Ghost Difference Imaging Using One Single-Pixel Detector

How can a target’s differential image of multiple components at a certain degree of freedom (DOF) of light (e.g., wavelength, polarization, and position) be obtained? Previous schemes often need two steps, i.e., first collecting each component of the target simultaneously with multiple detectors or sequentially with a single detector, then performing the difference operation in the digital postprocessing. Based on the principle of computational ghost imaging (GI), we here take advantage of the natural complementarity in space to design an illumination mode, which is no longer the spatiotemporal fluctuation of photon number (light intensity) but the photons with different components under the same DOF. By applying this engineered illumination to computational GI, termed ghost difference imaging (GDI), we can achieve differential imaging of multiple components of a certain DOF in a single-round acquisition using only one single-pixel detector with no extra digital-subtraction operations but higher SNR. Both multiwavelength-difference GDI and position-difference GDI are well demonstrated in simulations and experiments. Further, the GDI scheme reduces the number of sampling times for differential imaging while also increasing photons’ utilization. Our work, therefore, provides an insight for GI modality, along with a straightforward and low-cost update of the light path, which is suitable for all computational GI systems using the digital micromirror device.

Figure 1

(a) Original random illumination in CGI modality: spatial fluctuation of photons and light intensity; “+1” means the corresponding area is illuminated (filled with photons) while “0” means blocked (no photons). (b) Complementary illumination mode in our GDI scheme: spatial fluctuation of photons with different components under the same DOF; ${\mathrm{DOF}}_1$ and ${\mathrm{DOF}}_2$ represent two mutually orthogonal components under a specific DOF (e.g., wavelength, polarization, and position).

Figure 2

Schematic diagram of three schemes for obtaining the differential ghost image: scheme 1, use a SPD to sequentially collect the ghost image of each component and then perform digital subtraction (low-cost but requiring multiple-round acquisitions and digital operations); scheme 2, use multiple SPDs to simultaneously acquire images of each component and then perform digital-subtraction operations (single-round acquisition but high-cost, and requiring digital operations); scheme 3 (GDI scheme), use a SPD to directly obtain the differential ghost image with the complementary illumination mode (low-cost, single-round acquisition, and without extra digital operations).

Figure 3

(a) The flip of DMD micromirror: the deflection can lean towards the light source (on, “1”) or deviate from the light source (off, “0”) in the projection system, causing the pixel to be bright or dark on the surface of the object. (b) Schematic diagram of the configuration to generate a complementary illumination: light sources of different components under the same DOF (e.g. wavelength, polarization, and position or size) are incident on the DMD from different paths.

Figure 4

Classic method to deal with negative numbers in Hadamard-based illumination patterns: each original Hadamard pattern $H_i(x,y)$ (a) is divided into two binary patterns $H_i^{+}$ (b) and $H_i^{-}$ (c) according to the positive and negative.

Figure 5

Generation of the compound Hadamard lighting pattern for multiwavelength-difference GDI (a): filling the black areas in the original Hadamard lighting pattern at ${\lambda }_1$ (b) with the complementary illumination pattern at another wavelength of interest ${\lambda }_2$ (c).

Figure 6

(a) A possible optical setup of position-difference GDI scheme: the light source illuminates onto the DMD, and two complementary illumination patterns will be generated along the two paths, then two slightly different imaging lens groups project the two illumination patterns onto the target plane at slightly different magnifications. A beam splitter superimposes two illumination patterns together inherently. $M$, mirror; $L_1$ and $L_2$, two lens group; BS, beam splitter; $T$, target. Generation of the superimposed Hadamard lighting pattern for position-difference GDI ($128\times 128$ pixels) (b) according to (a): filling the black areas in the original Hadamard lighting pattern ($128\times 128$ pixels resize to $256\times 256$ pixels harnessing the BLIM) (c) with the complementary lighting pattern of a different size ($128\times 128$ pixels resize to $254\times 254$ pixels using the BLIM, then zero padding to $256\times 256$ pixels) (d). (e) The spatial dithering strategy [27, 28] binarizes the gray illumination pattern, in which $3\times 3$ subpixels represent a pixel of the gray illumination pattern. As a simplified equivalent optical setup compared to (a), the binary compound illumination pattern is directly loaded onto the DMD or projection system.

Figure 7

Schematic diagram of the experimental setup of the GDI scheme using the Hadamard-based illumination. Note that the CGI modality allows the single-pixel detector to not stare directly at the target scene (that is, the path from the target scene to the single-pixel detector has no lens and allows the existence of scattering media or turbulence).

Figure 8

Numerical results ($128\times 128$ pixels) of the wavelength-difference GDI scheme: (a1) the color object; the ghost images of the target at three channels R (a2), G (a3), and B (a4), respectively; (b1)–(b3) the engineered Hadamard lighting patterns for wavelength-difference GDI at R-G (c1), G-B (c2), and 2/3G-1/2(R+B) (c3), respectively. This simulation case shows that the multiwavelength-difference GDI scheme has the potential to be applied to the imaging and extraction of characteristic substances of interest.

Figure 9

Experimental results ($64\times 64$ pixels) of wavelength-difference GDI scheme: the imaging results of the first scene (a1) using the original Hadamard-based lighting patterns at white (W) light (a2), R (a3), G (a4), and B (a5) band; (b) is the full-color ghost image of the scene captured by a full-color single-pixel camera using three of the same SPDs. For a fair comparison, (b) is used as the simulation object to verify our GDI scheme as well. (c1)–(c3) are the simulation results of the differential ghost images using the GDI scheme while (d1)–(d3) are the corresponding experimental results at G-B, G-R, and 1/2(G+R)-B using corresponding compound lighting modes, respectively; The experimental results of multiwavelength-difference imaging of the second scene (e1) at white light (e2) and 1/2(G+R)-B (e3). In this case, the color object can be extracted from a monochromatic background.

Figure 10

Results of the position-difference GDI for edge detection. Numerical simulations for target (a) “Rice” ($128\times 128$ pixels) at different values of $\mathrm{\Delta }\mathrm{pixel}$ (b1)–(b4) and their absolute values (c1)–(c4) (i.e., the numerical results of edge detection). Experimental imaging results and their absolute values for target (d) ($64\times 64$ pixels) at $\mathrm{\Delta }\mathrm{pixel}=4$ (e1),(e2) and $\mathrm{\Delta }\mathrm{pixel}=6$ (e3),(e4), respectively. As a comparison, (f1)–(f4) are the noise-free simulation results using the position-difference GDI scheme corresponding to (e1)–(e4).

Figure 11

A set of results of the adaptive compressive GDI scheme using a “zig-zag” mask at different sampling rates. Ghost images in the first row: dual-wavelength-difference imaging at G-B; ghost images in the second row: position-difference imaging at $\mathrm{\Delta }\mathrm{pixel}=6$. Note that we do not adopt additional compression sampling or other optimized algorithms here, so there is no additional time cost or computational burden.

Figure 12

Numerical results of the comparison concerning SNR: (a) standard image “Mandrill” ($128\times 128$ pixels); (b) the ground truth of the differential image at R-G; (c) SNR curves of the three cases as the noise power changes; each value is calculated 50 times to get the error bar.

Figure 13

Numerical results of the comparison concerning image quality: (a) PSNR curves and (b) SSIM curves of the two schemes (GDI scheme and the CGI scheme with the digital subtraction) with the change of relative noise power, respectively. Each value is calculated 50 times to get the error bar. (c) A set of reconstructed dual-wavelength-difference ghost images of the two schemes at different noise power.