On Detection Improvement in MC-CDMA Image Watermarking on Fading Channel

Abstract

Code division multiple access (CDMA) at its multicarrier version of image watermarking scheme that is robust against random gain operation as well as enables high embedding rate is developed in this paper. Watermark embedding is done on mutually orthogonal set of host image samples wherein data bits are embedded using an orthogonal set of code patterns. Watermark decoding is also proposed by exploiting the principle of minimum mean square error combining. The decision variable for the binary watermark is the resultant weighted decision statistics. A generalized model of attack that consists of multiplicative and additive degradation on the watermarked image is considered. The first one follows Rayleigh distribution and the second one Gaussian distribution. We mathematically show that the proposed decoder structure incorporates the effect of random gain (fading attack) unlike the widely used correlator decoder. The concept of multiuser detection principle which is widely used in CDMA is then explored to cancel the effect of multiple bit interference in watermarking. Two variants of combined interference cancellation (CIC), namely threshold combined interference cancellation (TCIC) and group combined interference cancellation (GCIC) are then developed. Mathematical analysis and the results through simulations show that TCIC and CIC offer similar decoding error rate performance. On the other hand, a significant improvement in decoding error performance and payload are achieved through GCIC. Finally, the proposed method is studied for image error concealment and performance comparison is reported over the existing methods.

Appendix A

The minimum mean square error combining method estimates the j-th embedded watermark bit by the linear sum

$$ D^{j}=\sum _{n=1}^{N}w_{n}^{j}r_{n}^{j} $$

(15)

where $w_{n}$ is the weight factor for MMSEC. The working principle of MMSEC dictates that the estimation error for watermark bit decoding must be orthogonal to all the components of the received marked data. Mathematically can be written as

$$ E\biggl \lbrace (b_{n}^{j}-\sum _{n=1}^{N}w_{n}^{j}r_{n}^{j}) .r_{n}^{j}\biggr \rbrace =0 $$

(16)

when $n=1,2 \ldots N$.

The weight factor $w_{n}^{j}$ can be found from the solution to (16) by applying Wiener filter theory [40] and may be written as

$$ w_{n}^{j}=C^{-1}A $$

(17)

where $C=E\bigl \lbrace r_{n}^{j}r_{n}^{j}\mid \alpha _{n}\bigr \rbrace$ and $A=E\bigl \lbrace b_{n}^{j}r_{n}^{j}\mid \alpha _{n}\bigr \rbrace$. The symbol $E\bigl \lbrace .\bigr \rbrace$ denotes the expected value. This operation, when applied to the $r_{n}^{j}$ yields

$$ C=\alpha _{n}^{2}\gamma ^{2}\sum _{j=1}^{k}var{(b_k)}\rho _{kj}^{2}+N_{0}/2 $$

(18)

and

$$ A=\alpha _{n} $$

(19)

The weight factors for MMSEC watermark decoding can be found by substituting (18) and (19) in (17)

$$ w_{nj}={\alpha _{n}\over \left( \alpha _{n}^{2}\gamma ^{2}\sum _{j=1}^{K}var(b_{k})\rho _{kj}^{2}+N_{0}/2\right) } $$

(20)

and from (15)

$$ D^{j}=\sum _{n=1}^{N} {\alpha _{n}\over {\alpha }^{2}_{n}\gamma ^{2}\sum _{j=1}^{K}\left( var(b_k){\rho }^{2}_{(kj)}+N_{0}/2\right) } r_{n}^{j} $$

(21)