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The FSPI system has the high imaging quality and imaging efficiency[19-22]. The scheme of the FSPI system is shown in Fig.1. In the FSPI system, the laser illuminates on the spatial light modulator after being expanded by the lens. The computer pre-generates the illumination speckle patterns and sends them to the spatial light modulator to modulate the distribution of speckle. The difference of FSPI and other SPI system is that the pre-generated speckle patterns are sinusoidal structured patterns, which are designed according to the measured position of the target spatial spectrum. The modulated speckle patterns are emitted by the transmitting antenna and travel through the water. After arriving the position where the targets are located, the speckle patterns are reflected by the targets. A receiving antenna is used to collect the reflected light. The collected light is then focused on a bucket detector which has no spatial resolution. The measurements of the bucket are sent to the computational system to reconstruct the distribution of targets spatial spectrum.
The pre-generated modulation patterns in FSPI system is generally designed as:
$${P_\phi }\left( {x,y;{f_x},{f_y}} \right) = a + b \cdot \cos \left( {2\pi {f_x}x + 2\pi {f_y}y + \phi } \right)$$ (1) Here, x and y represent Cartesian coordinates in the space domain. a is the average intensity of the modulation pattern, and b represent the amplitude of the modulation pattern which is equal to the contrast. ϕ is the initial phase and (fx, fy) is the spatial frequency. To simplify the derivation, the intensity of laser which illuminates on the spatial light modulator is uniform and the intensity is denoted as E0. Therefore, the distribution of speckle emitted by the transmitting antenna is:
$$E\left( {x,y} \right) = {P_\phi }\left( {x,y;{f_x},{f_y}} \right){E_0}$$ (2) The speckle patterns travel through the water, and illuminate the targets. The light is mainly absorbed and scattered by water in the process of propagation[23-24]. The absorption of water is mainly reflected in the attenuation of light energy in water. The scattering includes back scattering and forward scattering. The back scattering further attenuates the energy of the modulated speckles and enhance the noise in the bucket detector. The noise is denoted as n. The effect of forward scattering on FSPI system is mainly reflected in the distortion of the modulated speckles which illuminates on the targets. The underwater environment is relatively complex. It is difficult to express the distortion of light distribution accurately by mathematical model. It is generally believed that the point spread function of water is of Gaussian-like type[25-26]. The atmosphere turbulence model is usually utilized to approximate water degradation function. According to the degradation model of atmosphere turbulence, the degradation model of water can be expressed as:
$$H\left( {{f_x},{f_y}} \right) = \exp \left[ { - k{{\left( {{f_x}^2 + {f_y}^2} \right)}^s}} \right]$$ (3) where k is blurring factor and s is turbulent constant. In addition, α1 is the attenuation proportional coefficient, which can be used to represent the attenuation of light energy due to water absorption and back scattering. The distribution of light illuminating on the targets can be expressed as:
$$\begin{split} E'\left( {x,y} \right) =& {\alpha _1}{P_\phi }\left( {x,y;{f_x},{f_y}} \right) \cdot\\ &{{E_0} * {{\mathbb F}^{ - 1}}\left[ {H\left( {{f_x},{f_y}} \right)} \right]} \end{split} $$ (4) Here, * represents the convolution calculation, and
${{\mathbb F}^{ - 1}}\left( . \right)$ represents inverse Fourier transform.The distribution of reflectivity of targets is denoted as f(x, y). The light reflected by the targets travels through the water and is received by the receiving antenna. The bucket detector records the energy of focused light. The light in the optical path from the target to the optical receiving antenna is also be absorbed and scattered by the water, which is similar to the emitting optical path. However, the bucket detector has no spatial resolution and is not used for imaging, which leads to that the distortion of water forward scattering has no effect on the imaging quality. In the receiving optical path, only the effect of water on the attenuation of echo energy is considered. The total energy attenuation coefficient caused by the receiving optical path is denoted as α2. Therefore, the measurement of bucket detector is:
$$\begin{split} {I_\phi }\left( {{f_x},{f_y}} \right) =& n + \displaystyle\iint {{\alpha _1}{\alpha _2}{E_0}f\left( {x,y} \right)}\;\cdot \\ &{{P_\phi }\left( {x,y;{f_x},{f_y}} \right)} \;\cdot\\ &{ * {{\mathbb F}^{ - 1}}\left[ {H\left( {{f_x},{f_y}} \right)} \right]{\rm{d}} x{\rm{d}} y} \end{split} $$ (5) In FSPI system, the target spatial spectrum can be obtained by the four-step phase-shifting approach, where all four speckle patterns have the same spatial frequency (fx, fy), but different phases. The phase shift between the two adjacent patterns is a constant
$\pi $ /2. When the values of the phase ϕ are equal to 0,$\pi $ /2,$\pi $ and 3$\pi $ /2, the measurements of bucket detector are denoted as I0,$I_{{\pi}/2} $ ,$I_{\pi} $ and$I_{3{{\pi}/2}} $ respectively. The four bucket detector measurements are utilized to estimate the Fourier spectrum value of the targets at the spatial frequency (fx, fy), which can be expressed as:$$C\left( {{f_x},{f_y}} \right) = \left( {{I_{{{3\pi } / 2}}} - {I_{{\pi / 2}}}} \right) + j\left( {{I_\pi } - {I_0}} \right)$$ (6) By submitting (1) and (5) into (6), (6) can be represented as:
$$C\left( {{f_x},{f_y}} \right) = {\alpha _1}{\alpha _2}{E_0}H\left( {{f_x},{f_y}} \right)C'\left( {{f_x},{f_y}} \right)$$ (7) $$ \begin{split} &{C'\left( {{f_x},{f_y}} \right) = \displaystyle\iint {f\left( {x,y} \right)}}\cdot \\ &{\exp \left( { - j2\pi \left( {{f_x}x + {f_y}y} \right)} \right){\rm{d}} x{\rm{d}} y} \end{split} $$ (8) C'(fx, fy) represents the true spatial frequency of unknown targets at the position (fx, fy) of spatial spectrum. Under the influence of water back scattering and forward scattering, the obtained spatial frequency value is equal to the product of the true spatial frequency value of unknown targets, attenuation coefficient and water degradation function. Moreover, the influence of water degradation function in spatial domain is mainly reflected in the blurring and unclear image. Therefore, the effects of the water degradation function need to be eliminated to obtain the clear image.
Due to the relative complexity of water, it is rather difficult to measure or estimate the parameters k and s in the water degradation function. The way to decrease the influence of water forward scattering is equivalent to an ill-posed problem of blind image restoration. In order to solve the ill-posed problem, the water degradation function compensation method is implemented to deburr the reconstruction image of FSPI in the spectrum domain. By submitting (3) into (7), (7) can be simplified as:
$$ \begin{split} C\left( {{f_x},{f_y}} \right) =& {\alpha _1}{\alpha _2}{E_0}C'\left( {{f_x},{f_y}} \right)\cdot\\ &{\exp \left[ { - k{{\left( {{f_x}^2 + {f_y}^2} \right)}^s}} \right]} \end{split}$$ (9) Suppose σ is the maximum value of spatial spectrum, which can be obtained by the FSPI system. The left and right sides of (9) are divided by σ for normalization. Ĉ (fx, fy) and Ĉ'(fx, fy) are used to represent the normalized spatial spectrum of the blurred image and the true image respectively. Taking logarithmic operation on the left and right sides of (9), it can be expressed as:
$$\begin{split} \ln \hat C\left( {{f_x},{f_y}} \right) =& \ln {\alpha _1}{\alpha _2}{E_0} + \ln \hat C'\left( {{f_x},{f_y}} \right) -\\ &{ k{{\left( {{f_x}^2 + {f_y}^2} \right)}^s}} \end{split} $$ (10) According to the symmetry of Gaussian-like functions, any line passing through the origin in the spatial spectrum satisfies (10). It is reasonable to set the value of fy as 0 and represent lnα1α2E0+lnĈ'(fx, fy) with a constant A. (10) is reformulated as:
$$\ln \hat C\left( {{f_x},0} \right) = A - k{\left( {{f_x}^2} \right)^s}$$ (11) Based on the spatial spectrum obtained by the FSPI system, the unknown parameters k and s can be calculated by the least squares fitting. The approximate water degradation function H (fx, fy) can be obtained when the parameters k and s are known. The spatial spectrum is divided by the water degradation function H (fx, fy) to calculate the true spatial spectrum of targets. Combining the inverse Fourier transform, the image of underwater targets can be finally reconstructed.
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In order to more intuitively show the high-efficiency reconstruction of the proposed method, the gray values in the line 163 (the light blue line) of the reconstruction results with various methods are drawn into a two-dimensional curve, as shown in the Fig.5. The black curve in Fig.5 is the gray values in the line 163 of the original target image. The green one is the two-dimensional curve of DSPI, the blue one is the two-dimensional curve of the conventional FSPI, and the red one is the two-dimensional curve of the proposed method. It can be seen from Fig.5 that the reconstruction error of DSPI is relatively large, and the noise almost drowns out the useful signals. There is less noise in the reconstructed results of the conventional FSPI and the proposed method. However, the turbid water decrease the reconstruction quality of the two methods, which is reflected in the blurred edge of the reconstruction results. The curves in the area of the oval dotted line in Fig.5 are corresponding to the area of target edge. The target edge areas are enlarged and shown in the right side of Fig.5. Because the target edge area of original is the clearest, the curves of gray value in the original target image are the steepest. For the influence of water forward scattering, the reconstructed results of the conventional FSPI and the proposed method get blurry. Thus, the gradients of the gray value with the conventional FSPI and the proposed method at the edge are both smaller than that of the original target image, and the curves (the red and blue ones) are gentle. With the increase of water turbidity, the water forward scattering gets more serious and the curves get gentler. Because the proposed method compensates for the influence of forward scattering, the curve at the edge is less steep than the conventional FSPI. Therefore, the red curve at the edge is steeper than the blue one.
In order to quantitatively compare the influence of water turbidity on the quality of reconstruction results of various methods, the point spread function (PSF) of reconstructed results are calculated based on the slant edge method[27]. The process of calculating the PSF is shown in Fig.6. First of all, a region which contains the desired edge is selected, assuming that the selected region is circled out with the red box in the first image of Fig.6. Sobel operator is used for edge detection in the selected region. The equation of the line corresponding to the edge can be fitted according to the edge. The distances between the whole pixels in the selected region and the fitted line are calculated. The distances and gray values of each pixel are stored into the set of X and Y. According to the data in the set of X and Y, the edge spread function (ESP) can be obtained and the Fermi function is used for ESP fitting. The reason to adopt Fermi function is that it can be well approximate the ESP, and it can be differentiable everywhere, which is conducive to the differentiation operation when finding the line spread function (LSP) later. In order to fit the ESP more accurately, a summation of three Fermi functions is adopted for fitting in practice.
The summation of three Fermi functions is shown as:
$${\rm{F}} \left( x \right) = K + \sum\limits_{i = 1}^3 {\dfrac{{{a_i}}}{{\exp \left( {\dfrac{{x - {b_i}}}{{{c_i}}}} \right) + 1}}} $$ (12) where K, ai, bi and ci are the fitting coefficients, which can be calculated according to the data in set X and Y. Considering the influence of noise on the fitting results and the accuracy improvement of the fitting process, especially the image noise is serious in the DSPI, the set of X and Y can be expanded by the mean of cubic spline interpolation. The accuracy of the fitting can be improved by increasing the data set. Then, according to the ESF, the LSF is solved by the differential. After differentiating (12), it can be obtained:
$${\rm{LSF}} \left( x \right) = \dfrac{{{\rm{d}} {\rm{F}} \left( x \right)}}{{{\rm{d}} x}} = \sum\limits_{i = 1}^3 {\dfrac{{ - {a_i}\exp \left( {\dfrac{{x - {b_i}}}{{{c_i}}}} \right)}}{{{c_i}{{\left[ {\exp \left( {\dfrac{{x - {b_i}}}{{{c_i}}}} \right) + 1} \right]}^2}}}} $$ (13) Because of the isotropic distribution, the PSF can be expressed as:
$${\rm{PSF}} \left( x \right) = {\rm{LSF}} \left( x \right) \times {\rm{LSF}} \left( y \right)$$ (14) In order to guarantee the fairness of PSF calculation, the calculation region of each method should be the same when calculating the PSF with slant edge method. The calculated PSF of reconstructed results with different methods are shown in Fig.7.
The different columns in Fig.7 represent the results with different reconstruction methods. The used methods from the left to the right are DSPI, FSPI and the proposed method respectively. Different rows represent the different imaging environments, and the added milk amounts are 20 mL, 10 mL and 5 mL respectively. As the calculated PSF shows in the Fig.7, the turbidity of water affects the reconstruction results, which is embodied in the broadening of the point diffusion function of the reconstruction results. When the amount of added milk is relatively small, the turbidity of water is relatively low. The full width at half maximum (FWHM) of the PSF reconstructed by various methods are relatively narrow. With the increase of the amount of added milk, the water gradually becomes turbid. All the FWHMs of the PSF with various methods become wider, and the imaging results become blurry. On the other hand, in the case of the same turbidity of water, the PSF of the reconstruction results with the DSPI is the widest, and the image gets the most blurry and the resolution is relatively low. The PSF of the reconstruction results with the conventional FSPI is slightly better than that of the DSPI, especially when the water is relatively turbid. For instance, when 20 mL milk is added, the FWHM of the PSF of FSPI is significantly smaller than that of the DSPI. The reconstructed result is clearer than that of the DSPI. The proposed method has the best imaging effect among the three methods. The FWHM of the PSF of the reconstructed results is the narrowest of all under the same turbidity degree condition. The resolution of the proposed method is the highest. Therefore, it can be concluded that forward scattering causes the least blurring to the proposed method.
In addition, the FWHM of PSF in the Fig.7 is calculated quantitatively. In Fig.7, the total number of pixels in the x direction is equaled to the total number of pixels in the y direction. Hence, this pixel number is denoted as N. For some certain PSF figure, the pixel value of different pixel coordinates (x, y) is P. Supposing that the maximum value of the PSF figure is Pmax and the corresponding coordinate is (xmax, ymax). Therefore, all the point on the line of y= ymax, which satisfy the condition of P>Pmax/2, can be found. The number of these points can be expressed as Nx. According to the isotropic distribution of PSF, the FWHM of PSF is defined as:
$$FWHM = \dfrac{{{N_x}}}{N}$$ (15) Based on (15), the FWHM of PSF with various methods under different turbidity conditions can be calculated. The calculated results is shown in Tab.1. The same conclusion can be drawn from Tab.1 as in Fig.7. From the quantitative calculation results, it can be obviously seen that the proposed method has high quality imaging performance than the other methods.
Table 1. FWHM results of PSF with various methods under different turbidity degrees
Milk amounts/mL DSPI FSPI Proposeed 20 0.0997 0.0565 0.0381 10 0.0764 0.0507 0.0366 5 0.0465 0.0496 0.0199
Underwater Fourier single pixel imaging based on water degradation function compensation method
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摘要: 水下光学成像的探测环境相对复杂。前向散射、后向散射和吸收极大地降低了水下光学成像的成像质量。单像素成像因其较高的抗噪声性而被认为是一种适合于水下光学成像的技术。对于水下单像素成像系统,目前存在的问题是需要采用结构光进行照明,当散斑在水下传播时,前向散射会使预先生成的散斑发生畸变。因此,重建结果的分辨率降低,重建结果模糊不清。为了减小前向散射对单像素成像系统的影响,对傅里叶单像素成像的重建过程进行改进。在空间谱域傅里叶单像素成像系统的水下退化函数进行估计,然后根据估计的退化函数实现目标空间谱反演。最后利用傅里叶变换对反演后的目标空间谱进行变换,最终获得目标的强度图像。理论分析和实验结果证明了该方法的有效性。利用该方法,能够有效地减小前向散射对成像质量的影响,提高了水下傅里叶单像元成像的重建结果质量。Abstract: The detection environment of underwater optical imaging is relatively complex. The forward scattering, back scattering and absorption greatly reduce the imaging quality of underwater optical imaging. Single pixel imaging is considered to be a suitable technique for underwater optical imaging because of its high noise resistance. The serious problem of underwater single pixel imaging system is that the structural light is required for illumination, and the forward scattering distorts the pre-generated speckle, when the speckle travels underwater. Therefore, the resolution of the reconstruction results decreases, making the reconstruction results blurry. In order to reduce the influence of the forward scattering on single pixel system, the reconstruction process of Fourier single pixel imaging should be improved. Underwater degradation function of Fourier single pixel imaging system was estimated in the spectrum domain, and then target spatial spectrum inversion was implemented based on the estimated degradation function. The image of the target can be obtained by transforming the target spatial spectrum with Fourier transform. The validity of the proposed method was proved by theoretical analysis and experimental results. Utilizing the proposed method, the influence of the forward scattering was reduced and the quality of reconstruction results of underwater Fourier single pixel imaging was improved.
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Table 1. FWHM results of PSF with various methods under different turbidity degrees
Milk amounts/mL DSPI FSPI Proposeed 20 0.0997 0.0565 0.0381 10 0.0764 0.0507 0.0366 5 0.0465 0.0496 0.0199 -
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