As an innovative imaging technique, single-pixel imaging, including ghost imaging^[1-9] or computational ghost imaging^[10-14], Fourier or Hadamard single-pixel imaging^[15-19], and single-pixel cameras^[20,21], which allows an image to be reconstructed using a detector without spatial resolution, has attracted increasing attention in recent years. So far, there have been many potential applications of single-pixel imaging for multi-spectral or hyperspectral imaging^[22,23], holography^[24], phase imaging^[25,26], optical encryption^[27], remote sensing^[28,29], infrared imaging^[30], THz imaging^[31,32], microscopy^[33], 3D imaging^[14], imaging through scattering media^[34], X-ray imaging^[35,36], etc.

Over the last decade or so, there are two main types of single-pixel imaging that have been demonstrated. Originating from ghost imaging^[1-9], one is to recover the image under random non-orthogonal illumination patterns including computational or compressive ghost imaging^[10-12]. The random patterns utilized are overcomplete and non-orthogonal, which need a great number of measurements and long data-acquisition time in this type. The other is to implement the image reconstruction via an algorithmic transform, using non-random orthogonal basis patterns such as the Fourier^[15,16], Hadamard^[17], or wavelet basis^[37]. Fast Fourier single-pixel imaging (FSI) has attracted much attention in recent years. Some demonstrations are performed by applying dithering strategies^[16,38] to binarize the gray-scale Fourier basis patterns. In addition, Liu et al.^[39] reported FSI with general basis patterns and proved that binary basis patterns can be used directly for illumination to implement FSI without the dithering strategy.

Since these basis patterns form a complete orthogonal set, it can solve the problems of long acquisition time and low image quality. In the previous studies, FSI is mainly realized via 2-, 3-, or 4-step phase-shifting approaches^{[15-17,40,41]}. Here, in this Letter, we investigate arbitrary n-step phase-shifting FSI and give general analytical expressions, and then we theoretically and experimentally compare the performances of different steps of phase-shifting FSI. The paper is organized as follows. In Section 2, we discuss n-step phase-shifting FSI and present general expressions. The numerical simulations and experiments of n-step phase-shifting FSI are provided in Section 3. Finally, we make the conclusion in Section 4.

FSI reconstructs the image of an object by applying an inverse Fourier transform on the measured Fourier spectrum of the object. The Fourier spectrum consists of a sequence of Fourier coefficients, each of which corresponds to a unique series of Fourier basis patterns. A Fourier coefficient can be acquired by applying the phase-shifting differential intensity measurement, in which the intensity, from the object illuminated by the corresponding Fourier basis patterns, is measured by a single-pixel detector. A two-dimensional (2D) Fourier basis pattern is expressed mathematically as^[15]P(x,y,fx,fy,ϕ)=a+bcos(2πfxx+2πfyy+ϕ),(1)where x and y are the 2D Cartesian coordinates, fx and fy are the corresponding spatial frequencies, a is the direct current term, b represents the pattern contrast, and ϕ is the initial phase. This pattern is projected onto the reflective object described by R(x,y), and a single-pixel detector collects the total intensity response Iϕ(fx,fy) of the reflected light, which can be expressed in the following form: I(fx,fy,ϕ)=Inoise+β∬SR(x,y)P(x,y,fx,fy,ϕ)dxdy,(2)where Inoise is the intensity of background noise, β is a constant dependent on the size and the location of the detector, and S is the illumination area.

In terms of the Fourier transform and phase-shifting algorithm, 2-, 3-, and 4-step phase-shifting FSI has been studied^[15-17,40]. Here, based on the principle of the phase-shifting algorithm, we establish a general n-step FSI mechanism. Each Fourier coefficient can be acquired, through an n-step phase-shifting approach, by the n intensity responses corresponding to the n illumination patterns, in which the n illumination patterns and the n intensity responses are defined, respectively, as Pl(x,y,fx,fy)=P(x,y,fx,fy,2(l−1)πn),(3)Il(fx,fy)=I(fx,fy,2(l−1)πn),(4)where the integer l satisfies 1≤l≤n.

Due to the Fourier basis patterns forming an orthogonal set, the intensity responses also possess orthogonality with respect to the initial phase. Therefore, the expression C(fx,fy) proportional to Fourier spectrum of R(x,y) can be represented as $$\begin{gathered} C(fx,fy)=∑l=1nglIl(fx,fy)=∑l=1ngl{Inoise+β∬SR(x,y) \\ ·[a+bcos(2πfxx+2πfyy+2πl−1n)]dxdy}, \end{gathered}$$(5)where n>2 and gl is a weight coefficient.

In terms of the trigonometric series theory and the orthogonality of the Fourier basis, the general analytical expressions C(fx,fy) can be achieved by selecting appropriate weight coefficients gl. Here we propose four cases for n-step phase-shifting.

In the first case (Case 1), the weight coefficients take gl=exp[j·2(l−1)π/n]. When n≥3, according to the following equation, ∑l=1nexp(j·2πl−1n)=0, the general analytical expression can be rewritten as $$\begin{gathered} C(fx,fy)=∑l=1nexp(j·2πl−1n)·Il(fx,fy) \\ =nbβ2·F{R(x,y)}, \end{gathered}$$(6)where the Fourier transform F{R(x,y)}=∬sR(x,y)·exp[−j2π(fxx+fyy)]dxdy, and the noise and DC terms are canceled due to properties of the trigonometric function. For Case 1, Eq. (6) is valid for the integer n≥3. By performing an inverse Fourier transform on Eq. (6), the object image via n-step phase-shifting FSI can be described by R(x,y)=2nbβ·F−1{C(fx,fy)}.(7)

In the same way, we can deduce the expressions for the following three cases.

In the second case (Case 2), we can choose the weight coefficients gl as g1=n−12, g2=12−jn4sin2πn, gn=12+jn4sin2πn, gm=−12 (3≤m≤n−1), such that $$\begin{gathered} C(fx,fy)=n−12I1(fx,fy)−(12−jn4sin2πn)I2(fx,fy) \\ −12∑l=3n−1Il(fx,fy)−(12+jn4sin2πn)In(fx,fy) \\ =nbβ2·F{R(x,y)}. \end{gathered}$$(8)

For the third case (Case 3), the weight coefficients gl can be given as, for n>3, $$\begin{gathered} g1=n(n−3)2n−8sin2πn,g2=jn4sin2πn, \\ g3,⋯,gn−1=−n2n−8sin2πn,gn=g2*. \end{gathered}$$(9)

Substituting the above weight coefficients gl into Eq. (5), we can express mathematically Eq. (5) as C(fx,fy)=∑l=1nglIl(fx,fy)=nbβ2·F{R(x,y)}.(10)

As for the last case (Case 4), we can describe n-step phase-shifting as a simple expression by selecting appropriate coefficients gl: $$\begin{gathered} C(fx,fy)={6I1−3Im+1−3Im+24+4cos(π/n)+j3I2−3In4sin(2π/n)I1−Im+2+jI2−Insin(2π/n) \\ ={32bβ·F{R(x,y)},ifn=2m+1,2bβ·F{R(x,y)},ifn=2m+2, \end{gathered}$$(11)where integer number m≥1, and Il(fx,fy) is abbreviated to Il.

When n=3, the first, second, and fourth cases [Eqs. (6), (8), (11)] give the same Fourier phase-shifting coefficients, C(fx,fy)=12(2I1−I2−I3)+j32(I2−I3).(12)

For n=4, the first, third, and fourth cases [Eqs. (6), (10), (11)] provide the same Fourier phase-shifting coefficients, C(fx,fy)=(I1−I3)+j(I2−I4),(13)which is usually utilized in the standard FSI^[15].

For the higher steps of phase-shifting, the above cases give more methods to achieve sequences of Fourier phase-shifting coefficients, which provide more strategies to measure the Fourier coefficient and balance the background noise via differential intensity measurements in practice. The symmetry and periodicity of the weight coefficients in Case 1 are better than those in other cases. The n-step (n≥6) FSI in Case 4 is more efficient than that in other cases since the n-step (n≥6) phase-shifting in Case 4 can be reduced into a 4-step or 5-step approach characterized by Eq. (11). Such results stem from both the changes in illumination patterns and the selection of the appropriate weight coefficients gl.

In this section, we will validate the proposed n-step phase-shifting approach by numerical simulations and experiments, taking n=3,4,5,6 as examples. (For more steps, the simulations and experiments can yield similar conclusions.) In the simulation, a natural image “Cameraman” of 64×64 pixels is used for testing. The simulation results are presented in Fig. 1. As seen in Fig. 1, the numerical simulations correspond with the theoretical results and are consistent. The numerical simulations indicate that the image reconstruction can be performed with the proposed n-step phase-shifting FSI. Indeed, as the theory suggests, the simulation results without noise in Fig. 1 for different steps and cases appear identical. However, in real-world environments, FSI will inevitably be influenced by environmental noise. To more accurately reflect the performance of these n-step phase-shifting approaches under noise, we will consider noise in the following experiments rather than simulations.

In order to validate the effectiveness of the proposed n-step phase-shifting approach in real-world scenarios, without loss of generality, we initially designed experiments for relatively simple objects. The schematic diagram of the experimental setup is shown in Fig. 2. A commercial digital light projector (Sony, VPL-EW575) controlled by a computer is employed to produce n-step Fourier basis patterns, which are of 256 gray-scale levels and successively projected. The speed of the projector modulating the patterns is 20 frames per second. The object is printed on white paper with a size of 64×64 pixels and is illuminated by the Fourier basis patterns. Due to each of the reconstructed images being 64×64 pixels in size, in n-step phase-shifting approaches of Cases 1, 2, and 3, there are n×64×64 Fourier coefficients to be obtained in the physical experiments, and the overall acquisition time is 5×10−2×n×64×64 s. For Case 4, in n=2m+1-step phase-shifting approaches, there are up to 5×64×64 Fourier coefficients to be obtained, and the overall acquisition time is up to 5×10−2×5×64×64s. In n=2m+2-step phase-shifting approaches of Case 4, there are up to 4×64×64 Fourier coefficients to be obtained, and the overall acquisition time is up to 5×10−2×4×64×64s.

The light intensity reflected from the object is detected by a single-pixel detector (Thorlabs, PDA100A2) and subsequently transmitted to the computer through a data acquisition card system (National Instruments, PCIe-6251 and BNC-2110). The sampling rate of the data acquisition card is set at 60 kS/s.

In the experiment, the n-step phase-shifting FSI technique is adopted for Fourier spectrum acquisition, where each Fourier coefficient is acquired by one of the sinusoidal patterns with n-step phase-shifting, according to one of cases described by Eqs. (6), (8), (10), and (11), respectively. Figures 3 and 4 show the reconstructed images of a binary object (Chinese character) and a gray-scale object (Mars ring), respectively, for 3- to 6-step FSI, where the 3-step FSI image for Case 3 is left blank because Case 3 is only applicable for n>3 in terms of Eq. (10).

To quantitatively compare the image quality using a 3- to 6-step phase-shifting approach, we employ a peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM)^[39] to assess the quality of reconstructed images. For 3–6 steps, we select the optimal image, which is reconstructed in the absence of noise via the 6-step phase-shifting approach presented by Case 1, as the reference image to calculate the PSNR and SSIM in real experimental results. Quantitative analysis of the PSNR of the image reconstructed by 3- to 6-step FSI for a binary object and a gray-scale object is illustrated in Tables 1 and 2, respectively.

It is shown in Tables 1 and 2 that the PSNRs in Cases 1, 2, and 4 for the 3-step approach are equal due to their coincidence with the same theoretical form for the phase-shifting Fourier coefficients discussed in the Section 2. For the same reason, the PSNRs in Cases 1, 3, and 4 for 4-step FSI behave identically as shown in Tables 1 and 2. In reality, there will be the influence of noise, and the objects will also be more complicated. In order to further distinguish the distinctive features of these n-step phase-shifting approaches, in our experiment, we increase the noise relatively by placing two neutral density filters (NDFs, GCC-301115A, Daheng) before the detector to differentiate the results. Figures 5(a) and 5(b) show the reconstructed images of the “astronaut,” each sized 64×64 pixels, under strong light and weak light, respectively. In this experiment, for strong light, the amplification gain of the PDA utilized is adjusted to 40 dB, resulting in an output voltage of approximately 2960 mV. Conversely, for weak light, the amplification gain is set to 0 dB, and the light intensity is attenuated about 70% with NDFs, leading to an output voltage of around 30 mV. We present the PSNR and SSIM of the reconstructed images for different noise levels in Tables 3 and 4, respectively.

By comparing these images in Figs. 5(a) and 5(b), it is evident that, with the increase of noise in illumination light, the 6-step FSI approach yields relatively better imaging results compared to the 4-step FSI approach. When considering the evaluation index for image quality, the regularity is more clearly observed in Tables 3 and 4. The above results show that these cases are not essentially equivalent to each other for multi-step phase-shifting. Furthermore, these results indicate that the FSI approaches with a larger number of steps are more noise-robust compared to the 4-step FSI approach.

The comparison of the image quality evaluation indices in Tables 1–4 implies that the phase-shifting approach of Case 1 is generally superior to the others. This is precisely because of the periodicity of the weight coefficients gl given in Eq. (6) in Case 1, making the elimination of the background noise more easily achieved through differential detection measurement in experiment. This results in a better image quality for this case compared to the others. Regardless of whether there is noise in FSI or the intensity of the noise is strong or weak, for each case from 3-step to 6-step, the experimental results in Figs. 3–5 show that the 6-step approach is consistently the most optimal. Moreover, this conclusion is reinforced by the data presented in Tables 1–4. Therefore we can infer that FSI approaches with a larger number of steps (n>6) could achieve better image quality compared to the 2- to 6-step FSI approaches.

In addition, it is known that a larger number of steps in phase-shifting require a longer data acquisition time. However, for more steps (n≥6), the Fourier phase-shifting in Case 4 can be simplified into a 4-step or 5-step approach, according to the theoretical formula of Eq. (11). From the results above, we can deduce that this alternative even-step approach in Case 4 not only ensures the same data-acquisition time as the 4-step approach but also achieves better image quality than the 4-step approach. This is evident in Fig. 5 and supported by the data in Tables 1–4. In general, these approaches provide us with more strategies to perform FSI with different steps. Thus, in terms of efficiency and practicality, studying Case 4 would be more meaningful.

In conclusion, we have investigated arbitrary n-step phase-shifting FSI both theoretically and experimentally. We obtain general analytical expressions of different types for n-step phase-shifting approaches, which also include 4-step phase-shifting utilized in standard FSI. The reconstructed images and the corresponding image quality evaluation indices are demonstrated for 3–6 steps in four phase-shifting cases. The experimental results show that, even in the presence of noise, Case 1 is generally better than the others due to the periodicity of the weight coefficients. This suggests these cases are not equivalent to each other for multi-step phase-shifting. In the presence of noise in n-step phase-shifting FSI, the distinctions among these methods become more pronounced. Additionally, these experimental results indicate that FSI approaches with a larger number of steps exhibit a noise-robust feature when compared to the 4-step FSI approach. More importantly, for more steps (n≥6), due to the even-step Fourier phase-shifting in Case 4 being simplified into a 4-step approach, this alternative even-step approach in Case 4 not only ensures the same data-acquisition time as the 4-step approach but also achieves better image quality than the 4-step approach. In general, these approaches give us more strategies to perform FSI for different steps, which could provide a guideline to balance the tradeoff between the image quality and the number of steps usually confronted in the application of FSI. Alternative optimized strategies to perform FSI could be more valuable in complex scenes.