Open Access
Add to BibSonomy
Abstract
A single-pixel compressive imaging technique that uses differential modulation based on the transformation of discrete orthogonal Krawtchouk moments is proposed. In this method, two sets of Krawtchouk basis patterns are used to differentially modulate the light source, then the Krawtchouk moments of the target object are acquired from the light intensities measured by a single-pixel detector. The target image is reconstructed by applying an inverse Krawtchouk moment transform represented in the matrix form. The proposed technique is verified by both computational simulations and laboratory experiments. The results show that this technique can retrieve an image from compressive measurements and the real-time reconstruction. The background noise can be removed by the differential measurement to realize the excellent image quality. Moreover, the proposed technique is especially suitable for the single-pixel imaging application that requires the extraction of the characteristics at the region-of-interest.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single-pixel imaging (SPI) technique [1–3], which is essentially different from traditional array imaging technique in imaging mechanism, has an obvious advantage in hardware complexity and industrial cost. In recent years, SPI has achieved successes in various applications such as infrared imaging [4], terahertz imaging [5], lidar [6], bioluminescence microscopic imaging [7,8] and gas detection [9]. The compressive sensing theory can restore original signals almost perfectly by a small amount of non-adaptive linear measurements for sparse or compressible signals, and breaks the limits of the Nyquist-Shannon sampling theorem [10]. Therefore, the single-pixel imaging based on compressive sensing (CS-SPI) algorithm [1,6,10] is widely applied. However, the CS-SPI algorithm needs to be iterative to solve a class of convex optimization problem, and the reconstruction result depends on the robustness and accuracy of reconstruction algorithm [11–13], which presents great challenges in the stability of computing time and imaging quality.
Another type of SPI algorithm is based on the deterministic orthogonal basis patterns and the matrix inverse algorithm. Hadamard single-pixel imaging (Hadamard-SPI) [14,15], Fourier single-pixel imaging (Fourier-SPI) [15–19] and Discrete cosine single-pixel imaging (Discrete cosine-SPI) [20] are representative SPI techniques using deterministic orthogonal basis patterns. Hadamard-SPI method uses the Hadamard basis patterns to modulate the light source, acquires the Hadamard spectrum of the target image, and reconstructs the target image through the inverse Hadamard transform. Similarly, Fourier-SPI method uses the Fourier basis patterns for modulation while Discrete cosine-SPI method uses the discrete cosine basis patterns to recover images via the corresponding inverse transform. Two advantages are brought by these types of SPI algorithms. The first advantage is the perfect reconstruction in principle when the image is fully sampled in the transformation domain because the basis patterns form a complete orthogonal set [15]. The second advantage is the rapid image reconstruction calculation through the corresponding inverse transform, thus the huge measurement matrix does not need to be stored and the reconstruction time can be greatly reduced.
In recent years, with the rapid development of the algebraic invariant theory, the image moment technology has been widely used in the field of image compression, computer vision, pattern recognition, digital watermark and so on [21–24]. The Krawtchouk moments are a set of discrete orthogonal moments, which were introduced in image analysis by Pew-Thian Yap et al [25]. There is no calculation error caused by the process of discretization. The Krawtchouk moments have the special ability to extract local characteristics from any region-of-interest in the image. According to the comprehensive analysis of the image moments technique such as Zernike, Legendre, Tchebichef moments and so on, the Krawtchouk moments can better express image information and have small noise sensitivity [25]. The basis function of the Krawtchouk moments is orthogonal in the image coordinate space, and the transformation of Krawtchouk moments can be invertible. Therefore, it is easy to reconstruct the target image. On the other hand, the image moments of each order are independent, thereby having the minimum information redundancy.
Considering the above advantages of Krawtchouk moments, a new single-pixel compressive imaging method based on the transformation of Krawtchouk moments (KM-SPI) is proposed in this paper. Two sets of orthogonal basis patterns calculated by the transform kernel of Krawtchouk moments are used to differentially modulate the light source. Then the Krawtchouk moments of the target image are acquired from the light intensities measured by a single-pixel detector. Finally, the target image is recovered through the inverse Krawtchouk moment transform. The simulations and laboratory experiments demonstrate that the proposed method can realize high quality imaging with compressive measurements of a few low-order moments. Any region-of-interest in an image can be reconstructed by the proposed method due to the special feature extraction ability of the Krawtchouk moments. On the other hand, the proposed method can realize rapid real-time reconstruction, because the reconstruction calculation is based on the inverse Krawtchouk moment transform represented in the matrix form. The layout of this paper is organized as follows. In Section 2, a brief description about Krawtchouk moments is given, and the imaging principles and deduction methods are introduced. In Section 3, simulations and laboratory experiments are performed to evaluate the proposed methods. In Section 4, the conclusions of this work are summarized.
2. Principles and methods
2.1 Krawtchouk moments
The Krawtchouk moments are discrete orthogonal moments derived from the Krawtchouk polynomials. The $n$-th order Krawtchouk polynomial is defined as [25–27]
The set of Krawtchouk polynomials $\{K_n (x;p,N)\}$ forms a complete discrete orthogonal basis with weight function
and satisfies the orthogonality condition where $n,\;m=1,2,\ldots ,N$; $\delta$ is the Kronecker delta function and $\rho (n;p,N)$ is the squared-norm defined as To achieve the numerical stability of classical Krawtchouk polynomials caused by the hypergeometric function, a set of weighted Krawtchouk polynomials ${\bar {K}_n (x;p,N)}$ is used as such that the orthogonality condition becomes In order to make the computation of the weighted Krawtchouk polynomials less demanding the processor resource, the three term recursive algorithm [28,29] is usually applied asSimilarly, $\omega (x;p,N)$ can be calculated recursively using the following formula.
with ${\omega (0;p,N)} = {{(1-p)}^N}$.Based on the above definitions, the orthogonal discrete Krawtchouk image moments of order $(n+m)$, for an ${N \times M}$ image with intensity function, $f(x,\;y)$, is defined as
For the special case $p_1$ = $p_2$ = $p$ = 0.5, the weighted Krawtchouk polynomials have the following symmetry [25,31]:
In calculating the Krawtchouk moments of an image, it is possible to simplify the operation by using the symmetry property, not only to save the computational cost, but also greatly reduce the cumulative error caused by iterative operation and improve the accuracy of image reconstruction. For the case $p \ne$ 0.5, some algorithms [29,31] have been proposed based on the symmetry property of Krawtchouk polynomial coefficients along the diagonals of the polynomial array, and the symmetry property can also be used to simplify operations.2.2 Proposed single-pixel compressive imaging method
The proposed method is based on the theorem of the Krawtchouk moments. The imaging steps are as follows:
- (1) Calculating and generating a series of Krawtchouk basis patterns. It can be found from Eq. (11) that the Krawtchouk moments are in fact the inner product of intensity function $f(x,\;y)$ and ${\bar {K}_{n} (x;{p_1},{N-1})} {\bar {K}_{m} (y;{p_2},{M-1})}$. The Krawtchouk moments of the object consist of a set of Krawtchouk polynomials coefficients. Each coefficient corresponds to a unique Krawtchouk basis pattern.
- (2) According to the positive and negative values of the pixels, the basis patterns are decomposed into corresponding pattern pairs so as to realize differential measurement to remove the noise generated by background lighting.
- (3) Loading the Krawtchouk basis pattern clusters onto the spatial optical modulator to modulate the object in turn, and concentrating and measuring the reflected light intensities by a single-pixel detector.
- (4) Generating the Krawtchouk moments based on the measured intensities which are equivalent to the products of the interaction between the Krawtchouk basis patterns and the object.
- (5) Reconstructing the image by the inverse moment transform in the matrix form.
The principle and deduction process are explained below. The two-dimensional Krawtchouk moment transform and the inverse transform of an ${N \times M}$-pixel image are defined as Eq. (11) and Eq. (12) respectively. Using the transform kernel as the Krawtchouk basis pattern for spatial optical modulation, its expression is as Eq. (15). Fig. 1(a) shows part of the Krawtchouk basis patterns generated from Eq. (15) for $N$ = $M$ = 128 and $p_1$ = $p_2$ = 0.5.
Fig. 1. Part of the Krawtchouk basis patterns for $N$ = $M$ = 128 and $p_1$ = $p_2$ = 0.5. (a) Part of the patterns $P(x,\;y;n,\;m)$ generated from Eq. (15); (b) The pair of patterns ${P^+}(x,\;y;n,\;m)$ and ${P^-}(x,\;y;n,\;m)$ split from $P(x,\;y;n,\;m)$ by Eq. (17).
For a specified order $(n_i+m_j)$, the Krawtchouk basis pattern $P(x,\;y;n_i,\;m_j)$ is used to modulate the object and converge the reflected light intensity. The response of the single-pixel detector $E(n_i,\;m_j)$ is
The background lighting may produce some noise. The differential measurement is used to solve this problem. $P(x,\;y;n,\;m)$ is splitted into a pair of patterns ${P^+}(x,\;y;n,\;m)$ and ${P^-}(x,\;y;n,\;m)$, which satisfy the following relation:
For pattern ${P^+}(x,\;y;n,\;m)$, under the specified order $(n_i+m_j)$ and the spatial coordinate $(x_k,\;y_l)$, if the value of $P(x_k,\;y_l;n_i,\;m_j)$ is positive, the value of ${P^+}(x_k,\;y_l;n_i,\;m_j)$ is equivalent to the value of $P(x_k,\;y_l;n_i,\;m_j)$, otherwise, the value of ${P^+}(x_k,\;y_l;n_i,\;m_j)$ is zero. Similarly, for pattern ${P^-}(x,\;y;n,\;m)$, under the specified order $(n_i+m_j)$ and the spatial coordinate $(x_k,\;y_l)$, if the value of $P(x_k,\;y_l;n_i,\;m_j)$ is negative, the value of ${P^-}(x_k,\;y_l;n_i,\;m_j)$ is the absolute value of the value of $P(x_k,\;y_l;n_i,\;m_j)$, otherwise, the value of ${P^-}(x_k,\;y_l;n_i,\;m_j)$ is zero. Fig. 1(b) shows the pair of patterns ${P^+}(x,\;y;n,\;m)$ and ${P^-}(x,\;y;n,\;m)$ which is split from $P(x,\;y;n,\;m)$ shown in Fig. 1(a).After differential operation, the following equation is obtained, and it can be seen that the response of the detector to background lighting $E_0$ has been eliminated.
Fig. 2. Procedure of single-pixel imaging based on Krawtchouk moments ($N$ = 128, $M$ = 128; $p_1$ = 0.5, $p_2$ = 0.5). (a) Complete sampling ($N_{max}$ = 128, $M_{max}$ = 128); (b) Compressive sampling ($N_{max}$ = 41, $M_{max}$ = 41).
The matrix representation is very useful in software packages such as MATLAB. Therefore, the proposed method uses the matrix form to reconstruct the object image, and the matrix form of the reconstruction equation shown in Eq. (20) can be written as
where $I_R$ is the reconstructed image; $K_m$ and $K_n$ are the matrix forms of the generated Krawtchouk polynomial; ${(.)}^T$ denotes the transpose of the matrix; $E$ is the measurement results of the single-pixel detector; the operator $(\ast )$ denotes the matrix multiplication.3. Experiments
3.1 Computational simulations
3.1.1 KM-SPI with different orders
Two sets of images (Peppers and Cameraman) with resolution of $128\times 128$ pixels were simulated to evaluate the imaging effect of the proposed method. The Krawtchouk moments of the images were calculated separately under order (29+29), (41+41), (64+64), (91+91) and (128+128), corresponding to the sampling rates at 5%, 10%, 25%, 50% and 100%. The three term recursive algorithm shown in Eq. (9) was applied to simplify the calculation of the weighted Krawtchouk polynomials with parameters $p_1$ = 0.5 and $p_2$ = 0.5. The symmetry property shown in Eq. (14) was used to calculate the Krawtchouk moments under order (91+91) and (128+128) for reducing the cumulative error caused by iterative operation. The percentage of root mean squared error (RMSE) was used to further quantitatively evaluate the quality of the reconstructed image, which was calculated by
Fig. 3. The Krawtchouk moments and the reconstructed images of Peppers and Cameraman images at different sampling rates.(KM: Krawtchouk Moments; RI: Reconstructed image).
3.1.2 KM-SPI with different region-of-interest parameter $p_1$ and $p_2$
The Krawtchouk moments are proved to be effective local descriptors. This locality property is controlled by appropriate adjustment of the parameters $p_1$ and $p_2$. The parameter $p_1$ is used to shift the region-of-interest vertically, while the parameter $p_2$ is used to shift the region-of-interest horizontally. The $128\times 128$ pixels original image consists of four English letters in four quadrals. The parameters $p_1$ and $p_2$ were set separately at (0.5,0.5), (0.2,0.2), (0.2,0.8), (0.9,0.1) and (0.9,0.9). The image reconstruction was carried out by using the KM-SPI method with the low order (29+29) at the sampling rate of 5%. The simulation results is shown in Fig. 4.
Fig. 4. The region-of-interest image reconstruction results of the krawtchouk moments with different parameters $p_1$ and $p_2$.
When the parameters $p_1$ and $p_2$ are set at (0.5,0.5), which means that the region-of-interest is located in the central region of the image. Since four letters are scattered over four corners of the image, there is no effective goal in the central region of the image. So the quality of image in Fig. 4(b) is not satisfactory. In case of $p_1$ = 0.2 and $p_2$ = 0.2, the region-of-interest is located in the upper left corner of the image. As shown in Fig. 4(c), the letter $K$ in the upper left corner is extracted and the reconstructed image quality is excellent. Similarly, when $p_1$ and $p_2$ are set at (0.2,0.8), (0.9,0.1) and (0.9,0.9), the region-of-interest is located in the upper right, lower left and lower right corner of the image, respectively. The corresponding simulation results are exhibited in Figs. 4(d)-(f), where the letters $M,S$ and $I$ in three corresponding corners are extracted, respectively. The simulation experiments demonstrate that when the region of centralized target distribution can be regarded as a prior knowledge, the key sampling in any region-of-interest of target can be directly performed by adjusting the values of parameters $p_1$ and $p_2$, and the region-of-interest in the image can be retrieved with good reconstruction quality in a few low-order moments. Thanks to the special feature extraction ability of the Krawtchouk moments, the proposed method is especially suitable for the situation in which an object occupies a partial region in the scene.
The KM-SPI method may be effective for the following application scenarios. Firstly, the distribution characteristics of imaging targets can be known in advance through experience. Secondly, a low-sampling blurred image has been obtained firstly to understand the approximate distribution of the target. Thirdly, for the application scenario of target recognition or trajectory tracking, the target can only or probably appear in a certain area.
3.1.3 Comparison of different single-pixel compressive imaging methods
There are two popular types of SPI technologies. The first type is based on the random sampling and the iterative solution, such as the CS-SPI method [7]. The process of CS-SPI reconstruction is a complex iterative equation calculation and solution process, which causes the problems of reconstruction time and storage of the measurement matrix especially in conditions of large scale signal. The other type is based on the deterministic orthogonal basis sampling and the matrix inversion, such as the Fourier-SPI method [16] and the proposed KM-SPI method. This type of orthogonal basis method can quickly reconstruct object images by using their respective inverse transform, and the complexity of the calculation does not increase with the scale of the signal. By using gray-scale modulation, the sampling speed of Fourier-SPI and KM-SPI is much slower than that of CS-SPI which uses 2-bit modulation. Therefore, the compressed sampling is usually performed to reduce sampling time. In the case of low sampling rate, the orthogonal basis method gives priority to sampling the low-frequency information to achieve better imaging effect. The CS-SPI method is based on the random sampling and lacks sufficient information when the sampling rate is low, resulting in unsatisfactory imaging effect. However, with the increase of the sampling rate, the imaging effect of CS-SPI will catch up with the orthogonal basis method.
In order to maintain the fairness of the comparison between different SPI methods, the Fourier-SPI method [16] was used for comparative experiments with the KM-SPI method. Both of the two methods belong to the orthogonal basis sampling method based on gray-scale basis patterns, and both reconstruction methods are based on inverse orthogonal transformation. Two sets of images (USAF and Dog) with resolution of $128\times 128$ pixels were chosen as the experimental objects. The sampling rates were set separately at 3%, 5%, 10% and 20%. In Fourier-SPI, with the prior knowledge that Fourier spectrum of any real-valued image is symmetrically conjugated, only half of the spectrum was obtained and the other half was calculated by using the conjugate symmetry property. In KM-SPI, three term recursive algorithm was applied to calculate the weighted Krawtchouk polynomials. For the test image USAF, the parameters $p_1$ and $p_2$ were set at (0.5,0.5). For the test image Dog, with the prior knowledge that the region-of-interest is located in the upper left corner of the image, $p_1$ and $p_2$ were set at (0.1,0.1).
The comparison of the simulation results is shown in Fig. 5, and RMSEs of reconstructed images are listed in Table 1. Under the same sampling rate, RMSEs of KM-SPI is smaller, which indicates that the imaging quality of KM-SPI is better than Fourier-SPI. Especially in the cases of the sample rate less than 5%, the advantage of the proposed KM-SPI method is more obvious. This is because the KM-SPI method has the special ability to extract the characteristics from region-of-interest in the object even with a few low-order moments, which means that compression imaging ability of the KM-SPI method is excellent.
Fig. 5. Comparison of reconstruction results of USAF and Dog images by KM-SPI and Fourier-SPI method.
Table 1. Comparisons of RMSEs of the reconstruction results at different sampling rates.
When the sampling rate reaches 20%, the advantage of KM-SPI is not as obvious as the low sampling rate. It may be due to the increasing cumulative errors introduced in the three term recursive calculation of the weighted Krawtchouk polynomials. If some high precision computing methods can be used to calculate more accurate polynomial values (complex calculations in Eq. (1)), a better single-pixel imaging results may be obtained by the proposed KM-SPI method.
Table 2 shows the reconstruction calculation time comparison of the two methods. It can be seen that the KM-SPI method can take less time under the same sampling rate. This is because that the reconstruction calculation of KM-SPI is based on the matrix form of the inverse Krawtchouk moment transform shown in Eq. (21). By contrast, the coefficients in the Fourier spectrum are complex numbers, and the real parts and the imaginary parts need to be obtained respectively. Thus, the number of measurements by Fourier-SPI will be twice as many as KM-SPI. In order to reduce the number of measurements, only half of a Fourier spectrum is obtained, but the conjugate symmetry coefficient recovery operation needs to be carried out before the image reconstruction. Although both KM-SPI and Fourier-SPI can quickly reconstruct object images by using their respective inverse transform, the reconstruction run time of KM-SPI is shorter than that of Fourier-SPI. Therefore, it can be concluded that the KM-SPI method is capable of achieving high quality single-pixel compressive imaging in the real-time reconstruction.
Table 2. Comparisons of reconstructed computational time (Unit: s).
3.1.4 Selection of quantization bits for high-speed binary modulation
In the SPI system, the digital micro-mirror device (DMD) is a widely used high-speed spatial optical modulator. Currently, the binary modulation frequency of DMD can reach 22kHz, but the modulation frequency in 8-bit gray pattern is about 250Hz [15]. The Krawtchouk basis patterns belong to the gray-scale patterns. Thus, to improve the imaging efficiency of the DMD-based proposed KM-SPI system, a transformation is needed. The gray-scale basis patterns are equivalently transformed to the corresponding binary basis patterns. In recent years, some binary modulation strategies for the gray-scale basis patterns of DMD-based Fourier-SPI system have been proposed, such as the spatial dithering strategy [18] and the signal dithering strategy [19]. The spatial dithering strategy is based on upsampling, which improves the imaging efficiency but reduces the imaging spatial resolution. The signal dithering strategy is based on a detected signal computational-weighted, which makes a tradeoff between the spatial resolution and the temporal resolution without sacrificing the imaging spatial resolution [19].
In this subsection, the binarization is realized by reference to the binary Fourier-SPI method based on signal dithering strategy [19]. By using different quantization bits $b$, the original gray-scale Krawtchouk basis patterns are decomposed to different quantities of binary patterns, which will result in a reconstruction error. The influence of the quantization bits in the imaging results was simulated and analyzed by using the $128\times 128$ pixels image Peppers. The image reconstruction was carried out by using the KM-SPI method with order (70+70) at the sampling rate of 30%, with parameters $p_1$ = 0.5 and $p_2$ = 0.5. The values of $b$ were in the range from 2 to 8. The simulation results are displayed in Fig. 6. The reconstructed image quality is improved with an increase in the $b$ value. The RMSEs under different quantization bits are shown in Fig. 7.
It can be seen that RMSEs decreases with the increase of quantization bits $b$. Moreover, when $b <$6, the gradient of RMSEs declines sharply. When $b \geqslant$ 6, the gradient of RMSEs tends to be stable, but the quality of the reconstructed images is not obviously improved. Since the more the quantization bits are, the more time it takes to modulate, the appropriate compromise quantized bits can be selected for binary conversion according to the actual demand. When the imaging accuracy is required to be high, $b$ could be 7 or even 8. In general, $b$ = 6 can meet the requirements of imaging efficiency and quality.
3.2 Laboratory experiments
The schematic of the experimental system is shown in Fig. 8. A film consisted of an English letter $K$ in the lower left corner was used as the imaging target. A halogen lamp (Thorlabs OSL2) was used as the light source. To demonstrate the imaging effect of KM-SPI at very low sampling rate, four sets of $128\times 128$-pixels gray-scale Krawtchouk basis patterns were generated respectively using three term recursive algorithm under order (13+13), (18+18), (23+23) and (29+29), corresponding to the sampling rates at 1%, 2%, 3% and 5%. With the prior knowledge that the region-of-interest was located in the upper left corner of the test target, the parameters $p_1$ and $p_2$ were set at (0.9,0.1). The difference and the binarization of the gray-scale Krawtchouk basis patterns were performed with quantized bits $b$ = 6. The high speed modulations for the obtained binary Krawtchouk basis patterns were carried out by DMD DLP7000 operated at 100Hz, which contains $1024\times 768$ individually addressable mirrors. The size of each mirror is 13.68 ${\mu }$m ${\times }$13.68 ${\mu }$m. The numbers of measurements were 2028, 3888, 6348 and 10092, respectively. The modulated light intensity signals were measured by an avalanche photodiode (APD, IDQ ID100). The measured intensities were calculated to generate the Krawtchouk moments of the object. The object images were reconstructed by using inverse Krawtchouk moment transform.
Using the above experimental system, four contrast experiments using the Fourier-SPI method were carried out at 100Hz modulation rate of the DMD with the sampling rates at 1%, 2%, 3% and 5%, respectively. In order to maintain the fairness of the comparison between the two SPI methods, the differential measurement was also used to restrain the noise, and the binarization of the gray-scale Fourier basis patterns were also performed based on signal dithering strategy [19] with quantized bits $b$ = 6. The experimental results were compared with that of the proposed KM-SPI method. With the prior knowledge that Fourier spectrum of any real-valued image is symmetrically conjugated, only half of the spectrum was obtained and the other half was calculated by using the conjugate symmetry property. 2016, 4320, 6336 and 10080 measurements were taken respectively in the experiments.
The reconstruction results are shown in Fig. 9, where images in the top line are reconstructed by Fourier-SPI at different sampling rates, and images in the bottom line are reconstructed by KM-SPI. It can be seen that the KM-SPI method can effectively realize single-pixel imaging of the target object with the binary differential modulation of DMD. Even with the sampling rate of 1%, the object can be imaged clearly by KM-SPI. It demonstrates the unique characteristic extraction capability of the KM-SPI method, which is especially suitable for the situation in which an object occupies a partial region in the scene. By contrast, the reconstructed images of Fourier-SPI are full of burrs in the case of the sampling rate 1%, 2% and 3%. Based on the comparison results, binary KM-SPI outperforms binary Fourier-SPI in terms of reconstruction quality in the case of the sampling rate lower than 5%. Specially, the advantage of KM-SPI method is apparently observable for the sampling rate lower than 3%. The results in these experiments coincides with those derived in the numerical simulations.
Fig. 9. Reconstructed images of single-pixel compressive imaging experiments. (a) Reconstruction images of Fourier-SPI method; (b)Reconstruction images of KM-SPI method.
4. Conclusion
In this paper, a new single-pixel compressive imaging method based on the transformation of discrete orthogonal Krawtchouk moments is proposed. The effectiveness of the method has been demonstrated by the simulation and the experimental results. The advantages of this method are concluded as follows. Firstly, it is based on the transformation of Krawtchouk moments and the deterministic discrete orthogonal Krawtchouk basis patterns are applied for modulation, thus it is a transform domain imaging technique. Due to the strong energy compression characteristics of the transformation of Krawtchouk moments, the proposed method can achieve efficient compressive imaging with the sampling rate at 5% or even lower. Secondly, the reconstruction calculation is based on the inverse Krawtchouk moment transform represented in the matrix form, which means that the rapid real-time reconstruction can be realized by the proposed method. Thirdly, the Krawtchouk moments are effective local descriptors. When the region of centralized target distribution can be regarded as a prior knowledge, the key sampling in any region-of-interest of target can be directly performed by adjusting the values of parameters, and the region-of-interest in the image can be retrieved with good reconstruction quality in a few low-order moments. It is especially suitable for the situation in which an object occupies a partial region in the scene. Finally, the proposed method realizes imaging through single-pixel point detector with the global sampling characteristic, so it inherits the advantages of single-pixel imaging such as high throughput, low measurement dimensions and low cost. In general, the proposed method can realize high quality and high efficiency imaging, which provides an alternative imaging approach for single-pixel compressive imaging, and has good adaptability to future applications such as extremely weak light imaging, invisible wavelength imaging, target recognition, trajectory tracking and others.
Funding
National Natural Science Foundation of China (61575207, 61601442, 61605218); National Basic Research Program of China (973 Program) (2016YFE0131500, 2018YFB0504302); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019154).
Acknowledgments
The authors thank Dr. Rong-Ping Deng for linguistic assistance.
References
1. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008). [CrossRef]
2. S. M. M. Khamoushi, Y. Nosrati, and S. H. Tavassoli, “Sinusoidal ghost imaging,” Opt. Lett. 40(15), 3452–3455 (2015). [CrossRef]
3. X. Liu, W. Yu, X. Yao, B. Dai, L. Li, C. Wang, and G. Zhai, “Measurement dimensions compressed spectral imaging with a single point detector,” Opt. Commun. 365, 173–179 (2016). [CrossRef]
4. K. Fan, J. Y. Suen, and W. J. Padilla, “Graphene metamaterial spatial light modulator for infrared single pixel imaging,” Opt. Express 25(21), 25318–25325 (2017). [CrossRef]
5. C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with meta material spatial light modulators,” Nat. Photonics 8(8), 605–609 (2014). [CrossRef]
6. G. A. Howland, P. B. Dixon, and J. C. Howell, “Photon-counting compressive sensing laser radar for 3d imaging,” Appl. Opt. 50(31), 5917–5920 (2011). [CrossRef]
7. W. Yu, X. Yao, X. Liu, R. Lan, L. Wu, G. Zhai, and Q. Zhao, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016). [CrossRef]
8. N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014). [CrossRef]
9. G. M. Gibson, B. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. A. Malcolm, and M. J. Padgett, “Real-time imaging of methane gas leaks using a single-pixel camera,” Opt. Express 25(4), 2998–3005 (2017). [CrossRef]
10. F. Magalhães, F. M. Araújo, M. V. Correia, M. Abolbashari, and F. Farahi, “Active illumination single-pixel camera based on compressive sensing,” Appl. Opt. 50(4), 405–414 (2011). [CrossRef]
11. D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal 26(3), 301–321 (2009). [CrossRef]
12. C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” (Masters of Science thesis), Rice University, Houston: TX, USA (2010).
13. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). [CrossRef]
14. L. Martínez-León, P. Clemente, Y. Mori, V. Climent, J. Lancis, and E. Tajahuerce, “Single-pixel digital holography with phase-encoded illumination,” Opt. Express 25(5), 4975–4984 (2017). [CrossRef]
15. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Hadamard single-pixel imaging versus Fourier single-pixel imaging,” Opt. Express 25(16), 19619–19639 (2017). [CrossRef]
16. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015). [CrossRef]
17. B. Xu, H. Jiang, H. Zhao, X. Li, and S. Zhu, “Projector-defocusing rectification for Fourier single-pixel imaging,” Opt. Express 26(4), 5005–5017 (2018). [CrossRef]
18. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017). [CrossRef]
19. J. Huang, D. Shi, K. Yuan, S. Hu, and Y. Wang, “Computational-weighted Fourier single-pixel imaging via binary illumination,” Opt. Express 26(13), 16547–16559 (2018). [CrossRef]
20. B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017). [CrossRef]
21. V. A. Coutinho, R. J. Cintra, F. M. Bayer, P. A. M. Oliveira, R. S. Oliveira, and A. Madanayake, “Pruned discrete Tchebichef transform approximation for image compression,” Circuits Syst. Signal Process 37(10), 4363–4383 (2018). [CrossRef]
22. G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image and Vision Comput. 28(3), 414–423 (2010). [CrossRef]
23. R. Benouini, I. Batioua, K. Zenkouar, A. Zahi, S. Najah, and H. Qjidaa, “Fractional-order orthogonal Chebyshev moments and moments invariants for image representation and pattern recognition,” Pattern Recognit. 86, 332–343 (2019). [CrossRef]
24. S. Golabi, M. S. Helfroush, and H. Danyali, “Non-unit mapped radial moments platform for robust, geometric invariant image watermarking and reversible data hiding,” Inf. Sci. 447, 104–116 (2018). [CrossRef]
25. P. T. Yap, R. Paramesran, and S. H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. on Image Process. 12(11), 1367–1377 (2003). [CrossRef]
26. R. Koekoek and R. F. Swarttouw, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,” arXiv preprint math 9602214 (1996).
27. X. Wang, M. L. Xu, B. Q. Li, H. L. Zhai, J. J. Liu, and S. Y. Li, “Prediction of phosphorylation sites based on Krawtchouk image moments,” Proteins 85(12), 2231–2238 (2017). [CrossRef]
28. I. Batioua, R. Benouini, K. Zenkouar, A. Zahi, and E. F. Hakim, “3D image analysis by separable discrete orthogonal moments based on Krawtchouk and Tchebichef polynomials,” Pattern Recognit. 71, 264–277 (2017). [CrossRef]
29. S. H. Abdulhussain, A. R. Ramli, S. A. R. Al-Haddad, B. M. Mahmmod, and W. A. Jassim, “Fast recursive computation of Krawtchouk Polynomials,” J. Math. Imaging Vis. 60(3), 285–303 (2018). [CrossRef]
30. S. P. Priyal and P. K. Bora, “A robust static hand gesture recognition system using geometry based normalizations and Krawtchouk moments,” Pattern Recognit. 46(8), 2202–2219 (2013). [CrossRef]
31. G. Zhang, Z. Luo, B. Fu, B. Li, J. Liao, X. Fan, and Z. Xi, “A symmetry and bi-recursive algorithm of accurately computing Krawtchouk moments,” Pattern Recognit. Lett. 31(7), 548–554 (2010). [CrossRef]
