Quantitative characterization of micro-scanning imaging aliasing and optical parameter optimization

Chao Zhang; Fafa Ren; Xiaorui Wang; Yangyang Li; Zhenshun Zhao; Yue Li

doi:10.1364/OE.516183

1. Introduction

Most imaging systems suffer from sampling aliasing problems. Imaging aliasing occurs due to under-sampling and intensity averaging that results from limited pixel size. Smaller pixels and denser arrays significantly increase costs. Additionally, simply increasing the sampling density cannot effectively improve the imaging resolution when the cutoff frequency of the optical system is lower than the sampling Nyquist frequency [1]. The cutoff frequency limitation of the optical system results in blurring of the signal sampled by the detector. Optimization of the optical modulation transfer function (MTF) is therefore typically regarded as a key indicator for system design. However, tradeoffs exist between the optical MTF and the sampling size to maximize the imaging quality [2].

When the cutoff frequency of the optical system is above the Nyquist sampling frequency, the imaging signal passing through the optical system is under-sampled by the detector, resulting in aliasing blur [3,4]. Under-sampling occurs in the range above the Nyquist frequency and below the optical cutoff frequency, thereby the amount of aliasing is closely related to the sampling frequency and the MTF of the optical system.

Micro-scanning is a technique that provides a final higher resolution image by combining multiple offset images of a lower resolution [5]. Micro-scanning imaging effectively reduces under-sampling aliasing by increasing the sampling frequency in the same field of view. The Nyquist frequency increases as the sampling interval shortens, reducing the under-sampled frequency range. However, since the detector size is fixed, in the actual micro-scanning process, the imaging signal originally captured by one detector unit is collected by different detector units after sub-pixel offset, as shown in Fig. 1(a). The sub-pixel components are shared among the multiple sampling results after sub-pixel shifts, as shown in Fig. 1(b). Micro-scanning acquires additional information by reducing the sampling interval, but the presence of overlapping sampling regions introduces redundant information.

Fig. 1. Offset and overlap of micro-scanning sampling.

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The effectiveness of micro-scanning imaging in reducing aliasing and improving resolution has been verified [6,7]. Schade [8] introduced spurious responses to quantify aliasing in general imaging. However, relevant studies fail to explain the intrinsic relationship between micro-scanning sampling and the mechanism of aliasing reduction, as well as the quantification of aliasing in micro-scanning sampling. Furthermore, there is a lack of insight into the redundant information introduced by micro-scanning on imaging aliasing and resolution.

Wang [9,10] used the pixel transfer function at the Nyquist frequency to analyze the micro-scanning imaging quality under different micro-scanning modes and detector fill factors. However, the influence of the optical system parameters on the micro-scanning imaging performance is still uncertain [11]. Furthermore, the mechanism of the optical MTF on the formation of micro-scanning aliasing is not well understood. The constraining relationships among full-chain micro-scanning imaging parameters, such as micro-scanning mode, fill factor, F-number, and post-filtering are not yet sufficiently discussed.

To address the quantitative description of imaging aliasing in micro-scanning sampling, this paper innovatively combines the spurious response model of general imaging with the micro-scanning sampling mechanism. An accurate micro-scanning aliasing analysis model is proposed based on the sampling squeeze properties, which can quantitatively describe the aliasing composition and aliasing removal mechanism of micro-scanning imaging. Meanwhile, an innovative approach to calculate the transfer function of the micro-scanning imaging system based on aliasing is introduced, which overcomes the difficulty of existing micro-scanning models in accurately characterizing and coupling the micro-scanning imaging process.

The transfer functions of the optical system, detector, and digital filter are coupled with the micro-scanning sampling process. First, the spurious responses under different sampling modes are evaluated based on the coupling and constraint between the transfer functions, and the out-of-band integral of the spurious responses is used to quantify imaging aliasing. The MTF determined by the detector size is included as part of the pre-sampling transfer function to characterize the redundant information introduced by the micro-scanning sampling process. Then, the stretch of the transfer function under micro-scanning sampling is calculated based on the amount of aliasing, and the stretch factor is used to represent the performance improvement achieved by micro-scanning sampling. Second, different optical transfer functions are substituted into the micro-scanning aliasing model, the amount of micro-scanning aliasing under different optical system parameters is calculated, and the system performance under different F-numbers is predicted to find the optimal F-number of micro-scanning imaging. The coupling relationships between optical system parameters, detector fill factors, and micro-scanning modes are analyzed. Finally, the simulation of micro-scanning imaging is performed by the actual imaging transfer and sampling process. The transfer function of micro-scanning systems with different optical parameters is measured by simulation experiments, the accuracy of the micro-scanning aliasing model is verified, and the test results of optimal F-numbers are given. This paper provides theoretical support for the matching relationship between micro-scanning imaging parameters, which is helpful for the trade-off and optimal design of micro-scanning imaging systems.

2. Accurate characterization of micro-scanning imaging aliasing and performance

Micro-scanning imaging helps reduce aliasing effects by increasing the sampling frequency. However, the increased sampling of micro-scanning introduces redundant information compared to conventional imaging. The key parameters in micro-scanning imaging, such as fill factor, detector size, micro-scanning mode, and optical F-number, have complex coupling relationships, which makes it difficult to characterize micro-scanning aliasing and analyze micro-scanning imaging performance. To determine matching relationships between parameters, it is necessary to analyze and characterize the physical mechanisms involved. Existing methods fail to quantify micro-scanning aliasing and accurately describe the coupling relationship between parameters.

This paper innovatively introduces aliasing quantification to characterize the performance of micro-scanning imaging. First, the spurious responses of micro-scanning imaging with different micro-scanning modes are analyzed based on the traditional imaging spurious response model. The out-of-band spurious response is used as an indicator, and the imaging aliasing in different micro-scanning modes and the mechanism for aliasing reduction are quantitatively analyzed. Then, based on the sampling squeeze property, the micro-scanning transfer function stretch factor is proposed to accurately characterize the complex coupling relationship among the optical system, detector, and digital filter transfer function in micro-scanning sampling.

Imaging aliasing is unavoidable when a continuous signal is sampled discretely. As shown in Fig. 2, the original imaging signal is blurred by the atmosphere, optical system, and motion, then sampled by the detector to obtain the output image. The optical system is similar to a low-pass filter that reduces the high-frequency information focused on the detector. The size of the detector determines the spatial sampling frequency. When the optical cutoff frequency is higher than the sampling frequency, the original imaging information within that frequency range is insufficient to be accurately captured, resulting in aliasing in the imaging results. The transfer performance of most imaging systems is primarily influenced by the optical system and detector sampling characteristics.

Fig. 2. Imaging transfer process with optical blur and detector sampling.

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The imaging process is divided into three stages: pre-sampling, sampling, and post-sampling. The pre-sampling transfer stage mainly includes the optical systems, motion blur, detectors, etc. In which the detector MTF in pre-sampling is determined by the detector size. The sampling process is determined by the sampling interval. The post-sampling transfer process involves amplifiers, digital filters, displays, human eyes, etc.

First, the formation mechanism of imaging aliasing in the sampling process is analyzed on the basis of spurious responses, as shown in Fig. 3. The pre-sampling MTF is the superposition of the diffraction limited optical transfer function and the aperture function of the detector unit [12]. The post-sampling process mainly involves post-filtering. In the frequency domain, periodic sampling causes responses above the Nyquist frequency to fold back to lower frequencies, forming replicated frequencies. The post-filtering is superimposed on the replicated pre-sampled signal to produce an aliasing signal. The spurious response region is below the aliasing signal. The Nyquist frequency defines the frequency range in which the sampling process can accurately reconstruct the complete signal. Therefore, the replicated frequencies are divided into the in-band and the out-of-band by the Nyquist frequency. If the sampled Nyquist frequency is lower than the cutoff frequency of the input signal, spurious responses will be produced, resulting in aliasing of the image result. Signals with high-frequency information that above the Nyquist frequency primarily affect the imaging resolution. Therefore, the integration of the out-of-band spurious response is used to quantify imaging aliasing.

Fig. 3. Imaging system transfer function composition and aliasing analysis.

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According to the spurious response defined by Schade, the spurious response of the first fold-back replicated frequency is expressed as

(1)$$SR = \frac{{\int_0^\infty {MT{F_{pre}}({u_s} \pm u)} O({{u_s} \pm u} )MT{F_{post}}(u)du}}{{\int_0^\infty {MT{F_{pre}}(u)O(u )MT{F_{post}}(u)} du}}$$

where ${u_s}$ is the sampling frequency, $MT{F_{post}}$ is the transfer function of post-sampling and mainly considers the digital filtering after sampling. $O(u )$ is the original imaging signal. $MT{F_{pre}}$ is the transfer function before sampling, including the transfer characteristics of the optical system and detector, expressed as,

(2)$$MT{F_{pre}}(u) = MT{F_{opt}}(u)MT{F_{det}}(u)$$

where $MT{F_{opt}}$ is the transfer function of circular aperture diffraction-limited optical system [13], and $MT{F_{det}}$ is the aperture function based on the size of the detector unit.

Assuming that the imaging content includes all spatial frequencies, with $O(u )= 1$. Since imaging resolution is primarily affected by the spurious response above the Nyquist frequency, sampling aliasing is defined as the sum of the out-of-band spurious responses induced by imaging sampling. For simplicity, only the first replica is considered, and the sum of the out-of-band spurious responses is calculated,

(3)$${A_{SR}} = \frac{{\int_{{u_n}}^{{u_c}} {MT{F_{pre}}({u_s} - u)} MT{F_{fliter}}(u)du}}{{\int_0^{{u_c}} {MT{F_{pre}}(u)MT{F_{fliter}}(u)} du}}$$

where ${u_n}$ is the Nyquist frequency, and ${u_c}$ is the cutoff frequency of the post-filtering, which is typically $2.5{u_s}$ in practical applications. $MT{F_{fliter}}$ is the transfer function of the digital filter.

It should be noted that proper digital filtering can effectively filter out frequency components above the Nyquist frequency, which reduces aliasing, but the high frequency information is also removed. For tasks such as target identification, recognition, and super-resolution imaging, the drawbacks outweigh the benefits. Therefore, a higher filter cutoff frequency is chosen to preserve more detailed information, but results in more aliasing. Fortunately, the aliasing can be reduced by micro-scanning imaging.

Secondly, the composition of imaging aliasing is also different under different sampling methods. The micro-scanning imaging system reduces aliasing by increasing the sampling frequency. To understand the mechanism of aliasing reduction, the micro-scanning sampling process is analyzed. In 2 × 2 micro-scanning sampling, the sampling interval is shortened by sub-pixel offset sampling. The sampling frequency is doubled and the Nyquist frequency is shifted to the right, which reduces the range of imaging aliasing frequencies. As shown in Fig. 4, the aliasing range of the 2 × 2 micro-scanning is significantly reduced. Under ideal conditions, signals can be accurately reconstructed under the 2 × 2 sampling Nyquist frequency, and the imaging aliasing is removed in the frequency range between the 2 × 2 sampling Nyquist frequency and the single sampling Nyquist frequency. Similarly, the range in which aliasing occurs is further reduced with 3 × 3 micro-scanning.

Fig. 4. Imaging aliasing under different sampling methods.

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In the case of k × k micro-scanning mode sampling, the sampling interval is reduced to 1/k of the original, resulting in the Nyquist frequency being k times that of the original. The aliasing of k × k sampling is expressed as,

(4)$$A_{SR}^{\textrm{k} \times \textrm{k}} = \frac{{\int_{k{u_n}}^{{u_c}} {MT{F_{pre}}(k{u_s} - u)MT{F_{fliter}}(u)} du}}{{\int_0^{{u_c}} {MT{F_{pre}}(u)MT{F_{fliter}}(u)} du}}$$

As the sampling frequency increases, the shape of the fold-back replicated frequency curve remains constant but shifts to the right with the sampling frequency, as shown in Fig. 4(b). The higher the sampling frequency, the fewer out-of-band replicated frequencies there are, resulting in less spurious response and aliasing.

However, the increased sampling rate of micro-scanning introduces redundant information compared to conventional imaging. In micro-scanning imaging, the size of the detector unit remains unchanged as the sampling frequency increases. The sampling range of each pixel is determined by the photosensitive size of the detector unit. As the sampling interval is reduced, the imaging signal originally captured by one detector unit is collected by multiple detector units after sub-pixel offset. Partial sub-pixel components are shared between the multiple sub-pixel shifted sampling results. There is a small amount of overlapping content that is sampled repeatedly, as shown in Fig. 1.

The repeatedly sampled content is mixed on adjacent micro-scanning sampling results, which reduces the resolution of micro-scanning reconstruction results. The micro-scanning acquires additional information through multiple offset sampling, but the exist of overlapping sampling regions introduces redundant information. Therefore, in the pre-sampling transfer function of the micro-scanning sampling process, the $MT{F_{\det }}$ is the same as that of single sampling. The MTF determined by the detector size is included as part of the pre-sampling transfer function to characterize the redundant information introduced by the micro-scanning sampling process.

Therefore, micro-scanning sampling still cannot achieve the multiple improvement in resolution corresponding to the reduced sampling interval and is limited by the detector size. The performance improvement of micro-scanning sampling can be quantitatively described by the squeeze property of the sampling process.

Under-sampling tends to blur the edges of the imaging results. This spatial spreading corresponds to a contraction in the frequency domain. Thus, the sampling process squeezes the pre-sampled transfer function. The amount of squeezing is determined by the out-of-band spurious response, which depends on the sampling frequency, the pre-sampling transfer function and the post-filtering.

Therefore, the effect of aliasing on the imaging results is essentially the squeezing of the system transfer function during the sampling process. Numerous recognition and identification experiments suggest that the squeezing of out-of-band spurious responses to the transfer function can be represented by squeeze factors [14,15]. For the same MTF value, the squeezed spatial frequencies are calculated from the squeeze factors as follows,

(5)$${u_{squeeze}} = {(1 - \alpha {A_{SR}})^\beta }u$$

where $\alpha$ and ${\beta _{}}$ are empirical constants. In the NVTherm model [16], the constants and are typically set to 0.58 and 1 respectively.

According to the definition of squeeze factors, the relationship between the MTF of pre-sampled and sampling-squeezed is expressed as,

(6)$$MT{F_{sam}}({u_{squeeze}}) = MT{F_{pre}}(u)$$

where $MT{F_{sam}}$ is sampling-squeezed transfer functions. Substitute the squeeze factor,

(7)$$MT{F_{sam}}(u) = MT{F_{pre}}\left[ {\frac{u}{{{{(1 - \alpha {A_{SR}})}^\beta }}}} \right]$$

However, micro-scanning imaging reduces aliasing, which is a stretching of the system transfer function during micro-scanning sampling. Compared to the squeezed MTF of single sampling, micro-scanning imaging achieves a reduction in the MTF squeezing under multiple offset sampling. In that sense, micro-scanning imaging stretches the transfer function of single sampling, thereby improving the imaging performance. Although micro-scanning imaging increases the sampling frequency, the transfer function is still limited by the photosensitive size of the detector. Therefore, the stretching of the transfer function by micro-scanning sampling limited by the $MT{F_{\det }}$, and the upper limit of the stretching depends on the pre-sampling transfer function.

In practice, the impact of the detector array on the imaging result is characterized in two parts during the micro-scanning imaging process. One is the sampling region, which describes the MTF degradation caused by sampling redundancy, and the other is the sampling interval, which describes the MTF squeeze caused by under-sampling.

For the k × k micro-scanning mode, the squeezing of the transfer function caused by sampling is expressed as follows,

(8)$$MTF_{sam}^{k \times k}(u) = MT{F_{pre}}\left[ {\frac{u}{{{{(1 - \alpha A_{SR}^{\textrm{k} \times \textrm{k}})}^\beta }}}} \right]$$

Figure 5(a) shows the aliasing regions corresponding to different micro-scanning sampling modes with an F-number of 1.4 and a fill factor of 50%. The squeezed transfer functions for the micro-scanning sampling modes of 2 × 2, 3 × 3, and 4 × 4 are calculated based on the amount of aliasing, as shown in Fig. 5(b). It can be seen that the stretching of the transfer function decreases with higher micro-scanning steps, and the small micro-scanning step shows a more significant improvement. The stretched transfer function is closer to the pre-sampling transfer function as the number of micro-scanning steps increases. According to the composition of the pre-sampling transfer function, the optical system and the photosensitive size of the detector determine the upper limit of the stretching.

Fig. 5. Aliasing and stretching of transfer functions under different micro-scanning modes.

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Micro-scanning imaging achieves the stretching of the MTF for single sampling by increasing the sampling frequency. For the same MTF value, the relationship between micro-scanning imaging and single sampling transfer functions is expressed as,

(9)$$u_{stretch}^{k \times k} = {\left( {\frac{{1 - \alpha A_{SR}^{\textrm{k} \times \textrm{k}}}}{{1 - \alpha {A_{SR}}}}} \right)^\beta }{u_1}$$

where ${u_1}$ is the spatial frequency of the single sampling transfer function, $u_{stretch}^{k \times k}$ is the spatial frequency of the micro-scanning imaging transfer function. The transformed expression is,

(10)$$u_{stretch}^{k \times k} = {\left( {1 + \frac{{\alpha ({{A_{SR}} - A_{SR}^{\textrm{k} \times \textrm{k}}} )}}{{1 - \alpha {A_{SR}}}}} \right)^\beta }{u_1}$$

Define ${e^k}$ as the micro-scanning imaging stretch factor, which represents the MTF stretching of micro-scanning sampling relative to single sampling. The stretch factor is calculated by the aliasing spurious response, which characterizes the complex coupling relationship between the transfer functions of the optical system, detectors, and digital filters with micro-scanning sampling, and is expressed as,

(11)$${e^k} = \frac{{\alpha ({{A_{SR}} - A_{SR}^{\textrm{k} \times \textrm{k}}} )}}{{1 - \alpha {A_{SR}}}}$$

Then, the MTF for different micro-scanning modes is expressed in terms of the MTF of single sampling as follows,

(12)$$MTF_{ms}^{k \times k}(u) = MT{F_{sig}}[{{{({1 + {e^k}} )}^\beta }u} ]$$

where $MT{F_{sig}}$ is the transfer function of the single sampling imaging system, $MTF_{ms}^{k \times k}$ is the transfer function of the k × k micro-scanning imaging system.

3. Quantitative analysis of micro-scanning performance improvement under different parameters

In the design of micro-scanning imaging systems, there are challenges in parameter optimization and trade-offs. Additionally, complex coupling relationships exist among the key parameters such as fill factor, detector size, micro-scanning mode, and optical F-number. Although studies have discussed several parameters, the analysis of optical system parameters is insufficient, and the trade-offs for optimizing these parameters are not well addressed. Therefore, the matching relationship of parameters between optical F-number, fill factor, and micro-scanning mode is analyzed based on the micro-scanning imaging characterization model proposed in this paper.

As a key component of the imaging transfer process, optical diffraction results in the loss of high frequency information. Typically, signals within the optical cutoff frequency and above the Nyquist frequency are aliased due to under-sampling. The level of imaging aliasing is affected by the combined effects of the sampling frequency and the optical cutoff frequency.

To understand the coupling relationship between micro-scanning sampling and optical systems, micro-scanning sampling results at different F-numbers are analyzed. The imaging blur caused by the optical system is represented as a point spread function in the spatial domain. Optical diffraction causes the image information to be spread out. For different diffraction radii, the micro-scanning system produces different sampling results during the spatial offset sampling process.

The diffraction spots of two points through the optical system must satisfy the Rayleigh criterion to be resolved. When the optical system has a small F-number, the point spread function's Airy disk size is relatively small. Typically, the sampling frequency is within the cutoff frequency of the optical system and the imaging resolution is limited by sampling. Points that are originally indistinguishable by single sampling can be distinguished by micro-scanning offset sampling, as shown in Fig. 6(a). However, the greater the interval between the Nyquist frequency and the optical cutoff frequency, the more aliasing occurs.

Fig. 6. Micro-scanning offset sampling under different F numbers.

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When the optical system has a large F-number, it usually provides poorer transmission characteristics and greater blur sizes. The sampling frequency exceeds the optical cutoff frequency, and the optical system limits the imaging resolution. Micro-scanning offset sampling is unable to distinguish the points that are initially indistinguishable by single sampling, as shown in Fig. 6(b). Micro-scanning sampling cannot effectively reduce aliasing.

Analysis of the coupling relationship between optical diffraction and sampling shows that different optical diffraction spot sizes result in different amounts of aliasing during the sampling process. The performance improvements achieved by micro-scanning sampling are also different. When the system is optical limited, the aliasing is small and micro-scanning imaging has little benefit. The lower the f-number, the more aliasing occurs. When the system is sampling limited, micro-scanning can effectively reduce aliasing. However, the ability to reduce aliasing for micro-scanning is also limited by the sampling redundant information, thus the smallest F-number will not provide the maximum benefit.

To quantitatively describe the trade-off optimization relationship between parameters, the coupling between optical parameters and fill factor, micro-scanning mode, detector size is achieved based on the aliasing model.

The MTF of the optical system mainly includes aberrations and diffraction. The primary design parameters of the optical system are considered so that aberrations are neglected. The MTF of a diffraction limited optical system is expressed as follows,

(13)$$MT{F_{opt}}(u )= \frac{2}{\pi }\left[ {\arccos \left( {\frac{u}{{{u_{\textrm{o}c}}}}} \right) - \frac{u}{{{u_{\textrm{o}c}}}}\sqrt {1 - {{\left( {\frac{u}{{{u_{\textrm{o}c}}}}} \right)}^2}} } \right]$$

Where ${u_{oc}}$ is the optical cutoff frequency, expressed as ${u_{oc}} = \frac{D}{\lambda } = \frac{f}{{\lambda F}}$, D is the diameter of the optical system, $\lambda$ is the wavelength, f is the focal length, and F is the F-number.

The optical transfer function directly influences the pre-sampling transfer function of the system, and thus indirectly influences the aliasing frequency distribution during sampling. The distribution of aliasing frequencies determines the amount of aliasing. By expanding the pre-sampling transfer function, the Eq. (13) is substituted into Eq. (2), and then Eq. (2) is substituted into Eq. (4). Aliasing under different optical MTF is calculated as,

(14)$$A_{SR}^{\textrm{k} \times \textrm{k}} = \frac{{\int_{k{u_n}}^{{u_c}} {MT{F_{opt}}(k{u_s} - u)MT{F_{\det }}(k{u_s} - u)MT{F_{fliter}}(u)} du}}{{\int_0^{{u_c}} {MT{F_{opt}}(u)MT{F_{det}}(u)MT{F_{fliter}}(u)} du}}$$

The optical transfer function is substituted into the formula for calculating aliasing spurious response. The aliasing spurious response is calculated under different F-numbers at 50% fill factor. Figure 7(a) and Fig. 7(b) show the aliasing response for single sampling and 2 × 2 micro-scanning sampling under different F-numbers.

Fig. 7. Aliasing for single sampling and micro-scanning sampling under different F-numbers.

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Obviously, as the F-number increases, the amount of sampling aliasing gradually decreases. The amount of sampling aliasing is positively correlated with the optical F-number. However, the performance improvement of micro-scanning sampling over single sampling under different F-numbers requires further analysis.

The micro-scanning stretch factor is used to describe the MTF improvement of micro-scanning over single sampling, and the coupling relationship between the optical F-number and other imaging parameters is discussed based on the micro-scanning aliasing calculation model. This analysis aims to find the proper F-number to achieve optimal micro-scanning imaging enhancement.

The stretch factor calculation results for three micro-scanning modes under different F-numbers with detector fill factors of 50%, 75%, 83% and 100% are shown in Fig. 8. For smaller fill factors, the micro-scanning stretch factor calculation results are relatively larger, and as the fill factor increases, the benefits of micro-scanning sampling gradually decrease. As the F-number increases, the micro-scanning stretch factor first increases and then decreases, and this rule is suitable for all fill factor values.

Fig. 8. Stretch factor calculation results under different F-numbers, fill factors, and micro-scanning modes.

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The results show that there is an optimal F-number that maximizes the stretch factor, and the optimal F-number increases as the fill factor increases. Specifically, at a fill factor of 50%, the optimal F-number is 3.7. At a fill factor of 75%, the optimal F-number is 4.5. At a fill factor of 83%, the optimal F-number is 5.2. At a fill factor of 100%, the optimal F-number is 5.5. The optimal optical F-number for a micro-scanning imaging system is typically around 5.

The performance of micro-scanning imaging is affected by several factors, including the micro-scanning mode, detector interval, fill factor, optical F-number, and filter cutoff frequency. For different micro-scanning modes, as the micro-scanning steps increases, the remaining aliasing gradually decreases, which reduces the ability to remove aliasing for micro-scanning.

A larger detector interval leads to a lower sampling frequency and increased aliasing, while the micro-scanning imaging performance is better. For the same sampling interval, a smaller fill factor means a smaller detector photosensitive size, which results in increased sampling aliasing and greater improvement by micro-scanning. However, when the post-filter cutoff frequency is relatively low, the aliasing is reduced, but the micro-scanning sampling cannot effectively improve the performance. Therefore, an appropriate higher cutoff frequency should be selected for the post-filter of the micro-scanning imaging system.

4. Simulation experiment and verification

To verify the reliability and accuracy of the imaging aliasing calculation and parameter optimization methods, micro-scanning imaging simulation experiments are performed. However, existing micro-scanning simulation methods lack the ability to simulate under multiple parameters. Therefore, this paper accurately simulates the micro-scanning imaging results based on the end-to-end actual imaging process. The analysis results of the aliasing characterization model are verified, and the performance of the micro-scanning imaging system is evaluated under different optical parameters.

During the micro-scanning imaging process, the imaging field remains unchanged. And the sampling positions of the detector shift by a sub-pixel distance, resulting in sampling differences. To accurately reflect the imaging differences caused by the offset sampling, it is necessary to construct a detailed distribution of the imaging information.

First, the accurate radiation field is obtained by coupling all global radiation components, including incident radiation, reflected radiation, and self-radiation. As shown in Fig. 9(a), the radiation distribution at the entrance pupil is obtained based on the imaging geometric relations, and atmospheric transmission attenuation and scattering are superimposed. Then, the radiation distribution at the entrance pupil is superimposed with the diffraction and transmission effects of the optical system to obtain the pre-sampled fine grid data, as shown in Fig. 9(b). Finally, the fine distribution is integrated and combined within the sampling range according to the sampling interval and the photosensitive size of the detector unit, and the imaging results at different sampling positions are obtained. The result for single sampling is shown in Fig. 9(c). For different micro-scanning modes, the results of different offset samplings are combined according to the sampling positions and reconstructed to obtain simulated micro-scanning imaging results. The 2 × 2 micro-scanning imaging result is shown in Fig. 9(d).

Fig. 9. Micro-scanning imaging simulation process illustration and the simulation results of each step.

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4.1 MTF test under different micro-scanning modes

The knife-edge, ISO-12233, and USAF patterns are commonly used to evaluate the transfer performance and resolution of imaging systems. This paper measures the transfer function of the imaging system by calculating the spatial frequency response (SFR) of the knife edge. Research shows that when the tilt angle of the edge is set to 5 degrees, the test results show relative stability compared to other angles [17]. Thus, the tilt angle is set at 5 degrees to minimize simulation test errors.

The knife-edge target is used as the input for the micro-scanning imaging simulation model, multiple micro-scanning sampling results are obtained under different micro-scanning modes. By calculating the SFR of the knife edge in the reconstruction results, the system transfer function under different micro-scanning steps is obtained, as shown in Fig. 10. The system parameters are a 50% fill factor and an F-number of 2.3.

Fig. 10. MTF test and the measured results based on knife-edge target.

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Simulation experiments are performed to obtain the transfer functions and stretch factors of different micro-scanning modes. The theoretical value of the stretch factor is calculated by using the aliasing model in section 2. After normalizing the theoretical and measured values separately, the results are shown in Fig. 11. The results show that the values calculated by the theoretical model are consistent with the changing trends in the experimental measured values for the performance improvement brought by different micro-scanning modes. The aliasing quantification model based on spurious response can be used to analyze the imaging performance of micro-scanning.

Fig. 11. Comparison of measurement results and calculation results.

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4.2 Analysis of micro-scanning imaging characteristics under different F-numbers

The knife-edge target is used as the input for the micro-scanning imaging simulation model, and different optical F-numbers are set to obtain the imaging results for 2 × 2, 3 × 3, and 4 × 4 micro-scanning modes. The MTF of the knife edge in the imaging results is calculated, and the corresponding experimental measured values of the stretch factors are calculated, which represent the practical performance of micro-scanning imaging under different optical parameters.

The measured stretch factors of micro-scanning imaging are shown as scattered dots in Fig. 12. The fluctuations in the calculation results are due to the fact that the measured MTF is based on sampled images and the pixel size is close to the diffraction spot size of the optical system. Gaussian filtering is applied to smooth the scattered point results, and the trend of the measured stretch factors of the transfer functions for the three micro-scanning modes under different F-numbers is shown in Fig. 12. The fill factors considered are 50%, 75%, 83%, and 100%.

Fig. 12. Measured results of the stretch factor under different F-numbers, fill factors, and micro-scanning modes.

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According to the experimental results, the optimal F-number is 4.3 when the fill factor is 50%. The optimal F-number is 5.3 when the fill factor is 75%. The optimal F-number is 6.0 when the fill factor is 83%. The optimal F-number is 7.2 when the fill factor is 100%.

The test results are compared with the calculated results in section 3, and the varying trend of the stretch factor remains consistent. The result confirms the existence of an optimal F-number for micro scanning imaging. As the stretch factor increases, the optimal F-number also increases. However, experimental measurements of the optimal optical F-number are overall a little larger than the theoretical value. The variation of the optical system parameters typically affects the sampling within a few pixels, and the micro-scanning reconstruction results are sensitive to the sampling phase [18]. Therefore, some errors are introduced in the calculation of the edge transfer function for a knife edge image. In addition, the calculation model has taken some approximations and only considered the aliasing caused by the first foldback frequencies. The error is within a reasonable range, does not affect the overall trend, and is sufficient to verify the conclusion in section 3.

In summary, the optimal optical F-number for a micro-scanning imaging system is typically between 4 and 7, when the micro-scanning imaging system achieves the maximum transfer function improvement through micro-scanning offset sampling. Specific discussions can be made by considering the actual system parameters.

5. Conclusions

Micro-scanning imaging systems achieve reduced aliasing and high-resolution imaging by increasing the sampling rate. However, existing models struggle to accurately describe micro-scanning aliasing and lack a comprehensive understanding of the coupling mechanisms in the micro-scanning imaging process. In addition, the coupling and trade-offs between system parameters are not clearly defined. To address the quantitative description of micro-scanning imaging performance improvement, this paper innovatively combines the traditional imaging spurious response model with micro-scanning sampling mechanism. Then, a micro-scanning aliasing analysis model is proposed that couples the transfer functions of the optical system, detectors, and digital filters to the micro-scanning sampling process. Furthermore, an innovative approach to calculate the transfer function of the micro-scanning imaging system based on aliasing is introduced, which overcomes the difficulty of existing micro-scanning models in accurately characterizing the micro-scanning sampling and coupling mechanism.

First, based on the constraints between the transfer functions of different components, the imaging aliasing under different sampling modes is evaluated, and the transfer function stretching of micro-scanning sampling for different parameters is obtained by aliasing calculations. Second, the amount of aliasing in micro-scanning imaging under different optical system parameters is calculated, and the performance improvement of micro-scanning imaging under different F-numbers is predicted. The results show that there exists an optimal F-number that maximizes the benefits of micro-scanning sampling, and this optimal F-number increases with the fill factor. Furthermore, the optimal micro-scanning imaging F-numbers for different fill factors are given, and the matching relationship between optical parameters, fill factors, and micro-scanning modes is discussed. Finally, a micro-scanning imaging simulation method based on the actual imaging transfer and sampling process is established. Through simulation experiments, the accuracy of the micro-scanning aliasing model is verified, and test results of optimal F-numbers are given. This study clarifies the mechanism of micro-scanning for aliasing removal and analyzes the matching relationships among optical system parameters, micro-scanning fill factor, and micro-scanning modes based on calculation and experimental results. It provides valuable insights for the refined optimal design of micro-scanning imaging systems.

Funding

National Natural Science Foundation of China (62005204, 62005206, 62075176); Fundamental Research Funds for the Central Universities.

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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Quantitative characterization of micro-scanning imaging aliasing and optical parameter optimization

Abstract

1. Introduction

2. Accurate characterization of micro-scanning imaging aliasing and performance

3. Quantitative analysis of micro-scanning performance improvement under different parameters

4. Simulation experiment and verification

4.1 MTF test under different micro-scanning modes

4.2 Analysis of micro-scanning imaging characteristics under different F-numbers

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Equations (14)

Optics Express