Machine Vision Guide - Lens Fundamentals

Lens Fundamentals

Thin-Lens Model

To understand machine vision lenses, we start with the thin-lens model. It is not an exact description of any real lens, but illustrates lens principles. It also provides terms with which to discuss lens performance. A ray, called the chief ray, follows a straight line from a point on the object, through the center of the lens, to the corresponding point on the image (figure 1). The lens causes all other rays that come from this same object point and that reach the lens to meet at the same image point as the chief ray. Those rays which pass through the edge of the lens are called marginal rays.

Focal Length

f-Number (F/#)

Numerical Aperture (NA)


Figure 1. Thin-lens model

The distance from the object plane to the lens is called the object conjugate. Likewise, the distance from the lens to the sensor plane is called the image conjugate. These conjugates are related by the lens maker�s formula:

Focal Length If we let the object conjugate, get very large, we see

In other words, the focal length is the distance between the lens and the sensor plane when the object is at infinity. For photographic lenses, the objects are usually far away, so all images are formed in nearly the same plane, one focal length behind the lens. From figure 1 and geometry, we can see that

The magnification is the ratio of the image to the object conjugates. If the focal length of a lens increases for a specified magnification, both object and image conjugates increase by the same ratio.
Thin-Lens Example

We need a magnification of 0.5x, with a working distance of 50 mm. We want to find the correct lens focal length and total system length (TSL). From the equations (after some algebra), we get:

so


Therefore, we need a lens with focal length of approximately 17 mm. The total system length is approximately 75 mm.
f-Number (f/#)
The f-number describes the cone angle of the rays that form an image (figure 2). The f-number of a lens determines three important parameters:
The brightness of the image The depth of field The resolution of the lens For photographic lenses, where the object is far away, the f-number is the ratio of the focal length of the lens to the diameter of the aperture. The larger the aperture, the larger the cone angle and the smaller the f-number. A lens with a small f-number (large aperture) is said to be �fast� because it gathers more light, and photographic exposure times are shorter. A well-corrected fast lens forms a high-resolution image, but with a small depth of field. A lens with a large f-number is said to be �slow�. It requires more light, but has a larger depth of field. If the lens is very slow, its resolution may be limited by diffraction effects. In this case, the image is blurred even at best focus. The f-number printed on a photographic lens is the infinite conjugate f-number. It is defined as:

where f is the focal length of the lens and A is the diameter of the lens aperture. When the lens is forming an image of a distant object, the cone half-angle of the rays forming the image is:


Figure 2. f-number
This infinite conjugate f-number is only applicable when the lens is imaging an object far away. For machine vision applications, the object is usually close and the cone angle is calculated from the working f-number.
f-Number (Working) In machine vision, the working f-number describes lens performance:

where s₂ and s₁ are the image and object conjugates, respectively. f/#_image is called the working f/-number in image space, or the simply image side f-number. Similarly, f/#_object is the object side f-number. For close objects, f/#_image is larger than f/#_infinity, so the lens is �slower� than) the number given on the barrel. For example, a lens shown as f/4 on its barrel (i.e, an f-number of 4) will act like an f/8 lens when used at a magnification of 1. The object-side f-number determines depth of field. It is given by:

Numerical Aperture (NA) For lenses designed to work at magnifications greater than 1 (for example, microscope objectives), the cone angle on the object side is used as the performance measure. By convention, this angle is given as a numerical aperture (NA). The NA (figure 3) is given by:


Figure 3. Numerical aperture (NA)
NA is related to f-number by these exact relationships:

For N/A < 0.25 (f-number >2), these simplify to:

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