5 wide angle lens design layouts

Equidistant projection (F-Theta type of lens) vs Perspective projection (F-TanTheta type of lens)

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也称为等态）

通常我们的透视投影方法ed 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:

在给定视野的图象格式取决于图像建筑方程
Space distortion
Object distortion
在图像上的光分布
Theoretical and achievable lens field of view
1. 在给定视野的图象格式取决于图像建筑方程

1.1 Types of projections and image formation equation

Table 1.1 Image height depends on projection law

- 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

Fig 1.2 Simplified tracing of principal ray for different projections.Focal length 12.5 mm was previously calculated based on perspective projection according with full frame diagonal format y’=21.635 as f′=y^′/tan⁡(ω)Then calculated focal length f’… — Fig 1.2 Simplified tracing of principal ray for different projections.

Focal length 12.5 mm was previously calculated based on perspective projection according with full frame diagonal format y’=21.635 as f′=y^′/tan⁡(ω)

Then calculated focal length f’ =12.5 mm was used for all other types of projections for image height calculations.

The lens is drawn by coinciding principal planes

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.

表1.2相应的标准传感器格式，具有用于给定角度FOV的图像高度和等于12.5 mm的焦距。

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]

根据投影的类型的空间变形图如下所示。

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

There is essential stretching space for Perspective projection after 140 deg.

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

There is small stretching space for Stereographic projection up to 280 deg.

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

There is small stretching space for Equidistant projection after 300 deg.

2.5面积平方的相对变化与赤道投影视角

There is even space imagination for Equisolid projection.

2.6面积值与正交投影视角的相对变化

There is compression space for 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.

- 是正交投影的压缩空间。

- 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.对象失真

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

只能拯救小物体的形状。

立体投影保留圈子。

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

4. Distribution of illumination

4.1 Light distribution in perspective projection

最大视野受图像边缘的强度下降的限制。

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.

Note: this equation is true if ω is the same for object space as for image space. Image illumination can be different if the ω is different for image space, a common occurence in real optical systems. This topic requires additional explanation — *Note: this equation is true if*ω.is the same for object space as for image space. Image illumination can be different if theω.is different for image space, a common occurence in real optical systems. This topic requires additional explanation

4.2 Distribution of illumination for different projections

Ratio of areas for Orthographic projection

赤道投影区域的比例

For Equidistant

For Stereographic

- Orthographic projection has perfectly even image illumination.

- 等距具有非常缓慢的强度下降。

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

5.最大理论和可行的FOV

Maximal theoretical FOV based on projection equation, acceptable space distortion, and decreasing of image illumination depends on 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

Lens layout with perspective (ortoscopic) projection

Lens image simulation with perspective projection

6.3 Example of lens with stereographic projection

Lens distortion with stereographic projection

Lens image simulation with stereographic projection

6.4 Example of lens with equidistant projection

Lens distortion with equidistant projection

Lens image simulation with equidistant projection

6.5 Example of lens with equisolid projection

Lens distortion with equisolid projection

Lens image simulation with equisolid projection

6.6 Example of lens with orthographic projection

Lens layout with orthographic projection

Lens image simulation with orthographic projection

7. Comparison of Image height and distortion
(F-TanTheta in Zemax) for all types of projections

This table shows value of image height and perspective distortion (referring to perspective projection) of different 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
如果需要在图像平面上保留小对象的形状，则立体投影是有用的。FOV可以超过180°。这广泛用于机器视觉系统。
如果需要在需要在图像平面上保持对象的角度尺寸，则等距投影是有用的。FOV可以超过180°。这广泛用于鱼眼镜镜片和天文相机。
如果需要在物体和图像空间中保持固识恒定的常数，则Equisolid投影是有用的。FOV可以超过180°。这用于科学摄影。
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.