We present two methods for the determination of the parameters of motion of a sensor, given the vector flow field induced by an imaging system governed by a perspective transformation of a rigid scene. We assume that the flow field V = (u(x, y), v(x, y)) is given. Both algorithms are new, and both integrate global data to determine motion parameters. The first algorithm (the flow circulation algorithm) determines the rotational parameters. It uses the curl of the flow field (curl (V)), which under many conditions is approximately a linear function of the form g(x, y) = ax + by + c. The coefficients of the linear function, a, b, and c, which may be determined by simple regression, are proportional to the desired rotational parameters of motion. Circulation values may be used in place of curl values, resulting in less noise. The second algorithm (the FOE search algorithm) determines the translational parameters of the motion independently of the first algorithm. This algorithm extends a recent method of Heeger and Jepson, giving a method for searching for the image focus of expansion. For every location (x0, y0) in the image plane, we compute a function u . (-y + y0) + v . (x - x0). When (x0, y0) is located at the focus of expansion, this function will be a quadratic polynomial (of a special form). We suggest several methods for determining when the function has the appropriate form; one method involves filtering the function by a collection of circular-surround zero-mean receptive fields. The other methods project the function onto a linear space of quadratic polynomials and measures the distance between the two functions. The error function for the first two methods is a quadratic polynomial of the candidate position, yielding a very ra id search strategy.
