Perspective distortion

In photography and cinematography, perspective distortion is a warping or transformation of an object and its surrounding area that differs significantly from what the object would look like with a normal focal length, due to the relative scale of nearby and distant features. Perspective distortion is determined by the relative distances at which the image is captured and viewed, and is due to the angle of view of the image (as captured) being either wider or narrower than the angle of view at which the image is viewed, hence the apparent relative distances differing from what is expected. Related to this concept is axial magnification – the perceived depth of objects at a given magnification.

Perspective distortion takes two forms: ''extension distortion'' and ''compression distortion,'' also called ''wide-angle distortion'' and ''long-lens'' or ''telephoto distortion,'' when talking about images with the same field size. Extension or wide-angle distortion can be seen in images shot from close using a wide-angle lens (with an angle of view wider than a ''normal lens''). Objects close to the lens appear abnormally large relative to more distant objects, and distant objects appear abnormally small and hence farther away – distances are extended. Compression, long-lens, or telephoto distortion can be seen in images shot from a distance using a long focus lens or the more common telephoto sub-type (with an angle of view narrower than a ''normal lens''). Distant objects look approximately the same size – closer objects are abnormally small, and more distant objects are abnormally large, and hence the viewer cannot discern relative distances between distant objects – distances are compressed.

Note that linear perspective changes are caused by ''distance,'' not by the lens per se – two shots of the same scene from the same distance will exhibit identical perspective geometry, regardless of lens used. However, since wide-angle lenses have a wider field of view, they are generally used from closer, while telephoto lenses have a narrower field of view and are generally used from farther away. For example, if standing at a distance so that a normal lens captures someone's face, a shot with a wide-angle lens or telephoto lens from the same distance will have exactly the same linear perspective geometry on the face, though the wide-angle lens may fit the entire body into the shot, while the telephoto lens captures only the nose. However, crops of these three images with the same coverage will yield the same perspective distortion – the nose will look the same in all three. Conversely, if all three lenses are used from distances such that the face fills the field, the wide-angle will be used from closer, making the nose larger compared to the rest of the photo, and the telephoto will be used from farther, making the nose smaller compared to the rest of the photo.

Outside photography, extension distortion is familiar to many through side-view mirrors (see " objects in mirror are closer than they appear") and peepholes, though these often use a fisheye lens, exhibiting different distortion. Compression distortion is most familiar in looking through binoculars or telescopes, as in telescopic sights, while a similar effect is seen in fixed-slit strip photography, notably a photo finish, where all capture is parallel to the capture, completely eliminating perspective (a side view).

Optics

Consider an idealised Gaussian optical system, with the image and the object in the same medium. Thus, for an object in focus, the distance between the lens and image plane the distance between lens and the object and the focal length $f$ are related by the thin-lens equation:

$\frac + \frac = \frac\,.$

The transverse magnification $M$ is related by

$M = \frac\,.$

The ''axial magnification'' $M_\text$ of an object at $s_\mathrm$ is the rate of change of the lens–image distance $s_\mathrm$ as the lens–object distance $s_\mathrm$ changes. For an object of finite depth, one can conceive of the axial magnification as the ratio of the depth of the image and the depth of the object:

$\begin M_\text &= \left, \frac \ = \left, \frac \left(\frac - \frac\right)^ \ \\ &= \left, -\left(\frac - \frac\right)^ s_\mathrm^ \ \\ &= \frac = M^2\,. \end$

As the magnification increases, the ratio of axial to transverse magnification also increases. For example, if then and the image will be twice as tall and four times as deep as the object.

Influencing factors

Perspective distortion is influenced by the relationship between two factors: the angle of view at which the image is captured by the camera and the angle of view at which the photograph of the subject is presented or viewed.

Angle of view of the capture

When photographs are viewed ''at the ordinary viewing distance'', the angle of view at which the image is captured accounts completely for the appearance of perspective distortion. The general assumption that "undoctored" photos cannot distort a scene is incorrect. Perspective distortion is particularly noticeable in portraits taken with wide-angle lenses at short camera-to-subject distances. They generally give an unpleasant impression, making the nose appear too large with respect to the rest of the face, and distorting the facial expression. Framing the same subject identically while using a moderate telephoto or long focus lens (with a narrow angle of view) flattens the image to a more flattering perspective. It is for this reason that, for a 35 mm camera, lenses with focal lengths from about 85 through 135 mm are generally considered to be good portrait lenses. It does however make difference, whether the photograph is taken landscape or portrait. A 50 mm lens is suitable for photographing people when the orientation is landscape. Conversely, using lenses with much longer focal lengths for portraits results in more extreme flattening of facial features, which also may be objectionable to the viewer.

Photograph viewing distance

Photographs are ordinarily viewed at a distance approximately equal to their diagonal. When viewed at this distance, the distortion effects created by the angle of view of the capture are apparent. However, theoretically, if one views pictures exhibiting extension (wide angle) distortion at a closer distance, thus widening the angle of view of the presentation, then the phenomenon abates. Similarly, viewing pictures exhibiting compression (telephoto) distortion from a greater distance, thus narrowing the angle of view of the presentation, reduces the effect. In both cases, at some critical distance, the apparent distortion disappears completely.

Examples

Below, a series of three photos shows the same scene shot from the same position with three different lenses: a wide-angle lens, a normal lens, and a telephoto lens. Notice that because the proportions in the image do not change with the angle of view, if the photos made with the wide-angle lens and the normal lens were cropped to contain the same scene as the photo made with the telephoto lens, then the image content would be exactly the same. The apparent difference in proportions results solely from the content added around the edges of the frame in the normal lens photo and the wide-angle photo.

35 mm