## 2 Background
### Existing Quality Assessment Methods  
The problem of assessing image reconstruction quality in Electrical Capacitance Tomography (ECT) has received little attention. The researchers commonly use Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), and the Correlation Coefficient (CC) to evaluate the accuracy and reliability of the reconstructed images [3; 23].  
Other less frequent measures include the relative Image Error (IE) between ground truths and reconstructed permittivity distributions [24] or the entropy of the reconstructed image [25].  
While PSNR, MSE, the Correlation Coefficient (CC), entropy, and relative error are valuable metrics for evaluating ECT image quality, they have several limitations that must be carefully considered. Unlike some other metrics, PSNR, MSE, and entropy are not normalized, which can hinder cross-dataset comparisons and potentially lead to misleading interpretations. Furthermore, entropy calculations are sensitive to small changes in the reconstructed image, potentially resulting in disproportionately large changes in entropy values, due to their non-linear nature. Similarly, while relative Image Error is normalized, variations in signal strength across different ECT imaging conditions can introduce bias in the assessment. This sensitivity can complicate the interpretation of results, and it may not accurately reflect meaningful differences in quality. Additionally, inconsistencies in calculation methods or reporting practices can introduce variability in results across studies. Moreover, PSNR, MSE, CC, entropy, and relative error may not always align with human visual perception. Images with high PSNR or CC values might still exhibit significant artifacts that are not adequately reflected by these metrics, potentially overstating the reconstruction quality. Ultimately, these metrics offer numerical values but lack contextual insights into the structural integrity or functional relevance of the reconstructed ECT images. They often fail to capture the structural information present in ECT images that extends beyond pixel-wise differences.  
While traditional image Quality Assessment metrics considering structural information tailored for raster images, such as the Structural Similarity Index Measure (SSIM), are not used for ECT image reconstruction evaluation, due to their reliance on pixel-based comparisons, our research aimed to explore the potential of graph-based methods for addressing these challenges. By leveraging the graph representation of ECT data and considering structural similarities between nodes and edges, these methods can provide more accurate and informative assessments of reconstruction quality. The SSIM, a metric designed to capture perceptual similarities between images, has demonstrated effectiveness in various image-processing applications. By adapting the SSIM to graph-based representation of ECT images and incorporating node neighborhoods via node grouping into PSNR and MSE calculations, we aim to address the structural information in ECT images, leading to more meaningful and accurate quality assessments  
Finally, the SSIM is less sensitive to image shifts than PSNR or MSE, due to its focus on structural similarity. The SSIM's local analysis, which considers luminance, contrast, and structural information, is more robust against small shifts than global metrics. By mim icking the SSIM's local analysis through node grouping, our proposed Group SSIM and Group PSNR metrics are also resilient against image shifts.