Aligning machine representations with human understanding is key to improving interpretability of machine learning (ML) models. When classifying a new image, humans often explain their decisions by decomposing the image into concepts and pointing to corresponding regions in familiar images. Current ML explanation techniques typically either trace decision-making processes to reference prototypes, generate attribution maps highlighting feature importance, or incorporate intermediate bottlenecks designed to align with human-interpretable concepts. The proposed method, named COMIX, classifies an image by decomposing it into regions based on learned concepts and tracing each region to corresponding ones in images from the training dataset, assuring that explanations fully represent the actual decision-making process. We dissect the test image into selected internal representations of a neural network to derive prototypical parts (primitives) and match them with the corresponding primitives derived from the training data. In a series of qualitative and quantitative experiments, we theoretically prove and demonstrate that our method, in contrast to post hoc analysis, provides fidelity of explanations and shows that the efficiency is competitive with other inherently interpretable architectures. Notably, it shows substantial improvements in fidelity and sparsity metrics, including 48.82% improvement in the C-insertion score on the ImageNet dataset over the best state-of-the-art baseline.