# Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging
## 3 Results
### Simulation Validations  
In the simulation validations, a letter \"E\" is used as the target, modulated by the spatial frequency of 0.2 mm\({}^{-1}\), as shown in Figure 3a. The pixel resolution of the image is 128 \(\times\) 128. The target in piecewise uniform motion along X and Y directions and in circular motion with a constant angular acceleration are simulated, respectively. 500 sampling patterns are used to reconstruct the reflected image, with P\({}_{x}\) and P\({}_{y}\) measured every 10 samplings (\(K\) = 10) to estimate target displacements. A single frame is measured in the piecewise uniform motion simulation, while two complete frames are measured in the circular motion simulation. Then, 30 dB Gaussian noise is added to the simulated data. The structural similarity index measure (SSIM) is used to evaluate the effects of the method.  
The results of the uniform motion simulation are shown in Figure 3. The target successively moves 90 pixels in the X direction and 60 pixels in the Y direction within 600 measurements. Figure 3b,c show the results obtained using the proposed method and conventional single-pixel SFDI, respectively. In the conventional single-pixel SFDI method, the reflected image reconstructed is severely degraded due to the target motion in the measurements. The targets in the demodulated DC and AC images are completely illegible. The proposed method restores the degraded image for both the DC and AC components and obtains the demodulated images with clear details. The SSIM of the DC and AC images between Figure 3c and Figure3a are 0.91 and 0.86, compared to the values of 0.53 and 0.49 between Figure 3b and Figure 3a, respectively. Figure 3d shows the target displacements in the X and Y directions estimated from the measurement values. The sinusoidal modulation reduces the accuracy of estimated displacement in the X direction to some extent, but the results are within the error of <2%.  
Figure 3: Simulation results of the target in piecewise uniform motion. (**a**) True target images. Red arrows show the moving path and directions of the target. Results obtained using (**b**) conventional single-pixel SFDI and (**c**) the proposed method. Target displacements in (**d**) X and (**e**) Y directions, calculated from motion estimation.  
The results of circular motion simulation are shown in Figure 4. The target moves clockwise in a circle with a constant angular acceleration, moving one turn in 1200 measurements, as shown in Figure 4a. A total of 50 frames were obtained using the dynamic reconstruction scheme. Figure 4b,c show the results of Frame #30 reconstructed by the conventional single-pixel SFDI and the proposed method, respectively. The mean SSIM of DC and AC images of all frames is increased from 0.58 and 0.55 to 0.91 and 0.86 after the restorations. The estimated displacements in the X and Y directions are also within the error of 2% of the true values. The angular velocities are also calculated from the estimated displacements and are fitted using linear regression, as shown in Figure 4d. The angular acceleration (i.e., the slope of the fitting line) is within an error of <0.2%, proving the capability of the proposed method to dynamically locate the moving target with varying speeds and directions.