# Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging
## 4 Discussion  
Despite its success, there are limitations of the method. The motion degradation model derived by the phase-shift property works well in dealing with the target translation within the FOV, but for other motions, such as target rotation or target moving out of the FOV, the model requires to be further improved. The current work is limited to process a single target. If multiple targets move within the FOV, the degradation model will be ineffective. Our future work will extend the method to multi-target motion deblurring. In addition, the method can be less effective when the spatial frequency used is low, because the phases of the DC and AC components can overlap with each other to hinder the phase-shift compensation. The low-frequency modulation will also reduce the efficacy of single-snapshot demodulation. We find that when the spatial frequency used is less than 0.1 mm\({}^{-1}\), the performance of the method is greatly reduced. In practice, the typically used spatial frequency is between 0.1-0.2 mm\({}^{-1}\), so the method is applicable to most scenarios. The motion estimation is important for image deblurring. But the accurate tracking of the target requires more frequently measuring two additional Fourier coefficients, which can further increase the number of samples for reconstructing the image. For the uniform motion and the circular motion validated in the numerical and phantom experiments, \(K\) value between 10 and 20 can give consideration to both imaging speed and quality. The number of measurements is only increased by less than a fifth. However, if the target moves completely randomly (its trajectory is hard to fit), a very small \(K\) value will be required, which could multiply the total number of measurements. In fact, the Fourier coefficients for image reconstruction can also be used for motion estimation. In the dynamic reconstruction, each updated Fourier coefficient has the phase-shift relation with its previously measured value, which can be used to calculate the target displacement between these two measurements. At the beginning stage of imaging, the errors are inevitable because only the total displacement within each frame is obtained, and the complex frequency components of the Fourier coefficients can further increase the difficulty and error in the calculation. However, as the displacements are updated constantly during imaging, the target locations are able to be tracked and fitted, thus improving the results. Similar motion estimation approaches using Fourier phase shifts have been reported in previous works. However, these works aimed to achieve an image-free tracking of moving target. Their ultimate goal was to obtain the location of the target, rather than to obtain the deblurred image. In these methods, Fourier phase shifts are only used to estimate target displacements. In contrast, the novelty of our work is to derive the motion degradation model for image deblurring. Our goal is to obtain a high-quality image of a moving target in single-pixel SFDI. Because the existing deblurring methods for SPI cannot handle the modulation background in the SFDI image and no motion deblurring method for single-pixel SFDI is reported, this study is not compared with other suitable methods. However, the comparison with the traditional single-pixel SFDI method can prove the effectiveness of the proposed method. The dynamic reconstruction strategy, while improving the speed of single-pixel SFDI, requires a trade-off between the frame rate and information redundancy as the measurement data is shared between frames. A more reasonable way to improve the imaging speed is to increase the DMD refresh rate. However, the grayscale sampling patterns required by FSI technique impose a limitation on this implementation as DMD generates grayscale patterns at a low refreshing rate. In this work, the DMD used supports a refresh rate of 120 Hz for grayscale modulation, but up to 4 kHz for binary modulation.