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Scanned images of patent or historical documents often contain localized zigzag noise introduced by the digitizing process; yet when viewed as a whole image, global structures are apparent to humans, but not to machines. Existing denoising methods work well for natural images, but not for binary diagram images, which makes feature extraction difficult for computer vision and machine learning methods and algorithms. We propose a topological graph-based representation to tackle this denoising problem. The graph representation emphasizes the shapes and topology of diagram images, making it ideal for use in machine learning applications such as classification and matching of scientific diagram images. Our approach and algorithms provide essential structure and lay important foundation for computer vision such as scene graph-based applications, because topological relations and spatial arrangement among objects in images are captured and stored in our skeleton graph. In addition, while the parameters for almost all pixel-based methods are not adaptive, our method is robust in that it only requires one parameter and it is adaptive. Experimental comparisons with existing methods show the effectiveness of our approach.