Foveated panoramic ghost imaging

Huan Cui; Jie Cao; Jie Cao; Qun Hao; Qun Hao; Qun Hao; Dong Zhou; Haoyu Zhang; Yingqiang Zhang

doi:10.1364/OE.482168

1. Introduction

Ghost imaging (GI) is an unconventional imaging technique that obtains object information by the correlation calculation between the modulated light pattern and the backscattered light intensity using a bucket detector [1–5]. GI has the advantages of high detection efficiency, wide spectrum range and low cost. Thus, it has been considered for using in many fields, especially in areas where the traditional imaging methods are technically unavailable or costly, such as infrared imaging [6], X-ray imaging [7], and terahertz imaging [8]. Recently, the proposal of panoramic ghost imaging (PGI) using only a curved mirror [9,10] extends the field of view (FOV) of GI to 360°, making GI a breakthrough in the related applications of wide FOV, such as security monitoring, visual navigation, panoramic situation awareness, etc. Due to the required multiple samples, most conventional GIs are degraded by low resolution or long sampling times [11]. There are already many works on GI optimization to improve the imaging quality with reduced measurements from reconstruction algorithms (such as compress sensing [12–15], deep learning [16–18]) to measurement matrices [19–21]. However, high-resolution GI and particularly high-resolution PGI with a wide FOV of 360° with reduced measurements and less time remains a serious challenge because of the large amount of data. Therefore, it is urgent to develop methods that achieve the coexistence of a wide FOV, and high resolution and high efficiency on GI used for wide FOV and high-resolution applications.

In practical applications of the wide FOV (such as monitoring [22], reconnaissance [23], VR display [24]), there is no need to acquire high-resolution information of all the scenes in the whole FOV. The region of interest (ROI), which is required to be of high resolution, occupies only a part of the whole FOV, while the other region of non-interest (NROI) does not need such a high resolution to provide the required information. Taking the monitoring of sea ship safety as an example, the sea occupies most of the area on the FOV, but we focus on ship information. Then, the imaging of the sea can maintain lower resolution, while the ships, even the ship's personnel, need for the higher-resolution imaging, which can make an efficient monitoring function. In other words, high-resolution imaging of only the ROIs is sufficient to meet the imaging requirements for most applications of wide FOV, and the resolution redundancy on the NROIs can be reduced to improve the imaging efficiency. One effective way to reduce the resolution redundancy is foveated imaging [25]. Inspired by the variant-resolution retina structure of the human eye, foveated imaging has been proposed with high resolution on the fovea area surrounded by the peripheral area with low resolution and has been widely used in many fields [26–29]. Recently, foveated imaging has also been introduced into the research on pattern design to enhance the data gathering capacity of traditional GI [30–37].

Based on the above observation and our previous work on PGI [9], we report a foveated panoramic ghost imaging (FPGI) to obtain foveated panoramic ghost images with high imaging efficiency by designing variant-resolution annular patterns flexibly and suitably. Specifically, first of all, to better fit the rectangular scenes in real applications, especially panoramic scenes that we focus on, and the ROIs having different lengths and widths, a flexible variant-resolution allocation structure via log-rectilinear transformation [38,39] is proposed, which can allocate the resolution of the ROI and NROI by setting parameters in the horizontal and vertical directions independently to flexibly meet different resolution requirements of GI in practical application. Later, since the pattern used for the projection in the PGI system is designed with a log-polar annular structure to match the convex geometry of the curved mirror [9], the variant-resolution annular pattern structure should be designed via the log-polar mapping from the proposed variant-resolution allocation structure to project on the curved mirror in the FPGI system, as shown in Fig. 1. In addition, since the FOV of the FPGI is 360°, the annular pattern structure is borderless without a fixed start-stop boundary. If the ROI is closed to the traditionally defined boundary, the traditional variant-resolution allocation structure would make the resolution loss of the NROI around the ROI but out of the boundary. Therefore, the variant-resolution allocation structure is further optimized to keep the ROI at any position in the center of the 360° FOV, called as a real fovea, by flexibly changing the initial position of the start-stop boundary of the annular pattern structure. The real fovea can further develop the central advantage of the fovea, especially in the applications where all the areas close to the ROI need to be of high resolution, reasonably reducing the resolution redundancy to avoid the resolution loss. In summary, the FPGI not only improves the imaging quality on the ROI with a high resolution and remain the lower-resolution imaging on the NROI with different required resolution reductions flexibly, but also reduces the reconstruction time due to the reduction of resolution redundancy, which provides an effective way to achieve the coexistence of a wide FOV, high resolution and high efficiency on GI, and further to promote the practical applications of GI with wide FOV.

Fig. 1. Schematic of the FPGI system. Given the designed variant-resolution annular pattern sequence via log-rectilinear transformation, the light modulated by the DMD is projected on the curved mirror through a projecting lens. After the multi-reflection of light between the curved mirror and the omnidirectional scene, the backscattered light intensity is collected by two single-pixel detectors to collect uniformly more and more light intensity from all directions. And then the unwrapping-free foveated panoramic ghost image can be reconstructed by the correlation calculation between the collected light intensity and the variant-resolution rectangular patterns via the log-polar mapping.

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2. Methodology

2.1 Variant-resolution annular pattern structure via log-rectilinear transformation

The conventional geometric transformation to obtain a retina-like variant-resolution allocation structure on GI is log-polar transformation [30,32], which reduces the resolution of the periphery along only a radial direction with a circular fovea. However, this conventional variant-resolution structure cannot be fully adapted to the rectangular scenes in real applications, especially panoramic scenes. For one thing, the circular geometry of fovea cannot be particularly suitable for the ROI where the length and width differ significantly. For another, the reduction of resolution only along the radial directional cannot meet the imaging requirements of some applications with different imaging resolution around the ROI. Therefore, we propose a variant-resolution allocation structure via log-rectilinear transformation to obtain more flexibility in the resolution allocation to meet different resolution requirements of GI in practical application. The proposed variant-resolution allocation structure via log-rectilinear transformation has several properties. First, the fovea can be designed as a rectangle according to the size of the required ROI to void the sampling redundancy of the extra area designed by the conventional circular fovea. Second, according to the imaging requirements for the NROI, the reduction of the resolution redundancy can be flexibly designed with different parameters in the horizontal and vertical directions independently based on the rectilinear transformation. Third, the resolution decay of the NROI in each direction is still exponential from the center point of the ROI to the sides based on the log transformation.

The variant-resolution allocation structure via log-rectilinear transformation is shown in Figs. 2 (b)-(c). Assume the uniform full resolution of PGI is P × Q, and the required variant resolution of FPGI is U × V. In the p(u)-q(v) coordinate system of one-to-one variant-resolution allocation, given that the central point of the ROI is (p₀, q₀), the radius of the ROI in the p direction is m_0 and that in the q direction is n_0; the resolution decay factor of the NROI in the p direction is α₁ and that in the q direction is α₂. Then, the model of the proposed variant-resolution allocation via log-rectilinear transformation can be expressed as:

(1)$$\left\{ \begin{array}{l} R_1^u = \left\{ \begin{array}{l} u,\quad \quad \quad \quad \quad \quad \quad \quad \;\;\;\;\;\;\;when\;1 \le u \le m\_0,\;\;\;\;\;\;\;\;\;\;\;\;\\ m\_0 \cdot \alpha_1^{u - m\_0},\quad \quad \quad \quad \quad \;\,\,\,when\;m\_0 \le u \le {{(U - 1)} / 2},\; \end{array} \right.\\ R_2^v = \left\{ \begin{array}{l} v,\quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,\,\,\;when\;1 \le v \le n\_0,\;\;\;\;\;\;\;\;\;\;\;\;\\ n\_0 \cdot \alpha_2^{v - n\_0},\quad \quad \quad \quad \quad \quad \,when\;n\_0 \le v \le {{(V - 1)} / 2},\; \end{array} \right.\\ p(u) = \left\{ \begin{array}{l} \max (1,round({p_0} - R_1^u)),\,\,\;\;\;\;when\;1 \le u \le {{(U - 1)} / 2},\;\;\;\,\\ {p_0},\quad \quad \quad \quad \quad \quad \quad \quad \,\quad \;when\;u = {{(U + 1)} / 2},\;\;\;\;\;\;\;\,\\ \min (round({p_0} + R_1^u),P),\,\,\;\;\;\;when\;{{(U + 1)} / {2 < u \le U}}, \end{array} \right.\\ q(v) = \left\{ \begin{array}{l} \max (1,round({q_0} - R_2^v)),\,\;\,\quad when\;1 \le v \le {{(V - 1)} / 2},\quad \\ {q_0},\quad \quad \quad \quad \quad \quad \quad \quad \;\,\quad when\;v = {{(V + 1)} / 2},\\ \min (round({q_0} + R_2^v),Q),\;\;\quad when{{\;\;(V + 1)} / {2 < v \le V}}, \end{array} \right.\\ U \in 2N + 1,V \in 2N + 1 \end{array} \right.$$

where $R_1^u$ represents the radius of the uth layer extending from the central point of the ROI in the p direction and $R_2^v$ represents the radius of the vth layer extending from the central point of the ROI in the q direction. p(u) provides the uth value of variant-resolution allocation in the p direction mapped in the uniform-resolution pattern structure, and q(v) provides the vth value of variant-resolution allocation in the q direction mapped in the uniform-resolution pattern structure. Note that, given a fixed ROI, the larger the value of α₁ or α₂ is, the smaller the corresponding values of U or V is, and the more resolution redundancy of the NROI is reduced to improve the imaging efficiency of the FPGI.

Fig. 2. Schematic of the variant-resolution annular pattern structure via log-rectilinear transformation. (a) Uniform-resolution rectangular pattern structure in log-polar coordinate with P × Q pixels. (b) Variant-resolution allocation structure with U × V pixels. (c) Detail of the one-to-one variant-resolution allocation structure in the p(u)-q(v) coordinate system. (d) Uniform-resolution annular pattern structure via log-polar mapping. (e) Variant-resolution annular pattern structure via log-polar mapping. (f) Detail of part of the variant-resolution annular structure.

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Furthermore, to match the convex geometry of the curved mirror used in the FPGI system shown in Fig. 1, the previous pattern used for the projection is designed with a log-polar annular structure [9]. The variant-resolution pattern used for the projection in the FPGI system is also designed with a log-polar annular structure accordingly. Therefore, based on the above variant-resolution allocation structure, the variant-resolution annular pattern structure in the FPGI with U rings and V non-uniform cells in each ring can be obtained via log-polar mapping, as shown in Fig. 2 (d)-(f). Combining Eq. (1), as p(u) represents the uth ring in the variant-resolution annular pattern structure mapped in the uniform-resolution annular pattern structure with P rings and q(v) represents the vth cell in each ring, and the variant-resolution annular pattern structure can be obtained with the following model:

(2)$$\left\{ \begin{array}{l} \varepsilon = \frac{{1\textrm{ + }\sin ({\pi /Q} )}}{{1 - \sin ({\pi /Q} )}}\\ r{c_1} = \frac{{{r_0}}}{{1 - \sin ({\pi /Q} )}}\\ {r_{p(u)}} = {r_0} \cdot {\varepsilon^{p(u)}}\quad \\ r{c_{p(u)}} = r{c_1} \cdot {\varepsilon^{p(u)\textrm{ - 1}}}\\ {\theta_{q(v)}} = q(v) \cdot \frac{{2\pi }}{Q}\textrm{ } \qquad (v = 1,2,3\ldots V)\\ {\xi_{p(u)}} = {\log_\varepsilon }(r{c_{p(u)}}) = {\log_\varepsilon }(r{c_1}) + p(u) - 1\textrm{ } \quad(u = 1,2,3\ldots U)\; \end{array} \right.,$$

where r₀ is the radius of the fovea blind hole in the annular pattern structure used to cover the center of the curved mirror with the self-reflection, ɛ is the increasing coefficient of the radius between adjacent rings in the uniform-resolution annular pattern structure, r_p_(u) is the outer radius of the uth ring in the variant-resolution annular pattern structure, rc_p_(u) is the center radius of the uth ring, θ_q_(v) is the angle of the vth cell of each ring starting at 0, and ξ_p_(u) is the mapped value of rc_p_(u) in the log-polar coordinate.

The above variant-resolution annular pattern structure is based on the design idea of the traditional foveated structure, which has a fixed start-stop boundary of FOV. However, the annular 360° FOV of the FPGI is borderless. Given that the designed annular pattern always starts at 0 in the angular direction of θ, there is an imaginary boundary from 0 to 2π. If the angular position of ROI is near 2π, in the log-polar coordinate, the areas of NROI near 0 would be the one with the farthest distances from ROI. With the variant-resolution allocation structure with traditional fovea, the area near 0 is allocated the lowest resolution to obtain the least information, filled in green in Fig. 3 (a). In the Cartesian coordinate system, however, the areas near 0 surrounding the ROI are also close to the boundary 2π, because the cyclic period is 2π, as shown in Fig. 3 (b), which should provide a higher resolution to obtain more information closely around the ROI. Therefore, considering the arbitrariness of the required ROI position, the variant-resolution allocation can be optimized by flexibly changing the initial position in the log-polar coordinate to center the ROI, as shown in Fig. 3 (c). Since the ROI is always kept in the center of the whole FOV flexibly, the optimized fovea is called as a real fovea. Based on the rotation invariance of the log-polar transformation [40], if the abscissa of the log-polar coordinate system shifts horizontally, the position of the Cartesian coordinate system will be rotated (anticlockwise is positive). Given that the central point of the required ROI is (p₀, q₀), the central point of the whole FOV is (p_c, q_c), and the corresponding rotated angle φ₀ to center the ROI can be calculated as follows:

(3)$$\left\{ \begin{array}{l} {\varphi_0} = ({{q_c} - {q_0}} )\cdot \frac{{2\pi }}{Q}\\ \theta^{\prime} = \theta + {\varphi_0} \end{array} \right.,$$

where θ represents the angle in the initial coordinate system and θ’ represents the angle in the rotated coordinate system where the ROI is in the center of the whole FOV. Then, the variant-resolution allocation can be optimized, as shown in Fig. 3 (e), where the area near 0 is given a higher-resolution allocation than that on the traditional allocation structure in Fig. 3 (a). The variant-resolution annular structure with a real fovea shown in Fig. 3 (f) can develop the central advantage of the fovea, reasonably reducing the resolution redundancy to avoid the resolution loss, especially in the applications where all the areas close to the ROI need to be of a high resolution.

Fig. 3. Schematic of variant-resolution annular pattern structure with real fovea. (a) The variant-resolution rectangular structure with traditional fovea in the log-polar coordinate, where the ROI near 2π is at the edge and the area in green is the farthest from the ROI. (b) The variant-resolution annular structure with traditional fovea in the log-polar coordinate, where the area in green is close to the ROI. (c) Variant-resolution structure with the real fovea via the horizontal shifting of the abscissa of the log-polar coordinate system to center the ROI in the whole FOV. (d) Variant-resolution annular structure with the real fovea with a rotated Cartesian coordinate system. (e) After optimization with the real fovea, the variant-resolution allocation structure in the original log-polar coordinate, where the area in green has a higher-resolution allocation than the one in (a). (f) Variant-resolution annular pattern structures with the real fovea in the original Cartesian coordinate system. Note that ξ'-θ’ is the shifted coordinate to center the ROI in the log-polar coordinate system from φ₀ to φ₀+ 2π and X'-Y’ is the rotated coordinate axis in the Cartesian coordinate system with the rotated angle φ₀.

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2.2 Unwrapping-free foveated panoramic ghost imaging

Based on our previous work on unwrapping-free PGI [9], Fig. 1 illustrates a schematic of the FPGI system. Given the imaging requirements of ROI and NROI in the omnidirectional scene, a variant-resolution annular pattern can be obtained according to the design method in Section 2.1. With the variant-resolution annular pattern projected onto the curved mirror, the back-scattered light intensity after the multi-reflection between the curved mirror and the omnidirectional scene, denoted by I, can be written as follows:

(4)$$I = \sum\limits_x {\sum\limits_y {{P_{v - annular}}(x,y) \cdot {O_{omnidirectional}}(x,y)} } ,$$

where x and y index the position of the omnidirectional scene in the Cartesian coordinate system, P_v-annular represents a variant-resolution annular pattern with U rings and V non-uniform cells of each ring, and O_{omnidirectional}represents the omnidirectional scene reflected on the curved mirror.

Although the variant-resolution allocation structure is designed with U × V pixels, the variant-resolution pattern is created by reformatting into the uniform high-resolution grid with P × Q pixels to ensure high-resolution imaging on the ROI. Therefore, given the two transformation matrices in the p direction and the q direction respectively, the variant-resolution rectangular pattern via log-polar mapping can be obtained as follows:

(5)$${P_{v - rectangular}} = {A_p}{P_n}A_q^T,$$

where P_n is the variant-resolution allocation structure matrix with U × V, P_{v-rectangular} is the variant-resolution rectangular pattern reformatted in the high-resolution grid with P × Q, A_p is a P × U transformation matrix mapping the U-element variant-resolution structure to the P-element high-resolution grid in the p direction, and A_q is a Q × V transformation matrix mapping the V-element variant-resolution structure to the Q-element high-resolution grid in the q direction. A_p and A_q are binary matrices where the locations of the ‘ones’ in Column n denote the high-resolution pixels that belong to the nth variant-resolution structure in the corresponding direction.

According to the previously proposed unwrapping-free reconstruction of PGI, the variant-resolution panoramic ghost image can be directly reconstructed by the correlation between the collected light intensity using the projected variant-resolution annular pattern and the variant-resolution rectangular pattern via log-polar mapping. To obtain better imaging quality with fewer measurements, an efficiency compress sensing reconstruction algorithm of total variation (TV) is used on the unwrapping-free reconstruction of the FPGI.

(6)$$\begin{aligned} \min \quad \textrm{ }&{||c ||_{{l_1}}}\\ s.t.\textrm{ }\quad&H{O_{v - panoramic}} = c\quad \quad \quad ,\\& \textrm{ }{{\boldsymbol P}_{v - rectangular}}{O_{v - panoramic}} = {\boldsymbol I} \end{aligned}$$

where H represents the gradient calculation matrix, c is the corresponding coefficient vector, l₁ represents the l₁ norm, I ∈R^T^× 1 denotes the measurement matrix of the projected variant-resolution annular patterns (T is the number of measurements), P_{v-rectangular} ∈R^T^× M represents the variant-resolution rectangular pattern sequence matrix via log-polar mapping (M = P × Q is the pixel number of the variant-resolution pattern on the high-resolution grid), and O_v-panoramic represents the required variant-resolution panoramic ghost image of the omnidirectional scene.

3. Results

3.1 Experimental setup

The experimental setup of the FPGI system is the same as that used in our previous work on PGI system [9]. Based on the schematic of FPGI system shown in Fig. 1, the light emitted from a light emitting diode (LED) operating at 500-700 nm (@20W) is modulated with the proposed variant-resolution annular pattern, which is designed based on the imaging requirement and generated by a DMD (Texas Instruments DLP4100; refresh rate, 10KHz). Through a projecting lens (focus length, 150 mm), the modulated light illuminates the curved mirror made of a plane-convex lens coated with a strong reflective film (diameter, 50.8 mm; central thickness, 16.3 mm; radius of curvature, 30.91 mm), and then generates the multi-reflection between the curved mirror and the omnidirectional scene. Later, the backscattered light intensity is collected by two photodetectors (Thorlabs PDA36A; active area, 13 mm²) to collect uniformly more and more light intensity from all directions and converted to a digital signal by a data acquisition board (DAQ, PICO6404E). Finally, the unwrapping-free foveated panoramic ghost image is reconstructed on the computer (6226R CPU × 2, 128GB RAM). Note that, in order to observe the omnidirectional scene conveniently, the object is attached to a hollow cylinder (height, 50 mm; base diameter, 75 mm).

Moreover, since the FPGI focuses on improving the imaging quality of the required ROI on the wide FOV by reducing the resolution redundancy of the NROI, the two classic evaluation indexes to analyze the imaging quality, peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM), are used to compare quantitatively the difference on the ROI between the uniform-resolution PGI and variant-resolution FPGI, which are defined as follows:

(7)$$\left\{ \begin{aligned} &{\text{PSNR}} = 10{\log _{10}}\frac{{{{\left( {{2^k} - 1} \right)}^2}}}{{{\text{MSE}}}} \hfill \\ &{\text{MSE}} = \frac{1}{S}\sum\limits_{i,j} {{{\left( {F'(i,j) - F(i,j)} \right)}^2}} \; \hfill \\ &\;{\text{SSIM}}{{\text{M}}_{i,j}} = \frac{{\left( {2\mu \mu ' + {c_1}} \right)\left( {2w + {c_2}} \right)}}{{\left( {{\mu ^2} + u{'^2} + {c_1}} \right)\left( {{\sigma ^2} + \sigma {'^2} + {c_2}} \right)}} \hfill \\ \end{aligned} \right.,$$

where F(i, j) is the ground truth of the ROI, F'(i, j) is the fovea in the reconstructed image, MSE represents the mean square error; S is the total pixels of the fovea; k is the number of bits and set as 8. µ and µ’ are the average value of F(i, j) and F'(i, j); σ and σ’ are the variance of F(i, j) and F'(i, j). w is the covariance between F(i, j) and F'(i, j). c₁= (k₁×L)² and c₂= (k₂×L)² are the constant with k₁= 0.01, k₂= 0.03 and L = 1. Note that, the higher the PSNR, the better the imaging quality of the fovea. In addition, the larger the value of SSIM is from 0 to 1, the reconstructed fovea is closer to the object of ROI.

3.2 FPGI with variant resolution

With the experimental setup of the FPGI system, FPGI experiments with variant-resolution are conducted. First, the simple object is a string of letters “PANORAMIC” attached to a hollow cylinder to make an omnidirectional 360° FOV, as shown in Fig. 4 (a). Accordingly, the uniform-resolution annular pattern structure is designed with 20 rings and 102 cells in each ring, as shown in Fig. 4 (b). To observe the effect of the FPGI with different reductions in the resolution redundancy, the three sets of parameters on the variant-resolution pattern structure can be seen in Table 1. Given the position of the required ROI “ORA”, highlighted by a red box in Fig. 4(a), and the different resolution decay factors of NROI in the p direction and q direction, the variant-resolution annular pattern structures via log-rectilinear transformation can be designed based on Eq. (1) and Eq. (2), and the corresponding pattern samples randomly filled by 0 or 1 are given in Fig. 4 (c)-(e). As seen in Table 1, as the resolution decay factors of the NROI increase, the total pixels of the three variant-resolution pattern structures decrease in the corresponding direction, resulting in a greater reduction in the resolution redundancy on the NROI. For example, for ‘Variant-resolution 2’, the resolution decay factor in the q direction α₂ increases from 1.08 to 1.16, and the pixel number of the variant-resolution pattern in the q direction is reduced from 66 to 50.

Fig. 4. FPGI with variant resolution. (a) A sample object to make an omnidirectional 360° FOV, where the ROI is highlighted by the red box. (b) A uniform-resolution annular pattern sample randomly filled by 0 or 1 with 20 × 102 pixels and the corresponding uniform-resolution rectangular pattern via log-polar mapping. (c) With the parameters of ‘Variant-resolution 1’ in Table 1, a random variant-resolution annular pattern sample with 19 × 66 pixels and the log-polar mapped variant-resolution rectangular pattern. (d) With the parameters of ‘Variant-resolution 2’ in Table 1, a random variant-resolution annular pattern sample with 19 × 50 pixels and the log-polar mapped variant-resolution rectangular pattern. (e) With the parameters of ‘Variant-resolution 3’ in Table 1, a random variant-resolution annular pattern sample with 16 × 38 pixels and the log-polar mapped variant-resolution rectangular pattern. (f) With the same measurements of full sampling, the unwrapping-free reconstructed panoramic ghost images using the four annular pattern sequences with different structures of (b)-(e) respectively, and the corresponding magnified views of the ROIs. (g) Quantitative comparison on the PSNR of fovea of the reconstructed image using a uniform-resolution annular pattern structure and the three variant-resolution pattern structures with various parameters shown in (b)-(e) at different sampling rates. (h) Quantitative comparison on the reconstruction time of unwrapping-free PGI using the four different pattern structures at different sampling rates. Note that, the sampling ratio refers to the uniform-resolution panoramic ghost image with 20 × 102 pixels, because the variant-resolution panoramic ghost image is also reformatted into the uniform-resolution grid to ensure high-resolution imaging on the ROI.

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Table 1. Parameters of the four pattern structures in Fig. 4

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Using the four annular pattern sequences with different parameters given in Table 1, the corresponding unwrapping-free panoramic ghost images were reconstructed. Figure 4 (f) shows the four reconstructed images with the same measurements of full sampling and the corresponding magnified views of the ROIs. According to the observation, both subjectively and quantitatively, the imaging quality of the three foveated variant-resolution panoramic images is better than that of the uniform-resolution panoramic image with noisy background. In addition, the quantitative curves of the PSNR at different sampling rates are given in Fig. 4 (g) to further analyze the imaging quality using the three variant-resolution annular pattern structure. With the greater reduction in the resolution redundancy of NROI from ‘variant-resolution 1’ to ‘variant-resolution 3’, the PSNR of the fovea is mainly higher at the same sampling ratio, because the obtained redundant information on the NROI is less; however, the resolution on the NROI is too low to be distinguished, especially in ‘variant-resolution 3’. Therefore, it is necessary to balance the PSNR of the ROI and the required information on the NROI by setting the parameters of the variant-resolution pattern structure flexibly in accordance with the various imaging requirements in different situations. Moreover, due to the reduction in the total resolution, the reconstruction time on the FPGI is always less than that on the traditional PGI using the same unwrapping-free reconstruction algorithm, as shown in Fig. 4 (h). In particular, at a sampling rate of 1, all the reconstruction time of the three variant-resolution foveated panoramic ghost images are closed to 0.4 s, while the reconstruction time of the traditional uniform-resolution panoramic ghost image is larger than 1.2 s, which is 3 times more than that of the image used on the FPGI.

A complex scene is also used to conduct the experiments to observe the effect of the FPGI once more, as shown in Fig. 5. Here, the uniform-resolution annular pattern structure is designed with 40 rings and 204 cells in each ring to obtain higher-resolution images. The required ROI is highlighted by the red box in Fig. 5 (a), and the random pattern samples with different pattern structures designed via the parameters in Table 2 are shown in Figs. 5 (b)-(e) respectively.

Fig. 5. FPGI with variant resolution. (a) A complex object to make an omnidirectional 360° FOV, where the ROI is highlighted by the red box. (b) A uniform-resolution annular pattern sample randomly filled by 0 or 1 with 40 × 204 pixels and the log-polar mapped uniform-resolution rectangular pattern. (c) With the parameters of ‘Variant-resolution 1’ in Table 2, a random variant-resolution annular pattern sample with 29 × 98 pixels and the log-polar mapped variant-resolution rectangular pattern. (d) With the parameters of ‘Variant-resolution 2’ in Table 2, a random variant-resolution annular pattern sample with 20 × 98 pixels and the log-polar mapped variant-resolution rectangular pattern. (e) With the parameters of ‘Variant-resolution 3’ in Table 2, a random variant-resolution annular pattern sample with 20 × 50 pixels and the log-polar mapped variant-resolution rectangular pattern. (f) With the same measurements of full sampling, the unwrapping-free reconstructed panoramic ghost images with the four annular pattern sequences with different structures of (b)-(e) respectively, and the corresponding magnified views of the ROIs. (g) Quantitative comparison on the SSIM of fovea of the reconstructed image using a uniform-resolution annular pattern structure and the three variant-resolution pattern structures with different parameters shown in (b)-(e) at different sampling rates. (h) Quantitative comparison on the reconstruction time of PGI with the four different pattern structures at different sampling rates. Note that, the sampling ratio refers to the uniform-resolution panoramic ghost image with 40 × 204 pixels.

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Table 2. Parameters of the four pattern structures in Fig. 5

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Figure 5 (f) shows the four unwrapping-free reconstructed panoramic ghost images with the same measurements at the full sampling rate, which use the uniform-resolution annular pattern sequence and the three variant-resolution annular pattern sequences with different parameters given in Table 2 respectively. It is clear that the uniform-resolution panoramic ghost image is much noisier, and the letter ‘B’ on the ROI is flat. In contrast, the backgrounds of the three variant-resolution panoramic ghost images are clean, and the letter ‘B’ on the ROI is structurally fuller. In addition, the three variant-resolution foveated panoramic ghost images also show the flexible acquisition on the NROI according to the imaging requirements in the horizontal and vertical directions independently by the variant-resolution allocation via log-rectilinear transformation. If the NROI around the ROI in the vertical direction is required to obtain major information, such as ‘Variant-resolution 1’, the resolution decay factor in the p direction α₁ is set with a small value of 1.16. Conversely, such as ‘Variant-resolution 2’, α₁ is set with a larger value of 1.69 to reduce the resolution redundancy in the vertical direction that is not needed, while α₂ is set with a small value of 1.06 to independently obtain the required information on the NROI around the ROI in the horizontal direction.

Moreover, to quantitatively analyze the structural information of the ROI, the SSIM of fovea at different sampling rates are given in Fig. 5 (g). First, the SSIM of fovea of the uniform-resolution PGI is always lower than that of the variant-resolution FPGI, which proves that the imaging quality of the ROI in our proposed FPGI is always better than that of the traditional PGI to achieve GI with high imaging quality and wide FOV. In addition, from the trends of the three curves on the FPGI with different variant-resolution pattern structures, the SSIM of ‘variant-resolution 3’ increases most rapidly and then remains steady with increasing sampling rate, where the reduction in the resolution redundancy of NROI on ‘variant-resolution 3’ is the greatest with the fewest total pixels shown in Table 2. For example, for the SSIM of fovea value reaching 0.7, the SSIM of fovea on ‘variant-resolution 1’ reaches the value at the sampling rate of 0.6, the SSIM of fovea on ‘variant-resolution 2’ reaches the value at the sampling rate of 0.5, while the SSIM on ‘variant-resolution 3’ can reach the value at the sampling rate of 0.3. The lower sampling rate means the fewer measurements, therefore, if fewer measurements are required to obtain higher-quality imaging on the ROI, the obtained information on the NROI should be less with a greater reduction in the resolution redundancy. With the same conclusion as the one on the first experiments using a sample object, it is necessary to balance the imaging quality of the ROI and the required information on the NROI by setting the parameters of the variant-resolution pattern structure flexibly in accordance with the various imaging requirements in different situations. Moreover, the obvious difference in the reconstruction time in Fig. 5 (h) shows that the FPGI can reduce the imaging time by reducing the resolution redundancy to improve the imaging efficiency of PGI. Overall, the above experiments show that the proposed FPGI can achieve the coexistence of a wide FOV, high quality and high efficiency on GI to promote the practical applications of GI with a wide FOV.

3.3 FPGI with real fovea

The ROIs in the above experiments are chosen at the center of the whole 360° FOV in the original start-stop boundary of [0, 2π]. However, in the practical applications, the ROI can be in any position of the whole FOV according to the imaging requirements. If the ROI is in the position close to the original start-stop boundary, such as the little engine framed in red in Fig. 6 (a), with the traditional-fovea variant-resolution annular structure, the resolution of the areas circled in green in Fig. 6 (c) would be very low to lose the information, because the areas are supposed to be the farthest from the ROI in the original start-stop boundary from 0 to 2π. However, the areas are close to the ROI in the actual position because it is an annular structure with a period 2π. Therefore, in order to keep the ROI in the center of the whole FOV, the variant-resolution annular pattern structure with a real fovea is optimized to avoid the resolution loss of the areas close to the ROI by flexibly changing the initial position of the start-stop boundary. Given that the central point of the ROI is (19,26), the central point of the whole FOV is (21,103); according to the Eq. (3), the optimized variant-resolution annular pattern structure of this ROI can be obtained, and one random annular pattern sample is shown in Fig. 6 (c).

Fig. 6. FPGI with a real fovea. (a) The object with an ROI located in the position close to the original start boundary 0. (b) A uniform-resolution annular pattern sample randomly filled by 0 or 1 with 40 × 204 pixels. (c) A traditional-fovea variant-resolution annular pattern sample with 39 × 86 pixels, where the resolution of the areas circled in green is lost. (d) A real-fovea variant-resolution pattern sample with 39 × 101 pixels, which avoids the resolution loss of the areas circles in green. (e) The unwrapping-free uniform-resolution panoramic ghost image with full sampling in the original coordinate and the corresponding image where the ROI is at the center. (f) The traditional-fovea variant-resolution panoramic ghost image with full sampling in the original coordinate and the corresponding image with centering the ROI, where the areas marked by the green box are not as clear as the areas marked by the blue box (Quantitatively, the SSIM of the green box is 0.35 while the SSIM of the blue box is 0.61). (g) The real-fovea variant-resolution panoramic ghost image with full sampling and the corresponding image with centering the ROI, where the areas marked by the green box are as clear as the areas marked by the blue box (Quantitatively, the SSIM of the green box is improved to 0.60, which is close to the SSIM of the blue box of 0.61).

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Using the annular pattern sequences shown in Figs. 6 (b)-(d), the reconstructed panoramic ghost images in the original log-polar coordinate from 0 to 2π and the corresponding image in the changed log-polar coordinate centering the ROI are given in Figs. 6 (e)-(g). First, in the original log-polar coordinate, the areas marked by the blue box are visually very far from the ROI; however, the distance to the ROI is actually the same as the distance between the areas marked by the blue box with the ROI in the omnidirectional FOV. With the traditional-fovea variant-resolution annular patterns, as shown in Fig. 6 (f), the areas marked by the green box are not as clear as the areas marked by the blue box which makes the resolution loss close around the ROI, especially in applications that require simultaneous attention to the information closely surrounding the ROI. With the real-fovea variant-resolution annular patterns, as shown in Fig. 6 (g), the areas marked by the green box are as clear as those marked by the blue box. The proposed variant-resolution annular pattern structure with a real fovea can be proven to reasonably reduce the resolution redundancy to avoid the resolution loss.

3.4 FPGI with multiple foveae

In practical applications, there may be more than one required ROI in the wide FOV of 360°, and the variant-resolution annular pattern structure of FPGI needs to be designed with multiple foveae to meet the imaging needs. Given the required multiple ROIs, the corresponding variant-resolution allocation structures with individual fovea can be designed independently according to the imaging requirements. Then, the whole NROI is divided into parts for surrounding each ROI based on the position relation with the required ROIs. Finally, the variant-resolution annular pattern structure with multiple foveae can be obtained by merging the corresponding parts on each variant-resolution allocation structure with the individual fovea.

Here, we present the two experimental examples to show the FPGI with multiple foveae: one with dual foveae and the other with treble foveae, as shown in Fig. 7. As seen from the comparison on the imaging quality and the reconstruction time between the reconstructed uniform-resolution panoramic ghost images and the variant-resolution panoramic ghost images with multiple foveae, although there are multiple ROIs, the FPGI is still of higher quality on each ROI than the one on the traditional PGI and retains the lower-resolution imaging on the surrounding NROIs; and the reconstruction time of the FPGI with multiple foveae is still less than that of the traditional uniform-resolution PGI, especially with the increase of sampling rate, the difference of the reconstruction time is more obvious. Certainly, the reduction of resolution redundancy on the NROIs for different requirements can be achieved by flexibly designing different variant-resolution annular pattern structures with multiple foveae.

Fig. 7. FPGI with multiple foveae. (a) The object with the two ROIs marked in red and green respectively. (b) A random variant-resolution annular pattern sample with double foveae. (c) Reconstructed uniform-resolution panoramic ghost image with full sampling and the magnified views of the two ROIs following the quantitative evaluation of the PSNR and SSIM. (d) Reconstructed variant-resolution panoramic ghost image using the patterns shown in (b) with full sampling and the magnified views of the two ROIs following the quantitative evaluation of the PSNR and SSIM. (e) The object with the three ROIs marked in red, green and blue respectively. (f) A random variant-resolution annular pattern sample with treble foveae. (g) Reconstructed uniform-resolution panoramic ghost image with full sampling and magnified views of the three ROIs following the quantitative evaluation of the PSNR and SSIM. (h) The reconstructed variant-resolution panoramic ghost image using the patterns shown in (f) with full sampling and magnified views of the three ROIs following the quantitative evaluation of the PSNR and SSIM. (i) Comparison on the reconstruction time between the traditional uniform-resolution PGI and the FPGI with the two ROIs at different sampling rates. (j) Comparison on the reconstruction time between the traditional uniform-resolution PGI and the FPGI with the three ROIs at different sampling rates. Note that, the sampling ratio refers to the uniform-resolution panoramic ghost image with 40 × 204 pixels.

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4. Discussion and conclusion

Here, we report an efficient panoramic ghost imaging method called the FPGI, which provides a way to achieve the coexistence of a wide FOV, high resolution and high efficiency on GI. In the FPGI, the high-resolution imaging is only on the ROIs of the whole FOV, while the NROIs are imaged with low resolution to reduce resolution redundancy to improve the imaging efficiency in the 360° wide FOV. To achieve the FPGI, we perform some work in this paper. First, we propose a variant-resolution allocation structure via log-rectilinear transformation to flexibly meet different resolution requirements of GI, which can allocate the resolution of ROIs and NROIs by setting parameters in the horizontal and vertical directions independently. Then, the variant-resolution annular pattern structure used for projection on FPGI is designed via log-polar mapping from the proposed variant-resolution allocation structure. Next, in order to reasonably reduce the resolution redundancy and avoid the resolution loss, the real-fovea variant-resolution allocation structure is further optimized to keep the ROI at any position in the center of the 360° FOV by flexibly changing the initial position of the start-stop boundary of the annular pattern structure. In addition, we also describe the design of an FPGI with multiple foveae to meet imaging needs with multiple ROIs in practical applications with a wide FOV. The experimental results demonstrate that, compared to the traditional PGI, the proposed FPGI can improve the imaging quality on the ROIs with a high resolution and flexibly remain the lower-resolution imaging on the NROI with different required resolution reductions in the horizonal and vertical direction. Simultaneously, the FPGI can reduce the reconstruction time to improve the imaging efficiency due to the reduction of the resolution redundancy.

Furthermore, the proposed FPGI can potentially be further enhanced by combining existing interesting ideas and mature technologies. For example, with deep learning, first, the pattern optimization with a priori knowledge of the ROIs can be achieved by training with neural networks, such as a deep Q-learning network [17]. Second, the intelligent selection of the ROIs can be achieved by using an object detection algorithm with deep learning, such as generative adversarial nets [34]. Third, the end-to-end deep-learning approach [16] can be introduced to directly correspond to each pixel of the variable-resolution structure to reduce measurements and reduce the imaging time. Besides, we have demonstrated the proposed FPGI in static scenes, an efficient and high-quality FPGI in a dynamic scene can be obtained in the near future by further optimizing patterns and reconstruction algorithms to enhance the functions of FPGI system to open up possibilities for more applications with a wide FOV, such as the motion tracking with an adaptive fovea [30] in a wide FOV. Moreover, the proposed variant-resolution allocation structure and the designed variant-resolution annular structure are not only for achieving the FPGI, but also provide new ideas and references for the practical process of GI and the applications on special scenes, such as circular FOV.

Funding

Beijing Municipal Natural Science Foundation (4222017); Ensan Foundation (2022011); National Natural Science Foundation of China (62275017, 62275022); Funding of foundation enhancement program under Grant (2019-JCJQ-JJ-273).

Acknowledgments

The authors thank the editor and the anonymous reviewers for their valuable suggestions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Foveated panoramic ghost imaging

Abstract

1. Introduction

2. Methodology

2.1 Variant-resolution annular pattern structure via log-rectilinear transformation

2.2 Unwrapping-free foveated panoramic ghost imaging

3. Results

3.1 Experimental setup

3.2 FPGI with variant resolution

3.3 FPGI with real fovea

3.4 FPGI with multiple foveae

4. Discussion and conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (2)

Equations (7)

Optics Express

Pattern structure	Uniform-resolution	Variant-resolution 1	Variant-resolution 2	Variant-resolution 3
Position of ROI	Central point (13,53), Radius m_0 = 6 (p direction), n_0 = 16 (q direction)
Resolution decay factors of NROI	-	α₁= 1.19 (p direction)α₂= 1.08 (q direction)	α₁= 1.19 (p direction)α₂= 1.16(q direction)	α₁= 1.67 (p direction)α₂= 1.79 (q direction)
Total pixels	20 × 102	19 × 66	19 × 50	16 × 38

Pattern structure	Uniform-resolution	Variant-resolution 1	Variant-resolution 2	Variant-resolution 3
Position of ROI	Central point (23,95), Radius m_0 = 7 (p direction), n_0 = 16 (q direction)
Resolution decay factors of NROI	-	α₁= 1.16 (p direction)α₂= 1.06 (q direction)	α₁= 1.69 (p direction)α₂= 1.06 (q direction)	α₁= 1.69 (p direction)α₂= 1.26 (q direction)
Total pixels	40 × 204	29 × 98	20 × 98	20 × 50