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The increasing application of deep learning in medical imaging has prompted the need for developing analysis tools to understand, evaluate, and enhance neural networks and their outcomes. However, image quality assessment of learning-based methods remains challenging due to their nonlinear and data-dependent nature. In this study, we introduce an analysis method for learning-based CT denoising algorithms based on the Jacobian matrix, which characterizes the response of the network to perturbations around its input. We trained a classic denoising network architecture using the normal dose images as label but using progressively lower dose images as input. For each network, the Jacobian matrices for an example input image were evaluated over 50 noise realizations in terms of their width and anisotropy. We further performed null space analysis of the Jacobians to investigate the preserved and null components of input perturbations. The variability in the network outputs and Jacobian matrices was found to increase with a reduction in dose levels. Such behavior, however, is only observed around edges in the anatomy. The width of the Jacobian matrix also increased as the dose decreased, suggesting an increase in the strength of the smoothing effect. The anisotropy of the Jacobian was more pronounced about the edges, indicating edge enhancement in the denoising process. The null space analysis showed a reduction in the preserved features of an input perturbation as the dose decreased, indicating greater misrepresentation of anatomical structures. The analysis proposed in this work aids in understanding and interpreting the denoising behavior of neural networks and can potentially serve as a means to regulate and optimize network performance.
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