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This paper proposes a novel data hiding method for authenticating binary images through establishing dense edge adaptive grids (EAG) for invariantly selecting good data carrying pixel locations (DCPL). Our method employs dynamic system structure with carefully designed local content adaptive processes (CAP) to iteratively trace new contour segments and to search for new DCPLs. By maintaining and updating a location status map, we re-design the fundamental content adaptive switch and a protection mechanism is proposed to preserve the local CAPs' contexts as well as their corresponding outcomes. Different from existing contour-based methods, our method addresses a key interference issue and has unprecedentedly demonstrated to invariantly select a same sequence of DCPLs for an arbitrary binary host image and its marked versions for our contour-tracing based hiding method. Comparison also shows that our method achieves better trade-off between large capacity and good perceptional quality as compared with several prior works for representative binary text and cartoon images.