  Image-free tracking methods based on single-pixel detectors (SPDs) can track
a moving object at a very high frame rate, but they rarely can achieve
simultaneous imaging of such an object. In this study, we propose a method for
simultaneously obtaining the relative displacements and images of a
translational object. Four binary Fourier patterns and two differential
Hadamard patterns are used to modulate one frame of the object and then
modulated light signals are obtained by SPD. The relative displacements and
image of the moving object can be gradually obtained along with the detection.
The proposed method does not require any prior knowledge of the object and its
motion. The method has been verified by simulations and experiments, achieving
a frame rate of 3332 Hz to acquire relative displacements of a translational
object at a spatial resolution of $128 \times 128$ pixels using a 20000-Hz
digital micro-mirror device. This proposed method can broaden the application
of image-free tracking methods and obtain spatial information about moving
objects.
