# Image Transmission through Scattering Media Using Ptychographic Iterative Engine
## 1 Introduction  
When the information-bearing light is disturbed by scattering media such as fog, haze [1], biological tissues [2], and the surfaces of walls [3], a common optical imaging system will fail to realize regular pixel-to-pixel mapping between the object space and the image space. The inherent reason is that the wavefront that carries the object information undergoes random disturbance as it propagates owing to the inhomogeneity of the refractive index distribution of the scattering media. To solve this problem, several methods have been proposed in recent years to transmit, focus, and image through scattering media based on different application scenarios [4; 5; 6; 7; 8; 9; 10; 11; 12]. Among them, transmission of image information through scattering media has received much attention. Typically, this task can be achieved with the help of the transmission matrix of the system [4], where a precise measurement of the system should be implemented. Wave front shaping [5] and phase conjugation [6] can be used when a reference point or object should be known a priori. Single pixel detection could be applied if the object could be projected with a series of fringe patterns [7; 8].  
On the other hand, the correlation property of the light field that passes through the scattering medium has also been explored to reconstruct the image of the object behind it. New progress in the memory effect makes it possible to image three-dimensional objects [13; 14], retrieve spectral information [15], and work under noisy conditions [16; 17]. The fundamental theory of the memory effect was developed by [PERSON] et al. [18] and [PERSON] [19]. The basic idea of this method is that for a scattering medium, there exists a maximum field-of-view (FOV) called the memory effect range, which is determined by the optical thickness of the medium. Within this range, the system is approximately translation-invariant, with the impulse response having the form of a speckle \(S(r)\). The autocorrelation of the speckle \(S(r)\) is a sharp-peaked function [11; 12] with a uniform background term. Therefore,with incoherent illumination, the camera image \(I(r)\) can be written as the convolution of the target function \(O(r)\) with the point spread function \(S(r)\) by:  
\[I(r)=O(r)*S(r), \tag{1}\]  
where the symbol \(*\) denotes convolution and \(r=(x,y)\) is the two dimensional spatial coordinate. The autocorrelation of the camera image reveals that:  
\[\begin{split} I(r)\star I(r)&=[O(r)*S(r)]\star[O( r)*S(r)]\\ &=[O(r)\star O(r)]*[S(r)\star S(r)]\\ &\approx O(r)\star O(r)+C,\end{split} \tag{2}\]  
where the symbol \(\star\) stands for correlation and \(C\) a constant value. Therefore, if the object is within the memory effect range, then the autocorrelation of the collected speckles is equivalent to the autocorrelation of the object. From the autocorrelation, a diffraction limited reconstruction of the target can be reconstructed using phase retrieval techniques [11; 12; 20].  
However, the memory effect imposes a limit of the FOV that can be achieved by this method [19]: \(\Delta\theta\approx\lambda/(\pi L)\), where \(L\) denotes the effective optical thickness of the scattering medium. For example, the FOV of a 0.5 mm thick chicken breast slice with the optical thickness being equal to 7 \(\upmu\)m is approximately 1.4 degrees when illuminated by a laser with a wavelength of 532 nm [12]. Here, we propose an approach to obtain large FOV imaging through scattering media with the help of a ptychographic iterative engine (PIE). Additionally compared with a conventional iterative phase retrieval algorithm, PIE is more robust without twin images and has a faster and more reliable convergence performance [21; 22; 23].