Profile map

The profile map provides a new way of handling the data acquired by light striping introduced in [III]. For each profile p illuminated by the laser and for every s_nth row i of the $N_1\times M_1$ coordinate set of the image of the camera, the column index j of the stripe seen in the image is measured with sub-pixel accuracy. The profile map is defined as a real valued image

$${\bf j}=\{ ({\bf s},j({\bf s}))\ \vert\ {\bf s}\in S=[1,\ldots ,N]\times [1,\ldots ,P]\},$$ (2)

where $N=\lceil N_1/s_n\rceil$ and P is the number of profiles recorded. The value $j({\bf s})=0$is assigned for locations outside the measurement coverage. The profile map is thus nothing but a convenient representation that supports the measuring technique. Its power in matching tasks is proven in the subsequent sections. Issues concerning multiple stripe observations on a single row and outliers are discussed in [III, VI]. The precision of the measuring is given by the image of sample variances $\hbox{Var}({\bf j})$ obtained by scanning the same view several times and using sample statistics [V]. Examples of real profile maps appear in [III, Figs. 4, 7; IV, Fig. 1; V, Fig. 2a] and synthetic ones in [VI, Figs. 2, 5]. The image of sample variances is illustrated in [V, Figs. 2b-2c].

In our light striping system [III-VI], the right-handed rectangular x,y,z coordinate system is fixed so that the xz-plane is parallel to the plane of the laser sheet and the first profile measured is given the value y=0. The object is moved stepwise with computer control in the direction of the negative y'-axis of a skewed x',y',z' coordinate system defined so that the x'- and z'-axes coincide with the x- and z-axes, respectively [VI, Fig. 1]. We have
 $$\begin{align}{\bf x'}&=(b_{11}{\bf i'}+b_{12}{\bf j}+b_{13})\ast (b_{31}{\bf i'}... ...{22}{\bf j}+b_{23})\ast (b_{31}{\bf i'}+b_{32}{\bf j}+1)^{-1},\notag \end{align}$$
where ${\bf i'}=s_n({\bf i}-1)+1$ is a scaled image of the row indices, ${\bf p}$ is the image of the profile indices, s_p defines the step size of the object movement, and the coefficients ${\bf b}=[b_{11}\ldots b_{32}]^T$ determine the projective transformation between the image plane and the plane of the laser sheet. The skewed frame is rectified by
 $$\begin{align}{\bf x}&={\bf x'}+{\bf y'}x_0,\notag\\ {\bf y}&={\bf y'}\sqrt{1-x_0^2-z_0^2}, \\ {\bf z}&={\bf z'}+{\bf y'}z_0,\notag \end{align}$$
where (x₀,z₀) is the orthogonal projection of the point (0,1,0) of the x',y',z' frame onto the xz-plane. The parameters ${\bf c}=[{\bf b}^T\ x_0\ z_0]^T$ are solved and $\hbox{Cov}({\bf c})$ estimated during calibration. The intrinsic parameters of the laser and camera are assumed known.