Analysis of Systems for Superimposing Projected Images

Many visualization applications require digital displays and projectors with very high resolution, and one well-known technique to increase resolution is to spatially arrange several projected images in a tiling configuration. Less known is the fact that projectors also allow a very different type of solution, called “wobulation,” which is the superimposition of several projections, each shifted from the others by displacements of a fraction of the pixel size. Such precise positioning can be achieved, for instance, moving the projection elements in time using optical methods. For optimal results this method requires sophisticated pre-processing of the images that are to be projected, taking into account all the combination effects caused by the superimposition, in order to properly produce the desired high-resolution projection. In this work we study the properties of image superimposition systems, and analyze the determination of the optimal signals to be projected and the techniques for their computation. We show that basically the behavior of the superimposition system can be decomposed in two components. The first is linear, and corresponds to the addition of light intensities. The second is defined by the upper and lower limits in light output, and is nonlinear. The first component can be modeled as a linear dynamic system, which is spatially-invariant if identically shifted images are used, and can be evaluated using basic signal processing techniques. This analysis shows potential problems concerning the existence of solutions and stability of the computations. Next, we develop more realistic models, incorporating the nonlinear behavior, which require optimization techniques and algorithms. While the resulting problems are common in optimization, the imaging problem has very large dimensions, with typically millions of variables and constraints. We study solution methods that can exploit the particularities and sparsity of the problems, and that can cope with the fact that the problems may be numerically ill-conditioned. Finally, we consider methods to improve numerical stability and convergence speed, and show numerical results. The numerical results indicate that typically, when the subframes are arranged for modest gains in resolution (e.g., twice the original resolution), the system’s behavior is dominated by the linear component, and the desired gains in resolution are mostly achieved for many typical types of image. On the other hand, the systems designed for obtaining more than three times the original resolution are much more severely limited by the nonlinear component, and for most images it quickly becomes more difficult to improve resolution.