Changes between Version 5 and Version 6 of DevWikiAffineParameters

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Timestamp:
09/13/11 10:58:24 (21 months ago)
Author:
bnordgren
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  • DevWikiAffineParameters

    v5 v6  
    6161 
    6262 1. scaling 
    63  1. clockwise rotation 
     63 1. clockwise rotation around the origin 
    6464 1. shearing parallel to the x axis 
    6565 1. shearing parallel to the y axis 
    … …  
    8888[[Image(aggregate_usage.png)]] 
    8989 
     90 * x' = o,,11,, x + o,,12,, y 
     91 * y' = o,,21,, x + o,,22,, y 
    9092 
     93== Constructing the affine transformation == 
    9194 
    92 Sx : scale in the x direction 
    93 Sy : scale in the y direction 
    94 Tx : translation in the x direction 
    95 Ty : translation in the y direction 
    96 Kx : shearing parallel to x axis 
    97 Ky : shearing parallel to y axis 
    98 theta : angle of rotation CLOCKWISE around the origin (not around the 
    99 x axis, not around the y axis: around the origin; or if you like, 
    100 around the invisible Z axis coming out of the screen and poking you in 
    101 the eye.) 
     95The 2x2 matrix '''O''' is the upper-left hand corner of the affine transformation, which is the 3x3 matrix, '''A'''. The right hand column contains the offsets, or translation, of the raster in the x and y directions. The bottom row is always filled with the numbers 0, 0, 1. The result of this is that there are six parameters to an affine transformation which can actually change, as shown:  
    10296 
    103 The next step is to define an order. I chose: "Scale followed by 
    104 rotation followed by shearing (skew)". 
     97[[Image(affinematrix.png)]]  
    10598 
    106 Multiplying these matrices together, as described here 
    107 (http://en.wikipedia.org/wiki/Transformation_matrix#Composing_and_inverting_transformations) 
    108 gives a 2x2 matrix, which is not an affine transform yet. Let's label 
    109 the coefficients of this matrix as follows 
     99 * a,,11,, = o,,11,, = s,,x,, ( (1 + k,,x,, k,,y,,) cosθ + k,,y,, sinθ ) 
     100 * a,,12,, = o,,12,, = s,,x,, ( k,,x,, cosθ + sinθ ) 
     101 * a,,13,, = t,,x,, 
     102 * a,,21,, = o,,21,, = s,,y,, ( -(1 + k,,x,, k,,y,,) sinθ + k,,y,, cosθ ) 
     103 * a,,22,, = o,,22,, = s,,y,, ( - k,,x,, sinθ + cosθ ) 
     104 * a,,23,, = t,,y,, 
    110105 
    111 | O11  O12 | 
    112 | O21  O22 | 
     106where:  
     107 * s,,x,, : scale factor in x direction 
     108 * s,,y,, : scale factor in y direction 
     109 * t,,x,, : offset in x direction 
     110 * t,,y,, : offset in y direction 
     111 * θ : angle of rotation clockwise around origin 
     112 * k,,x,, : shearing parallel to x axis 
     113 * k,,y,, : shearing parallel to y axis 
    113114 
    114 (I apologize for the ascii graphics throughout. Gmail is not using a 
    115 monospaced font even in "plain text" mode.) 
     115It is these six parameters (a,,11,,...a,,23,,)which are typically stored within a geospatial image file format to record the conversion from pixel index to geolocation. The actual conversion is described as follows: 
    116116 
    117 So what I get for these coefficients (after multiplying in the order 
    118 specified) is: 
     117[[Image(affineusage.png)]] 
    119118 
    120 O11 = Sx * (cos(theta) + Ky sin(theta)) 
    121 O12 = Sx * (Kx cos(theta) + sin(theta)) 
    122 O21 = Sy * (-sin(theta) + Ky cos(theta)) 
    123 O22 = Sy * (-Kx * sin(theta) + cos(theta)) 
     119 * E = a,,11,, i + a,,12,, j + a,,13,, 
     120 * N = a,,21,, i + a,,22,, j + a,,23,, 
    124121 
    125 And the final bit is to add translation by making this into an affine transform: 
     122where E is easting, N is northing, i is pixel column and j is pixel row. The last row of the matrix equation is always ignored, as it boils down to 1=1. It is there to make a square matrix used to calculate the inverse operation. 
    126123 
    127 | O11 O12 O13 | 
    128 | O21 O22 O23 | 
    129 |   0      0     1    | 
     124== Link to postgis raster ==  
    130125 
    131 where: 
     126The six parameters of the affine transform are given the following names in postgis raster:  
    132127 
    133 O13 = Tx 
    134 O23 = Ty 
     128 * ScaleX = a,,11,, 
     129 * SkewX =  a,,12,, 
     130 * OffsetX = a,,13,, = t,,x,, 
     131 * SkewY =  a,,21,, 
     132 * ScaleY = a,,22,, 
     133 * OffsetY = a,,23,, = t,,y,, 
    135134 
    136 To be pedantic, this is used as follows: 
     135With the exception of OffsetX and OffsetY, the names are somewhat arbitrary for the general case. 
    137136 
    138 | E |         | O11 O12 O13 |  | i | 
    139 | N |   =   | O21 O22 O23 |  | j | 
    140 | 1 |         |    0     0     1   |   | 1 | 
     137== Summary ==  
    141138 
    142 where: 
    143 E  = easting 
    144 N = northing 
    145 i   = pixel column 
    146 j   = pixel row 
    147  
    148  
    149 The coefficients map onto our jumbled named coefficients as follows : 
    150  
    151 ScaleX = O11 
    152 SkewX = O12 
    153 OffsetX = O13 
    154 SkewY = O21 
    155 ScaleY = O22 
    156 OffsetY = O23 
    157  
    158 The important thing to note is that the things we're rather loosely 
    159 calling Scale[XY] and Skew[XY] represent all of Scale, Rotation and 
    160 Shearing. This dictates that you cannot set these coefficients without 
    161 knowing all three. 
     139Throughout this page, the following symbols have been