Measure distance to object with a single camera in a static scene

Question

let's say I am placing a small object on a flat floor inside a room.

• First step: Take a picture of the room floor from a known, static position in the world coordinate system.
• Second step: Detect the bottom edge of the object in the image and map the pixel coordinate to the object position in the world coordinate system.
• Third step: By using a measuring tape measure the real distance to the object.

I could move the small object, repeat this three steps for every pixel coordinate and create a lookup table (key: pixel coordinate; value: distance). This procedure is accurate enough for my use case. I know that it is problematic if there are multiple objects (an object could cover an other object).

My question: Is there an easier way to create this lookup table? Accidentally changing the camera angle by a few degrees destroys the hard work. ;)

Maybe it is possible to execute the three steps for a few specific pixel coordinates or positions in the world coordinate system and perform some "calibration" to calculate the distances with the computed parameters?

Answer 1

Most vision libraries (including opencv) have built in functions that will take a couple points from a camera reference frame and the related points from a Cartesian plane and generate your warp matrix (affine transformation) for you. (some are fancy enough to include non-linearity mappings with enough input points, but that brings you back to your time to calibrate issue)

A final note: most vision libraries use some type of grid to calibrate off of ie a checkerboard patter. If you wrote your calibration to work off of such a sheet, then you would only need to measure distances to 1 target object as the transformations would be calculated by the sheet and the target would just provide the world offsets.

Answer 2

If the floor is flat, its equation is that of a plane, let

a.x + b.y + c.z = 1

in the camera coordinates (the origin is the optical center of the camera, XY forms the focal plane and Z the viewing direction).

Then a ray from the camera center to a point on the image at pixel coordinates (u, v) is given by

(u, v, f).t

where f is the focal length.

The ray hits the plane when

(a.u + b.v + c.f) t = 1, 

i.e. at the point

(u, v, f) / (a.u + b.v + c.f)

Finally, the distance from the camera to the point is

p = √(u² + v² + f²) / (a.u + b.v + c.f)

This is the function that you need to tabulate. Assuming that f is known, you can determine the unknown coefficients a, b, c by taking three non-aligned points, measuring the image coordinates (u, v) and the distances, and solving a 3x3 system of linear equations.

From the last equation, you can then estimate the distance for any point of the image.

The focal distance can be measured (in pixels) by looking at a target of known size, at a known distance. By proportionality, the ratio of the distance over the size is f over the length in the image.