## Abstract

An optical technique to detect the velocity of a moving object that has a brightness distribution is presented. We show theoretically and experimentally that the amplitude of the detected signal in transmission-grating velocimetry is proportional to the Fourier component of the object spectrum at the spatial frequency of the grating. This fact can be used advantageously to sense the velocities of moving objects of any shape in one pass through the detector’s field of view. Results of the application of this method to measure the velocities of some common outdoor moving objects are also presented.

© 1992 Optical Society of America

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### Equations (14)

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(1)
$$I\left(t\right)={\displaystyle \int f\left(x-\upsilon t\right)T\left(x\right)\mathrm{d}x,}$$
(2)
$$T\left(x\right)={a}_{0}+a\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi ux\right).$$
(3)
$$T\left(x\right)={a}_{0}+a\phantom{\rule{0.2em}{0ex}}\text{exp}\left(i2\pi ux\right),$$
(4)
$$I\left(t\right)=a\phantom{\rule{0.2em}{0ex}}{\displaystyle \int f\left(x-\upsilon t\right)\text{exp}\left(i2\pi ux\right)}\mathrm{d}x.$$
(5)
$${x}^{\prime}=x-\upsilon t.$$
(6)
$$\begin{array}{cc}I\left(t\right)& =a\phantom{\rule{0.2em}{0ex}}\text{exp}\left(i2\pi u\upsilon t\right){\displaystyle \int f\left({x}^{\prime}\right)}\text{exp}\left(i2\pi u{x}^{\prime}\right)\mathrm{d}{x}^{\prime}\\ & =a\phantom{\rule{0.2em}{0ex}}\text{exp}\left(i2\pi u\upsilon t\right)F\left(u\right),\end{array}$$
(7)
$$F\left(u\right)={\displaystyle \int f\left({x}^{\prime}\right)\text{exp}\left(i2\pi u{x}^{\prime}\right)\mathrm{d}{x}^{\prime}}.$$
(9)
$$T\left(x\right)=\frac{1}{2}+\frac{2}{\pi}{\displaystyle \sum _{u}^{\text{odd}}\frac{\text{cos}\phantom{\rule{0.2em}{0ex}}2\pi ux}{u}}.$$
(10)
$$I\left(t\right)=a\phantom{\rule{0.2em}{0ex}}{\displaystyle \int f\left(x-\upsilon t\right)\text{cos}\left(2\pi ux\right)}\mathrm{d}x.$$
(11)
$${x}^{\prime}=x-\upsilon t,$$
(12)
$$I\left(t\right)=a\phantom{\rule{0.2em}{0ex}}{\displaystyle \int f\left({x}^{\prime}\right)\text{cos}\left(2\pi u{x}^{\prime}+2\pi u\upsilon t\right)}\mathrm{d}{x}^{\prime}.$$
(13)
$$\begin{array}{cc}I\left(t\right)& =a\phantom{\rule{0.2em}{0ex}}{\displaystyle \int f\left({x}^{\prime}\right)[\text{cos}\left(2\pi u{x}^{\prime}\right)\text{cos}\left(2\pi u\upsilon t\right)}\\ & -\text{sin}\left(2\pi u{x}^{\prime}\right)\text{sin}\left(2\pi u\upsilon t\right)]\mathrm{d}{x}^{\prime}\\ & =a\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}2\pi u\upsilon t{\displaystyle \int f\left({x}^{\prime}\right)\text{cos}\phantom{\rule{0.2em}{0ex}}2\pi ux\phantom{\rule{0.2em}{0ex}}\mathrm{d}{x}^{\prime}}\\ & -a\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2\pi u\upsilon t{\displaystyle \int f\left({x}^{\prime}\right)}\text{sin}\phantom{\rule{0.2em}{0ex}}2\pi ux\phantom{\rule{0.2em}{0ex}}\mathrm{d}{x}^{\prime}\\ & =a\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}2\pi u\upsilon t{F}_{1}\left(u\right)-a\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2\pi u\upsilon t{F}_{2}\left(u\right),\end{array}$$
(14)
$$\begin{array}{c}I\left(t\right)=a\phantom{\rule{0.2em}{0ex}}F\left(u\right)\text{cos}\left(2\pi u\upsilon t+\varphi \right),\\ F\left(u\right)={\left\{{\left[{F}_{1}\left(u\right)\right]}^{2}+{\left[{F}_{2}\left(u\right)\right]}^{2}\right\}}^{1/2},\\ \text{tan}\phantom{\rule{0.2em}{0ex}}\varphi =-\frac{{F}_{2}\left(u\right)}{{F}_{1}\left(u\right)}.\end{array}$$