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# Distortion

This is Section 4.4 of the Imaging Resource Guide

The term distortion is often applied interchangeably with reduced image quality. Distortion is an individual aberration that does not technically reduce the information in the image; while most aberrations actually mix information together to create image blur, distortion simply misplaces information geometrically. This means that distortion can actually be calculated or mapped out of an image, whereas information from other aberrations is essentially lost in the image and cannot easily be recreated. Please note that in extreme high distortion environments, some information and detail can be lost due to resolution change with magnification or because of too much information being crowded onto a single pixel.

Distortion is a monochromatic optical aberration that describes how the magnification in an image changes across the field of view at a fixed working distance; this is critically important in precision machine vision and gauging applications. Distortion is distinct from parallax, which is a change in magnification (field of view) with working distance (more on parallax is provided in the section on telecentricity in The Advantages of Telecentricity). It is important to keep in mind that distortion varies with wavelength, as shown in Figure 1, and that when calibrating distortion out of a machine vision system the wavelength of the illumination needs to be taken into account. Curves like the one in Figure 1 are very helpful in determining how to calibrate out distortion.

As with other aberrations, distortion is determined by the optical design of the lens. Lenses with larger fields of view will generally exhibit greater amounts of distortion because of its cubic field dependence. Distortion is a third-order aberration that, for simple lenses, increases with the third power of the field height; this means that larger fields of view (a result of low magnification or short focal length) are more susceptible to distortion than smaller fields of view (high magnification or long focal length). The wide fields of view achieved by short focal length lenses should be weighed against aberrations introduced in the system (such as distortion). On the other hand, telecentric lenses typically have very little distortion: a consequence of the way that they function. It is also important to note that when designing a lens to have minimal distortion, the maximum achievable resolution can be decreased. In order to minimize distortion while maintaining high resolution, the complexity of the system must be increased by adding elements to the design or by utilizing more complex optical glasses.

### How is Distortion Specified?

Distortion is typically specified as a percentage of the field height. Typically, ±2 to 3% distortion is unnoticed in a vision system if measurement algorithms are not in use. In simple lenses, there are two main types of distortion: negative, barrel distortion, where points in the field of view appear too close to the center; and positive, pincushion distortion, where the points are too far away. Barrel and pincushion refer to the shape a rectangular field will take when subjected to the two distortion types, as shown in Figure 2.

##### Figure 2: An Illustration of Positive and Negative Distortion

Distortion can be calculated simply by relating the Actual Distance (AD) to the Predicted Distance (PD) of the image using Equation 1. This is done by using a pattern such as dot target shown in Figure 3. More on distortion targets in our Choosing the Correct Test Target application note.

(1)$$\text{% Distortion} = \left( \frac{\text{AD} - \text{PD}}{\text{PD}} \right) \times 100 \%$$
##### Figure 3: Calibrated Target (Red Circles) vs. Imaged (Black Dots) Dot Distortion Pattern

It is important to note that while distortion generally runs negative or positive in a lens, it is not necessarily linear in its manifestation across the image for a multi-element assembly. Additionally, as wavelength changes, so does the level of distortion. Finally, distortion can change with changes in working distance. Ultimately, it is important to individually consider each lens that will be used for a specific application in order guarantee the highest level of accuracy when looking to remove distortion from a system.

### Example of Distortion Curves

Figure 4 shows negative, or barrel, distortion in a 35mm lens system. In this specific example, all of the wavelengths analyzed carry almost identical levels of distortion, thus wavelength-related issues are not present. In Figure 5, an interesting set of distortion characteristics can be seen: first, there is separation in the level of distortion for the different wavelengths, and second, both negative and positive distortion is present in this lens. Distortion of this nature is referred to as wave, or moustache, distortion. This is often seen in lenses that are designed for very low levels of distortion, such as those designed for measurement and gauging applications. In this scenario, calibrating the system so that distortion is removed can require special consideration for applications where different wavelengths are used.

### Geometric Distortion vs TV Distortion: An Important Difference

In lens datasheets, distortion is usually specified in one of two ways: radial, geometric distortion or RIAA TV distortion. Geometric distortion describes the distance between where points appear in the distorted image and where they would be in a perfect system. In practice, this can be measured using a distortion dot target. The difference between the distance from the center of the target to any dot in the field of view and the distance from the center of the image to the same, now misplaced dot (shown in Figure 3), provides the radial distortion percentage calculated with Equation 1.

The measurement of TV distortion is specified by an RIAA imaging standard, and is determined by imaging a square target that fills the vertical field of view. The difference in height between the corners and the center edge of the square is used to calculate the TV distortion with Equation 2; this describes the apparent straightness of a line which appears at the edge of the image, which is essentially the geometric distortion at a single field point. By only specifying distortion at one point in the field, it is possible to misrepresent a non-zero distortion lens as having 0% distortion. In Figure 5, a 0% intercept can be found for any of the wavelengths shown. However, when the full image circle is considered, it is obvious that the lens has non-zero distortion. An example of how TV distortion can be found is shown in Figure 6.

(2)$$\textbf{DTV} = \frac{\Delta H}{\boldsymbol{H}} \times 100 \%$$
##### Figure 6: TV Distortion with both Barrel and Pincushion Distortion

As shown in Figure 5, in real world, compound imaging lens assemblies, distortion is not necessarily monotonic and can change signs across the field of view, which is why a radial distortion plot is preferable to the single RIAA value. Due to the way it is specified, the TV distortion value can be much lower than the maximum geometric distortion value of the same lens, thus it is important to be aware of what type of distortion is being specified when choosing the most appropriate lens for an application.

### Keystone Distortion

In addition to the previous distortion types mentioned, which are inherent to the optical design of a lens, improper system alignment can also result in keystone distortion, which is a manifestation of parallax (shown in Figure 7a and 7b).

##### Figure 7: Examples of Keystone Distortion

When calibrating an imaging system against distortion, keystone distortion should be considered in addition to radial geometric distortion. Although distortion is often thought of as a cosmetic aberration, it should be carefully considered against other system specifications when choosing the right lens. In addition to the potential for a loss in image information, algorithmic distortion correction takes additional processing time and power, which may not be acceptable in high speed or embedded applications.

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