HELP SOLVE_DILATION                             David Young
                                                April 1994

LIB * SOLVE_DILATION provides a single procedure that finds the focus of
expansion and the rate of dilation for a simple dilating dense
flow-field.

solve_dilation(U_array, V_array) -> (D, x0, y0)
solve_dilation(UVlist) -> (D, x0, y0)
        In the first form, U_array and V_array should be arrays of
        identical dimensions containing numbers representing the
        components of the optic flow field, perhaps as returned by the
        procedure provided by *horn_schunck.

        In the second form, UVlist should be a list of vectors or other
        explodable structures. Each vector should contain 4 elements,
        specifying x, y, u and v in that order, where u and v are the
        components of the flow vector for the point (x, y) in the image.

        The three results are as described below.

This is intended mainly for teaching demonstrations rather than for
practical applications. The procedure assumes that the flow is of the
form

    u = D (x - x0)
    v = D (y - y0)

where x and y are image coordinates, u and v are the flow vector
components for the image point at (x, y), x0 and y0 are the coordinates
of the focus of expansion, and D is the rate of dilation.

This assumption is valid for movement relative to a surface normal to
the line of sight, with no camera rotation.  It will be a reasonable
approximation if the visible objects in similar distances from the
camera and the amount of camera rotation is small.

The algorithm uses a least-squares approach. The quantity

                              2                      2
    E = SUM [ (u  - D x  - u0)   +   (v  - D y  - v0) ]
         i      i      i               i      i

where

    u0 = -D x0          and             v0 = -D y0

is minimised. Here (u , v ) is the flow vector at (x , y ).
                     i   i                          i   i

The least-squares solution is straightforward - see the program for the
formulae.


--- $popvision/help/solve_dilation
--- Copyright University of Sussex 1994. All rights reserved.