Example 35 Measuring real view angles


Starting from the eighth version of VideoCAD, the simulation accuracy of view areas of cameras in VideoCAD is greatly improved because of the possibility of simulating lens distortion.
Three view angles (horizontal, vertical, diagonal) are computed inside VideoCAD from the lens focal length and image sensor size without accounting distortion. For most longfocus lenses distortion can be ignored, but in case of shortfocus lenses the distortion introduces significant error in the calculation. To simulate distortion we need to know the lens focal length, image sensor size and at least one of the real angles. 
The value of the real angle can be taken from the manufacturer's specifications, as in the Example. But not all manufacturers measure the real view angles of their cameras.
Lets consider a simple technique of practical measurement of camera view angles. Many variants of the technique are possible depending on the conditions.
Problem
There is a camera with a lens. It is necessary to measure the real angles: horizontal, vertical and diagonal. To accurately modeling the view area, taking into account lens distortion.
The front nodal point of a lens and measurement inaccuracy
If the camera view area is in the form of a pyramid, the top of the pyramid will be in the front nodal point of the lens. The front nodal point of the lens is located on the front principal plane within the length of the lens on the main optical axis, but its exact position is unknown to us.
If we assume that the front nodal point of the lens is in the middle, the possible error of measuring the dimensions of the view area will be the ratio of half of the length of the lens by the distance from the middle of the lens to the screen. For example, if the physical length of the lens = 40mm, and the distance from the middle of the lens to the screen = 1 meter, then the inaccuracy of measuring the size of the field of view = 40/2/1000 = 0.02, ie + 2%. The inaccuracy of measurement of an angle will be less depending on the angle. For example, if the angle is 90 degrees, the inaccuracy will be+ 1.3%.
If this accuracy is not sufficient, you should use more distance from the camera to the screen, but it will require a larger screen, that is not always convenient.
Another way of reducing the inaccuracy is locating the front nodal point of the lens within the length. To do this, perform 2 measurements at different distances between the front edge of the lens and the screen, for example 0.5 and 0.25 meters. Then, assuming that the view angles are constant, solve the system of equations, where the unknown is the distance from the front edge of the lens to the front nodal point.
Omitting the course of the solution, we present the resulting formula for the calculation.
dL=(S2 * L1  S1 * L2) / (S1  S2);
where:

Order of work
1. Fully open the lens aperture, if possible.
2. Focus the lens on the camera working distance.
You should focus not on the distance to the screen but on a distance that will work the camera on. In most cases, this distance is the hyperfocal distance at maximum aperture. Focusing on the working distance is performed for a case, if the angles of view or the position of the front nodal point will depend on the focus distance.
3. Secure the camera in front of a flat screen.
4. Close the aperture as possible to obtain a sharp image from the camera.
5. Adjust the camera position so that the main optical axis of the camera will be strictly perpendicular to the screen. Mark the center of the image on the screen.
6. Watching the image, mark the extreme points on the screen width, height, and diagonal field of view.
7. Measure the distances on the screen between the points on the boundaries and the marked center of the image. Boundary points must be strictly symmetrical relative the center. If it is not the case, then adjust the position of the camera and go back to step 5
8. Measure the distances between the points (field of view size in width, height and diagonal).
9. Measure the distance from the front nodal point of the lens to the screen.
10. Calculate the view angles using the formulas:
Ah=2 * ARCTAN(h / (2 * L));
Aw=2 * ARCTAN(w / (2 * L));
Ad=2 * ARCTAN(d / (2 * L));
where (see image):
•  ARCTAN  arc tangent, inverse tangent function; 
•  Ah,Aw,Ad  obtained real view angles in height, width and diagonal respectively; 
•  h  distance between extreme points on height of the field of view; 
•  w  distance between extreme points on width of the field of view; 
•  d  distance between extreme points on diagonal of the field of view; 
•  L  distance between lens front nodal point and the screen. 
You can use the obtained angles in the Sensor and Lens box for precise modeling view area of this camera.
It is possible to make very simplified test on a camera already installed.
1.  Firstly measure sized and distances of real objects in cameras view area. 
2.  Then construct 3D model in VideoCAD, place a camera. 
3.  Then choose angles of view to obtain image model equal to real image from the camera. 
See also: Sensor and Lens>Lens distortion, About lens distortion,, Modeling lens distortion