Rectification (photography)

Under equalizing is understood in the photograph the correction of geometric errors of optical imaging . These can be caused by the lens (see aberrations ) or by the setup or alignment of the camera (see converging lines ).

Compensation of geometrical aberrations of the lens

Many lenses show a pincushion or barrel distortion . Straight lines are mapped as a curve if they do not run through the center of the image . Wavy distortions can occur in complex optics with many lenses.

The distortion can be compensated for by optical methods (for example back projection through the recording lens) or by calculations during image processing. Lateral color errors can also be compensated for in a similar way .

The compensation by means of electronic image processing makes use of the fact that the extent of the distortion mostly depends on the distance from the center of the image. In such a case, the image can be described by polar coordinates . The position of the image points is represented by the distance from the image center and an angle, for example from the horizontal. The computational correction depends on the distance from the center of the image and can be described as a mathematical function . A polynomial is used for the calculation :

${\displaystyle R_{n}=a+bR_{a}+cR_{a}^{2}+dR_{a}^{3}+eR_{a}^{4}}$

( = corrected distance from the center of the image, = uncorrected distance from the center of the image, = constants) ${\displaystyle R_{n}}$ ${\displaystyle R_{a}}$ ${\displaystyle a,b,c,d,e}$

Odd powers (phenomenological modeling)

In phenomenological modeling, the effects on the image are to be compensated for by choosing the mathematically most favorable model. For example, by introducing the camera constant as a virtual variable: ${\displaystyle r_{0}}$

${\displaystyle \Delta r=A_{1}(r^{3}-r_{0}^{2}r)+A_{2}(r^{5}-r_{0}^{4}r)+A_{3}(r^{7}-r_{0}^{6}r)}$

Even powers (physical modeling)

The physical modeling is mostly done relative to the center of symmetry, which is near the center of the image:

${\displaystyle \Delta r=q_{2}r^{2}+q_{4}r^{4}+q_{6}r^{6}}$

From a physical point of view, the radially symmetrical distortion errors mostly correspond to even functions .

Color channels

If this transformation is only used for individual color channels of a digital image, disruptive color fringes ( chromatic aberration ) can also be compensated.

Compensation by camera position or type

Crashing lines through the tilted camera

Image rectified

A photograph is created according to the laws of central perspective . For this reason, parallel lines in the subject will only run parallel in the image if they lie in a plane in the subject that is parallel to the plane of the film. So if the camera z. B. pivoted upwards in front of a house, converging lines appear , the house seems to tilt backwards. In the case of large format cameras or the tilt and shift lenses of small and medium format cameras, this is avoided by not pivoting the camera to select the image section, but shifting the image section.

Compensation using software

Almost all lenses have typical distortions due to their production. The determining factor here is the radial distortion, which results from the design of the lens and increases with increasing focal length. Another factor is tangential distortion, which is also referred to as decentering distortion. It results from the fact that the lens and CCD chip as well as the individual optical and mechanical components of a lens are not perfectly aligned with one another. Radial distortion is mainly determined by factors of lower order; influencing variables of higher order can be neglected. See the description of the calibration procedure for radial distortion according to Zhang. Mathematically, they can be described by the Brown-Conrady model (also known as the plumb bob model). The model makes it possible to compensate for both radial and tangential distortions.

${\displaystyle x_{\mathrm {d} }=x_{\mathrm {u} }(1+K_{1}r^{2}+K_{2}r^{4}+\cdots )+(P_{2}(r^{2}+2x_{\mathrm {u} }^{2})+2P_{1}x_{\mathrm {u} }y_{\mathrm {u} })(1+P_{3}r^{2}+P_{4}r^{4}\cdots )}$

${\displaystyle y_{\mathrm {d} }=y_{\mathrm {u} }(1+K_{1}r^{2}+K_{2}r^{4}+\cdots )+(P_{1}(r^{2}+2y_{\mathrm {u} }^{2})+2P_{2}x_{\mathrm {u} }y_{\mathrm {u} })(1+P_{3}r^{2}+P_{4}r^{4}\cdots )}$

Explanation:

${\displaystyle (x_{\mathrm {d} },\ y_{\mathrm {d} })}$ = distorted image point, projected onto the image plane with a specific lens,

${\displaystyle (x_{\mathrm {u} },\ y_{\mathrm {u} })}$ = undistorted image point, as projected by an ideal pinhole camera,

${\displaystyle (x_{\mathrm {c} },\ y_{\mathrm {c} })}$= Center of distortion ( assumed as focal point ),

${\displaystyle K_{n}}$= = radial distortion coefficient, ${\displaystyle n^{\mathrm {th} }}$

${\displaystyle P_{n}}$= = tangential distortion coefficient, ${\displaystyle n^{\mathrm {th} }}$

${\displaystyle r}$= , and ${\displaystyle {\sqrt {(x_{\mathrm {u} }-x_{\mathrm {c} })^{2}+(y_{\mathrm {u} }-y_{\mathrm {c} })^{2}}}}$

${\displaystyle ...}$ = an infinite sequence.

A barrel distortion typically results in a negative term for , while a pincushion distortion results in a positive value. ${\displaystyle K_{1}}$

The software can compensate for this distortion by distorting the image in the opposite direction. It calculates which pixel corresponds to the undistorted pixel, which is very complex due to the non-linearity of the equation. Lateral chromatic aberration (purple / green color fringes) can be significantly reduced by such image warping , separated for red, green and blue channels.

An alternative method calculates the undistorted image point by iteration .

Calibrated systems

Calibrated systems work with profiles of individual camera and lens data:

• Adobe Photoshop Lightroom and Adobe Photoshop can compensate for complex distortions.
• PTlens is a Photoshop plug-in or standalone application that compensates for complex distortions. It fixes not only linear distortions, but also second and higher nonlinear degree distortions and takes the focal length into account.
• Lensfun is a free database for compensating lens distortion.
• Optics Pro from DxO Labs can compensate for complex distortions and takes the focal length into account.
• proDAD Defishr includes a rectification tool and a calibration tool. In addition to the inclusion of the camera data, the distortion is calculated using a checkerboard pattern.

Many digital cameras perform automatic distortion compensation using parameters that are stored in the device's firmware. With newer digital camera systems , these parameters are usually transferred automatically from the lens to the camera housing. Depending on the provider, the lateral color error can also be reduced. The required parameters are automatically taken over by the image processing in the camera and suitable raw data converter software. These software functions are an integral part of corresponding camera systems and make it possible to design lenses that are less complex, more compact and easier. Since many of these lenses have clearly visible distortions, automatic consideration is useful when processing raw data .

Manual editing

Numerous software products allow manual editing of the distortion. Some examples:

• Adobe Photoshop and Adobe Photoshop Elements (from version 5) include a filter for simple (pillow / barrel) distortion.
• PhotoLine provides tools for manipulating chromatic aberration, distortion and perspective distortion.
• In addition to the actual purpose of automatically compensating for converging lines, ShiftN also allows manual processing of simple pincushion or barrel-shaped distortions.
• Corel PaintShop Pro Photo includes a lens distortion effect for simple pincushion / barrel or fisheye spherical distortion.
• The GIMP has lens correction from version 2.4.
• PhotoPerfect has functions for eliminating pincushion distortion and removing color fringes (chromatic aberration).
• Hugin can be used to compensate for distortion, although it was not primarily designed for this.

See also

• Rectification
• Shift (photography)

credentials

1. ↑ Geometric calibration and orientation of digital image recording systems, Dipl.-Ing. Robert Godding PDF
2. ↑ Camera modeling, PDF  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.  
3. ↑ JP de Villiers, FW Leuschner, R. Geldenhuys: Centi-pixel accurate real-time inverse distortion correction . In: 2008 International Symposium on Optomechatronic Technologies . SPIE.
4. ↑ Robert Godding: Geometric calibration and orientation, digital image recording systems P. 11 P.5.2.2 . .
5. ↑ Zhengyou Zhang: A Flexible New Technique for Camera Calibration . .
6. ↑ Janne Heikkilä, Olli Silvén: A Four-step Camera Calibration Procedure with Implicit Image Correction . .
7. ↑ PTlens . Retrieved June 21, 2014.
8. ↑ Lensfun . Retrieved June 21, 2014.
9. ^ Carlisle Wiley: Articles: Digital Photography Review . Dpreview.com. Retrieved June 21, 2014.
10. ^ Hugin tutorial - Simulating an architectural projection . Retrieved June 21, 2014.

Web links

In addition to some commercial tools , several free tools are also able to compensate for distortions:

• PanoTools
• hugin
• PTOpenGUI
• Digicam
• GIMP
• ShiftN