# Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging
## 3 Results
### Experimental Validations  
The experimental validations were performed using a custom-developed single-pixel SFDI system [17]. In the system, two DMDs are used as the spatial light modulator, one generating the sinusoidal illumination pattern for projection, and the other generating the Fourier basis patterns for spatially encoding the reflected images. The light signal is detected by a high-sensitivity photomultiplier tube. The DMD refresh rate used in the experiment is 40 Hz. The pixel resolution of the image is 128 \(\times\) 128, over the FOV of 40 \(\times\) 40 mm\({}^{2}\). The spatial frequency of the sinusoidal illumination is 0.2 mm\({}^{-1}\). The number of samplings for image reconstruction is 132, and the P\({}_{x}\) and P\({}_{y}\) are measured every 10 samplings (\(K\) = 10) for motion estimation. Then, 3-step phase-shifting FSI is adopted to acquire the complex-valued Fourier coefficients, resulting in a total of 474 measurements to obtain a deblurred image. A tissue-like phantom is used in the experiment. The phantom was fabricated using India ink as the main absorber, Intralipid-20% as the main scatterer, and was cured using agar. The phantom is 70 cm \(\times\) 70 cm \(\times\) 30 cm in size with the background absorption  
Figure 4: Simulation results of the target in circular motion. (**a**) True target images. Red line and arrows show the moving path and directions of the target. Results (Frame #30) obtained using (**b**) conventional single-pixel SFDI and (**c**) the proposed method. Target displacements in (**d**) X and (**e**) Y directions and (**f**) angular velocities, calculated from motion estimation. The red windows indicate the measurements used for reconstructing Frame #30.  
coefficient (\(\mu_{a}\)) of 0.01 mm\({}^{-1}\) and the reduced scattering coefficient (\(\mu^{\prime}_{s}\)) of 1.2 mm\({}^{-1}\). A cylindrical target with the diameter of 1 cm and the depth of 0.5 cm is in the middle of the phantom. The \(\mu_{a}\) and \(\mu^{\prime}_{s}\) of the target are 0.04 mm\({}^{-1}\) and 1.2 mm\({}^{-1}\), respectively. In the experiment, the phantom moves in different speeds within the FOV. The speed is controlled by manually operating a linear translation stage. Since manual control is hard to ensure a constant speed, we use the average speed as the true value, which is calculated according to the total measurement time and the total displacement of the target. The average speeds of the phantom are approximately \(v\) = 0, 0.5, 1, 1.5, 2 and 2.5 mm/s, respectively. The maximum validated speed is 2.5 mm/s, as the higher speed will force the target to move out of the FOV. The reflected images of the phantom at different speeds were obtained using the conventional reconstruction method and the proposed method, respectively. The optical properties were then extracted from the reflectance data. Figure 5 shows the optical property results of the static phantom and the phantom at different speeds, respectively. When the phantom moves, the absorption maps obtained by the conventional single-pixel SFDI have severe artifacts, resulting in significant reductions in the quantitative accuracy. The scattering maps do not have the motion artifacts, as the target has the same reduced scattering coefficient as the background. In contrast, the proposed method effectively deblurs the target and improves the quantitative accuracy. The estimated speeds of the phantom are within the error of 8%. The SSIM and root-mean-square error (RMSE) of the results are calculated. The results of the static phantom are used as the reference. The proposed method significantly improves the SSIM and RMSE of the results at all speeds compared with the conventional method, as shown in Table 1. The results indicate that the proposed method has a strong ability to image a moving target.