Analysis of noise sensitivity of Tchebichef and Zernike moments with application to image watermarking

In this paper we show that, contrary to the common belief found in some of the literature, Tchebichef moments are more sensitive to image additive noise than Zernike moments. We examine the problem of noisy image reconstruction by the method of orthogonal moments. We comparatively show this by imposing different types and levels of noise on various images and by measuring the error due to the added noise alone after image reconstruction. Here the error due to the added noise alone is defined, quantified and calculated by subtracting the noise-free image reconstruction error from the total noisy reconstructed error, which includes both reconstruction and added noise errors. The reconstruction error is with respect to a given original non-noisy image. A reconstruction measure metric for better evaluating the sensitivity of orthogonal moments towards noise added to images, namely accumulative relative error, is also introduced and proposed. As a result of this noise analysis study, we also present an empirical comparative study of Tchebichef and Zernike moments in image watermarking applications. In particular, we consider the case of moment-based watermarking schemes involving moment watermarks being embedded in a given carrier image moments. We show that the Tchebichef moments of a given image are more sensitive to image malicious and intended manipulations than Zernike moments, and hence are more capable of detecting tampering performed on watermarked images during transmission. However, this suggests and will most likely make Zernike orthogonal moments more suitable as moment descriptors employed as pattern features in scene registration, recognition, modeling, and data compression in noisy scenes, than Tchebitchef moments. (C) 2013 Elsevier Inc. All rights reserved.