# Super-Resolution Lensless Imaging of Cells Using Brownian Motion
## 2 Materials and Methods
### Super-Resolution Reconstruction Algorithm for Cell Image in a Lensless System  
#### 2.3.1 Motion Estimation Algorithm  
Due to the random direction of Brownian motion, we used multi-frame super-resolution technology to reconstruct the super-resolution image, and extracted the multi-frame image of Brownian motion for super-resolution image reconstruction. To examine super-resolution in a lensless system, we set up an observation model of a low resolution image, and then reconstructed a high resolution image using a multi-frame low resolution image using this model. In this study, the super-resolution algorithm of the image can be analyzed using the observation model shown in Figure 2.  
This system symbolizes the cell transmitted to the surface of the sensor shadow image, and is similar to the image captured by a high-rate optical microscope. The acquisition of low-resolution image sequences can be represented as follows:  
\[y_{k}=M_{k}B_{k}DX+N_{k}. \tag{1}\]  
In this system, the blur matrix \(B_{k}\) and the down-sampling matrix \(D\) are known. The blur matrix is mainly caused by diffraction, and the down-sampling matrix is determined by the pixel size and pixel spacing of the CMOS image sensor. The \(M_{k}\) of each cell in this system is unknown, but the motion vector can be estimated by the motion estimation algorithm. Therefore, the key problems of the cell image super-resolution algorithm are motion estimation and high-resolution image reconstruction. Motion vectors must be known before high-resolution image reconstruction. Therefore, we first introduced the motion estimation algorithm in this system. Based on the research on Brownian motion displacement, we decided to use the two-step method to estimate the motion. In the first step, the Fourier-Mellin method was used to coarsely estimate the integer pixel displacement [30; 31]. Assuming the reference image is \(I_{1}\) and the image to be registered is \(I_{2}\), and \(I_{1}(x\), \(y)\) and \(I_{2}(x,y)\) are the pixel grayscale of the two images at coordinates \((x,y)\). If the Fourier transform of \(I_{1}\) and \(I_{2}\) are \(F_{1}(u,v)\) and \(F_{2}(u,v)\), then \(F_{2}\) is  
\[F_{2}(u,v)=F_{1}(u,v)e^{-\beta(ux_{0}+vy_{0})}. \tag{2}\]  
Its mutual power spectrum is  
\[Corr(u,v)=\frac{F_{2}(u,v)F_{2}^{*}(u,v)}{\left|F_{1}(u,v)F_{1}^{*}(u,v)\right| }=\varphi_{1}(u,v)-\varphi(u,v)=e^{\beta(ux_{0}+vy_{0})}, \tag{3}\]  
where \(corr(u,v)\) becomes a phase correlation function, in which the image \(I_{1}(x,y)\) is translated (\(x_{0},y_{0}\)) and rotated by \(\theta_{0}\) to obtain the image \(I_{2}(x,y)\), which is shown as follows:  
\[I_{2}=I_{1}(x\cos\theta-y\sin\theta+x_{0},y\cos\theta+x\sin\theta+y_{0}). \tag{4}\]  
This was Fourier transformed:  
\[F_{2}(u,v)=e^{\beta(ux_{0}+vy_{0})}F_{1}[(u\cos\theta_{0}-v\sin\theta_{0}),(v \sin\theta_{0}+u\cos\theta_{0})], \tag{5}\]  
If the amplitude spectra of \(F_{1}(u,v)\) and \(F_{2}(u,v)\) are \(M_{1}(u,v)\) and \(M_{2}(u,v)\), then  
\[M_{2}(u,v)=M_{1}(u,v)[(u\cos\theta_{0}-v\sin\theta_{0}),(v\sin\theta_{0}+u \cos\theta_{0})], \tag{6}\]  
Equation (6) shows that the amplitude of the spectrum is related to its rotation angle \(\theta_{0}\) and is independent of the translation amount (\(x_{0},y_{0}\)) of the image. Therefore, \(\theta_{0}\) and (\(x_{0},y_{0}\)) can be calculated separately. First, the rotation angle \(\theta_{0}\) is calculated in the frequency domain, and then (\(u,v\)) in the image amplitude spectrum is transformed into polar coordinates (\(\rho,\theta\)). The relationship between them is  
\[u=\rho\cos\theta, \tag{7}\]  
\[v=\rho\sin\theta, \tag{8}\]  
Obtained from Equations (7) and (8) is  
\[M_{2}(\rho,\theta)=M_{1}(\rho,\theta-\theta_{0}), \tag{9}\]  
\[y_{k}=M_{k}B_{k}DX+N_{k}. \tag{1}\]0 Multi-frame super-resolution observation model, where \(X\left(x,y\right)\) is the original image and this image is a high-resolution image.