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It has been a challenging but significant research topic to extract the centerlines of the coronary arteries of the CTA volume in clinical applications. A new method is proposed to full-automatically extract and recognize the centerlines of the major branches in this paper. This method, which is called the directional minimal path, originates from an improved minimal path algorithm. In this method, the cost function is redefined, and the starting point of the coronary is automatically detected. The proposed method is validated by extracting and recognizing the three major coronary branches, and the average overlap percentage was 81.0%.