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Imatest Documentation The upper plot shows the distortion-caused change in radius ∆r (normalized to the center-to-corner distance, i.e., the half-diagonal) as a function of the distorted (input) radius r(distorted) = rd. ∆r = r(corrected) - r(distorted) = ru - rd where r(corrected) = ru is the corrected ( undistorted) radius. ∆r is a function of the scaling (i.e., magnification) of the corrected image. The bold solid lines —— in the upper plot show ∆r for no scaling, i.e., for the straight correction formulas: ru = rd + k1 rd Image showing complex ("wave") distortion: barrel near center; pincushion near corners. 3 (3rd order; blue); ru = rd + h1 rd 3 + h2 rd 5 (5th order; green); or the arctan/tan equations (red). With these equations ∆r tends to increase as a function of r(distorted), i.e., it tends to be largest near the image corners. The dashed lines - - - - show ∆r when ru (the corrected image) is scaled for no shift (∆r = 0) at the image corners (rd = 1). The scaling factor is shown in the Legend box, which also shows the error function (described in the Algorithm; different for the 5th order calculation) and the number of iterations. The upper plot is difficult to relate to the perceived distortion type (pincushion or barrel); the only difference between the solid and dashed lines is the scaling factor (image magnification). The lower plot, which contains Curvature and Distortion measurements (explained in the green box below) is directly related to the distortion type . Equations for distortion and curvature The standard distortion equations— ru = rd + k1 rd 3 (3rd order), ru = rd + h1 rd 3 + h2 rd 5 (5th order), or the arctan/tan equations— do not give a clear picture of whether distortion takes the barrel or pincushion form. And to complicate matters, some images like the one above have barrel distortion at some radii and pincushion at others. The key to determining the type of distortion is to recall that radius ru is the position of a point, its derivative dru /drd is the slope, and its second derivative d 2 ru /drd 2 is the curvature— local curvature, which is a 358 of 451
Imatest Documentation function of radius r closely related to distortion. It is instructive to look at the first and second derivatives of the 3rd and 5th order equations. 3rd order: dru /drd = 3 k1 rd 2 ; d 2 ru /drd 2 = Curvature = 6 k1 rd . 5th order: dru /drd = 3 h1 rd 2 + 5 h2 rd 4 ; d 2 ru /drd 2 = Curvature = 6 h1 rd + 20 h2 rd 3 . The problem with Curvature as defined here is that it tends to be proportional to r (it's exactly proportional for the 3rd order equation). To get a more consistent measurement, we define Distortion as Curvature divided by r. Distortion = Curvature/r = 6 k1 (3rd order) = 6 h1 + 20 h2 rd 2 (5th order) Distortion is a single number for the 3rd order case. Distortion and Curvature is only visible in lines that have a tangential component. Tangential edges are illustrated in the page on Chromatic Aberration. Curvature ( d 2 ru /drd 2 ) and Distortion (Curvature/r) are plotted as dashed ( - - - ) and bold solid —— lines, respectively, in the lower plot. Interpretation: Positive Distortion (and Curvature) represents local barrel distortion; Negative represents local pincushion distortion. In the example above, distortion goes from barrel to pincushion around rd = 0.8 (the diameter at the sides of the image). This is visible near the corners, where the (barrel) curved lines straighten out. The blue lines (3rd order equation) are the simplest but least accurate approximation to distortion. The 5th order equation (red lines), which has two parameters (h1 and h2 ), is more accurate. If the 3rd and 5th order curves are close, the 3rd order curve is sufficient. In the above example, which is not typically, they are very different; the third order equation is completely inadequate. The arctan/tan equation is also characterized by a single parameter. It behaves differently from the 3rd order equation for large ∆r. The 3rd order distortion value in the figure is a constant solid blue blue equal to 6 k1 . Links Real-Time Lens Distortion Correction by Michael R. Bax (of Stanford) contains the standard distortion equation, ru = rd (1 + k1 rd 2 ) and its inverse (more complex). Distortion by Paul van Walree, who also has excellent descriptions of several of the lens (Seidel) aberrations and other sources of optical degradation. Lens aberrations, including distortion, are discussed in pages from James R. Graham of the U.C. Berkeley Astronomy Department, based on Applied Optics & Optical Engineering, Vol XI by J. C. Wyant & K. Creath, which goes into real depth. Worth checking out if you're mathematically inclined. Optical Metrology Center A UK consulting firm specializing in photographic metrology. (Site must be viewed with Internet Explorer.) They have a large collection of interesting technical papers emphasizing distortion and 3D applications, for example, Extracting high precision information from CCD images. PTlens is an excellent program for correcting distortion. It uses the equation from Correcting Barrel Distortion by Helmut Dersch, creator of Panorama Tools, who states, Photogrammetry is the science of making geometrical measurements from images. George Karras has sent me some interesting links with material relevant to Imatest development: http://www.vision.caltech.edu/bouguetj/calib_doc/ | http://nickerson.icomos.org/asrix/index.html. The correcting function is a third order polynomial. It relates the distance of a pixel from the center of the source image (rsrc) to the corresponding distance in the corrected image (rdest) : rsrc = ( a * rdest 3 + b * rdest 2 + c * rdest + d ) * rdest The parameter d describes the linear scaling of the image. Using d=1, and a=b=c=0 leaves the image as it is. Choosing other d-values scales the image by that amount. a,b and c distort the image. Using negative values shifts distant points away from the 359 of 451
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