How To Calculate Dynamic Range Camera

This page describes Imatest's distortion calculations, comparing the different distortion formulas and modules.

Baloney formulas – Modules – Idiot box Distotion and Field of View – Image, Geometry, Distortion, FoV display
Radial distortion plot – Distortion Profile plot – Compare results – Links

Lens distortion has 2 bones forms, barrel and pincushion , as illustrated below.

None

Barrel

Pincushion

In addition to these three, at that place can exist "mustache" or "wave" distortion, which is barrel near the center and pincushion near the edges, or vice-versa.

Image Quality Video Series: Distortion

Distortion formulas

In the equations which utilise to several modules, r_d is the distorted (measured) radius normalized to the center-to-corner distance. r_u is the undistorted radius. r is normalized to the center-to-corner altitude.

N*	Distortion model	Standard	Sectionalisation (Power is 1 less than standard model)	Notes
one	tertiary guild polynomial	\(r_u = r_d + k_1 r_d^3\)	\(r_u = r_d / (1 + k_1 r_d^ii)\)	[1][A]
2	5th lodge polynomial	\(r_u = r_d + k_1 r_d^3 + k_2 r_d^5\)	\(r_u = r_d / (i + k_1 r_d^ii + k_2 r_d^four)\)	[two]
3	Tangent (for butt distortion)	\(r_u = \tan(x p_1 r_d )/(10 p_1 ) ;\;\; p_1 > 0\)		[1][A]
3	Arctangent (for pincushion distortion)	\(r_u = \arctan(10 p_1 r_d )/(x p_1 ) ;\;\; p_1 < 0\)		[1][A]
iv	The all-time of the above choices (third order, 5th gild, and arctangent/tangent)
6	5th order ALL polynomial	\(r_u = r_d + k_1 r_d^2 + k_2 r_d^3 + k_3 r_d^iv + k_4 r_d^5\)	\(r_u = r_d / (1 + k_1 r_d + k_2 r_d^2 + k_3 r_d^3 + k_4 r_d^4)\)	[3]
7	The best of seventh guild polynomial and arctangent/tangent: Bachelor in Checkerboard.
8	7th order polynomial	\(r_u = r_d + k_1 r_d^3 + k_2 r_d^v + k_3 r_d^7\)	\(r_u = r_d / (1 + k_1 r_d^2 + k_2 r_d^iv + k_3 r_d^six)\)	[4]
ix	7th order ALL polynomial	\(r_u = r_d + \sum_{i=2}^7 k_{i-1} r_d^i\)	\(r_u = r_d /( \sum_{i=1}^6 k_i r_d^i)\)	[4]
ten	9th order polynomial	\(r_u = r_d + k_1 r_d^3 + k_2 r_d^v + k_3 r_d^vii + k_4 r_d^9\)	\(r_u = r_d / (1 + k_1 r_d^2 + k_2 r_d^iv + k_3 r_d^vi + k_4 r_d^8)\)	[iv]
11	11th society polynomial	\(r_u = r_d + \sum_{i=1}^5 k_i r_d^{2i+1}\)	\(r_u = r_d / (r_d + \sum_{i=1}^five k_i r_d^{2i})\)	[four]

Notes: [one] All modules. [ii] All except eSFR ISO. [three] Checkerboard and SFRplus. [four] Checkerboard-only.
[A] arctan/tan and tertiary order models are inadequate for measuring wave (mustache) baloney. A minimum of a 5th lodge polynomial is required.

Checkerboard distortion settings for
standard (odd coefficient) and division (even coefficient) models

*Due north is the index of the baloney adding dropdown menu in More settings. N = fovcalc in ini files for the above modules. It is displayed past the INI File Monitor. North = fovcalc = 5 is for no distortion (no FoV) adding. four is for the best of ane-3. 7 is for the best of three, half-dozen.

The third-order equation is one of the textbook Seidel Aberrations, which are depression-guild polynomial approximations to lens degradations. It only works well for small amounts of baloney
The Division baloney model (for polynomials) appears to exist slightly more authentic than the standard model for the same number of coefficients. When chosen, due norththursday club (odd) polynomials are replaced by (n-1)th society (fifty-fifty) polynomials, equally shown above.
Fifth (and college) lodge coefficients produce more than accurate results, peculiarly for "wave" or "mustache" baloney, which might resemble barrel near the heart of the paradigm and pincushion near the corners (or vice-versa).
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 fifty-fifty-only, depending on how the model). We have non observed much advantage to these settings.
Higher order polynomials (seventh society or higher; available only with Checkerboard) should be used with great care because results can get unstable, particularly at the outer parts of the image. The epitome used to calculate the coefficients should accept valid corner points most the edge of the image and at that place should exist 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 photographic camera calibration). Distortion center adding is very fast and should always be selected. Also calculates MTF and LCA. Framing and alignment are non critical. Wide range of working distances. Works well with strongly barrel-distorted images.	Works simply with checkerboard pattern. Checkerboard detection may be slow, but remaining calculations are fast.
Dot pattern	Based on CPIQ Part 2 document.	CIPQ and ISO-compliant. Also measures Lateral Chromatic Aberration (LCA).	Intolerant to misalignment. Somewhat ho-hum. Nosotros may add together 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 Abnormality, colour and tonal response. Works with pre-distorted charts (useful with strongly barrel-distorted images). Run into SFRplus Baloney and Field of View measurements for more particular, especially near using pre-distorted charts.	Slightly less accurate than Checkerboard. The prototype should have a small amount of white space above and below the summit and bottom bars. This limits the range of working distances.
Baloney *Non recommended for new projects.*	Imatest's original (legacy) module for computing distortion. May exist deprecated in a future release.	Works with filigree patterns, lines and edges, as well equally checkerboard patterns.	Often fails with strongly butt-distorted images. Less accurate than Checkerboard . Just measures distortion (no MTF, etc.). Filigree patterns can be difficult to piece of work with because grid lines tin can exist lost if they're too thin, and precision is lost if they're too thick.
eSFR ISO	Limited distortion adding. No distortion center.	Distortion is calculated forth with other results.	Express distortion formulas. Does not work with higher-club polynomials.

Telly distortion and Field of view

SMIA* Idiot box Distortion is calculated from the distortion model equation (and the Distortion parameters for pre-distorted charts). [*SMIA is the at present-defunct "Standard for Mobile Imaging Compages", started past Nokia and STMicroelectronics in 2004.]

Television receiver Distortion from the SMIA specification, §v.twenty. Referring to the image on the correct,

SMIA TV Distortion = 100( A-B )/B ; A = ( A₁+A_two)/2

The box on the right is described in the SMIA spec every bit "about filling" the image. Since the test chart grid may non exercise this, Baloney uses a false box whose height is 98% that of the paradigm. Note that the sign is opposite of k_ane and p_one . SMIA Television receiver Distortion > 0 is pincushion; < 0 is barrel.

Algorithm: SMIA Tv set Distortion is not really calculated from the upper and lower confined, whose locations can vary considerably in different images. Instead it is calculated from the baloney coefficients using the selected equation, and using virtual horizontal lines located 1% of the epitome meridian below the top and above the bottom of the image.

TV distortion: traditional definition SMIA vs. traditional (ISO) TV baloney

SMIA TV distortion is twice as big (2X) as traditional Television baloney, 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 flick and boob tube 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 self-consistent. In the traditional definition, Boob tube distortion is the modify (Δ) of the center-to-top distance divided by by the bottom-to-elevation altitude. In the SMIA definition, both A and B are lesser-to-acme distances.

Imatest five.1+ allows you to select between traditional TV distortion (now a part of several ISO standards, including ISO 16505) and SMIA Television receiver distortion. Y'all can make the option on the left side of the Options 2 window (push button on the lower-right of the Imatest main window). The selection affects only graphic displays (ISO Tv set distortion = SMIA TV baloney ⁄ 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 past applying the distortion model equation to the top, side, and diagonal (e'er r = one) of the image. In order to calculate FoV in units of distance (cm) a nautical chart geometrical distance has to be entered into the Rescharts More than settings window.

Module	Value
SFRplus	Bar-to-bar chart top in cm
Checkerboard	Foursquare spacing (cm)
eSFR ISO	Reg. marker vertical spacing cm

If the Lens-to-chart distance (in cm) is entered, angular FoV will also be calculated.

Image, Geometry, Baloney, FoV display

An image brandish is available in most modules that calculate distortion. In Rescharts it contains a large amount of information (not all baloney-related). Here is an case showing arrows and lines to the corrected epitome, available for Checkerboard-only. (The arrows & lines tin be turned on or off in the More settings window.)

Paradigm, Geometry brandish showing arrows and corrected prototype locations

Corrected prototype showing arrows to original image
Click on the image to view full-sized.

Here is an example of the corrected image, besides showing arrows and lines.

This display contains

Dimensions andROI information (for MTF calculations)
Baloney coefficients (9th order sectionalization model here)
SMIA Tv set distortion
Distortion center shift (in pixels)
Epitome (fundamental square) shift
Field of View (Diagonal, Horizontal, and Vertical; degrees and cm if appropriate settings accept been entered)
Convergence angles (perspective distortion)

Radial distortion plot

The radial distortion plot is bachelor in modules that summate distortion. This plot has 4 display options: 1. Delta-r, or two. Lens Geometric Distortion (LGD), iii. r undistorted (r _u), or d(LGD)/d(r _u). LGD is shown below.

SFRplus Radial Distortion plot showing Lens Geometric Baloney 100% ( r_d – r_u )/r_u

The display options are

the change in radius Δr (normalized to the center-to-corner distance, i.eastward., the one-half-diagonal) 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 Stage 2 specification, and is equivalent to Optical Baloney (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 exist useful for determining the visual deposition caused by distortion, which is likely to be proportional to the maximum-minimum values. A modify in sign may be a ameliorate indicator of mustache (wave) distortion than a change in fifth social club polynomial sign (which isn't always calculated).

The solid lines show results for correction formulas: r_u = r_d + one thousand₁r_d ³ (3rd gild polynomial; blueish ); r_u = r_d + h₁r_d ^three+ h₂r_d ⁵ (5th order ploynomial; greenish ); or the arctan/tan equations ( red ). The best fit (5th lodge in this case) is shown in assuming. With these equations |Δr| oftentimes increases every bit a part of r(distorted), i.e., information technology tends to exist largest well-nigh the prototype corners. The selected value (or the one with the least mistake, err) is shown in boldface.

Baloney Contour plot – Checkerboard-only

New in 2021.2. Described in Rescharts Slanted-Border modules Role 4: Distortion profile plot.

Comparison of results for different modules

A set of images for comparison dissimilar Imatest distortion calculations tin can exist downloaded by clicking on distortion_comparison_barrel_pin.zilch. These images were created by the Exam Charts module, converted to bitmaps of equal size, then every bit distorted. The zip file includes barrel and pincushion-distorted images for Distortion, Checkerboard, Dot Pattern, SFRplus, and eSFR ISO. Equally the Lens Geometrical Baloney figures for the modules evidence, agreement is first-class. The Dot Blueprint module uses the algorithm specified in the Photographic camera Telephone Prototype Quality (CPIQ) specification, but the other modules produce equivalent results.

Images are shown reduced. Click on the paradigm to view full-sized.

Notes: the third club calculations (Distortion and SFRplus) are less accurate than the fifth order and arctan/tan calculations (i.e., they cannot be every bit practiced a fit to the actual distortion.) The green line (SMIA TV Baloney) in the Dot Design figure cannot be compared with the other figures.