How are images formed in fisheye lenses? why can a fisheye lens have a wider field of view than an ordinary lens module? What is the difference between an ordinary lens module and a fisheye lens?
When fisheye lenses were first invented, the scientist obtained the design inspiration from the vision of real fish. The first concept of a fish eye lens is based on simulating how fish look up underwater.
In 1906, an American physicist named Robert W. Wood coined the term “fish eye.” He described the experiment in detail in a paper: He assembled a camera in a bucket filled with water and shot pictures starting from the bottom in the upward direction. The experiment was designed to simulate how fish perceive an ultra-wide hemisphere view from underwater.
Most of the ordinary lenses can be simplified as a pinhole camera model. In this model, the light beam propagates along a straight line, and the image and the object are similar. Or, in other words, a perspective transformation is performed between the image and the object. Under perspective transformation, a straight line is still a straight line after transformation, a curve is still a curve after transformation, and the intersection point of two straight lines is still the point where the two straight lines intersect after transformation, etc.
In a sense, the use of the camera lens is to transform the object space into an image space ( i.e. the optical space coordinatizing the visual representation or component of a scene). The imaging plane can be thought of as cutting a slice in the image space and intercepting a plane to form a captured image.
However, the lens based on the pinhole camera model has a drawback – the line propagates in a straight line, making it difficult for the lens to capture objects at the edge. As shown in the figure below, for red line arrows with identical lengths, as the lines get closer to the edge of the lens, the length of the arrow also becomes longer after imaging. However, the size of our camera sensor is limited, so it will not be possible for an ordinary lens module to capture images of objects that are very close to the edges. If a field of view of 180° is to be reached, the imaging plane is required to be infinite, which is not feasible.
Figure 1. Pinhole Imaging Model
The fundamental principle of a fish eye lens is to make the exit angle of the light smaller than the incident angle so that space within a broad range of FOV can be projected onto an imaging plane of limited size. Think of how a fish looks up underwater. When the fish in the water are close to the water and observe the scene above water, the field of view of the fish can reach about 180°.
Imagine when an incident light ray travels from air to water, because the refractive index of water is greater than that of air, the light rays bend toward the normal, and light rays close to the water’s surface will also enter the water at an angle equal to the refraction angle. Based on the above, it can be understood that when fish in the water are close to the water surface, the fish can see the scenes in the 180° angle range above the water surface.
Figure 2. Fisheye Lens Imaging Model
The crystalline lens (the word lens here means the biological structure of the fish eye) of the fish eye and the optical medium can be regarded as an optical structure as a whole. The front surface of the lens and the horizontal plane of the water form a plano-concave lens, and water is the medium of light propagation. If the water medium is replaced by an optical material with a high refractive index, and by forming an optical lens, it is possible to achieve a wide range of vision.
To further expand the field of view, the front plane surface of the plano-concave lens is modified and made into a convex plane, at the same time, the curvature of the back surface is also increased to ensure that the change in optical power is not large, which makes the plano-concave lens transform into a meniscus-shaped lens. This is how the first lenses of fisheye lens modules evolved. The first lenses of fisheye lens modules are all negative lenses that are similar to a parabola and protrude outward a lot.
Figure 3. The evolving of a fisheye lens
The Structure of Fisheye Lenses:
A fisheye lens is an inverted telephoto lens because of the large FOV and short focal length. Fisheye lenses often contain several lenses with negative focal lengths in the front, serving as components to gather light from an entire hemisphere into the aperture stop. The rear lens group, therefore, must have positive focal lengths to counterbalance the total power of the lens module.
Figure 4. The structure of fisheye lenses