There are two primary methods of image formation in lens design.

Perspective projection (F-Tan Theta lenses also called Rectilinear or Orthoscopic).

Equidistant projection (F-Theta lenses also called Equiangular)

The Perspective projection method is most often used during the design of lenses with field of view in the 40-60 degree range. This formation method maintains straight lines in images while stretching space. For some photography types such as satellite imaging, this stretching is acceptable even at wide angles. For many applications however, it is not acceptable. The photo on the right above shows a 130 degree image with distortions.

Equidistant projection is used in wide-angle lenses like a fisheye lens. It bends straight lines but can provide more than 180 degrees of lens FOV (left photo). Moreover, this projection saves angles. This is useful in astronomy photography and many applications on earth.

There are 3 less common methods of image formation or projection these are:

Stereographic

Equisolid

Orthographic projections

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The key criteria used to evaluate which methods are best include:

Image format at given field of view depending on image construction equation

Space distortion

Object distortion

Light distribution on image

Theoretical and achievable lens field of view

**Image format at given field of view depending on image construction equation**

**1.1 Types of projections and image formation equation **

?- angle of object FOV

*y’* – image height

f – focal length

Next table shows how image size depends on type of projection for given angular FOV

** 1.2 Image size calculation of different types of projection for 120° FOV**

Different projections give different image height for given angular FOV (see below for details)

**1.3 Corresponding standard sensor formats with image height calculated for given angular FOV and equal 12.5 mm Focal Length. **

**2. Space distortion **

**2.1 Definition**

Space distortion is defined as the ratio of paraxial value of areas forming by small solid angle at image plane for given angle ω of FOV to the area forming by equal value of solid angle at center FOV (ω=0).

Derivation of values of Space distortion was implemented in [1]

Plots of space distortion depending on the type of projection are shown below.

**2.2 Relative change of area value with angle of view of perspective projection **

**2.3 Relative change of area value with angle of view of stereographic projection **

**2.4 Relative change of area value with angle of view of equidistant projection **

**2.5 Relative change of square of area with angle of view for Equisolid projection **

**2.6 Relative change of area value with angle of view of orthographic projection **

### 2.7 Summary

– There is image stretching for Perspective, Stereographic and Equidistant projections.

– Distortion of Perspective projection limits FOV to 120-140 deg.

– Stereographic projections can provide maximal angular FOV 210-250 deg.

– Equidistant projections can provide maximal angular FOV 300-350 deg.

-There is compression space for Orthographic projection.

– Orthographic projection has 180 deg. limit of angular FOV.

– There is no space distortion for Equisolid projection.

– Equisolid projection can provide maximal angular FOV up to 360 deg.

**3. Object distortion **

Perspective, Equidistant, Equisolid and Orthographic projections give deformation of shape of small objects through the field of view.

Only stereographic projection saves shape of small objects.

Stereographic projection preserves circles.

Stereographic projection is conformal –preserves angles of intersects of two curves

**4. Distribution of illumination **

**4.1 Light distribution in perspective projection **

Maximal field of view is limited by intensity drop-off at edge of image.

A well known formula for perspective projection describes decreasing light distribution from center to edge is

It can be used for other projections if consider changing size of square in plane of image.

**4.2 Distribution of illumination for different projections **

Ratio of areas for Orthographic projection

Ratio of areas for Equisolid projection

For Equidistant

For Stereographic

– Orthographic projection has perfectly even image illumination.

– Equidistant has a very slow intensity decreasing.

– Equidistant, stereographic, and equi-solid provide very wide FOV because of slow intensity drop.

**5. Maximal theoretical and feasible FOV **

**Specific properties of particular projections **

**6. Examples of lens design with different image projections **

Below we present examples of lenses for each projection. The lenses with close parameters for each projection were found and then additionally optimized by Zemax to match the image formation equation for each projection.

Angular FOV is 120 deg. for all samples

**6.1 Example of lens provided perspective projection**

**6.2 Example of lens with stereographic projection **

**6.3 Example of lens with equidistant projection**

**6.4 Example of lens with equisolid projection **

**6.5 Example of lens with orthographic projection **

**7. Comparison of Image height and distortion**

(F-TanTheta in Zemax) for all types of projections

(F-TanTheta in Zemax) for all types of projections

**8. Conclusion**

All image formation or projection types can be useful. Which one depends on the application

Perspective projection is useful to preserve straight lines in an image. The maximal field of view should however be less than 140 degrees. This is widely used in photography lenses and aero photo lenses

Stereographic projection is useful if preserving the shape of small object on image plane is required. FOV can be more than 180 deg. This is widely used in machine vision systems.

Equidistant projection is useful if preserving angular sizes of object on image plane is required. FOV can be more than 180 deg. This is widely used in fish eye lenses and astronomical cameras.

Equisolid projection is useful if preserving constant ration of solid angles in object and image spaces is required. FOV can be more than 180 deg. This is used in scientific photography.

Orthographic projection is useful if evenness of illumination through the entire image plane is required. FOV can be up to 180 degrees. This is used in cheap cameras and door eye viewers.

**9. References of theoretical materials **

1. Field of View – Rectilinear and Fishye Lenses

http://www.bobatkins.com/photography/technical/field_of_view.html

2. Margaret M. Fleck. Perspective Projection: The Wrong Imaging Model. 1995, Technical report 95-01

http://mfleck.cs.illinois.edu/my-papers/stereographic-TR.pdf

3. Models for the various classical lens projections

http://michel.thoby.free.fr/Fisheye_history_short/Projections/Models_of_classical_projections.html

4. About the various projections of the photographic objective lenses

http://michel.thoby.free.fr/Fisheye_history_short/Projections/Various_lens_projection.html

**10. References of lens design **

1. U.S.Patent 3661447

2. JP #: 04-267,212

3. Imaging lens and imaging device US 20090009888 A1

4. Wide-Angle Objective. Zeiss #1058 page #0550.

5. JP Patent 4238312