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.