Assessing the Fidelity and Noise Resilience of Quantum Fourier, Wavelet, and PCA Transforms in Amplitude-Encoded Image Reconstruction

The fusion of quantum computing with image processing has significantly improved both computational efficiency and fidelity in amplitude-encoded image reconstruction. While prior investigations have delved into advancements in quantum-to-classical data decoding, there remains a dearth of research addressing hybrid quantum-classical architectures and strategies to enhance noise resilience in quantum image processing. This study evaluates the implementation of three quantum transforms: Quantum Fourier Transform (QFT), Quantum Wavelet Transform (QWT), and Quantum Principal Component Analysis (QPCA), focusing on their efficacy in reconstructing images subjected to Gaussian and salt-and-pepper noise. Reconstruction quality is quantified through metrics such as error rates, similarity indices, and uncertainty quantification metrics, supported by a sensitivity analysis to ascertain robustness under varying noise conditions. The results demonstrate that QWT consistently achieves the highest Structural Similarity Index Measure (SSIM), while QFT often yields negative histogram similarity values, indicating significant alterations in color distribution. For instance, when analyzing one image, QWT exhibited a moderate decrease in Peak Signal-to-Noise Ratio (PSNR) from 37.83 to 37.71 with noise introduced, whereas QFT's PSNR experienced a more pronounced decline from 38.70 to 38.48. The study elucidates an inverse relationship between Mean Squared Error (MSE) and PSNR, underscoring the adverse effects of noise on image quality. Notably, QPCA was observed to produce the lowest Euclidean distance, signifying a strong retention of image integrity. These findings underscore the importance of judiciously selecting quantum methodologies tailored to specific application requirements, particularly in critical domains such as medical imaging, where precision and accuracy are paramount.