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5 <TITLE>Camera Calibration Tool - Camera Model</TITLE>
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16 <CENTER><TABLE CELLSPACING=0 CELLPADDING=5 WIDTH="90%" BGCOLOR="#6A9BED" NOSAVE >
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19 <CENTER><I><FONT COLOR="#000000"><FONT SIZE=+4>Camera Calibration Tool - Camera Model</FONT></FONT></I></CENTER>
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29 <H1>Description of the intrinsic camera model</H1>
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32 <P>After calibration, the intrinsic camera model may be saved into a text file by clicking on the "Save" button in the <a href="doc_calib.html#properties" target="application window">property control window</a>. An example of such a file is <a href="CalibResult.txt">CalibResult.txt</a>:
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38 M[0,0]= 286.2791138 M[0,1]= 0.0000000 M[0,2]= 156.6844177<BR>
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39 M[1,0]= 0.0000000 M[1,1]= 287.7630615 M[1,2]= 130.9805145<BR>
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40 M[2,0]= 0.0000000 M[2,1]= 0.0000000 M[2,2]= 1.0000000<BR>
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51 This complete intrinsic camera model is encoded with the matrix <b>M</b> (also known as camera matrix) and the vector <b>D</b> (distortion vector).
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53 <p><br><font size=+1>Definition of the intrinsic parameters:</font>
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55 Let us make slight changes in notation: (numerical values are included for clarity purposes only - when running calibration on your camera, you will naturally obtain a different set of numerics)<BR>
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59 <b><font size=+0>Focal length:</font></b> The focal length in pixels is the 2x1 vector <b>fc = [ M[0,0] ; M[1,1] ] = [286.3 ; 287.8]</b>.
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60 <!-- The quantities <b>fc(1)</b> and <b>fc(2)</b> are the horizontal and vertical focal length (in x and y) in pixels. --></li>
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63 <b><font size=+0>Principal point:</font></b> The principal point in pixels is the 2x1 vector <b>cc = [ M[0,2] ; M[1,2] ] = [156.7 ; 131]</b>.</li>
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66 <b><font size=+0>Skew coefficient:</font></b> The skew coefficient related to the angle between the x and y pixel axes is the scalar <b>s = M[0,1] = 0</b>.</li>
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69 <b><font size=+0>Distortion:</font></b> The image distortion coefficients
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70 are <b>k1 = D[0] = -0.416691</b>, <b>k2 = D[1] = 0.250142</b>, <b>p1 = D[2] = -0.000386</b> and <b>p2 = D[3] = -0.001894</b>. The coefficients <b>k1</b> and <b>k2</b> are associated to radial distortions (up to the fourth order), and <b>p1</b> and <b>p2</b> are associated to tangential distortions.</li>
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74 A zero skew (<b>s=0</b>) means that the pixels are rectangular on the
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75 sensor. In that present release of the calibration software, the
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76 camera model will always assume rectangular pixels (or zero
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77 skew). This assumption is valid for most current imaging sensors. In future
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78 releases, a complete estimation of the skew coefficient will be included.
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81 The next section provides detailed definitions of the entire set of intrinsic parameters.
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85 <p><br><font size=+1>Description of the pixel projection operator:</font>
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87 Let us make slight changes in notation: (numerical values are included for clarity purposes only - when running calibration on your camera, you will naturally obtain a different set of numerics)<BR>
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89 <p>Let <b>P</b> be a point in space of coordinate vector <b>XX<sub>c</sub>
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90 = [X<sub>c</sub>;Y<sub>c</sub>;Z<sub>c</sub>]</b> in the camera reference
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92 <br>Let us project now that point on the image plane according to the intrinsic
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93 parameters (<b>fc</b>,<b>cc</b>,<b>s</b>,<b>k1</b>,<b>k2</b>,<b>p1</b>,<b>p2</b>).
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94 <br>Let <b>x<sub>n</sub></b> be the normalized (pinehole) image projection:
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98 <td><b>x<sub>n</sub></b></td>
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120 <center><table><tr><td>
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121 <b>X<sub>c</sub>/Z<sub>c</sub></b>
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122 </td></tr></table></center>
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128 <center><table><tr><td>
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129 <b>Y<sub>c</sub>/Z<sub>c</sub></b>
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130 </td></tr></table></center>
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151 <td><b>= </b></td>
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172 <center><table><tr><td>
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174 </td></tr></table></center>
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180 <center><table><tr><td><center><table><tr><td>
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182 </td></tr></table></center></td></tr></table></center>
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205 <p>Let <b>r<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup></b>.
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206 <br>After including lens distortion, the new normalized point coordinate
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207 <b>x<sub>d</sub></b>
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208 is defined as follows:
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240 <td><center><b>x<sub>d</sub>(1)</b></center></td>
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243 <td><center><b>x<sub>d</sub>(2)</b></center></td>
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268 <b>(1 + k1*r<sup>2</sup> + k2*r<sup>4</sup>) * x<sub>n</sub> + dx</b>
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276 <p>where <b>dx</b> is the tangential distortion vector:
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281 <td><b> = </b></td>
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302 <center><table><tr><td>
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303 <b>2*p1*x*y + p2*(r<sup>2</sup> + 2*x<sup>2</sup>)</b>
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304 </td></tr></table></center>
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310 <center><table><tr><td>
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311 <b>p1*(r<sup>2</sup> + 2*y<sup>2</sup>) + 2*p2*x*y</b>
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312 </td></tr></table></center>
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336 It is worth noticing that this distortion model was first introduced
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337 by Brown in 1966 and called "Plumb Bob" model (radial polynomial +
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338 "thin prism" ). The tangential distortion is due to "decentering", or imperfect centering of the lens components and other manufacturing defects in a compound lens.
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340 <p>Once distortion is applied, the final pixel coordinates <b>x_pixel = [x<sub>p</sub>;y<sub>p</sub>]</b>
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341 of the projection of <b>P</b> on the image plane is:
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346 <align="left"><b>x<sub>p</sub> = fc(1)*x<sub>d</sub>(1) + s*x<sub>d</sub>(2) + cc(1)</b></align>
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352 <align="left"><b>y<sub>p</sub> = fc(2)*x<sub>d</sub>(2) + cc(2)</b></align>
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359 Therefore, the pixel coordinate vector <b>x_pixel</b> and the normalized (distorted) coordinate vector <b>x<sub>d</sub></b> are related to each other through the linear equation:
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385 <td><center><b>x<sub>p</sub></b></center></td>
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388 <td><center><b>y<sub>p</sub></b></center></td>
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391 <td><center><b>1</b></td>
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416 <center><b>=</b></center>
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421 <center><b>M</b></center>
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424 <center><b>.</b></center>
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448 <td><center><b>x<sub>d</sub>(1)</b></center></td>
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451 <td><center><b>x<sub>d</sub>(2)</b></center></td>
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454 <td><center><b>1</b></center></td>
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482 where <b>M</b> is known as the camera matrix, and defined as follows:
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493 <center><b>M</b></center>
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497 <center><b>=</b></center>
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528 <center><table><tr><td><center><table><tr><td>
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529 <center><b>fc(1)</b></center>
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530 </td></tr></table></center></td></tr></table></center>
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533 <center><table><tr><td><center><table><tr><td>
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534 <center><b>s</b><center>
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535 </td></tr></table></center></td></tr></table></center>
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538 <center><table><tr><td><center><table><tr><td>
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539 <center><b>cc(1)</b><center>
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540 </td></tr></table></center></td></tr></table></center>
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545 <center><table><tr><td><center><table><tr><td>
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546 <center><b>0</b><center>
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547 </td></tr></table></center></td></tr></table></center>
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550 <center><table><tr><td><center><table><tr><td>
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551 <center><b>fc(2)</b><center>
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552 </td></tr></table></center></td></tr></table></center>
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555 <center><table><tr><td><center><table><tr><td>
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556 <center><b>cc(2)</b><center>
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557 </td></tr></table></center></td></tr></table></center>
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562 <center><table><tr><td><center><table><tr><td>
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563 <center><b>0</b><center>
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564 </td></tr></table></center></td></tr></table></center>
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567 <center><table><tr><td><center><table><tr><td>
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568 <center><b>0</b><center>
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569 </td></tr></table></center></td></tr></table></center>
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572 <center><table><tr><td><center><table><tr><td>
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573 <center><b>1</b><center>
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574 </td></tr></table></center></td></tr></table></center>
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607 <BR>Observe that this matrix is exactly the same as the one appearing in the original text file generated by the calibration tool
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608 Notice that <b>fc(1)</b> and <b>fc(2)</b> are the focal
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609 distance (a unique value if expressed in mm) expressed in units of horizontal and vertical
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610 pixels. Therefore, the camera model naturally handles non-square
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611 pixels. In addition, the coefficient <b>s</b> encodes the angle
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612 between the x and y sensor axes. Consequently, this generic model allows for non-rectangular pixels (altough this release of the software does not explicitly estimate the skew coefficient - future ones will). Some authors refer to that generic type of model as
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613 "affine distortion" model.
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615 <p><i>Convention: </i>The pixel coordinates are defined such that <b>[0;0]</b> is the
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616 upper left pixel of the image (the center of the pixel),
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617 <b>[nx-1;0]</b> is the upper right corner pixel, <b>[0;ny-1]</b> is
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618 the lower left corner pixel and <b>[nx-1;ny-1]</b> is the lower right
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619 corner pixel where <b>nx</b> and <b>ny</b> are the width and height of
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620 the image (for example <b>nx=640</b> and
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625 For more information, refer to the <a href="calib_doc/index.html">tutorial page</a> on Camera Calibration and try the Matlab implementation provided with it.
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633 <FONT SIZE=+3><a href="doc_calib.html" target="application window">Back to main camera calibration page</A></FONT>
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projects / opencv / blob