# Image Motion Measurement and Image Restoration System Based on an Inertial Reference Laser
## 4 Principles of Image Restoration  
Non-blind restoration algorithms have made considerable progress in the field of image restoration. Several representative non-blind restoration algorithms have achieved ideal image restoration effects in their respective application fields, such as the Wiener filtering algorithm [34], total variation regularized image restoration algorithm [35; 36; 37], hybrid spatio-spectral total variation image restoration algorithm [38],image restoration algorithm based on the natural image gradient distribution model [39], and fast non-blind restoration algorithm based on natural image gradient constraints [40]. The authors in [41] proposed a local piecewise regularization Richardson Lucy (RL) method and constructed a new regularization term that effectively controlled the noise and edge ringing effect in the restored image. According to the theory of linear systems, the image degradation process can be modeled as follows:  
\[g(x,y)=f(x,y)\times h(x,y)+n(x,y) \tag{1}\]  
where \(g(x,y)\) is the degraded image; \(f(x,y)\)is the clear image; \(h(x,y)\) is the degeneration transfer function of the imaging system, namely, the degradation point spread function PSF; and \(n(x,y)\) represents noise. According to the image degradation model, after the image degradation, PSF is reconstructed based on the available measurement information, the image can be restored directly using algorithms such as the deconvolution method, Wiener filtering method, or RL method. The frequency spectrum of the restored image can be formulated as:  
\[F(u)=\frac{G(u)}{H(u)}-\frac{N(u)}{H(u)} \tag{2}\]  
where \(G(u)\) is the degraded image, \(H(u)\) is the degraded PSF, and \(N(u)\) is the Fourier transform of noise. The local piecewise RL algorithm was used for the image non-blind restoration, that is, based on the RL algorithm, a new regularization term was constructed to suppress noise and ringing and to improve the quality of image restoration. The Bayesian maximum posterior probability of the restored image can be formulated as follows:  
\[\mathbf{f}=argmin_{\mathbf{f}}\{-lnP[(\mathbf{g}|\mathbf{f})]-ln[P(\mathbf{f})]\} \tag{3}\]  
Figure 5: Power spectral density curve of the background noise of the vibration isolation platform. The noise in the imaging process mainly includes readout noise and photoelectric shot noise. The readout noise is independent of the signal and follows Gaussian distribution. The photoelectric shot noise is related to the signal, which is the photoelectric noise caused by the photoelectric conversion process of the sensor and obeys Poisson distribution. Because a CCD imaging system with a low readout noise was used in the test, the noise of the image data approximately conforms to the Poisson noise model, that is, \(P\left(g|f\right)\) approximately conforms to the Poisson noise model, then:  
\[-ln[P(g|f)]=\sum_{i}\{(h\!f)_{i}-g_{i}ln[(h\!f)_{i}]+ln[(g_{i})!]\} \tag{4}\]  
where \(i\) is the pixel index. To construct the regular term \(-ln[P(f)]\), a spatial neighborhood system \(S\) was initially defined, followed by a series of displacement matrix \(C_{k}^{j}\quad(j=1,2,\cdots)\), in order for the result of \(C_{k}^{j}\!f\) to move the entire image for \(|k_{j}|\) pixels along the direction of \(K_{j}\), which represents the vector from the center pixel to the other pixels in \(S\). Therefore, the regular term can be expressed as:  
\[J_{r}(f)=-ln[P(f)]=f^{\!T}M\!f \tag{5}\]  
\[M=\sum_{j}\left(I-C_{k}^{j}\right) \tag{6}\]  
where \(I\) is the identity matrix. Assuming that \(\lambda\) is the regularization coefficient, Equations (5) and (6) can be substituted into Equation (3), then:  
\[f=argmin\left(\sum_{i}\{(hf)_{i}-g_{i}ln[(hf)_{i}]\}+\frac{\lambda}{2}f^{T}M \!f\right) \tag{7}\]  
By using expectation-maximization (EM) algorithm, Equation (7) can be transformed into the following iterative equation:  
\[f^{\!+1}=\frac{f}{1+\left.\frac{\lambda}{2}\frac{dJ_{r}(f)}{df}\right|_{f}} \cdot\left(h^{T}\frac{g}{h\!f}\right) \tag{8}\]  
where \(t\) is the number of iterations.