The main source of noise in computed tomography (CT) images is a quantum noise, which results from statistical fluctuations of X-ray quanta reaching the detector. This paper proposes a neural network (NN) based hybrid filter for removing quantum noise. The proposed filter consists of bilateral filters (BFs), a single or multiple neural edge enhancer(s) (NEE), and a neural filter (NF) to combine them. The BFs take into account the difference in value from the neighbors, to preserve edges while smoothing. The NEE is used to clearly enhance the desired edges from noisy images. The NF acts like a fusion operator, and attempts to construct an enhanced output image. Several measurements are used to evaluate the image quality, like the root mean square error (RMSE), the improvement in signal to noise ratio (ISNR), the standard deviation ratio (MSR), and the contrast to noise ratio (CNR). Also, the modulation transfer function (MTF) is used as a means of determining how well the edge structure is preserved. In terms of all those measurements and means, the proposed filter shows better performance than the guided filter, and the nonlocal means (NLM) filter. In addition, there is no severe restriction to select the number of inputs for the fusion operator differently from the neuro-fuzzy system. Therefore, without concerning too much about the filter selection for fusion, one could apply the proposed hybrid filter to various images with different modalities, once the corresponding noise characteristics are explored.