# On Ghost Imaging Studies for Information Optical Imaging
###### Abstract
These significantly limit the design and development of further imaging capabilities from the perspective of information theory, and make it difficult to fully exploit the capability of the Shannon channel capacity for transmitting image information in an efficient and accurate way. Optical Imaging from the Viewpoint of Optical Coherence Theory and the Connection between Ghost Imaging and Information Optical Imaging  
Generally speaking, the aim of an imaging system is to transmit the image information of imaging objects to realize spatial domain \"point-to-point\" signal reproduction, and can be mathematically considered as accomplishing the vector transform from the object Hilbert space to the image Hilbert space [33]. In an optical system, the image information is essentially carried by propagating light fields \(E(\mathbf{r},t)\), which, according to the optical coherence theory [34], is a random process that can be well described by its statistical properties. Namely, the entirety of the information that it carries is contained in its joint moments of all orders (i.e., optical correlation functions of all orders)  
\[ \begin{split}& G^{(M,N)}(\mathbf{r}_{1},\mathbf{r}_{2},\cdots,\mathbf{r}_{M+N };t_{1},t_{2},\cdots,t_{M+N})\\ =&\left\langle\begin{array}{l}E^{*}(\mathbf{r}_{1},t_{ 1})E^{*}(\mathbf{r}_{2},t_{2})\cdots E^{*}(\mathbf{r}_{M},t_{M})\\ \times E(\mathbf{r}_{M+1},t_{M+1})E(\mathbf{r}_{M+2},t_{M+2})\cdots E(\mathbf{r}_{M+N},t_{ M+N})\end{array}\right\rangle.\end{split}  \tag{3}\]  
It is worth noting that the optical correlation function has different appellations and notations; for the case \(M=N\) involving light fields at \(2M\) different spatiotemporal positions, the correlation function \(G^{(M,M)}\) is also referred to as being of order \(M\) rather than \(2M\), and is written as \(G^{(M)}\) instead. Below in this article, the appellation and notation corresponding to \(G^{(M)}\) are applied. Since the coherent transfer function (CTF) of an imaging system relates detected light fields with light fields from the object [35], by properly designing paraxial-approximate CTFs of various imaging systems and combining with the coherence property of light sources, the desired image information can be retrieved with varied accuracy from the correlations of different orders of light fields at the detection plane. From this perspective, traditional imaging systems can be understood as acquiring the desired image information from the first-order correlation of light fields at the detection plane [36]. Specifically, in a typical camera system, a spatial point-like transfer function is designed under the paraxial approximation so that the light intensity (the first-order auto-correlation of light fields) recorded at the detection plane is  
\[I_{\rm t}(\mathbf{r})=\langle E_{\rm t}^{*}(\mathbf{r})E_{\rm t}(\mathbf{r})\rangle=h_{\rm I }(\mathbf{r})*T_{\rm o}(\mathbf{r}), \tag{4}\]  
and the point-to-point reproduction of the desired spatial image information \(T_{\rm o}(\mathbf{r})\) is realized. In the holographic imaging system [37], the hologram generated by interfering the object beam and the reference beam is distributed as  
\[ \begin{split} I(\mathbf{r})=&\langle E_{\rm r}^{*}( \mathbf{r})E_{\rm r}(\mathbf{r})\rangle+\langle E_{\rm o}^{*}(\mathbf{r})E_{\rm o}(\mathbf{r}) \rangle\\ &+\langle E_{\rm r}^{*}(\mathbf{r})E_{\rm o}(\mathbf{r})\rangle+\langle E _{\rm o}^{*}(\mathbf{r})E_{\rm r}(\mathbf{r})\rangle^{\prime}\end{split}  \tag{5}\]  
and the complex transmittance information of the imaging object \(t(\mathbf{r}^{\prime})\) is essentially contained in the first-order cross-correlation between the light fields of two beams, i.e., the last two terms of Equation (5), where \(\langle E_{\rm r}^{*}(\mathbf{r})E_{\rm o}(\mathbf{r})\rangle\propto\int t(\mathbf{r}^{ \prime})\exp\left[\frac{k}{2\pi}(\mathbf{r}^{\prime}-\mathbf{r})^{2}\right]{\rm d} \mathbf{r}^{\prime}\). The image information can also be extracted from the high-order correlation of light fields [38].