Simultaneous imaging and element differentiation by resonant neutron ghost imaging

Abstract

We present a proof-of-principle demonstration of energy-resolved resonant neutron ghost imaging. Based on the resonant absorption dips of different elements, we simultaneously image and distinguish the composition of three differently shaped components of an object. The initial neutron beam is spatially and energy selectively modulated by a series of Hadamard matrix masks of pixel width 100 μm. The spectral intensity transmitted through an object is measured by a ⁶Li glass single-pixel detector. Through integration of the total counts within resonant dips and correlating them with the corresponding Hadamard patterns, isotope-specific images of In, Ag and W objects are obtained at an effective spatial resolution of ~200 μm. Reconstruction algorithms based on compressed sensing or convolutional neural networks can greatly reduce the data acquisition time by ~70% with respect to the full set of 1024 patterns, as well as enhance the image quality. Incorporating ghost imaging into energy-resolved neutron imaging thus has great potential for the simultaneous realization of fine spatial and spectral resolution, which has important value for the noninvasive analysis of material composition and distribution not only in basic research but also in industrial applications.

Graphical abstract

Keywords

ghost imaging / neutron imaging / energy-resolved

1 Introduction

Ghost imaging (GI) is a technique that reconstructs the image of an object via the correlation between the illumination distribution and the intensity transmitted or reflected from the object. In its first demonstration [1], two beams from a parametric down-conversion crystal were used to produce the correlation. One of the photon beams interacted with the object and the total transmitted intensity was collected by a bucket detector (single-pixel detector). Simultaneously, an x−y scanning fiber single-photon detector was used to record the spatial distribution of the other beam. After many exposures, an image of the object was observed in the coincidence counting rate. Equivalent schemes in which a set of predesigned intensity patterns is projected onto or placed after the object in a single beam path are known as computational GI [2] or single-pixel imaging, respectively. These various forms of intensity correlation imaging have already been performed throughout the spectrum from visible to microwave [3] to X-rays [4–9], and recently even with atoms [10], electrons [11], and neutrons [12, 13]. Their potential benefits are wide-ranging and include spatial resolution enhancement [14], dose reduction [4, 6], and robustness in turbulence [15].

Neutron imaging, due to its distinctively high sensitivity to light elements and high penetration, is usually regarded as a non-destructive inspection technique complementary to x-ray and gamma-ray radiography. With the construction of spallation neutron sources, the short pulse and high flux characteristics of the output beam has made energy-resolved neutron imaging feasible, including Bragg edge imaging [16, 17], resonant neutron imaging [18, 19] and polarized neutron imaging [20]. In the epithermal energy range ({10}^{-2}−{10}^2 eV), resonant absorption dips in the neutron spectrum hold rich potential for nuclide specific imaging [18, 19, 21], which, combined with time-of-flight (ToF) measurements, enable a multi-dimensional measurement of a material’s physical or chemical properties.

However, the main difficulty is to simultaneously measure the spatial and temporal data sets with high resolution. One intuitive solution is to exploit the energy resolving capability of a two-dimensional pixelated ToF detector, but this poses considerable technological challenges [22–25]. Intense studies have been conducted in the past decade and much progress has been made. Cutting-edge micro-pattern detectors based on a micro-pixel chamber [26] or a thin-foil gas electron multiplier [27] provide sub-millimeter spatial and sub-microsecond temporal resolution, but still have to strike a balance between spatial and temporal resolution, detection efficiency, and effective count rate. Detectors combined with {}^{10}B doped micro-channel plates and Timepix readout chips [18] can now achieve higher spatial (~55 μm) and temporal (~20 ns) resolutions via neutron transmission spectroscopy at resonant energies even exceeding 10 keV. Based on these detectors, element-specific imaging has been realized for several materials, at a spatial resolution of 150 μm with 25–65 min integration times [18].

An alternative solution for energy-resolved neutron imaging can be provided by computational GI [2]. This avoids the need for pixelated detectors and beam splitting, which is very difficult for GI with particles. Thus, to realize energy-resolved neutron GI, we only need the well-established single-pixel ToF detectors for spectral measurements. The imaging system will be greatly simplified, thus allowing more freedom to choose different types of detectors and readout electronics to optimize the spatial and temporal resolution, detection efficiency and count rate at the same time. Neutron GI has already been realized in the thermal energy range with a reactor neutron source [12], and with improved spatial resolution at a spallation neutron source [13]. In the latter experiment, we also performed a proof-of-principle demonstration of spectroscopic neutron GI with a spatial resolution of 100 μm and a time resolution of 10 μs (corresponding to 0.4% at 1 Å), but the energy bandwidth was minimal. Here we present the first demonstration of energy-resolved resonant neutron ghost imaging (RNGI) in the epithermal energy range, which was also performed at the No. 20 beamline of the China Spallation Neutron Source. To the best of our knowledge, this is the first experiment using RNGI for element differentiation.

2 Method

The experimental setup is shown schematically in Fig.1. Poly-energetic neutrons emitted from the spallation source at a repetition rate of 25 Hz emerge from the end of a 20 mm diameter collimator at a rate of around 10⁷/(cm²·s), and are first passed through a 1.6 mm square aperture then through a mask plate engraved with a series of Hadamard matrix patterns to illuminate an object, after which the total transmitted intensity is measured by a single-pixel detector. The lower left inset shows the beam spot at the mask plane which has a quasi-Gaussian distribution with a full-width-at-half-maximum spot size of 2 mm. The aperture was adjusted to ensure that only one pattern was illuminated in each measurement.

The middle inset is an example of a modulated neutron spot immediately behind the mask plate, which was composed of 1024 Hadamard matrix patterns, each containing 32 × 32 pixels of size ~100 μm × 100 μm which determines the theoretical spatial resolution of the GI reconstruction. The plate was fabricated by etching the matrices on a silicon wafer to a depth of (310 ± 10) μm [13], which were then filled with a mixture of powdered indium, silver and tungsten in equal proportion by weight. Of course, the powder ingredients should correspond to the isotopic composition of the object because no single material has a sufficiently broad absorption energy spectrum in the epithermal energy range, so certain prior knowledge of the object is necessary. The object (top inset) consisted of a stack of three 0.5 mm thick metal plates composed of pure In, Ag and W, respectively, stenciled with the letters N, G, and I of stroke width 0.5 mm. These three elements were selected on account of their resonant absorption in the energy range of interest. The object was placed 1.2 cm behind the mask, the closest possible, to minimize blurring of the modulation patterns, which were projected one by one onto the sample by means of precisely controlled translation stages. A complete set of 1024 spectra was recorded by a 6 mm thick {}^6Li enriched glass scintillator coupled to a photomultiplier tube (Hamamatsu Model R329-02). However, only the first 512 modulation patterns were used to reconstruct the images, since the matrices were already rearranged beforehand in an optimum order [28], and then compressed sensing (CS) [29, 30] was adopted. A convolutional neural network algorithm was also used [31] to further improve the image quality.

From ToF measurements, the transmission spectra of the object through three randomly chosen Hadamard patterns were plotted as a function of energy, as shown in Fig.2, where each spectrum is normalized with respect to that of the neutron beam with the mask plate removed. Here we can clearly see the resonant absorption dips, and also observe that the transmission is almost identical for the three patterns except for specific dips, which are marked with blue, orange and green windows, for {}^{115}In 1.46 eV, {}^{109}Ag 5.2 eV and {}^{186}W 18.8 eV, respectively. The other two resonant absorption dips for {}^{115}In at 3.82 eV and 9.07 eV are not considered because of their much weaker absorption compared to that at 1.46 eV [21]. The variations in the resonant absorption dips for {}^{115}In 1.46 eV, {}^{109}Ag 5.2 eV and {}^{186}W 18.8 eV indicate the object’s different transmission response under specific illumination patterns, and the spectra can be expressed as

(1)s^i(E)={\displaystyle \sum _{m,n}{O}_{mn}(E)\cdot M_{mn}^i(E)\cdot T_{mn}(E),i=1,\cdots ,M,}

where O_{mn}(E) is the neutron distribution at the mask plane, M_{mn}^i(E) the ith mask pattern, T_{mn}(E) the object transmission, M\le 1024 the number of measurements, and m,n are the integers indicating the spatial dimensions of the image. Note that M_{mn}^i(E) only generates a certain spatial modulation within the absorption dips, so the variation within each spectral window can only come from the same isotope in the object. This forms the basis of RNGI. We can thus integrate the spectra to yield the bucket intensities for each isotope,

(2)b^{ij}={\displaystyle {\int }_{E^{j_1}}^{E^{j_2}}{s}^i(E)\mathrm{d}E,}

where the integral limits E^{j_{1,2}} for the jth isotope are chosen according to the corresponding transmission window, specifically, 1.25−1.66 eV for In, 4.8−5.6 eV for Ag, and 17.48−19.46 eV for W. The variations in the other dips or out-of-resonance energy regions are negligible because of their much smaller attenuation cross-section. We also set an extra out-of-resonance window (gray band) near 5.9 eV for comparison.

The image reconstruction of RNGI is performed by solving for the object transmission T_{mn}(E) at resonance from Eq. (1). This can be achieved, with the traditional GI algorithm, by correlating b^{ij} with the mask pattern M_{mn}^i [1]. Note that Eq. (1) can also be regarded as the linear sampling of T_{mn} under the matrices M_{mn}^i. By utilizing compressed sensing [29] we can reduce the number of measurements M required and improve the image quality. The CS optimization can be translated to a convex optimization problem, and here we adopt the augmented Lagrangian and alternating direction based total variation algorithm TVAL3 [32] to minimize the cost function:

(3)\underset{t}{min}{\displaystyle \sum _i{\Vert D_{i}t\Vert }_2+{\displaystyle \frac{\mu }{2}}{\Vert At-b\Vert }_2^2,\mathrm{s}.\mathrm{t}.t\ge 0,}

where {\Vert x\Vert }_2 is the l_2 norm of vector x, t is the flattened vector of the estimated object transmission, b is the bucket intensity integrated for a given isotope, A is the sub-Hadamard matrix of size M\times 1024,\mu >0 is the penalty parameter of the quadratic fidelity term, D_{i}t is the discrete gradient vector of t at position i, and \sum {\Vert D_{i}t\Vert }_2 is the total variation. The fidelity term {\Vert At-b\Vert }_2^2 minimizes the difference between the estimation and measurements while the total variation ensures the image noise removal and edge preservation at the same time [32]. The non-negative constraint is applied to the transmission t, and the TVAL3 solver iteratively optimizes t to obtain a faithful reconstruction. The penalty parameter expresses the relative importance of the fidelity term and total variation, which is found to be a major factor in the reconstruction. The value of \mu can be determined from system simulations that take noise statistics, detector structure and response into account; here it is iteratively optimized in the continuum mode of TVAL3.

To remove the effect of the original neutron distribution O_{mn} (E), we acquired the data sets in two steps. After inserting the object, the bucket intensities were again measured for each modulation pattern then normalized with respect to the reference spectra, and were additionally normalized for intensity fluctuations with the neutron beam data continuously recorded by the spallation source control monitor.

3 Results and discussion

Only the data for the first 512 frames of the optimally ordered set of 1024 Hadamard matrices were taken to reconstruct the images, and the corresponding normalized bucket intensities b_n^{ij} (j = 1−4) within the above four windows are shown in Fig.3(a). The colored shadows indicate the three-standard-deviations (3\sigma ) range for each series of b_n^{ij}, corresponding to the depth of modulation of the object in each resonant energy band and are closely related to the contrast-to-noise ratio (CNR) of the reconstructed images; the larger the 3\sigma, the higher the CNR of the image. Also, the 3\sigma of the out-of-resonance energy (5.9 eV) represents the noise level of this experiment, which is normalized to 1.00 as the reference.

The corresponding ground truths and their binarized images are shown in Fig.3(b)−(d) and (e)−(g). The images reconstructed with the traditional GI algorithm and CS for each series of b_n^{ij} are shown in Fig.3(h)−(j) and (k)−(m), respectively. Isotopic distributions are extracted from the stacked object (see top inset in Fig.1). However, the edges of the images are quite noisy. The spatial resolution of RNGI depends upon the pixel size of a Hadamard pattern, the sensitivity of the bucket detector, and the noise level. The pixel size is usually the key factor since it determines the highest spatial frequency of the object that can be measured. Using a Hadamard mask plate of 100 μm pixel size we can achieve a theoretically spatial resolution of ~100 μm [13]. We can see that the former are barely decipherable, while the latter are quite clear. To quantify the reconstruction quality, we adopt the CNR defined as

(4)\mathrm{C}\mathrm{N}\mathrm{R}={\displaystyle \frac{⟨G_1⟩-⟨G_0⟩}{\sqrt{{\sigma }_1^2+{\sigma }_0^2}}},

where G_1, G_0 are the reconstructed pixel values when the ground-truth object transmissions are 1 and 0, respectively, and {\sigma }_1, {\sigma }_0 are the corresponding standard deviations of G_1,G_0. The binarized images in Fig.3(e)−(g) are used to calculate the CNR in this paper.

The CNR of the images reconstructed by the GI algorithm decreases at higher resonant energies, i.e., in Fig.3(h) and (j), from 0.57 for the letter “N” ({}^{115}In 1.46 eV) to 0.35 for letter “I” ({}^{186}W 18.8 eV), respectively. This is attributed to the decreasing natural abundance of {}^{115}In (95.7%), {}^{109}Ag (48.2%) and {}^{186}W (28.4%), which reduces the depth of modulation and the number of absorbed neutrons, thus the ratio of neutrons to gamma ray noise is also reduced. In addition, the respective CNRs of the images reconstructed by the GI algorithm (0.57, 0.4, 0.35) are smaller than those reconstructed by CS (0.87, 0.85, 0.81). This shows the great ability of CS to eliminate noise. Precise determination of the weighting of each factor is important for quantitative analysis or crystallographic characterization [18] but is beyond the scope of our proof-of-principle experiment. In our case, the variation of the bucket intensities is the result of all the above factors. The fluctuation of the out-of-resonance energy (E_n=5.90 eV) reflects the noise level and we take this as a reference ({\sigma }_r=1). The normalized standard deviation {\sigma }_r then describes the signal-to-noise ratio (SNR) of the bucket intensities, and is a good estimation of the CNR, i.e., the {\sigma }_r for {}^{115}In, {}^{109}Ag and {}^{186}W are 2.37, 1.52, and 1.09, respectively.

We note that to scan all the modulation matrices is extremely time consuming, which is the most severe limitation of GI. For instance, to obtain good data statistics and minimize the effect of the spectral jitter, we set an exposure time as long as 60 s for each pattern, as shown in Fig.3, so the total acquisition time was about 10 h (including mechanical movement time) for the first 512 frames of the optimally ordered set of 1024 Hadamard matrices.

Compressed sensing offers the possibility to shorten the acquisition time by reducing the number of measurements required, which does not compromise the image quality and so is widely used in GI [29, 33]. According to the Nyquist criterion we need at least N^2 measurements to generate a faithful reconstruction. We define the sampling rate as S=M/N^2, where M is the number of measurements used for image reconstruction and N = 32 is the dimension of the square image. The order of the original Hadamard matrices was reorganized according to \beta =\Vert H_a{H}^i\Vert descending order, according to the l_1 norm of each row H^i under the Haar wavelet transform H_a [28].

The images of “N” reconstructed with sampling rates of S = 1/8, 1/4, 1/2 (corresponding to the number of measurements 128, 256 and 512, respectively) and their corresponding cross-sections are shown in Fig.4. The CNR increases with higher sampling rate, from 0.79 for S = 1/8, to 0.82 for S = 1/4, and to 0.89 for S = 1/2, as shown in Fig.4(a)−(c). This is consistent with the decrease of the background noise (minimum value of the cross-sections) in Fig.4(d)−(f) and can be explained by the fact that the reordering of the matrices puts patterns with relatively low spatial frequencies at the beginning of the sequence; while the later patterns of finer structure (higher spatial frequency) give incremental information of the object in reconstruction.

Another possibility for reducing acquisition time with enhanced image quality is to use deep learning [34], which processes data statistically and automatically using multilayered artificial neural networks [35–38]. As described by Lyu et al. [36], superior quality images can be generated with deep learning compared to traditional CGI and CS algorithms, even at a significantly reduced sampling rate of S ~1/20. To further enhance the quality of our images, we utilized the multi-level wavelet convolutional neural network algorithm (MW-CNN) [31], which gives a better balance between the computational accuracy and efficiency. The results are shown in Fig.5, where the number of measurements used for reconstruction is 300 (S = 29.3%), acquired over a total duration of ~6 h.

We can see that the image quality is significantly enhanced in Fig.5 with less noise. The width of the strokes of the letters is close to that of the ground truth object (500 μm), the size of each image box is 32 × 32 pixels and that of each pixel 100 × 100 μm², thus the pixel resolution can be ~100 μm. However, the actual spatial resolution should be calculated by measuring the half-width of the slope of the cross-section near the edge of the object [marked by the red rectangle boxes in Fig.5(d)−(f)]. As shown in Fig.5(g)−(i), the spatial resolution is estimated to be approximately 2 pixels (200 μm). This result is caused by the blurring effects of the uncollimated beam, gamma ray interference (gamma-rays induced by neutron activation will be included in the detector output), randomly scattered neutrons, etching imperfections, and other experimental issues. We can also see that deep learning is advantageous for obtaining high image quality. For example, for the isotopes {}^{115}In, {}^{109}Ag and {}^{186}W, the CNR of their images reconstructed by CS is also rather poor, as can be seen in Fig.3(k)−(m), but when CNN is employed their CNR values much improved, as shown in Fig.5(a)−(c).

With regard to spatial resolution, in traditional neutron imaging this is mainly determined by the pixel size of the detectors. Conventional pixelated detectors based on {}^6Li possess a rather large pixel size (3 mm × 3 mm in Ref. [23]), but the {}^6Li glass scintillator detector in our experiment was chosen because of its high count rate and efficiency. In our RNGI setup, the spatial resolution is independent of the pixel size of the detector and mainly depends on that of the mask pattern. Furthermore, efficient algorithms and a priori knowledge of the object also enable reduction of the neutron irradiation, which is attractive for applications where the total incident energy on the sample is of major concern. Besides resonant neutron imaging, GI can also be incorporated into other energy resolving protocols, such as Bragg edge imaging and polarized neutron imaging, to simplify the system as well as optimize performance.

4 Conclusion

To summarize, based on resonant neutron GI we have experimentally demonstrated the simultaneous imaging and element diagnosis of the different components in an object by their corresponding relative transmissivities at their resonant absorption dips in different epithermal energy bands. The neutron beam was modulated spatially within the resonant absorption spectrum of various metals ({}^{115}In, {}^{109}Ag and {}^{186}W) by a specially designed Hadamard mask plate, with a {}^6Li glass scintillator as a single-pixel detector. Image reconstruction algorithms based on CS and convolutional neural network algorithms could reduce the data acquisition time by ~70%, while greatly improving the image quality. Combining GI with time-of-flight techniques not only avoids the need for complex pixelated ToF detectors, but also enables the simultaneous optimization of spatial and temporal resolution, count rate, and detection efficiency for energy-resolved neutron imaging. With improved single-pixel detectors of higher sensitivity, neutron GI could even be performed with portable neutron sources of much lower flux. Our method thus has great potential for the noninvasive analysis of material composition in all fields of science as well as in industrial applications, such as in-vivo imaging of the vascular system of plants, studies of geological samples, diagnosis of storage batteries, and product inspection.

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