Blind digital watermarking of rational Bézier and B-spline curves and surfaces with robustness against affine transformations and Möbius reparameterization

  • Authors:

  • Song-Hwa Kwon;Tae-wan Kim;Hyeong In Choi;Hwan Pyo Moon;Sung Ha Park;Heon-Ju Shin;Jung Kyo Sohn

  • Affiliations:

  • Department of Mathematics, The Catholic University of Korea, Bucheon 420-743, Republic of Korea;Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, Republic of Korea and Research Institute of Marine Systems Engineering, Seoul National University, ...;Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea;Department of Mathematics, Dongguk University-Seoul, Seoul 100-715, Republic of Korea;Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, Republic of Korea;STX R&D Center, STX Offshore and Shipbuilding Co. Ltd., Changwon 641-060, Republic of Korea;Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea

  • Venue:
  • Computer-Aided Design
  • Year:
  • 2011

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Abstract

We present a blind watermarking scheme for rational Bezier and B-spline curves and surfaces which is shape-preserving and robust against the affine transformations and Mobius reparameterization that are commonly used in geometric modeling operations in CAD systems. We construct a watermark polynomial with real coefficients of degree four which has the watermark as the cross-ratio of its complex roots. We then multiply the numerator and denominator of the original curve or surface by this polynomial, increasing its degree by four but preserving its shape. Subsequent affine transformations and Mobius reparameterization leave the cross-ratio of these roots unchanged. The watermark can be extracted by finding all the roots of the numerator and denominator of the curve or surface: the cross-ratio of the four common roots will be the watermark. Experimental results confirm both the shape-preserving property and its robustness against attacks by affine transformations and Mobius reparameterization.