Moving object trajectory estimation using an optical Fourier processor

not easy to recover the full complex transform
S(k x , k y ,ω ) using an optical Fourier processor. If only
the real part is recovered, the frequency domain representation is given by,
reflection of the other in the kx-ky plane. The first plane represents an object moving along a trajectory with velocity vector v = [vx vy] while the second plane represents an object moving along the same path but in the opposite direction, v' = [-vx -vy] = -v. The target’s velocity can be determined by solving for the slope of one of the planes.
extension to that of Knudsen where by the 2D spatial component of the computationally intensive FT is determined using an optical processor rather than a conventional electronic processor. The present implementation estimates the trajectory of a single moving object; however, the computational savings realized through the optical processor would allow realtime multiple-target trajectory estimation.
3. EXPERIMENTAL
The trajectory estimation system shown in Figure 1 consists of a personal computer (PC) and an optical Fourier transform coprocessor. A sequence of images
Moving Object Trajectory Estimation Using an Optical Fourier Processor
Pierre Lane, K. Steven Knudsen†, and Michael Cada
DalTecha, Dalhousie University, Electrical and Computer Engineering Dept. P.O. Box 1000, Halifax, NS B3J 2X4, Canada
s(x, y, n)= δ (x - v x n - x0 ) δ (y - v y n - y0) (1)
where n is the integer frame number. The linear trajectory of a single object therefore describes a line in the 3D spatiotemporal domain. The mixed domain
π exp(− iφt )δ (ω + ωt ) + π exp(iφt )δ (ω − ωt ) (4)
and the energy of the FT is confined to a pair of planes defined by k x vx + k y vy ±ω = 0 . One plane is the
†Resolute Research Ltd., 24 Midridge Rise N.E., Calgary, AB T2X 1E3, Canada
ABSTRACT
A vision system that estimates the trajectory (velocity and direction) of moving targets in a two-dimensional field at video frame rates has been developed and constructed. The system uses an optical Fourier processor to calculate the frequency domain representation of moving objects from which their trajectory is estimated using conventional electronic processing techniques. In a series of experiments, target velocities were estimated to within 4 percent of their actual value and direction was estimated to within 3 degrees.
S(k x , k y ,ω ) =
∞
å
S(k x , k y , n)exp(− iω n)
(3)
n= -∞
= 2π exp(iφt )δ (ω − ωt )
The energy of the FT is confined to a plane in 3D frequency space defined by, k x vx + k y v y − ω = 0 . It is
representation is calculated by Fourier transforming the spatial dimensions,
[ ] ( ) ∞ ∞
(S k x , k y ,n) = ò ò s(x, y,n) exp i kx x + k y y dxdy (2) -∞ -∞ = exp[i(ωt n +φt )]
Keywords: Trajectory estimation; Fourier processor; Smartt interferometry; Mixed domain processing.
1. INTRODUCTION
We take for granted the ability to discern objects, estimate their motion, and navigate in threedimensional spaces using our biological vision system. Duplicating these capabilities in a machine vision system, even to a limited degree, has proved to be a very difficult task indeed1. The computational power required for even a basic vision system seems to transcend the capabilities of traditional electronic processors.
where ωt = k x vx + k y vy and φt = k x x0 + k y y0 . The
mixed domain representation of a moving object can be interpreted as a complex 1D sinusoidal sequence in n with temporal frequency ÿt and phase φt. The frequency is determined by the velocity of the object and the phase is determined by its initial position. The frequency domain representation is calculated by Fourier transforming the temporal dimension of the mixed domain signal,
Consider an object with velocity v = [vx vy] and initial position (x0, y0), moving along a line in the x-y plane. The absolute position of the object in space-time is described by,
2. THEORY
Trajectory estimation is typically performed in the temporal domain2-4or the frequency domain5-8. The method presented here is based on the theory of mixeddomain signal processing9 developed by Knudsen and Bruton. Knudsen has successfully applied their signal processing algorithms to the trajectory estimation10 of multiple targets. In general, the frequency domain approach to trajectory estimation involves the calculation of a three-dimensional (3D) spatiotemporal Fourier transform (FT). The work described here is an