I.Definition
Distortion is a geometric aberration characterized by the bending of straight lines in the actual scene, which disrupts the geometric similarity between the object and the image, despite not affecting image sharpness. It is determined by the behavior of the chief rays and manifests specifically as an inconsistency in actual lateral magnification across different fields of view.

II.Types
Based on variations in magnification, distortion is primarily classified into the following three types:

• Pincushion distortion: Also known as positive distortion. It is characterized by a lateral magnification at the image edges that is greater than that at the center, causing straight lines to curve inward. This distortion is commonly found in telephoto lenses.
• Barrel distortion: Also known as negative distortion. It is characterized by a lateral magnification at the image center that is greater than that at the edges, causing straight lines to curve outward. This distortion is particularly prevalent in wide-angle lenses.
• Moustache distortion: A complex distortion combining the characteristics of both pincushion and barrel distortions. It typically manifests as barrel distortion in the center of the image, gradually transitioning to pincushion distortion towards the periphery, resulting in complex, wave-like curves.

III.Causes
In an ideal lens, distortion is absent because the aperture stop, nodal point, and optical center coincide.This alignment ensures that the chief ray maintains a constant direction of propagation and that the transverse magnification remains constant as it passes through the optical center.

Real-world lenses, however, are composed of multiple lens elements.In these systems, the fundamental cause of distortion is a non-constant effective focal length (EFL).This variability causes the transverse magnification to change with the angle of the incident light (i.e., across different fields of view), ultimately resulting in a loss of geometric similarity between the image and the object.

The fundamental cause of distortion is the variation of the effective focal length (EFL) across the field of view, which results in a spatially non-uniform transverse magnification. This change in focal length is closely tied to the position of the aperture stop. Acting as a “window” that limits the light beam, the stop determines which specific aperture zone of the lens light from different fields of view traverses. Since the refractive power differs across the lens’s aperture zones (weaker in the center, stronger at the edges), the system’s EFL fluctuates as light passes through different zones, ultimately leading to distortion.

Taking a single convex lens as an example:

• Aperture at the image side (behind the lens): The chief rays from the edge of the field of view pass through the lens first.Due to the screening effect of the aperture, only rays with a steep exit angle can pass.This results in the actual image height at the edges being greater than the ideal image height, producing pincushion distortion (positive distortion).
• Aperture at the object side (in front of the lens): The chief rays from the edge of the field of view enter through the peripheral aperture zone of the lens and are excessively refracted towards the optical axis.This causes the actual image height at the edges to be less than the ideal image height, producing barrel distortion (negative distortion).

Note: If a convex (positive) lens is replaced with a concave (negative) lens, the observed distortion phenomena will be reversed.

IV.Distortion Calculation Principles Traditional TV Distortion (EBU Tech 3249) Originating from the era of analog television, this method was initially developed to quantify the visual effect of “straight lines becoming curved” on the edges of images caused by cathode ray tube (CRT) displays or early camera lenses. It is defined by measuring the degree of curvature of edge lines in the image, with the calculation formula as follows:
$$D = \frac{\Delta H}{H} \cdot 100$$ Where:

• $\Delta H$ is the difference between the actual image height and the ideal image height
• $H$ is the image height of the ideal undistorted image.
  

SMIA TV Distortion
SMIA TV Distortion is an industry standard evolved from traditional TV distortion, and its numerical magnitude is typically about twice that of traditional TV distortion. The calculation process is as follows:
1. Feature extraction: Use a high-precision test chart (such as a checkerboard or dot grid) to extract the actual imaging coordinates $(x_{real}, y_{real})$ of the feature points in the image.
2. Model fitting: Combined with the nominal ideal coordinates, use the least squares method to fit the radial distortion model of the lens (typically using a third-order or fifth-order polynomial, such as $r_{\text{real}} = r_{\text{ideal}} \left(1 + k_1 r^2 + k_2 r^4\right)$).
3. Virtual Frame Mapping: Based on the fitted model, establish a virtual test frame at 98% of the image’s field height to ensure that the calculation covers most of the field of view and avoids edge invalid areas.
4. Numerical output: Calculate the “ideal height (B)” and “distorted actual height (A)” of the virtual frame under the model, respectively, and substitute them into the SMIA standard formula for calculation. The calculation formula is as follows:

• Calculate the actual distortion heights of the left and right edges, $A_{1}$ (left edge) and $A_{2}$ (right edge) respectively, and take their average to obtain $A = \frac{A_{1}+A_{2}}{2}$
• $B$ is the vertical height of ideal undistorted image (the nominal height of the test frame).
• The formula is:

$$ SMIA= \frac{A-B}{B} \times100% $$ When SMIA >0 , it indicates pincushion distortion; when SMIA <0, it indicates barrel distortion.

See More
iso_17850_畸变测试, Distortion Test