# Underwater 2D Image Acquisition Using Sequential Striping Illumination
## 2 Approach of Image Recovery from Sequential Striping Illuminated Images  
### Image Formation Model  
A typical structured lighting technique is illustrated in Figure 1. The projector projects a light plane and sweeps across the object surface. By detecting the image position of the intersection curve of the light plane and the surface, the 3D coordinates of each point on this curve can be computed once the correspondence between the CCD coordinate and the projector coordinate is set up. It should be noted that, in real implementations, stripe lighting refers to a stripe-like illumination whose transverse intensity distributions have a very narrow support. Thus, the light plane is just an approximated model of structured lighting. Instead of the geometrical model of a light plane, we consider the transverse distribution of a light stripe projected onto an object surface, observed from the CCD image coordinate \(I(x)\), where \(x\) is situated along the scanning direction (Figure 1). Thus, we can represent the image formation with respect to a one-dimensional description associated with a single image row along the \(x\) coordinate. Suppose the albedo of the object surface is \(O(x)\). At a moment during the scanning duration, the center of the light stripe is positioned at \(s\). Thus, the one-dimensional image with respect to a change of \(s\) can be expressed as:  
\[f(x,s)=I(x-s)\mathrm{O}(x) \tag{1}\]  
In underwater imaging, however, the forward scattering occurs during light transmission from the object to the camera, whose imaging effect can be expressed by the point spread function (PSF) \(h(x)\), whereas back scattering induces added background component \(b(x)\). Thus, the general image formation is:  
\[f(x,s)=h(x)*[I(x-\tau)O(x)]+b(x,s) \tag{2}\]  
where \(*\) is the convolution operation. We now take the integration operation on both sides of Equation (2) with respect to the variable \(x\). We have:  
\[\mathrm{g}(s)=\int_{\Omega}f(x,s)dx=\int_{\Omega}h(x)*[I(x-s)\mathrm{O}(x)] dx+\int_{\Omega}b(x,s)dx \tag{3}\]  
where \(\Omega\) is the row image domain. Noting the equivalent expressions:  
\[\int_{\Omega}h(x)dx=1\text{ and }\int_{\Omega}h(x)*f(x)dx=\int_{\Omega}f(x)dx \tag{4}\]  
We can formulate Equation (3) as:\[\mathrm{g(s)}=\int_{\Omega}I(x-s)O(x)dx+\int_{\Omega}b(x,s)dx=I(s)*O(\tau)+B(s) \tag{5}\]  
In the above derivations, we have implied that \(I(x)\) is of symmetrical distribution. The form of Equation (5) suggests that the function \(O(x)\) can be obtained by the deconvolution operation once we generate the image \(\mathrm{g(s)}\) by integrating each of the sequentially striping illuminated frame images. Consequently, the albedo image is obtained by the recovered \(O(x)\) row by row. Some limiting factors have to be addressed to apply the model of Equation (5). It is noted that the illumination function \(I(x)\) is actually not the same as that originated from the projector. The function \(I(x)\) is the projected light distribution observed from the view of the camera, whose distribution is relevant to the surface shape and is affected by forward scattering during projection. It implies that the estimation of \(O(x)\) should be implemented through blind deconvolution by Equation (5), for which the exact distribution of \(I(x)\) is unknown from the observation of the camera. Obviously, however, the first illumination distribution from the projector provides a good initial estimate of \(I(x)\). In addition, attenuation during light transmission has to be considered in image formation. We omit this in Equations (1)-(5) because the attenuation factor, \(\exp(-cl)\), where \(c\) denotes attenuation coefficient and \(l\) the transmission length, is a multiplicative scalar that can be merged into the function of \(O(x)\).