Facial landmark detection has been studied over decades. Numerous neural network (NN)-based approaches have been proposed for detecting landmarks, especially the convolutional neural network (CNN)-based approaches. In general, CNN-based approaches can be divided into regression and heatmap approaches. However, no research systematically studies the characteristics of different approaches. In this paper, we investigate both CNN-based approaches, generalize their advantages and disadvantages, and introduce a variation of the heatmap approach, a pixel-wise classification (PWC) model. To the best of our knowledge, using the PWC model to detect facial landmarks have not been comprehensively studied. We further design a hybrid loss function and a discrimination network for strengthening the landmarks' interrelationship implied in the PWC model to improve the detection accuracy without modifying the original model architecture. Six common facial landmark datasets, AFW, Helen, LFPW, 300-W, IBUG, and COFW are adopted to train or evaluate our model. A comprehensive evaluation is conducted and the result shows that the proposed model outperforms other models in all tested datasets.