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Ghost Imaging is a fairly recently emerged imaging concept, which in its earliest forms was based on correlating the output of two photodetectors – a high-resolution pinhole detector scanning the light field that has not interacted with the object being imaged, and a bucket detector measuring light that has interacted with said object. While it was originally believed to be an effect with a purely quantum nature, as the first experiments were based on entangled photons, it was later understood to be generalizable to situations with much higher photon densities [Shapiro, 2008].
The revelation that using computer-generated light patterns can enable one to forgo the pinhole detector entirely led to the onset of Computational Ghost Imaging (CGI). Effectively, then, CGI uses an imaging scheme reversed from what is generally used in most applications, using a multi-pixel projector to form light patterns and measuring the object-distorted patterns with a single-pixel detector. The resolution of the acquired image is determined on the resolution of the used light patterns. The major advantage of such a scheme is that in recent years, ultrafast single pixel light detectors have been developed, giving CGI great potential for use in 3D-vision applications.
While initially randomized patterns were used in CGI, it was shown that it is far more efficient to use patterns that form an orthogonal basis, most commonly utilizing the Hadamard matrices, whereby the amount of measurements needed to form a good image of the scene of interest dropped significantly [Shibuya, 2015]. Using patterns based on the Hadamard matrices, however, introduces a dependency of the final quality of the registered image on the properties of the Hadamard matrix used to form the patterns. For instance, the Hadamard matrix required to achieve patterns of a given resolution may simply not exist, or the composition of the Hadamard matrix may introduce anomalies such as a certain pixel standing out of the final image.
In addition to the effects the pattern characteristics have on the final image, the image may also be distorted by the way the image is retrieved from patterns and the respective measurement results from the single-pixel detectors. Namely, for CGI, mainly 2 strategies can be taken for the assembly. One involves weighted addition, and the other assembles an equation system which is then solved. The two methods can lead to identical results, as they do in Differential Computational Ghost Imaging (DCGI), but don’t necessarily do so [Ferri, 2010].
We show, based on simulations, how one can utilize the different order Hadamard matrices in conjunction to one another as well as the Hadamard matrix properties to achieve patterns of a desired resolution in terms of final pixel count, and in which cases the image assembly methods are identical. Furthermore, limitations for further increasing the resolution accessibility will be shown. These results have the potential to directly reduce the number of measurements needed to acquire an image of a certain resolution, and can thus be of great value in CGI applications, ranging from 3D-vision in autonomous cars to optical cryptography.
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