Preklad z AJ a trosku matematiky

The second step involves solving for 3D motion. This process attempts to derive the motion of the camera by solving the inverse-projection of the 2D paths for the position of the camera. This process is referred to as Calibration.

To explain further: when a point on the surface of a three dimensional object is photographed its position in the 2D frame can be calculated by a 3D projection function. We can consider a "camera" to be an abstraction that holds all the parameters necessary to model a camera in a real or virtual world. Therefore a camera is a vector that includes as its elements the position of the camera, its orientation, focal length, and other possible paremeters that define how the camera focuses light onto the film plane. Exactly how this vector is constructed is not important as long as there is a compatible projection function "P".

The projection function "P" takes as its input a camera vector (denoted "camera") and another vector the position of a 3D point in space (denoted "xyz") and returns a 2D point that has been projected onto a plane in front of the camera (denoted "XY"). We can express this:

:"XY" = P("camera", "xyz")

The projection function transforms the 3D point and strips away the component of depth. Without knowing the depth of the component an "inverse" projection function can only return a set of possible 3D points, that form a line emanating from the center of the camera and passing through the projected 2D point. We can express the inverse projection as:

:"xyz" ∈ P'("camera", "XY")

or

:{"xyz" : P("camera", "xyz") = "XY"}

Let's say we are in a situation where the features we are tracking are on the surface of a rigid object such as a building. Since we know that the real point "xyz" will remain in the same place in real space from one frame of the image to the next we can make the point a constant even though we do not know where it is. So:

:"xyz""i" = "xyz""j"

where the subscripts "i" and "j" refer to arbitrary frames in the shot we are analyzing. Since this is always true then we know that:

: P'("camera""i", "XY""i") ∩ P'("camera""j", "XY""j") ≠ {}

Because the value of "XY""i" has been determined for all frames that the feature is tracked through by the tracking program, we can solve the reverse projection function between any two frames as long as P'("camera""i", "XY""i") ∩ P'("camera""j", "XY""j") is a small set. Set of possible "camera" vectors that solve the equation at i and j (denoted C"ij").

:C"ij" = {("camera""i","camera""j"): P'("camera""i", "XY""i") ∩ P'("camera""j", "XY""j") ≠ {})

So there is a set of camera vector pairs C"ij" for which the intersection of the inverse projections of two points "XY""i" and "XY""j" is a non-empty, hopefully small, set centering around a theoretical stationary point "xyz" .

In other words, imagine a black point floating in a white void and a camera. For any position in space that we place the camera, there is a set of corresponding parameters (orientation, focal length, etc) that will photograph that black point exactly the same way. Since C has an infinite number of members, one point is never enough to determine the actual camera position.

As we start adding tracking points, we can narrow the possible camera positions. For example if we have a set of points {"xyz""i,0",...,"xyz""i,n"} and {"xyz""j,0",...,"xyz""j,n"} where i and j still refer to frames and n is an index to one of many tracking points we are following. We can derive a set of camera vector pair sets {C"i,j,0",...,C"i,j,n"}.

In this way multiple tracks allow us to narrow the possible camera parameters. The set of possible camera parameters that fit, F, is the intersection of all sets:

:F = C"i,j,0" ∩ ... ∩ C"i,j,n"

The fewer elements are in this set the closer we can come to extracting the actual parameters of the camera. In reality errors introduced to the tracking process require a more statistical approach to determining a good camera vector for each frame, optimization algorithms and bundle block adjustment are often utilized. Unfortunately there are so many elements to a camera vector that when every parameter is free we still might not be able to narrow F down to a single possibility no matter how many features we track. The more we can restrict the various parameters, especially focal length, the easier it becomes to pinpoint the solution.

In all, the 3D solving process is the process of narrowing down the possible solutions to the motion of the camera until we reach one that suits the needs of the composite we are trying to create.

Pri tejto kalibracii 3D by ti pomohli obrazky