end{array}} \right]^{\rm{T}}}$

$ = \frac{1}{\lambda }\left\{ {f\sin \kappa [ - {a_3}(X - {X_{\rm{s}}}) - {b_3}(Y - {Y_{\rm{s}}}) - {c_3}(Z - {Z_{\rm{s}}})]} \right. - x[\sin \varphi \sin \omega (X - {X_{\rm{s}}}) - $

$\;\;\cos \omega (Y - {Y_{\rm{s}}}) - \cos \varphi \sin \omega (Z - {Z_{\rm{s}}})]\} $

$ = \frac{1}{{\bar Z}}\left\{ {f\sin \kappa [ - {a_3}({a_1}\bar X + {a_2}\bar Y + {a_3}\bar Z) - {b_3}({b_1}\bar X + {b_2}\bar Y + {b_3}\bar Z) - {c_3}({c_1}\bar X + {c_2}\bar Y + {c_3}\bar Z)]} \right. - $

$\;\;\left. {x[\sin \varphi \sin \omega ({a_1}\bar X + {a_2}\bar Y + {a_3}\bar Z) - \cos \omega ({b_1}\bar X + {b_2}\bar Y + {b_3}\bar Z) - \cos \varphi \sin \omega ({c_1}\bar X + {c_2}\bar Y + {c_3}\bar Z)]} \right\}$

$ = \frac{1}{{\bar Z}}[ - f\sin \kappa \bar Z + x(\bar X\sin \kappa + \bar Y\cos \kappa )]$

$ = - f\sin \kappa - \frac{1}{f}({x^2}\sin \kappa + xy\cos \kappa )。$

3) $\frac{{\partial x}}{{\partial \kappa }} = {\mathit{\boldsymbol{J}}_M} \cdot {\mathit{\boldsymbol{J}}_\kappa }$

$ = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial {m_{11}}}}}&{\frac{{\partial x}}{{\partial {m_{12}}}}} {\frac{{\partial x}}{{\partial {m_{13}}}}}&{\frac{{\partial x}}{{\partial {m_{14}}}}} {\frac{{\partial x}}{{\partial {m_{21}}}}}&{\frac{{\partial x}}{{\partial {m_{22}}}}} {\frac{{\partial x}}{{\partial {m_{23}}}}}&{\frac{{\partial x}}{{\partial {m_{24}}}}} {\frac{{\partial x}}{{\partial {m_{31}}}}}&{\frac{{\partial x}}{{\partial {m_{32}}}}} {\frac{{\partial x}}{{\partial {m_{33}}}}}&{\frac{{\partial x}}{{\partial {m_{34}}}}} \end{array}} \right] \cdot $

${\left[ {\begin{array}{*{20}{c}} {\frac{{\partial {m_{11}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{12}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{13}}}}{{\partial \kappa }}} {\frac{{\partial {m_{14}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{21}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{22}}}}{{\partial \kappa }}} {\frac{{\partial {m_{23}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{24}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{31}}}}{{\partial \kappa }}} {\frac{{\partial {m_{32}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{33}}}}{{\partial \kappa }}}&{\frac{{\partial {m_{34}}}}{{\partial \kappa }}} \end{array}} \right]^T}$

$ = \frac{1}{\lambda }\left[ {\begin{array}{*{20}{c}} X&Y&Z 1&0&0 0&0&{ - xX} { - xY}&{ - xZ}&{ - x} \end{array}} \right] \cdot $

${\left[ {\begin{array}{*{20}{c}} { - f{a_2}}&{ - f{b_2}}&{ - f{c_2}} {f{a_2}{X_{\rm{s}}} + f{b_2}{Y_{\rm{s}}} + f{c_2}{Z_s}}&{f{a_1}}&{f{b_1}} {f{c_1}}&{ - (f{a_1}{X_{\rm{s}}} + f{b_1}{Y_{\rm{s}}} + f{c_1}{Z_{\rm{s}}})}&0 0&0&0 \end{array}} \right]^T}$

$ = \frac{1}{\lambda }[ - f{a_2}(X - {X_{\rm{s}}}) - f{b_2}(Y - {Y_{\rm{s}}}) - f{c_2}(Z - {Z_{\rm{s}}})]$

$ = \frac{1}{{\bar Z}}\left[ { - f({a_2}{a_2} + {b_2}{b_2} + {c_2}{c_2})\bar Y} \right]$

$ = y$。

求导结果与解析方法^[1]的求导结果完全一致。

4 结论与贡献

共线方程线性化是摄影测量数据处理的前提, 基于解析方法建立的共线方程线性化过程及结果形式非常复杂, 不易掌握。利用矩阵分析方法能有效降低推导的复杂性, 求导过程清晰明了, 为摄影测量这一基本关系式处理提供全新的分析方法。本文主要结论和贡献如下。

1) 借鉴计算机视觉中“共线方程”的矩阵形式, 构造出摄影测量中共线方程的矩阵表达形式, 为利用矩阵分析方法奠定理论基础, 也为计算机视觉与摄影测量学科之间建立重要联系。

2) 详细推导建立共线方程线性化的矩阵分析方法, 并且对推导结果给予验证。

3) 矩阵分析方法建立的共线方程线性化过程对于其他形式构造的旋转矩阵都有较好的适用性, 如四元数、反对称矩阵及轴角方法等, 都可将共线方程线性化问题归结为旋转矩阵对角元素的求导, 思路清晰, 也利于数值解算。