This page describes Imatest’s distortion calculations, comparing the different distortion formulas and modules.
Distortion formulas – Modules – TV Distotion and Field of View – Image, Geometry, Distortion, FoV display
Radial distortion plot – Compare results – Links
Lens distortion has two basic forms, barrel and pincushion, as illustrated below.
None

Barrel

Pincushion

In addition to these three, there can be “mustache” or “wave” distortion, which is barrel near the center and pincushion near the edges, or viceversa. 
Distortion formulas
In the equations which apply to several modules, r_{d} is the distorted (measured) radius normalized to the centertocorner distance. r_{u} is the undistorted radius. r is normalized to the centertocorner distance.
N*  Distortion model  Standard  Division (Power is one less than standard model) 
Notes 
1  3rd order polynomial  \(r_u = r_d + k_1 r_d^3\)  \(r_u = r_d / (1 + k_1 r_d^2)\)  [1][A] 
2  5th order polynomial  \(r_u = r_d + k_1 r_d^3 + k_2 r_d^5\)  \(r_u = r_d / (1 + k_1 r_d^2 + k_2 r_d^4)\)  [2] 
3  Tangent (for barrel distortion)  \(r_u = \tan(10 p_1 r_d )/(10 p_1 ) ;\;\; p_1 > 0\)  [1][A]  
3  Arctangent (for pincushion distortion)  \(r_u = \arctan(10 p_1 r_d )/(10 p_1 ) ;\;\; p_1 < 0\)  [1][A]  
4  The best of the above choices (3rd order, 5th order, and arctangent/tangent)  
6  5th order ALL polynomial  \(r_u = r_d + k_1 r_d^2 + k_2 r_d^3 + k_3 r_d^4 + k_4 r_d^5\)  \(r_u = r_d / (1 + k_1 r_d + k_2 r_d^2 + k_3 r_d^3 + k_4 r_d^4)\)  [3] 
7  The best of 7th order polynomial and arctangent/tangent: Available in Checkerboard. 

8  7th order polynomial  \(r_u = r_d + k_1 r_d^3 + k_2 r_d^5 + k_3 r_d^7\)  \(r_u = r_d / (1 + k_1 r_d^2 + k_2 r_d^4 + k_3 r_d^6)\)  [4] 
9  7th order ALL polynomial  \(r_u = r_d + \sum_{i=2}^7 k_{i1} r_d^i\)  \(r_u = r_d /( \sum_{i=1}^6 k_i r_d^i)\)  [4] 
10  9th order polynomial  \(r_u = r_d + k_1 r_d^3 + k_2 r_d^5 + k_3 r_d^7 + k_4 r_d^9\)  \(r_u = r_d / (1 + k_1 r_d^2 + k_2 r_d^4 + k_3 r_d^6 + k_4 r_d^8)\)  [4] 
11  11th order polynomial  \(r_u = r_d + \sum_{i=1}^5 k_i r_d^{2i+1}\)  \(r_u = r_d / (r_d + \sum_{i=1}^5 k_i r_d^{2i})\)  [4] 
Notes: [1] All modules. [2] All except eSFR ISO. [3] Checkerboard and SFRplus. [4] Checkerboardonly.
[A] arctan/tan and 3rd order models are inadequate for measuring wave (mustache) distortion. A minimum of a 5th order polynomial is required.
*N is the index of the distortion calculation dropdown menu in More settings. N = fovcalc in ini files for the above modules. It is displayed by the INI File Monitor. N = fovcalc = 5 is for no distortion (no FoV) calculation. 4 is for the best of 13. 7 is for the best of 3, 6.
 The thirdorder equation is one of the textbook Seidel Aberrations, which are loworder polynomial approximations to lens degradations. It only works well for small amounts of distortion
 The Division distortion model (for polynomials) appears to be slightly more accurate than the standard model for the same number of coefficients. When chosen, nth order (odd) polynomials are replaced by (n1)th order (even) polynomials, as shown above.
 Fifth (and higher) order coefficients produce more accurate results, especially for “wave” or “mustache” distortion, which might resemble barrel near the center of the image and pincushion near the corners (or viceversa).
 The two settings with ALL polynomial coefficients (5th order ALL and 7th order ALL) use all coefficients up to the maximum instead of alternate coefficients (odd or evenonly, depending on how the model). We have not observed much advantage to these settings.
 Higher order polynomials (7th order or higher; available only with Checkerboard) should be used with great care because results can become unstable, especially at the outer parts of the image. The image used to calculate the coefficients should have valid corner points near the edge of the image and there should be sufficient rows or columns.
Modules
Module  Comments  Advantages  Disadvantages 
Checkerboard  Extremely accurate. Recommended for new projects.  Fast (after pattern detection). Extremely accurate (good enough for camera calibration). Distortion center calculation is very fast and should always be selected. Also calculates MTF and LCA. Framing and alignment are not critical. Wide range of working distances.  Works only with checkerboard pattern. Checkerboard detection may be slow, but remaining calculations are fast. 
Dot pattern  Based on CPIQ Part 2 document.  CIPQ and ISOcompliant. Also measures Lateral Chromatic Aberration (LCA).  Intolerant to misalignment. Somewhat slow. We may add the algorithms used in Checkerboard, which are more flexible. 
SFRplus 
A highly versatile module with many image quality factor measurements  Fast and moderately accurate. Also calculates MTF, Lateral Chromatic Aberration, color and tonal response. 
Slightly less accurate than Checkerboard. The image should have a small amount of white space above and below the top and bottom bars. This limits the range of working distances. 
Distortion  Imatest’s original (legacy) module for calculating distortion. Will be deprecated in a future release, not recommended for new projects.  Works with grid patterns, lines and edges, as well as checkerboard patterns.  Does not work with strongly barreldistorted images. Less accurate than Checkerboard. Only measures distortion (no MTF, etc.). Grid patterns can be difficult to work with because grid lines can be lost if they’re too thin, and precision can be lost if they’re too thick. 
eSFR ISO  Limited distortion calculation. No distortion center.  Distortion is calculated along with other results.  Limited distortion formulas. Does not work with higherorder polynomials. 
TV distortion and Field of view
SMIA* TV Distortion is calculated from the distortion model equation (and the Distortion parameters for predistorted charts). [*SMIA is the nowdefunct “Standard for Mobile Imaging Architecture”, started by Nokia and STMicroelectronics in 2004.]
TV Distortion from the SMIA specification, §5.20. Referring to the image on the right,
SMIA TV Distortion = 100( AB )/B ; A = ( A_{1}+A_{2 })/2
The box on the right is described in the SMIA spec as “nearly filling” the image. Since the test chart grid may not do this, Distortion uses a simulated box whose height is 98% that of the image. Note that the sign is opposite of k_{1} and p_{1}. SMIA TV Distortion > 0 is pincushion; < 0 is barrel.
Algorithm: SMIA TV Distortion is not actually calculated from the upper and lower bars, whose locations can vary considerably in different images. Instead it is calculated from the distortion coefficients using the selected equation, and using virtual horizontal lines located 1% of the image height below the top and above the bottom of the image.
SMIA vs. traditional (ISO) TV distortion SMIA TV distortion is twice as large (2X) as traditional TV distortion, now included in several standards. The traditional definition, shown on the right, has been adapted from the publication “Optical Terms,” published by Fujinon. The same definition appears in “Measurement and analysis of the performance of film and television camera lenses” published by the European Broadcasting Union (EBU). 
At Imatest we have traditionally used the SMIA definition, which has been widely adopted in the mobile imaging industry, because it is selfconsistent. In the traditional definition, TV distortion is the change (Δ) of the centertotop distance divided by by the bottomtotop distance. In the SMIA definition, both A and B are bottomtotop distances.
Imatest 5.1+ allows you to select between traditional TV distortion (now a part of several ISO standards, including ISO 16505) and SMIA TV distortion. You can make the selection on the left side of the Options II window (button on the lowerright of the Imatest main window). The selection affects only graphic displays (ISO TV distortion = SMIA TV distortion ⁄ 2). Both results are included in CSV and JSON results (for all modules that calculate distortion).
Field of view (FoV) is calculated for SFRplus, Checkerboard, and eSFR ISO by applying the distortion model equation to the top, side, and diagonal (always r = 1) of the image. In order to calculate FoV in units of distance (cm) a chart geometrical distance has to be entered into the Rescharts More settings window.
Module  Value 
SFRplus  Bartobar chart height in cm 
Checkerboard  Square spacing (cm) 
eSFR ISO  Reg. mark vertical spacing cm 
If the Lenstochart distance (in cm) is entered, angular FoV will also be calculated.
Image, Geometry, Distortion, FoV display
An image display is available in most modules that calculate distortion. In Rescharts it contains a large amount of information (not all distortionrelated). Here is an example showing arrows and lines to the corrected image, available for Checkerboardonly. (The arrows & lines can be turned on or off in the More settings window.)
Image, Geometry display showing arrows and corrected image locations
Here is an example of the corrected image, also showing arrows and lines.
This display contains
 Dimensions andROI information (for MTF calculations)
 Distortion coefficients (9th order division model here)
 SMIA TV distortion
 Distortion center shift (in pixels)
 Image (central square) shift
 Field of View (Diagonal, Horizontal, and Vertical; degrees and cm if appropriate settings have been entered)
 Convergence angles (perspective distortion)
Radial distortion plot
The radial distortion plot is available in modules that calculate distortion. This plot has four display options: 1. Deltar, or 2. Lens Geometric Distortion (LGD), 3. r undistorted (r_{u}), or d(LGD)/d(r_{u}). LGD is shown below.
SFRplus Radial Distortion plot showing Lens Geometric Distortion 100% ( r_{d} – r_{u})/r_{u}
The display options are
 the change in radius Δr (normalized to the centertocorner distance, i.e., the halfdiagonal) as a function of the distorted (input) radius r_{d}.
Δr = r(undistorted) – r(distorted) = r_{d} – r_{u}  Lens Geometric Distortion (LGD), which is included in the CPIQ Phase 2 specification, and is equivalent to Optical Distortion (defined by Edmund Optics).
LGD = 100% ( r_{d} – r_{u})/r_{u}  Undistorted radius, r_{u}
 Curvature d(LGD)/d(r_{u}). This curvature (new in Imatest 5.0) may be useful for determining the visual degradation caused by distortion, which is likely to be proportional to the maximumminimum values. A change in sign may be a better indicator of mustache (wave) distortion than a change in 5th order polynomial sign (which isn’t always calculated).
The solid lines show results for correction formulas: r_{u} = r_{d} + k_{1 }r_{d}^{3} (3rd order polynomial; blue); r_{u} = r_{d} + h_{1 }r_{d}^{3 }+ h_{2 }r_{d}^{5} (5th order ploynomial; green); or the arctan/tan equations (red). The best fit (5th order in this case) is shown in bold. With these equations Δr often increases as a function of r(distorted), i.e., it tends to be largest near the image corners. The selected value (or the one with the least error, err) is shown in boldface.
Comparison of results for different modules
A set of images for comparing different Imatest distortion calculations can be downloaded by clicking on distortion_comparison_barrel_pin.zip. These images were created by the Test Charts module, converted to bitmaps of equal size, then equally distorted. The zip file includes barrel and pincushiondistorted images for Distortion, Checkerboard, Dot Pattern, SFRplus, and eSFR ISO. As the Lens Geometrical Distortion figures for the modules show, agreement is excellent. The Dot Pattern module uses the algorithm specified in the Camera Phone Image Quality (CPIQ) specification, but the other modules produce equivalent results.
Images are shown reduced. Click on the image to view fullsized.
Distortion module  Dot Pattern  SFRplus 
Checkerboard module 
Notes: the third order calculations (Distortion and SFRplus) are less accurate than the fifth order and arctan/tan calculations (i.e., they cannot be as good a fit to the actual distortion.) The green line (SMIA TV Distortion) in the Dot Pattern figure cannot be compared with the other figures.
Links
Wikipedia Distortion (optics) page. A good read (not long). Claims that the division model is more accurate.
Edmund Optics Distortion application note