Emission ghost imaging: Reconstruction with data augmentation

Ghost imaging enables two-dimensional reconstruction of an object even though particles transmitted or emitted by the object of interest are detected with a single pixel detector without spatial resolution. This is possible because for the particular implementation of ghost imaging presented here, the incident beam is spatially modulated with a nonconfigurable attenuating mask whose orientation is varied (e.g. via transverse displacement or rotation) in the course of the ghost imaging experiment. Each orientation yields a distinct spatial pattern in the attenuated beam. In many cases ghost imaging reconstructions can be dramatically improved by factoring the measurement matrix which consists of measured attenuated incident radiation for each of many orientations of the mask at each pixel to be reconstructed as the product of an orthonormal matrix $Q$ and an upper triangular matrix $R$, provided that the number of orientations of the mask ($N$) is greater than or equal to the number of pixels ($P$) reconstructed. For the $N<P$ case, we present a data augmentation method that enables $QR$ factorization of the measurement matrix. To suppress noise in the reconstruction, we determine the Moore-Penrose pseudoinverse of the measurement matrix with a truncated singular value decomposition approach. Since the resulting reconstruction is still noisy, we denoise it with the adaptive weights smoothing method. In simulation experiments our method outperforms a modification of an existing alternative orthogonalization method where rows of the measurement matrix are orthogonalized by the Gram-Schmidt method. We apply our ghost imaging methods to experimental x-ray fluorescence data acquired at Brookhaven National Laboratory.