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Abstract
Underwater imaging is a challenging task because of the large amounts of noise and the scattering nature of water. Conventional optical methods cannot realize clear imaging in underwater conditions owing to the limitations of low sensitivity, geometrical aberrations, and the narrow spectrum of photoelectric detectors. By contrast, single-pixel imaging (SPI) is a promising tool for imaging in poor-visibility environments. Nevertheless, this challenge is faced even when using traditional SPI methods in highly turbid underwater environments. In this work, we propose a Hadamard single-pixel imaging (HSI) system that outperforms other imaging modes in turbid water imaging. The effects of laser power, projection rate, and water turbidity on the final image quality are systematically investigated. Results reveal that compared with the state-of-the-art SPI techniques, the proposed HSI system is more promising for underwater imaging because of its high resolution and anti-scattering capabilities.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Underwater imaging is of great importance and has wide applications in underwater target identification, target tracking, and ocean exploration. However, in contrast to land photography, some intractable constraints exist in underwater imaging [1]. First, because of inhomogeneous media, scattering often causes a blurring effect; this effect rarely occurs in land-based imaging. Second, light attenuation usually leads to underexposure. Third, the sediment particles in turbid water also cause electronic noises and affect imaging fidelity. Consequently, traditional optics-based technologies that employ charge-coupled device (CCD) and complementary metal-oxide semiconductor (CMOS) technologies cannot be used to observe underwater targets under high-turbidity conditions [2]. Various methods have been reported for imaging objects through scattering media: photoacoustic imaging [3,4], optical coherence tomography [5–7], and vectorial light-detection system [8]. However, these methods have the disadvantage of being susceptible to environmental conditions and have a high degree of complexity. Therefore, the issues of high signal-to-noise ratio (SNR) and anti-scattering specific to underwater imaging still remain unsolved.
Until recently, single-pixel imaging (SPI) has been employed to retrieve the hidden information about an object from scattering media. SPI exploits the correlation between the object morphology and light intensity fluctuations. In principle, SPI is effective in low-contrast and even invisible environments where common optical imaging approaches fail to achieve the desired goal. Thus, SPI is more appropriate for underwater imaging. SPI, which has become increasingly popular, is based on ghost imaging (GI) [9–11] and is a branch of computational optics imaging. To overcome the drawbacks of low signal-to-noise ratio (SNR) and time-consuming nature, conventional GI is developed using a few advanced computational algorithms such as compressive sensing single-pixel imaging (CSSI) [12] and Fourier single-pixel imaging (FSI) [13]. Note that comparing with the non-deterministic convex optimization procedure of CSSI, FSI is a deterministic sampling model that enables reconstruction of the Fourier spectrum of the object being imaged. Because of the sparsity of the Fourier spectrum of the image, most of the image details are concentrated at the center of the Fourier spectrum. FSI can reduce the sampling time by capturing only the central part of the Fourier spectrum, thereby achieving high efficiency.
In addition to high speed, antiscattering property and high-fidelity imaging are desirable for contemporary technologies. In this context, the Hadamard matrix, which has the characteristics of a 2-bit value {−1, 1} and sparsity of its transform spectrum, has been successfully applied to illuminating patterns [14–16]. The core equipment in an SPI system, i.e., a digital micromirror device (DMD), is more suitable for 2-bit basis patterns to maximize the projection rates (up to 22 kHz). According to the Walsh–Hadamard transform algorithm [17], Zhang et al. clarified the principles of the Hadamard single-pixel imaging (HSI) technique and first compared HSI with FSI from different aspects [18]. Although HSI shares similar principles, a deterministic sampling model based on transform domain, with FSI, previous investigations revealed that HSI has excellent robustness to noise in an atmospheric environment. Nevertheless, few studies have focused on imaging in underwater environments, especially under turbid and blurry conditions. In addition, there is a lack of systematical analysis of the influence of turbidity on SPI system performance. Hence, we propose a HSI system and extend it to underwater scenarios.
In this work, we focus on underwater imaging in a turbid environment using the proposed HSI system. Because of low efficiency and the lack of antiscattering imaging capability, traditional techniques cannot be used for effective turbid water imaging. In particular, water with a turbidity of more than 50 nephelometric turbidity units (NTU) is considered highly turbid water [2]. Our proposed method can be used to observe underwater objects clearly under the highest turbidity of 90 NTU. Various influencing factors in turbid water are analyzed and optimized through experimental investigations. The Hadamard SPI system is a powerful tool for high-resolution, antiscattering underwater imaging.
2. Methods
2.1 Underwater experimental system
The schematic of the proposed experimental system is shown in Fig. 1. We used a 532 nm laser as the light source and a DMD (Texas Instrument DLP7000 V-7001) for light modulation. Illumination patterns of 64 × 64 pixels were sequentially projected with reflected light. These patterns carried the information of the object scene (a heart-shaped figure printed on a polyvinyl chloride (PVC) film, as shown in Fig. 1). The target object is placed in a water tank of dimensions 60 × 25 × 30 cm3 and is submerged in turbid water samples with different volumes of kaolin clay powder. A small amount of correlated light is recorded by a photodiode (Thorlabs PDA100A2, Si Switchable Gain Detector, 320–1100 nm) together with a 16-bit acquisition board (NI USB-6356). The focus lengths of L1 and L2 in Fig. 1 are 200 and 50 mm, respectively; the DMD and detector are set on the focal plane of the lens.
Fig. 1. Experimental setup of the Hadamard single-pixel imaging system. (LA: light source, either light-emitting diodes or lasers; BE, beam expander; O, the object under the water; P, particles; L1, L2, lenses; WT, water tank; DMD, digital micromirror device; D, single-pixel detector; DAQ, data acquisition system; and PC, personal computer). Light from the active laser passes through the BE to ensure that a larger visual field is obtained and then illuminates the object scene. The reflected light is converged into the DMD, which codifies a set of microstructured light patterns. The scattering light carries information of the object and the set patterns is collected by a single-pixel detector. The final signal is converted by the DAQ and processed in a PC.
We utilize the clock signal sent by the DMD board to synchronize the DMD projection rates and the DAQ system. The acquired data are processed on a laptop (Intel Core i5-8265U CPU 1.80 GHz, 8-GB RAM, and 64-bit Windows 10 operating system). On the basis of the reconstruction algorithms, we obtain the final experimental results.
2.2 Principles and reconstruction algorithm of HSI
In the HSI system, the Hadamard transform ($\textrm{H\{ \} }$) of an image I (x, y) is defined as [17]
The Hadamard matrix is composed of only two values, i.e., {−1, 1}. After a series of calculations and transforms, the Hadamard basis still comprises those two values. However, no negative value exists for the DMD or an image, and this may lead to a problem in its modulation. Hence, a differential operation is required for the HSI. In particular, to acquire a Hadamard coefficient, two measurements must be made. One is ${\textrm{P}_\textrm{H}}\textrm{(}x,y\textrm{)}$, whereby the value −1 in the Hadamard basis matrix becomes 0. The other is [1 − ${\textrm{P}_\textrm{H}}\textrm{(}x,y\textrm{)}$], whereby the value 1 in the Hadamard basis matrix becomes 0, and the value −1 becomes 1. Therefore, H(u, v) is obtained as
where D+1 and D−1 represent the two measurements ${\textrm{P}_\textrm{H}}\textrm{(}x,y\textrm{)}$ and [1 −${\textrm{P}_\textrm{H}}\textrm{(}x,y\textrm{)}$], respectively. The final image is recovered from the inverse Hadamard transform of the detected Hadamard spectrum.The flowchart of SPI is shown in Fig. 2, which is suitable for all state-of-the-art SPI systems. The total number of the image pixels is set as N, which means that at least N measurements are required to reconstruct the entire image. However, because of the sparsity of the image in some spectral fields (e.g., Fourier field or Hadamard field), we can recover the object using only m measurements (m << N).
Fig. 2. Flowchart of single-pixel imaging (SPI). The image resolution should be set at the beginning of the experiment. Considering the balance of efficiency of the algorithm and view of the laser spot under the experimental conditions, the image pixels are usually set as 64 × 64. Using a few number observations (m) and the same number of illumination patterns, the target object image can be reconstructed from the signals captured by a single-pixel detector.
The principle of HSI is based on the Hadamard transform of an image. HSI is the reconstruction process of the Hadamard spectrum, which is composed of a group of Hadamard coefficients. To determine each coefficient, we use the corresponding Hadamard basis pattern to illuminate the object. The final image is reconstructed using the inverse Hadamard transform algorithm.
Here, some details regarding the SPI experiment should be mentioned: The first point is the choice of light source. It is demonstrated that the absorption and scattering coefficients of the laser with wavelengths of round 500 nm are quite low according to the literatures [2] and [19]. And it is proved to have a more suitable penetration depth under the water. Considering the characteristics of high intensity and low divergence of the laser light, we finally choose a 532 nm laser as the illumination source in our experiment.
The second point is about the Hadamard basis patterns (e.g., N = 64 × 64). They are resized by an integral factor k. Hence, the real size of the illumination patterns is k64 × k64 (usually k${\in} $[2,8]). Thus, we can increase the intensity of the detected light to obtain high SNR. Because the value of Hadamard basis patterns is binary, the HSI is naturally suitable for SPI based on DMD projection. Meanwhile, the HSI reconstruction algorithm is much more efficient than other SPI reconstruction algorithms because of its 2-bit illumination patterns and low computational complexity.
Another significant factor is the imaging time, which mainly depends on the projection rates and the sampling ratio. In our experiment, the projection rate is mostly chosen as 50 Hz, and the sampling ratio is in the range of 10%−30%. Considering our 64 × 64 target object (50 Hz; sampling ratio is 0.1) as an example, the imaging time will be (0.1 × 64 × 64)/50 seconds, i.e., nearly 8.2 s. Moreover, because of the high projection rates, the quality of the reconstructed image would be low. Therefore, balancing the imaging time and the image quality based on different application requirements is a decisive factor.
2.3 Evaluation indexes of reconstruction image quality
When imaging the target object, we need to change the location of the object to meet the requirements of different SPI configurations. Hence, no ground truth images are available. Further, two no-reference image evaluation indexes are used to evaluate the reconstruction qualities of images: contrast noise ratio (CNR) and edge preservation index (EPI). We also calculate the noise level (NL) to evaluate the images on the other side.
To define the CNR, we first select several representative parts of the region of interest (ROI) and background in the image. CNR is calculated as:
EPI is commonly used to measure the noise suppression of a smoothing filter. In this paper, we choose block-matching and 3D filtering (BM3D) algorithm [20] for the image postprocessing. The whole process is comprised of two steps: basic estimate and final Wiener estimate. With the BM3D smoothing filter, we can obtain high-quality images with better CNR and calculate EPI to measure the ability to maintain edge details of an image. EPI is obtained using the following equation:
The noise level reported in Ref. [21] can be considered as an estimation parameter for the noise. It is claimed that their noise level algorithm is superior in terms of accuracy and stability compared with other state-of-the-art competitive algorithms. The principle of this evaluation index is based on the gradients and statistics of the noise-stained image. A high value of the NL means that the image is contaminated badly by noise. Therefore, we use this evaluation index to measure the qualities of the reconstructed images.
3. Experiments and evaluation
3.1 HSI performances in turbid water
To test the characteristics of HSI in turbid water, we examine some factors of HSI under different conditions. The details are given in the following subsections.
3.1.1 Effect of laser power
We test the HSI system with different laser powers (ranging from 40 to 120 mW) as a function of different sampling ratios under an underwater distance of 55 cm and turbidity of 10 NTU. The reconstruction results (the sampling ratios, from the first to the third row are 0.1, 0.15, and 0.2, respectively) are shown in the upper panels of Fig. 3, and the scatter diagram of the three evaluation indexes are shown in the bottom panels.
Fig. 3. HSI system under the following conditions: 55 cm underwater distance, 50 Hz projection rate, and 10 NTU turbidity with laser powers ranging from 40 to 120 mW. (a) The HSI reconstruction results. (b)-(d) The error bar diagrams of evaluation indexes. The three scatter diagrams at the bottom show the trends of CNR, EPI, and noise level as the laser power increases. The red, blue, and yellow scatter points represent sampling ratios of 0.1, 0.15, and 0.2, respectively.
From Fig. 3, the following interpretations are derived:
- 1. The intensity of light plays an important role in determining the results of HSI. An increase in laser power improves the qualities of the reconstructed images.
- 2. The sampling ratio has different effects on the reconstruction results. Without noise or scattering, the image quality will be improved with an increase in the sampling ratio. However, for underwater imaging, especially in turbid water, imaging is always affected by scattering and absorption. With the increase in the sampling ratio, the image details become clearer, however simultaneously, the noise is magnified.
3.1.2 Effect of DMD projection rate
Another important factor that affects both reconstruction quality and imaging time is projection rates. We set the HSI system for the following conditions: 10 cm underwater distance, 25 NTU turbidity, 65 mW laser power; and DMD projection rates in the range of 50–1000 Hz. The reconstruction results are shown in Fig. 4, from which the following inferences are made:
- 1. A high projection rate means less imaging time. HSI can achieve real-time imaging if the DMD projection rates are improved. Better image reconstruction is obtained even at low sampling ratios.
- 2. With the increase in the projection rate, the CNR and EPI image quality indexes decrease. This problem limits the real-time and high-resolution imaging implemented by SPI. However, under high scattering, HSI can recover better image quality compared with that achieved by traditional array-imaging methods.
Fig. 4. HSI system under the following conditions: 65 mW laser power, 10 cm underwater distance, 25 NTU turbidity, and projection rates in the range of 50–1000 Hz. (a) The HSI reconstruction results. (b)-(d) The error bar diagrams of evaluation indexes. The three scatter graphs (red, blue, and yellow points represent sampling ratios of 0.1, 0.15, and 0.2) show the trends of CNR, EPI, and noise levels, respectively, as the projection rates increase.
3.1.2 Influence of water turbidity
Detection under real underwater conditions always accompanies noises arising from various sources, especially from different water turbidities. Therefore, we simulate turbid water with different NTU values by adding different volumes of kaolin clay powder. From Fig. 5, we draw the following conclusions:
- 1. As the turbidity increases, both the reconstruction indexes degrade and the noise level increases. However, when the water turbidity reaches 80 NTU, we can still distinguish the object profile. The HSI system exhibits excellent performance in high-turbidity water imaging.
- 2. Despite the presence of some abnormal data points in Fig. 5, the evaluation indexes satisfy the expected trend. Furthermore, we can still improve the reconstruction quality by appropriately increasing the laser power or reducing the imaging distance to balance the factors that influence the final results.
Fig. 5. HSI system under the following conditions: 65 mW laser power, 50 Hz projection rate, 10 cm underwater distance, and different turbidities in the range of 5–80 NTU. (a) The HSI reconstruction results. (b)-(d) The error bar diagrams of evaluation indexes. CNR, EPI, and noise level scatter graphs of water turbidity are shown in the bottom panels. The scatter diagrams show the fluctuation of the evaluation indexes with an increase in sampling ratio from 0.1 to 0.2.
3.2 Comparison of different SPI modes
In this subsection, we present a comparison of the underwater imaging performance of the three mainstream SPI techniques (CSSI, FSI, and HSI) to highlight their advantages and disadvantages. Identical experimental conditions (see Fig. 1) are used for all the techniques. To identify the DMD loading time of different SPI, CSSI employs random Hadamard matrix as projection patterns; and the basis patterns of FSI are binarized by error diffusion dithering algorithm [22]. In addition, the reconstruction algorithm of CSSI, which employs the total variation regularization method, exhibit the best performance among the three reconstruction algorithms [23]. FSI is reconstructed using four-step phase-shifting algorithms [13].
Without loss of generality, the imaging object pattern is changed to a fine-pointed star. To demonstrate the better performance of the SPI modes compared with that of a traditional camera in an underwater environment, we consider three groups under different water conditions (see Fig. 6): turbidity, projection rate, underwater distance, and laser power of (1) 2 NTU, 50 Hz, 55 cm, and 65 mW, respectively; (2) 20 NTU, 50 Hz, 11 cm, and 95 mW, respectively; and (3) 90 NTU, 50 Hz, 7 cm, and 65 mW, respectively. Images are captured using a CMOS camera (Thorlabs DCC1645C) instead of the red dash part in Fig. 1 under identical conditions with SPI modes. Meanwhile, the propagation images of the laser beam in turbid water are also shown.
Fig. 6. Comparison of different imaging systems with an increase in water turbidity. The first row indicates the change in laser beam propagation in turbid water; rows from the second to the last show the performances of the CMOS camera, CSSI, FSI, and HSI, respectively.
From Fig. 6, we can infer the following: (1) With the increase in water turbidity, the laser beam is scattered to a greater extent. The scattered light is the main factor causing image noise. Meanwhile, as the imaging distance increases, the intensity of light decreases, thereby affecting the CNR of the final images. (2) The image quality of the three SPI modes is acceptable in slightly turbid water. However, as the water turbidity increases, the CMOS imaging and CSSI fail to image the object, whereas FSI and HSI can still distinguish the object. Moreover, subjective assessment shows that when the water turbidity reaches 90 NTU, the proposed HSI reconstruction method outperforms other methods. (3) Traditional CMOS underwater imaging is much more challenging than SPI, and the image is easily polluted by laser speckle, which is caused by the imaging characteristics of the coherent light source. However, three SPI modes seem to be less affected by laser speckle.
To show the specific evaluation parameters of different SPI results for underwater imaging, we compare CSSI, FSI, and HSI under the following three groups of conditions: (a) the laser power, DMD projection rate, and water turbidity are 90 mW, 50 Hz, and 30 NTU, respectively, while the underwater distance is varied from 14 to 55 cm; (b) the underwater distance, DMD projection rate, and turbidity are 18 cm, 50 Hz, and 30 NTU, while the laser power is varied from 65 to 140 mW; and (c) the underwater distance, DMD projection rate, and laser power are 11 cm, 50 Hz, and 90 mW, while the water turbidity is varied from 5 to 50 NTU. Further, we also simultaneously consider the running time of the reconstruction algorithm. The reconstruction results and detailed analysis are given in the following subsections.
The reconstructed images and related evaluation indexes are shown in Figs. 7–9 and Table 1, from which we can draw the following conclusions:
(1) It is demonstrated that HSI, FSI, and CSSI all perform well in slightly turbid water in the case of below 30 NTU. However, as the turbidity increases (Fig. 9), the quality of the reconstructed image deteriorates. When NTU comes to 50 or more, the object profile by means of CSSI and FSI both submerge in the noise, but HSI outperforms other methods since it obtains the highest CNR and EPI’s values. Therefore, HSI exhibits the best performance among the three SPI modes.
3.2.1 Effect of underwater distance
Fig. 7. Performances of three SPI systems under the conditions of 90 mW laser power, 50 Hz projection rate, and 30 NTU water turbidity for different underwater distances (14–55 cm). (a) The reconstruction results. (b)-(d) The scatter diagrams of evaluation indexes.
3.2.2 Effect of laser power
Fig. 8. Performance of three SPI systems under the conditions of 18-cm underwater distance, 50 Hz projection rate, and 30 NTU water turbidity for different laser powers (65–140 mW). (a) The reconstruction results. (b)-(d) The scatter diagrams of evaluation indexes.
3.2.3 Effect of water turbidity
Fig. 9. Performance of three SPI systems under the conditions of 11 cm underwater distance, 50-Hz projection rate, and 90 mW laser power for different water turbidity values (5–50 NTU). (a) The reconstruction results. (b)-(d) The scatter diagrams of the evaluation indexes.
3.2.4 Running time of SPI algorithms
Table 1. Average running times of three SPI reconstruction algorithms.
(2) A closer examination of the FSI images reveals some mosaic-like noise stripes that only exist in these images; they are caused by quantization errors [18]. These noises recognized as a part of foreground targets cannot be easily detected by the noise level algorithm, but can be noticed by subjective observation. Whereas continuous Fourier patterns can alleviate the mosaic-like strips, but at the cost of time consuming. Hence, even FSI exhibits excellent performance in terms of the noise level index; however, in terms of other two indexes and subjective evaluation, FSI does not perform well.
(3) Imaging quality of three SPI methods show a significant negative correlation with underwater distance and water turbidity, which is ascribed to the absorption, attenuation and scattering of the light underwater propagation. Laser power can improve the CNR by increasing the illumination intensity and meanwhile it also can increase the scattering noises. It can be found in Fig. 8 that the indexes of CNR and EPI would undergo a flat trend with the increase of laser power.
(4) HSI takes the least time to reconstruct the object image (Table 1). Thus, it will save more time when used under real conditions. In particular, HSI is more practical and robust for use under turbid water conditions, especially in the high turbid water.
4. Discussion
SPI outperforms traditional imaging methods in specific aspects such as scattering media and invisible spectra [24]. In our experiment, we apply HSI to an underwater environment with kaolin clay particles as the scattering particles and compare its performance with those of two other SPI modes under different conditions. To sum up, a systematic comparison among three SPIs is concluded in Table 2 for a intuitive understanding. To date, few attempts have been made to realize single-pixel underwater imaging with high resolution and efficiency. Our findings will be conducive to establishing a robust framework for underwater imaging.
Table 2. Comparison between CSSI, FSI and HIS.
4.1 Light source
A light source plays a significant role in the underwater imaging experiment. The wavelength, intensity, and working mode affect the final results. First, depending on the different working environments, we can choose different wavelengths to realize the minimum attenuation and scattering coefficients. Under some complicated conditions, we can take full advantage of the SPI system. For example, infrared rays [24,25] can be employed as a light source in invisible, rainy, or foggy weather cases. Second, the higher the power of the light source, the better is the responsivity of the signal. With an increase in the light source power, we can realize higher light intensity and SNR. Notice that the forward-scattering light (which is caused by particles in the turbid water), i.e., the noises, will also be amplified. Third, the working mode of the laser can be divided into continuous wave and pulse modes. In some underwater imaging studies [26–29], researchers not only utilize the pulse laser to obtain a peak signal by the instantaneous pump but also apply the time-gated imaging technique to remove forward-scattering light. Thus, we can distinguish the object information from the scattering background light to improve image quality.
4.2 Imaging environment
To minimize the ambient light that acts as noise and improve the SNR of the reconstructed image, the experiment is conducted at night and the water tank is covered by a black cloth. Moreover, the characteristics of the imaging object can determine the SNR. Regardless of the substance, the reflectivity of light is the main factor. Hence, the higher the reflectivity, the better is the SNR. In our experiment, we select PVC as the object material and printed a black shape to increase the reflectivity of object and obtain an acceptable signal intensity and contrast [30]. To improve the practical performance of the HSI system, a narrow band filter with the same wavelength as that of the light source and polarized light can also be employed.
4.3 Single-pixel detector
A photoelectric detector also plays a significant role in an SPI system, with light responsivity, dark current, and sensitivity affecting the final results. A photodiode can be utilized as the detector if low experimental requirements. By contrast, using a photomultiplier as the detector is helpful for improving the signal intensity under complicated conditions. However, the cost will increase. Considering the balance of practicability and price, a silicon-based detector is preferred. To enhance the signal intensity, we select a lens coated with an antireflection film [31] in front of our single-pixel detector. We could also set the DAQ card to capture greater signal values at a given DMD projection rate to calculate the average value as the corresponding intensity. Thus, we could minimize the noise signal with an appropriate trade-off with imaging speed.
4.4 Underwater conditions
It is much more complicated in the real underwater environment. Different sizes of particles and free-floating plankton underwater can cause different imaging results. The gaolin clay powder with different diameters from 0.02 µm to 10 µm can mimic real underwater circumstance. In this condition, the scattering phenomenon is mainly caused by both Rayleigh and Mie’s scattering. In addition, the absorption and attenuation of light propagation under the water will decrease the intensity of the light dramatically when the underwater distance increases. Moreover, the temperature, salty and turbulence would also affect the final imaging results, which need to be considered in the following research.
The quality of the final reconstructed image can be affected by the abovementioned factors. In addition, methods for eliminating forward-scattering light [2,32–34], such as Fourier filtering and polarization, can improve the SPI results. Further, postprocessing algorithms can also optimize the results.
5. Conclusions
Considering complex underwater environments that seriously restrict underwater imaging, a novel imaging framework based on the Hadamard transform is proposed in this study. This underwater HSI system incorporates a 532-nm laser and a signal-pixel detector. An experimental comparison of three SPI modes reveals that the proposed method outperforms the other SPI modes and the traditional imaging techniques from the aspects of high resolution and anti-interference capabilities in highly turbid water.
Funding
Science and Technology Projects of Fujian Province (2017FJSCZY02); Fundamental Research Funds for the Central Universities (20720180068); National Natural Science Foundation of China (61571377, 61771412, 61871336).
Acknowledgments
The authors thank Weiwei You for helpful comments on the manuscript.
Disclosures
The authors declare that they have no conflicts of interest that could have appeared to influence the work reported in this paper.
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