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.

Figure 1

(a) Experimental measurement matrix acquired at Brookhaven National Laboratory. (b) Scan of measurement matrix [corresponding to red dashed line in (a)].

Figure 2

The number of pixels is $P$ = 276. We equate the theoretical measurement matrix to 10 times the experimental measurement matrix shown in Fig. 1. Here we show reconstructions of noise-free data determined with data augmentation, $QR$ and TSVD steps where the relative threshold for the truncated SVD method, ${\kappa }_{\mathrm{SVD}}$, is $1.49\times {10}^{-8}$. (a)–(d) Reconstructions for $N=275,250,200,$ and 100. (e)–(h) Associated scatterplots of reconstructions and the digital phantom. Points that fall on the red line correspond to cases where the reconstruction and true value agree exactly.

Figure 3

We equate the theoretical measurement matrix to 10 times the experimental measurement matrix shown in Fig. 1. The number of pixels is $P=276$. We show reconstructions of simulated noisy data where $N$ = 275 where ${\kappa }_{\mathrm{SVD}}$ varies. (a) Digital phantom. (b)–(i) Reconstructions where ${\kappa }_{\mathrm{SVD}}={10}^{-6}, {10}^{-5}, 5\times {10}^{-5}, {10}^{-4}, 5\times {10}^{-4}, {10}^{-3}$, and $5\times {10}^{-3}$, respectively.

Figure 4

We equate the theoretical measurement matrix to 10 times the experimental measurement matrix shown in Fig. 1. The number of pixels is $P=276$. For our data augmentation method (with $QR$ and TSVD steps), we show RMSE of reconstructions of simulated noisy data as a function of ${\kappa }_{\mathrm{SVD}}$. (a) $N$ = 275, (b) $N$ = 250, (c) $N$ = 200, and (d) $N$ = 150.

Figure 5

We equate the theoretical measurement matrix to 10 times the experimental measurement matrix shown in Fig. 1. The number of pixels is $P=276$. (a)–(d) Reconstructions of simulated noisy data for $N=275,250,200,$ and 100. (e)–(h) Associated scatterplots of reconstruction and the digital phantom. Points that fall on the red line correspond to cases where the reconstruction and true value agree exactly.

Figure 6

Here we show denoised versions of the Fig. 5 reconstructions. (a)–(d) Denoised reconstructions for $N=275,250,200,$ and 100. (e)–(h) Associated scatterplots of denoised reconstruction and digital phantom. Points that fall on the red line correspond to cases where the reconstruction and true value agree exactly.

Figure 7

We equate the theoretical measurement matrix to 10 times the experimental measurement matrix shown in Fig. 1. The number of pixels is $P=276$. We show denoised reconstructions of simulated noisy data. (a), (c), and (e) Reconstructions determined with the method from [14] for $N$ = 275, 250, and 200. (b), (d), and (f) Reconstructions determined with our method for $N$ = 275, 250, and 200.

Figure 8

Experimental setup at the NIST Beamline for Materials Measurement at the National Synchrotron Light Source II at Brookhaven National Laboratory.

Figure 9

An x-ray fluorescence spectrum excited by a monochromatic x-ray beam at 12 keV incident on the physical phantom made with copper wire [see photograph in Fig. 11]. The bucket signal is obtained by summing the counts under the Cu ${\mathrm{K}}_{\alpha }$ and Cu ${\mathrm{K}}_{\beta }$ peaks. X rays that undergo elastic scattering or Compton scattering do not contribute counts to the bucket data.

Figure 10

The computer-designed barcode mask was fabricated by bottom-up gold filling of nominally 20.7-$\mu \mathrm{m}$-deep trenches patterned and etched into the surface of a silicon wafer. The “gold bars in the barcode pattern are actually arrays of 8-$\mu \mathrm{m}$-wide Au-filled trenches and 2-$\mu \mathrm{m}$-wide Si spacers that have been repeated as needed for the bar width. Illumination of the system was defined by stepping the barcode mask through a sequence of 50-$\mu \mathrm{m}$ displacements parallel to the barcode, generating a total of 442 spatial patterns that were captured by a CCD camera. The same sequence of illumination patterns was repeated with the specimen in place for x-ray fluorescence data acquisition. The patterns and the corresponding x-ray fluorescence signals were processed offline for ghost imaging reconstruction.

Figure 11

In the “Simulation” column, we show the digital phantom and reconstructions and denoised reconstructions of simulated data with signal-to-noise similar to experimental data. In the “Experiment” column, we show the physical phantom and reconstructions and denoised reconstructions of experimental data. For all cases the number of pixels is $P=276$. Except for (k), the number of mask orientations is $N=$ 442. In an effort to suppress systematic errors, we aggregate the experimental data so that $N$ is reduced from 442 to 147. To enable analysis with the $QR$ and TSVD steps, we augment the aggregated data so that $N=$ 294. In (k) we show the denoised reconstruction of the augmented data.