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Signal-to-noise ratio is a key factor in lensless imaging, particularly for low diffraction signal experiments in the single shot regime. We present our recent study of the noise impact on holography with extended references. Experimental data have been measured in single shot acquisition using an intense coherent soft X-ray high harmonic source. The impact of hardware and software noise under various detection conditions is discussed. A final comparison between single shot and multi-shot regimes is given.
© 2016 Optical Society of America
1. Introduction
In the last decade, lensless imaging has shown spectacular advances. Researchers have demonstrated the capability of imaging nano-scale objects with nanometer scale spatial resolution and on femtosecond time scales using coherent diffractive imaging (CDI) or holographic techniques. Powerful free-electron laser (FEL) sources are the best candidates to answer exciting challenges, such as tracking in real space and time the ultrafast motion of electrons, atoms and nanoscale objects. Thanks to the remarkable progresses in ultrafast laser sources, high order harmonics (HH) sources have proven to be good alternatives for lensless imaging experiments. Due to the low output of HH sources, first demonstrations of CDI on HH sources required hours of signal accumulation [1]. However, thanks to femtosecond laser developments, single shot femtosecond nanoscale imaging is now available using intense table-top HH source [2]. Subsequently, single-shot femtosecond X-ray holography with extended references has also been demonstrated using an HH source [3]. High resolution lensless imaging with HH sources is limited by the amount of photons (i.e. the dose), and requires the investigation of techniques that provide a sufficient robustness to noise. This is especially relevant for transient ultrafast nanoscale imaging, as signals tend to be very low while accumulation is not always possible [4]. Imaging techniques have various sensitivities to signal to noise ratio impact on the reconstruction process especially when the information tends to be low or sparse.
In this work, we analyze an extension of Fourier Transform Holography (FTH) to the case of extended references (HERALDO) in the low signal regime. Holography [5] is inspired by the idea of “full recording”: the spatial amplitude and phase of the sample are encoded in the interferences between light diffracted by the object and by a reference. Then, a direct and non-iterative mathematic transform is required to reconstruct the object image. This makes holography very robust to noise errors. The direct in-line scheme proposed by Gabor is now applied to X-ray imaging, reaching sub 50 nm spatial resolution [6]. In Fourier transform holography, the field diffracted by a point like reference in the vicinity of the sample interferes in the far field with the diffraction from the sample. A simple Fourier transform then gives directly the object complex image [7]. In HERALDO, the image reconstruction is based on a differential operator applied to the registered hologram [3,8–10 ]. Sharp edges of the extended references act as Dirac functions ensuring a high spatial resolution. The resolution is no longer limited by the reference size (like in FTH) but by the quality of its edges, so one can increase the diffraction signal without affecting the resolution. However, sources of noise, coming from the diffracted photons or due to the detector itself, can have an impact on the image reconstruction process. An analysis of the photon noise influence in the HERALDO technique can be found in the work of M. Guizar-Sicairos [11]. The findings, in photon-limited statistics, are that the HERALDO technique is robust to the photon noise. The bulk of noise contribution by the extended reference is filtered during the reconstruction procedure. However, in many experiments, the signal-to-noise ratio of the diffraction patterns is not only limited by the photon statistic, but includes also a noise contribution from the detector. This can be particularly critical for low photon energy, low scattering efficiency single particle imaging at FELs or HH beamlines. Using our soft X-ray HH beamline [12] in the single and multi-shot regimes, we have investigated the influence of detection noises on the image reconstruction and the best achievable spatial resolution.
2. Experimental arrangement
For a quantitative comparison, all the measurements have been conducted using the same test object. It consists in a geometrical pattern of 1 μm x 1 μm in size (see Fig. 1 ) with a significant level of details. Note that in the low signal regime, the weak contribution to the hologram from each detail of the object complicates significantly the possibility of reconstruction and the spatial resolution accessible [13, 14 ]. Various references (i.e. the squares in Fig. 1) are etched next to the test object at a distance larger than the minimum holographic separation (about 1µm, in accordance with the holographic separation condition [8]) so that the total pattern “object + reference” is contained in a box of about 3 µm x 3 µm. Two HERALDO configurations have been studied: two slits (one dimensional reference) or two squares (two dimensional references). For FTH, two pinholes of about 100 nm diameter act as holographic references. For all techniques, the diffraction patterns are obtained using the same harmonic beam in the femtosecond single-shot regime. The experimental setup is described in our previous work [2,3,12 ]. Briefly, the soft X-ray harmonic beam, monochromatized at a wavelength of 32 nm, is focused on a 5 µm diameter spot. The sample is illuminated with about 109 to 1010 photons in a single shot with pulse duration of 20 fs. The experimental data were acquired with a back-illuminated cooled CCD PI-MTE from Princeton Instruments, 19 mm after the sample. The CCD has two readout frequencies, 100 kHz and 1 MHz. The pixel size is of 13.5 µm. The hardware binning option enables the possibility to combine charges from adjacent pixels into a single effective pixel during the readout process. At the photon energy used for this experiment (~40 eV), taking into account the quantum efficiency (~40%), one photon hitting the CCD chip creates about 4 photon-electrons. With the amplifier gain setting and the 16 bits dynamics of the CCD, these photo-electrons are converted approximately to the same number of counts.
Fig. 1 (left) SEM image of the geometric grid object with two square references. White scale bar: 500 nm. (right) Corresponding experimental single-shot diffraction pattern (log. scale).
3. Impact of detector noises and reconstruction filters in HERALDO
In our experiment, the signal is not photon-limited due to the low photon energy and the modest flux. As a consequence, the detector noise has a large contribution to the noise in the hologram. In general, noise can be separated into four independent components according to their nature: 1) Photon noise, directly related to the diffraction signal and obeying a Poisson distribution with the signal-to-noise ratio , where N is the photon number; 2) Readout noise, related here to the CCD camera readout quality, which is the uncertainty introduced during the process of quantifying the electronic signal. The readout noise mainly arises from the on-chip preamplifier. It is characterized by its standard deviation (or its variance ); 3) Dark noise (also called dark current), introduced by the thermally generated electrons within the silicon layers of the CCD. The dark current describes the statistical variation of the thermal electrons at a given CCD temperature and a given exposure duration, obeying also a Poisson law. It is characterized by electrons/pixel/s; 4) Parasite light noise, due to scattering during the light transport from the source to the object.
Usually, with a low diffraction signal, the noise is dominated by the readout noise, which is called readout-noise limited; with a high diffraction signal, the photon noise is dominant, which is called photon-limited. Here the number of electrons generated by one photon is comparable to the readout noise at 1 MHz. As a consequence, the measured diffraction patterns are mainly influenced by the readout noise and the photon noise. The SNR of the diffraction pattern is then
Overall, we distinguish in the holograms two regions: the region of low spatial frequencies is photon limited and the region of high spatial frequencies is readout-noise limited. We will present here different strategies to optimize the image reconstruction process in such context.
HERALDO requires application of a linear differential operator associated to the reference shape in object space. This step, in practice, is realized by a point-by-point multiplicative filter, , in the Fourier domain (the hologram), at the p and q pixel coordinates. Then, the object image is reconstructed by applying an inverse Fourier transform on the filtered diffraction pattern. is a high-pass filter, which also amplifies the high spatial frequency region, dominated by the noise. An additional low-pass filter is used to eliminate the amplified readout noise. In the following example, we show that the low-pass filter is essential to the HERALDO reconstruction process, especially with diffraction patterns significantly influenced by noise.
The test object used in this study (Fig. 1 left) contains two squared references. In the HERALDO process, these two squares can provide eight independent reconstructions (associated with each of the eight corners) in a single hologram acquisition. Figure 1 (right) represents a typical single-shot diffraction pattern acquired with a readout frequency of 100 kHz and within a window size of 600 x 600 pixels with 2 x 2 hardware binning ratio. The measured diffraction signal is ~6 x 107 photons. The noise around the diffraction signal is here dominated by the readout noise, which leads to a background level of 14 counts and a noise level of 4 counts, measured in the red square (Fig. 1 right).
Figure 2 presents the effect of the low-pass filter during the reconstruction process. When no low-pass filter is used (top line), the high spatial frequencies of the hologram are significantly amplified after applying the multiplicative filter. The reconstructions are strongly affected by the noise. When applying a low-pass filter, whose diameter is too large (800 pixels, not shown) to eliminate all the readout noise, the object is reconstructed but with a low quality. Applying a suitable low-pass filter (bottom line), with a diameter (400 pixels) small enough to eliminate most detection noise at high spatial frequencies, the object is then clearly reconstructed. To quantify the noise contribution, we can use the power SNR of the reconstructed image [11]:
Fig. 2 Influence of the low-pass filter during the reconstruction process. (a,d) Multiplicative filter without (a) and with super Gaussian (order 3) low-pass filter (d: 400 pixels diameter). (b,e) Results of the multiplication (in Fourier space) of the measured diffraction pattern by the HERALDO multiplicative filter (a,d respectively). (c,f) Inverse Fourier transform of (b,e) respectively, giving the object reconstructions. While (c) is drowned in noise, the eight independent reconstructions in (f) are within the green and the yellow squares (each associated to a square reference, see text for details). The fact that the reconstructions in the yellow square are of low quality is due to a misalignment of the XUV beam (the corresponding reference was not sufficiently illuminated). Note that the multiplicative filter is slightly tilted to agree with the diffraction axis.
SNRr is the ratio between the signal energy and the noise energy presented in the reconstruction image. The signal energy is calculated by integrating the signal inside the object region No (blue square in Fig. 2), and the noise energy is the multiplication of No and its variance calculated in the region without signals (white square in Fig. 2). Here, the noise energy is the total contribution of photon noise and readout noise. Note that SNRr only accounts for the statistical noise and does not include the effects of resolution loss. In particular, SNRr does not integrate the local modulations of the hologram, i.e., the fringes.
Figure 3 emphasizes the effect of the low-pass filter diameter. We used the diffraction pattern presented in Fig. 1 and varied the filter diameter from 900 to 200. While an image reconstruction can be obtained without the low-pass filter, Fig. 3 shows that optimizing its diameter leads to a better contrast. This is a direct consequence of the competition between different factors. First, the smaller the filter diameter is, the more the SNRr at high spatial frequencies (region where the readout noise tends to dominate) is reduced. However, the measured contrast of the image, related to the resolution of the reconstructions, decreases for small values, meaning that the spatial resolution does not depend only on the SNRr. Note that the smallest filter diameter imposes a resolution limit of 113 nm, close to the period of the three horizontal slits of the object (190 nm), which explains the loss of resolution when the filter diameter gets smaller.
Fig. 3 (a) Evolution of the reconstruction SNRr and contrast with the low-pass filter diameter. (b) Illustration contrast measurements. For each filter diameter, a lineout of the object reconstruction is taken as shown in the inset. The contrast is the mean value of the successive peaks contrasts. (b) was obtained for a filter diameter of 400 pixels.
4. Effect of the readout frequency and hardware binning
We have shown that detector noises are a major constraint for low-signal experiments. However, reconstructions can be improved by applying either software or hardware procedures. In this section, we present experimental results and analysis of different signal to noise ratio improvement strategies during the detection stage. We first study the evolution of the noise with the CCD readout frequency. The diffraction pattern recorded from the object presented in Fig. 1(d) with 1 MHz and 100 kHz readout frequencies are shown in Fig. 4 . The two diffraction patterns have equivalent signals (4.8x106 at 1 MHz and 5x106 at 100 kHz). The other conditions are similar. After applying a super Gaussian filter, we clearly see the influence of readout noise in regions where the detected signal is low. This effect is reduced when working at 100 kHz. The background level, which can be adjusted by the CCD settings, is of around 30 counts in both cases. The noise level is 10 counts and 4 counts for 1 MHz and 100 kHz respectively. The SNRr is increased by a factor of about 2 for 100 kHz reconstructions, and the resolution is clearly improved. Thanks to the reduced noise level, the reconstructions associated to the second reference, which are difficult to be resolved due to bad illumination of the reference, are better reconstructed (Figs. 4(a) and 4(b)). In this configuration, we can estimate the relative influence from the photon noise and the readout noise in the reconstruction. It turns out that, for a readout frequency of 1 MHz and a large filter diameter, the influence of the readout noise is larger than that of the photon noise. An optimal reconstruction should then find a compromise between image quality and resolution as illustrated in Fig. 3.
Fig. 4 Effect of the CCD readout frequency. (a,b) Single shot diffraction pattern from the object of Fig. 1(d) with a 1 MHz and 100 kHz respectively, after applying the HERALDO multiplicative and super Gaussian (320 pixels diameter) filters. In inset, the corresponding reconstruction from one of the reference.
Another possibility to increase the signal-to-noise ratio, when the oversampling is larger than required, is to use the hardware binning option to increase the effective pixel size. In the ideal case, the SNR enhancement is equal to the binning ratio. Figure 5 shows the diffraction patterns for 1 x 1, 2 x 2 and 3 x 3 binning ratios of a test object (a grid similar to Fig. 3 with a single square reference that can provide three independent reconstructions). When no binning is applied, the weak diffraction signal associated with the object is buried in the noise (the white squares in the bottom right corners of Figs. 5(a)-5(c) show a zoom into the red squares of the diffraction pattern). When the binning factor is increased, more signal is extractable from the high frequencies content of the hologram, even when the number of photons on the detector is smaller. Keeping the same filter equivalent diameter for the three binning factors, the reconstruction SNRr is improved by a factor three, and the resolution is improved in general. Figure 6 shows the object reconstruction from the holograms presented in Fig. 5 (with 1 x 1 and 3 x 3 binning factors) and using optimum algorithm parameters for each case (low pass filter diameter of 400 pixels and 200 pixels, respectively). The three horizontal bars and the three small holes that are not reconstructed when no binning is used are clearly visible with binning. We conclude that a proper hardware binning can clearly improve the reconstruction quality. However, there is a binning ratio limit given by the fact that the sampling ratio of the diffraction pattern has to be respected. The other advantage of hardware binning is to reduce the long readout time when using 100 kHz readout frequency.
Fig. 5 Effect of hardware binning. Single shot diffraction pattern from the object in the inset (total size: 2 x 2 µm) for 1 x 1 (a), 2 x 2 (b) and 3 x 3 (c) hardware binning factor. The zoom-in insets correspond to the red square in each image and show the signal enhancement. The window size for each hologram is 1200 x 1200, 600 x 600 and 467 x 467 pixels from a) to c) respectively. All the images are presented with the same color scale.
Fig. 6 HERALDO reconstruction of the holograms from in Fig. 5(a) and 5(c). The white squares emphasize the best reconstructed image from one of the square reference. The size of the squares is 1 µm x 1 µm.
5. Comparison between various reference shapes
Using optimized data collection and data analysis, we now compare HERALDO with slits and square references to Fourier Transform Holography. The point reference has a diameter of 110 nm giving a resolution limit of 80 nm. In the HERALDO procedure, a slit reference requires a 1D derivation step along the slit direction. The resolution is then limited by the transverse size of the slit, as it is not affected by the derivation. In the present case, the slit width is 115 nm, limiting the resolution to 82 nm. In the case of a square reference, a 2D derivation along the two edges orthogonal directions is needed. The resolution is not related to the size of the square but to quality of the etched corners. All the data are taken using a hardware binning of 3 and an acquisition frequency of 1 MHz. Figure 7 shows the test samples, the measured diffraction pattern and the reconstructed images. The total number of photons measured on the CCD in the point (FTH), slit and square (HERALDO) reference configurations is respectively 3.3x105, 1.1x106 and 2.1x106. We used different filter diameters in order to independently optimize each reconstruction. As shown in Fig. 7(b), the slit reference clearly gives the best result.
Fig. 7 Comparison between FTH (a), HERALDO with slit (b) and square (c) references. The diffraction images were acquired in single shot with a times 3 hardware binning factor. In insets, corresponding SEM images of the sample and references and image reconstructions (the size of the white box is 1 µm x 1 µm). The filter parameters were optimized independently.
However, to get a more quantitative image reconstruction criterion, we plot the image profile along the direction of the object with the hole and three slits. It is show in Fig. 8 . The plot profiles confirm the qualitative observation: we get highest contrast in the image reconstruction for the slit reference (95%). The square references give a good contrast (80%) but the signal is lower in the three slits part of the image. The FTH point reference has even lower contrast (65%). This is mainly due to the fact that the hologram contains a lower total number of photons because of the limited size of the point reference compared to extended references. As a consequence, high frequency photons are in the background. Note that the derivative filter lowers the SNRr in the reconstruction by amplifying the high frequency noise from the hologram. It is even more pronounced in the case of a square reference as a second order polynomial is necessary. This is correlated to the fact that at low photon energy and low flux, the readout noise tends to dominate the far field diffraction pattern. The best compromise is found with a slit reference which geometry reduces the effect of noise amplification. However, the resolution given by a slit reference is limited by the slit width, while the resolution of the two-dimensional reference is theoretically non-limited.
6. Multiple-shot acquisition
As demonstrated in the previous examples, the reconstruction of the diffraction pattern is mainly limited by the amplified readout noise. When the sample is not damaged by the radiation or by the physical process under investigation, it is possible to increase the incident beam flux by shot accumulation. Figure 9 presents a comparison between single-shot and multiple-shot acquisition of the grid test object. The measured 5-shot diffraction pattern has 2.9 x 108 photons and the single-shot acquisition has 5.9 x 107 photons. Both holograms are recorded within a window size of 1200 x 1200 non-binning pixels and the readout frequency is 1MHz. The single-shot acquisition has equivalent photon noise and readout noise contributions in its reconstruction, while the 5-shot reconstruction is mainly influenced by the photon noise. Table 1 compares the 1-shot and 5-shot best reconstructions extracted from Fig. 9. The SNRr is increased from 2.95 to 4.5. The effect is particularly sensitive at high frequencies in the holograms which contributes to a better spatial resolution and image contrast. However, the disadvantage of the accumulation is the eventual blur of the diffraction pattern due to the instable beam position, and radiation damage for certain samples.
Fig. 9 Diffraction patterns (a,c) and best reconstructions (b,d) for respectively single-shot (a) and 5-shot (c) detections. The SEM image of the grid test object is shown as an inset, the size of the geometric pattern is 1 µm x1 µm.
Table 1. Analysis of the best reconstructions with 1-shot and 5-shot, respectively.
7. Conclusion
In summary, we have presented different strategies to increase the resolution and signal to noise in holography with extended reference when the signal is comparable to the detector readout noise. The basic principle of HERALDO makes it robust to photon noise, the latter coming essentially from the large signal diffracted by the extended reference. The application of the differential operator compensates for the influence of the reference, including its noise. When the signal is now, i.e., in the single shot regime, various sources of noise, especially the readout noise, will affect the image reconstruction. Indeed, the differential operator in HERALDO acts as a high-pass filter amplifying significantly the noise at high frequency. This increases the proportion of wrong information in the reconstruction. Our finding is that the readout noise is the main obstacle in low signal regime. Strategies can be applied to reduce this noise. On the detection side, hardware binning and readout frequency tuning improve the signal-to-noise ratio of the diffraction pattern. Additionally, a posteriori image processing, like low pass filtering to reduce the readout noise contribution, increases the SNRr of the reconstructed images.
It effectively compensates the amplification of the readout noise by the multiplicative filter introduced by the reconstruction procedure. We conclude that the reconstruction quality can be improved by the optimization of the detection stage and a carefully chosen low-pass filter diameter. The comparison between FTH, one-dimensional and two-dimensional HERALDO shows that the slit reference is the best holographic configuration for our harmonic beam. At the low flux and low photon energy used here, the noise amplification by the second order polynomial multiplicative step decreases more the spatial resolution than the lateral size of a slit.
Accumulation over multiple shots in a long acquisition time can also improve the diffracted signal, the image reconstruction quality and resolution. The signal is then photon noise limited. This imposes to work with high beam stability (in terms of pointing, intensity and phase spatial distribution, coherence properties, wavefront…) as it otherwise might reduce the visibility of the diffraction details. In addition, accumulation is not compatible with experiments that result in sample destruction if there is no possibility to provide multiple identical copies. Nevertheless, upcoming high repetition rate XUV sources are likely to meet the needed stability criteria in the near future and could contribute towards compact lensless X-ray imaging approaches.
Acknowledgments
We acknowledge discussions with Manuel Guizar-Sicairos. We acknowledge support from the Saclay SLIC laser team. Membranes were processed thanks to the SEM/FIB facility of the CSNSM (Centre de Sciences Nucléaire et de Sciences de la Matière) Univ. Paris-Sud/CNRS/In2P3. We acknowledge financial support from the European Union through the EU-LASERLAB (SLIC 001682) and the VOXEL FET (665207), from the French ministry of research through the 2013 ANR grant ”NanoImagine” (ANR-13-NANO-0005) and the joint ANR-DFG 2014 grant KO 3798/4-1 ”IPEX”, and from the C’NANO research program through the NanoscopiX grant, from the Swedish Research Council (Vetenskapsrådet), grant number 637-2013-439/D0043901 and the Swedish Foundation for International Cooperation in Research and Higher Education (STINT).
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