# Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging
## 2 Materials and Methods
### Motion Degradation Model for Single-Pixel SFDI  
In single-pixel SFDI, the reflected image is reconstructed from a sequence of sampling patterns and their corresponding measurement values. When the target moves, the reflected image at the \(i\)th measurement is  
\[I_{i}(x,y)=M_{\text{DC}}(x+\Delta x_{i},y+\Delta y_{i})+M_{\text{AC}}(x+\Delta x _{i},y+\Delta y_{i})\cdot\cos(2\pi f\cdot x+\varphi_{0}), \tag{1}\]  
where \(M_{\text{DC}}(x,y)\) and \(M_{\text{AC}}(x,y)\) are the DC and AC amplitude images; \(\Delta x_{\text{i}}\) and \(\Delta y_{\text{i}}\) are the target displacements in the X and Y directions at the \(i\)th measurement (taking the image at the first measurement as the true image), respectively; \(f\) is the spatial frequency of the sinusoidal illumination; and \(\varphi_{0}\) is the initial phase. The target moves between measurements, but the sinusoidal illumination in the imaging field of view (FOV) remains stationary during the imaging process. This requires that the motion deblurring only works on the moving target while retaining the background sinusoidal fringes. In this method, the motion degradation model is derived by the two properties of the Fourier transform that the sinusoidal modulation of an image in the spatial domain results in a frequency shift in the Fourier domain (frequency-shift property), and the translation of an image in the spatial domain results in a phase shift in the Fourier domain (phase-shift property). By transforming the reflected image into the Fourier domain, the sinusoidal fringes can be simplified as an impulse component that shifts the spectral center of the AC image to the frequency of \(f\). The Fourier spectrum of the reflected image is therefore separated into the DC and AC components. Then, according to the phase-shift property,the effects of the target displacements on the DC and AC components can be eliminated by the phase-shift compensation. To facilitate the processing in the Fourier domain, the FSI that acquires the Fourier coefficients of the image is adopted. Within the short time of a single measurement, the target can be treated as relatively static. The moving process is accordingly discretized into M periods, each corresponding to a measurement. The Fourier coefficient acquired at the \(i_{th}\) measurement satisfies the phase-shift and frequency-shift properties, as,  
\[F^{\prime}{}_{v}(u,v)=F_{v}(u,v)\cdot e^{-2\pi(\Delta x_{i}\cdot\frac{u-u_{0}} {N}+\Delta y_{i}\cdot\frac{v}{N})}+N(u,v),\ u\geq 0, \tag{2}\]  
where \(F_{v}(u,v)\) and \(F^{\prime}{}_{v}(u,v)\) are the DC or AC Fourier components of the true and blurry images, respectively; \(\{\mathrm{DC},\mathrm{AC}\}\in v\); \(N(u,v)\) is the noise spectrum; \(u\) and \(v\) are the spatial frequencies of the Fourier coefficients; \(u_{0}\) is the center frequency of the sinusoidal illumination, where \(u_{0}=0\) for the DC image; and N is the pixel resolution of the image. Here, it is assumed that the target moves within the FOV during the entire imaging procedure. Because the Fourier spectrum of a natural image is conjugate symmetric, only the Fourier coefficients for \(u\geq 0\) are required to measure. The coefficients for \(u<0\) are obtained using the conjugate symmetry. The imaginary term in Equation (2) is denoted as the degradation function for the DC or AC component,  
\[H_{v}(u,v)=e^{-2\pi(\Delta x_{i}\cdot\frac{v-u_{0}}{N}+\Delta y_{i}\cdot\frac {v}{N})},\ u\geq 0, \tag{3}\]  
The degradation function for the AC component has a frequency shift of \(u_{0}\) in the X direction due to the sinusoidal modulation. The phase shifts caused by the target displacements can be compensated by inverse filtering based on the degradation functions,  
\[\hat{F}_{v}(u,v)=\frac{F^{\prime}{}_{v}(u,v)}{H_{v}(u,v)}, \tag{4}\]  
Because the DC and AC components are separated and concentrated in their respective central frequencies, the phase-shift compensation works on them independently.