# “变换”(Coverage) 的工作原理

## 描述

“变换”工具的基本做法是比较“输入 Coverage”与“输出 Coverage”中对应控制点的坐标。Tic-ID 会确定要进行比较的控制点。所需控制点的最小数量取决于所使用的变换选项。这里所说的控制点被视为针对变换的控制点，并且不会由变换方程转换。输出控制点的坐标在变换后保持不变。不过，任何与输入控制点重合的要素通常都不会与输出控制点重合。因此，在变换之后可能需要对要素进行调整，以强制它们与“输出 Coverage”中的控制点重合。

`x’ = Ax + By + C`

`y’ = Dx + Ey + F`

“变换”工具将报告这六个参数及其几何解释。以下是仿射变换的输出报告的节选部分示例。

``````Scale (X,Y) = (246.140,255.702)  Skew (degrees) = (-0.061)
Rotation (degrees) = (0.334)  Translation = (2890.267,3679.906)
RMS Error (input, output) = (0.084,20.592)

Affine X = Ax + By + C
Y = Dx + Ey + F
A =   246.135   B =   -1.763        C =   2890.267
D =   1.434     E =   255.696       F =   3679.906
``````

``````A = mx · cos t
B = my · (k · cos t - sin t)
D = mx · sin t
E = my · (k · sin t + cos t)
C = translation in x direction
F = translation in y direction

where

mx = change of scale in x direction
my = change of scale in y direction
k = shear factor along the x-axis = tan (skew angle) (the skew angle is measured from the y-axis)
t = rotation angle, measured counterclockwise from the x-axis
``````

SIMILARITY 变换可以缩放、旋转和平移数据。但不会单独缩放轴，也不会产生任何倾斜。要进行相似变换至少需要两个控制点。以下是针对某一相似变换的报告，该变换使用之前经过了仿射变换的 coverage。

``````Scale (X,Y) = (249.927,249.927)
Rotation (degrees) = (0.362)  Translation = (2855.407,3715.168)
RMS Error (input, output) = (0.118,29.398)

Similarity X =  Ax + By + C
Y = -Bx + Ay + F
A =   249.922         B =   -1.578
C =   2855.407        F =   3715.168
``````

``````x’ =  Ax + By + C
y’ = -Bx + Ay + F

where

A = s · cos t
B = s · sin t
C = translation in x direction
F = translation in y direction

and

s = scale change (same in x and y directions)
t = rotation angle, measured counterclockwise from the x-axis
``````

PROJECTIVE 变换基于更加复杂的公式，至少需要四个控制点。

```x’ = (Ax + By + C) / (Gx + Hy + 1)
y’ = (Dx + Ey + F) / (Gx + Hy + 1)```

PROJECTIVE 报告包括近似比例、RMS 误差和方程参数。由于解释的复杂性，将不对投影变换参数进行解释。有关详细信息，请参阅本主题结尾处的参考中所列出的摄影测量文本之一。下面是投影变换输出报告的一个示例。

``````Approximate scale = 1479.087
RMS Error (input, output) = (0.040,60.878)

2D Projective  X = (Ax + By + C) / (Gx + Hy + 1)
Y = (Dx + Ey + F) / (Gx + Hy + 1)

A =         55.667   B =       -718.999   C =    2125052.558
D =       -199.525   E =       1385.541   F =     317759.475
G =         -0.001   H =          0.000

Principal point of input (xp,yp) = (2.000,16.946)
Exposure center of output(Xc,Yc) = (2127791.000,343183.000)
``````

RMS 误差将测量“输出 Coverage”控制点与变换后的“输入 Coverage”控制点位置之间的误差。

`RMS Error (input, output) = (0.031,37.465)`

```tic id  input x   input y   output x      output y     x error   y error
------------------------------------------------------------------------
1       2.000     16.946    2127791.000   343183.000    14.463    75.499
2       12.764    16.821    2143469.000   343326.000   -31.043   -85.363
3       2.052     1.976     2128000.000   320680.000   -36.290    -2.353
4       12.922    2.013     2143729.000   320912.000    20.245    -6.163
5       2.082     9.442     2127944.000   332015.000    22.016   -74.699
6       12.662    9.442     2143320.000   332015.000    10.609    93.079```

## 示例

```IDTIC    XTIC       YTIC
1       2.000     16.946
2      12.764     16.821
3       2.052      1.976
4      12.922      2.013
5       2.082      9.442
6      12.662      9.442```

```Tic-IDs	 X Coordinates	 Y Coordinates
1	       2,127,791     	343,183
2       	2,143,469     	343,326
3       	2,128,000     	320,680
4       	2,143,729     	320,912
5       	2,127,944     	332,015
6	       2,143,320     	332,015```

“输出 Coverage”的控制点文件必须包含要保留在“输出 Coverage”中的每个控制点的 x,y 坐标；这种情况下，控制点位置处于美国国家平面坐标系中。控制点坐标的调整可以在 ArcMap 中进行，也可以在 ArcInfo Workstation 的 TABLES 或 INFO 模块中进行。

``````Scale (X,Y) = (1452.317,1508.433)  Skew (degrees) = (0.416)
Rotation (degrees) = (0.218) Translation = (2124994.654,317664.385)
RMS Error (input, output) = (0.048,71.614)

Affine  X = Ax + By + C
Y = Dx + Ey + F
A =     1452.230   B =       -5.526   C =   2124994.654
D =       15.858   E =     1508.462   F =    317664.385

tic id  input  x      input  y
output x      output y   x error   y error
--------------------------------------------------
1          2.000        16.946
2127791.000    343183.000    14.463    75.499
2         12.764        16.821
2143469.000    343326.000   -31.043   -85.363
3          2.052         1.976
2128000.000    320680.000   -36.290    -2.353
4         12.922         2.013
2143729.000    320912.000    20.245    -6.163
5          2.082         9.442
2127944.000    332015.000    22.016   -74.699
6         12.662         9.442
2143320.000    332015.000    10.609    93.079
``````

``````Approximate scale = 1479.087
RMS Error (input, output) = (0.040,60.878)

2D Projective  X = (Ax + By + C) / (Gx + Hy + 1)
Y = (Dx + Ey + F) / (Gx + Hy + 1)
A =         55.667   B =       -718.999   C =    2125052.558
D =       -199.525   E =       1385.541   F =     317759.475
G =         -0.001   H =          0.000

Principal point of input (xp,yp) = (2.000,16.946)
Exposure center of output(Xc,Yc) = (2127791.000,343183.000)

tic id  input  x      input  y
output x      output y   x error   y error
--------------------------------------------------
1         2.000         16.946
2127791.000     343183.000    -4.438    45.252
2        12.764         16.821
2143469.000      343326.000   -11.447   -36.202
3         2.052          1.976
2128000.000      320680.000   -17.300    46.421
4        12.922          2.013
2143729.000      320912.000     1.704   -36.962
5         2.082          9.442
2127944.000      332015.000    21.787   -93.410
6        12.662          9.442
2143320.000      332015.000     9.694    74.901
``````

``````Scale (X,Y) = (1483.794,1483.794)
Rotation (degrees) = (0.377)  Translation = (2124800.900,317942.729)
RMS Error (input,output) = (0.162,240.958)

Similarity  X =  Ax + By + C
Y = -Bx + Ay + F
A =         1483.762   B =           -9.765
C =      2124800.900   F =       317942.729

tic id  input  x      input  y
output x      output y   x error   y error
--------------------------------------------------
1          2.000        16.946
2127791.000    343183.000  -188.053   -76.916
2         12.764        16.821
2143469.000    343326.000   106.378  -300.277
3          2.052         1.976
2128000.000    320680.000  -173.717   214.680
4         12.922         2.013
2143729.000    320912.000   225.411   143.724
5          2.082         9.442
2127944.000    332015.000  -146.109   -42.262
6         12.662         9.442
2143320.000    332015.000   176.089    61.051
``````

``````Scale (X,Y) = (1.000,-1.000)  Skew (degrees) = (0.000)
*** Negative Y scaling indicates reflection around X axis. ***
Rotation (degrees) = (180.000)  Translation = (800.000,0.000)
RMS Error (input, output) = (0.000,0.000)

Affine  X = Ax + By + C
Y = Dx + Ey + F
A =       -1.000   B =        0.000   C =      800.000
D =        0.000   E =        1.000   F =        0.000

tic id  input  x      input  y
output x      output y   x error   y error
--------------------------------------------------
1        700.000      100.000
100.000      100.000      0.000     0.000
2        700.000      800.000
100.000      800.000      0.000     0.000
3        100.000      800.000
700.000      800.000      0.000     0.000
4        100.000      100.000
700.000      100.000      0.000     0.000
``````

### 参考书目

Maling, D.H., Coordinate Systems and Map Projections.George Philip., 1973.

Maling, D.H., “Coordinate systems and map projections for GIS.”In:Maguire D.J., M.F. Goodchild, and D.W. Rhind (eds.), Geographical Information Systems:principles and applications. Vol. 1, pp.135-146.Longman Group UK Ltd., 1991.

Moffitt, F.H., and E.M. Mikhail, Photogrammetry.Third Edition.Harper and Row, Inc., 1980.

Pettofrezzo, A.J., Matrices and Transformations.Dover Publications, Inc., 1966.

Slama, C.C., C. Theurer, and S.W. Henriksen (eds.), Manual of Photogrammetry.4th Edition.Chapter XIV, pp.729-731.ASPRS, 1980.

5/10/2014