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Computed tomography is widely used as an imaging tool to visualize three-dimensional structures with expressive bone-soft tissue contrast. However, CT resolution and radiation dose are tightly entangled, highlighting the importance of low-dose CT combined with sophisticated denoising algorithms. Most data-driven denoising techniques are based on deep neural networks and, therefore, contain hundreds of thousands of trainable parameters, making them incomprehensible and prone to prediction failures. Developing understandable and robust denoising algorithms achieving state-of-the-art performance helps to minimize radiation dose while maintaining data integrity. This work presents an open-source CT denoising framework based on the idea of bilateral filtering. We propose a bilateral filter that can be incorporated into a deep learning pipeline and optimized in a purely data-driven way by calculating the gradient flow toward its hyperparameters and its input. Denoising in pure image-to-image pipelines and across different domains such as raw detector data and reconstructed volume, using a differentiable backprojection layer, is demonstrated. Although only using three spatial parameters and one range parameter per filter layer, the proposed denoising pipelines can compete with deep state-of-the-art denoising architectures with several hundred thousand parameters. Competitive denoising performance is achieved on x-ray microscope bone data (0.7053 and 33.10) and the 2016 Low Dose CT Grand Challenge dataset (0.9674 and 43.07) in terms of SSIM and PSNR. Due to the extremely low number of trainable parameters with well-defined effect, prediction reliance and data integrity is guaranteed at any time in the proposed pipelines, in contrast to most other deep learning-based denoising architectures.