Mastering machine vision involves a wide range of complex knowledge—from image-processing algorithms to camera calibration—each requiring a deep understanding in order to design effective solutions. Recently, I’ve been delving into the topic of camera selection and have briefly explored the principles behind camera lens imaging. Today, I’ll summarize these concepts using geometric optics diagramming techniques.

• At the focal point of a camera lens (a convex lens) that converges parallel light rays, the light source passes through the optical center of the camera lens and intersects with a line passing through the focal point at a single point. The plane containing this point is called the focal plane—the image plane of the light source—as shown in the figure below.

• Object distance, image distance, focal length

According to the Gaussian imaging formula:

Among them, ff is the focal length, u is the object distance, and v is the image distance. The object distance is the distance from the object to the optical center o of the camera, while the image distance is the distance from the imaging plane to the optical center o.

Among these cases, when the object distance is between two times the focal length and infinity, the image distance falls between one time and two times the focal length, producing an inverted, reduced image. This is precisely the imaging principle behind cameras, as illustrated in the figure below.

When an object is located between twice the focal length and once the focal length, the image distance lies between twice the focal length and infinity, producing an inverted, magnified image. This is precisely the principle behind projector imaging, as illustrated in the figure below.

When an object is located at twice the focal length, the image distance equals the object distance, and the image distance is also at twice the focal length, forming an inverted image of the same size as the object.

When an object is located at one focal length, no image is formed; instead, parallel rays of light are produced, and the light does not converge. See the figure.

When an object is placed within one focal length of a lens, a virtual, upright, and magnified image is formed on the same side as the object—this is the principle behind a magnifying glass.

• The relationship between depth of field and focal length, aperture size, and object distance.

Depth of field (DOF) refers to the range of distances in front of and behind the subject that can be brought into sharp focus by a camera lens or other imaging device.

From the relationship among object distance, image distance, and focal length mentioned above, we know that the image distance varies with the object distance. Therefore, when the camera’s imaging plane (on which light forms an image) happens to be located exactly at the image distance—the focal plane—then the image of the object is focused precisely onto a single point on the imaging plane, resulting in the sharpest possible image. If the imaging plane is not at the focal plane, the image will appear blurry. Within a certain range, the distance between the object and the imaging plane—both in front of and behind the focal plane—is referred to as the depth of field, as shown in the figure.

At the imaging end, the red area represents the position where the image is sharp. When the target plane lies between the two green lines, the resulting image blur is acceptable. Correspondingly, at the object end, the distance between the two green lines defines the depth of field. At the imaging end, two green lines will appear.The circle of confusion—the circle of confusion refers to the phenomenon where, due to optical aberrations, the light rays from a point object fail to converge into a single point when forming an image, resulting in a diffused circular projection on the image plane. As the light rays begin to converge and diverge around the focal point, the image of a point gradually becomes blurred, forming an enlarged circle. The circle of confusion corresponding to the edge of the depth of field is called the permissible circle of confusion. The radius of this circle is typically provided by the camera manufacturer and is related to the camera's resolution.

Several rules:

The longer the focal length, the shallower the depth of field;

The larger the aperture, the shallower the depth of field;

The closer the object is to the camera, the shallower the depth of field.

• Choosing a camera lens

Determine the lens's focal length:

Magnification F = Pixel size / Image width (Once the camera is selected, the pixel size is also determined.)

Focal length = Working distance × Magnification / (Magnification + 1)

The lens size is larger than the camera’s sensor diameter.

What do “far” and “near” mean on the lens? They’re used for focusing at different working distances. When the working distance is close, turn the focus ring toward “near”; when the working distance is far, turn it toward “far.”

For a lens whose far end is located at the focal length, there is a working distance point A where the image remains sharp from A all the way to infinity. However, for some lenses, the far end is not located exactly at the focal length; therefore, as the working distance increases beyond this point, the image will become blurred.