# Improvements of Computational Ghost Imaging by Using Sequenced Speckle
## 2 Experiment Setup and Measurement Principle
### Measurement Principle  
In SGI, a DMD is used to sequentially project N different illuminations of random speckle patterns, \(S1(x,y)\), \(S2(x,y)\),..., \(S_{n}(x,y)\), onto the object \(O(x,y)\). The corresponding sequence of intensity signals, \(I1(p)\), \(I2(p)\),..., \(I_{n}(p)\), is recorded by the bucket detector. The intensity signal IN for the nth speckle pattern can be calculated as follows:  
\[I_{n}=\iint O(x,y)S_{n}(x,y)dxdy. \tag{1}\]  
The object image can be reconstructed using common correlation reconstruction algorithms [1; 2; 3; 32; 33]. \[G(x,y)=\frac{1}{N}\sum_{n=1}^{N}(I_{n}-\langle I\rangle)S_{n}(x,y). \tag{2}\]  
Here, the ghost image \(G(x,y)\) depends on the designed intensity distribution \(S_{n}(x,y)\) of sequenced random speckle patterns and the intensity signals IN. In each experiment and simulation, we used a total of 9000 patterns. To quantitatively evaluate the retrieved ghost image quality, we employ the contrast-to-noise ratio (CNR) and resolution. The CNR is commonly calculated by comparing the object signal strength to the background signal strength and normalizing it to the image noise [18; 19]. This measure helps distinguish object signals from background noise. Since there is a trade-off between CNR and resolution, we estimate the resolution of the retrieved ghost image using a method originally proposed by [PERSON] et al. [34]. The principle of resolution estimation is depicted in Figure 2. An area of the retrieved ghost image is selected such that it is equally covered by the signal and background. By  
Figure 1: Conceptual diagram of the computational SGI system. A He–Ne laser illuminates a high-speed DMD, on which rapidly-changing sequenced random speckle patterns are displayed. A bucket detector synchronized with the DMD acquires and transfers the measured data to a computer for image reconstruction. fitting the measured line-out in Figure 2 with a Gaussian curve and determining the fitting parameters \(\omega_{1}\), \(\omega_{2}\), and \(\Delta t\), the resolution R can be calculated using Equation (3). \[R=\frac{2\Delta t}{\omega_{1}+\omega_{2}}. \tag{3}\]  
Based on the findings from previous pseudo-thermal ghost imaging experiments [19], we have confirmed a significant degradation in the quality of ghost images as the speckle size increases. However, it is observed that with larger speckle sizes, the speed of image appearance during reconstruction also increases. Considering this trade-off scenario, we introduce a novel approach called sequential ghost imaging (SGI) as a solution. To investigate the relationship and patterns of sequenced random speckle patterns in SGI, we divided them into three distinct groups based on different ordering methods. These groups were created by arranging speckles of varying sizes to form multiple sequence combinations, as illustrated in Figures 3 and 4. The three ordering rules were named Different Step (DS), Different Directions (DD), and Different Group (DG) for easy understanding and differentiation. In DS, speckle sequences were designed with increasing sizes, incrementing in different steps. Specifically, DS-1 to DS-5 featured speckles ranging from 27 \(\upmu\)m to 540 \(\upmu\)m with average step sizes of 26.68 \(\upmu\)m, 54.39 \(\upmu\)m, 108 \(\upmu\)m, 256 \(\upmu\)m, and 513 \(\upmu\)m, respectively. The average step sizes were calculated to account for practical limitations while maintaining the increment of speckle size. In DD, we examined two sets of speckle sequences: one with increasing speckle sizes (DD-1) and another with decreasing speckle sizes (DD-2). These sequences were compared to DS-3 and DS-4, respectively, which followed a stepwise decrease in speckle sizes within the same range as DS-1 and DS-2. For DG, we explored different speckle composition groups, as shown in Figure 4. Eight groups were devised with varying speckle patterns for ghost imaging. DG-1 to DG-4 comprised four speckle patterns each, while DG-5 and DG-6 contained six speckle patterns per group.