## Abstract

A 3D sensing method to retrieve an entire shape from many segmented profiles is described. Image registration is not required in this method. Advantages of this method also include (1) very high integration accuracy, (2) improved robustness, (3) reduced computational time, (4) very low computation cost for the data fusion, and (5) capability of compensating distortions of the optical system at every pixel location.

© 2008 Optical Society of America

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### Equations (7)

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(1)
$$\left[\begin{array}{c}{x}_{g}\\ {y}_{g}\end{array}\right]=\left[\begin{array}{c}\frac{\phi \xb7d}{2\pi}\\ 0\end{array}\right],$$
(2)
$$\left[\begin{array}{c}x+\Delta x\\ y+\Delta y\\ z+\Delta z\end{array}\right]=\left[\begin{array}{cc}{r}_{11}^{\left(p\right)}& {r}_{12}^{\left(p\right)}\\ {r}_{21}^{\left(p\right)}& {r}_{22}^{\left(p\right)}\\ {r}_{31}^{\left(p\right)}& {r}_{32}^{\left(p\right)}\end{array}\right]\left[\begin{array}{c}\phi \xb7\frac{d}{2\pi}\\ \phantom{\rule{.2em}{0ex}}\\ 0\end{array}\right]+\left[\begin{array}{c}{t}_{1}^{\left(p\right)}\\ {t}_{2}^{\left(p\right)}\\ {t}_{3}^{\left(p\right)}\end{array}\right].$$
(3)
$$\left[\begin{array}{c}{x}_{d}+\Delta {x}_{d}\\ {y}_{d}+\Delta {y}_{d}\end{array}\right]=\left[\begin{array}{ccc}{r}_{11}^{\left(c\right)}& {r}_{12}^{\left(c\right)}& {r}_{13}^{\left(c\right)}\\ {r}_{21}^{\left(c\right)}& {r}_{22}^{\left(c\right)}& {r}_{23}^{\left(c\right)}\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]+\left[\begin{array}{c}{t}_{1}^{\left(c\right)}\\ {t}_{2}^{\left(c\right)}\end{array}\right].$$
(4)
$$\{\begin{array}{c}x={a}_{0}+{a}_{1}z\\ y={b}_{0}+{b}_{1}z\end{array},$$
(5)
$$z=\sum _{n=0}^{N}{c}_{n}{\phi}^{n}.$$
(6)
$$R(x,y)=\frac{1}{2}+\frac{1}{4}\mathrm{cos}\left(\frac{2\pi}{{d}_{x}}x\right)+\frac{1}{4}\mathrm{cos}\left(\frac{2\pi}{{d}_{y}}y\right),$$
(7)
$$\{\begin{array}{c}z=\sum _{n=0}^{N}{c}_{n}^{\left(l\right)}\xb7{\left({\phi}^{\left(l\right)}\right)}^{n}\\ x={a}_{0}^{\left(l\right)}+{a}_{1}^{\left(l\right)}z\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\\ y={b}_{0}^{\left(l\right)}+{b}_{1}^{\left(l\right)}z\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\end{array},$$