The Continuous Wavelet Transform (CWT) of a finite energy 1-D signal f(t) is given by:

where ψ* (.) is the complex conjugate of ψ (.). The above equation can be viewed as convolution of the signal with dilated band-pass filters. A single level decomposition by wavelet transform gives a set of approximate and detailed coefficient specifying the low and the high frequency components respectively. For a 2-D signal such as an image, first level decomposition will result into four components as shown in Fig 3. The first output (‘A’ as shown in Fig 3 also known as the approximate image) gives the image at a lower resolution, while the other three outputs correspond to horizontal, vertical, and diagonal details (‘B’, ‘C’ and ‘D’ as shown in Figure 3 also known as the detailed image) of the image. The second level of wavelet decomposition takes the first output‘A’ and further decomposes it into four sub-images.

For watermarking an image, Lth-level wavelet decomposition is applied yielding 3L detail images at each of the L resolution levels, and a gross approximation of the image at the coarsest resolution level. The value of L is user defined and ‘Daubechies 6’ mother wavelet is used for decomposition. The kth detail image component at the lth-esolution level of the host is denoted by, f_k,l (m,n), where k = h, v, d (which stands for “horizontal,” “vertical,” and“diagonal” detail coefficients, respectively), l = 1, ........ L and (m,n) is the particular spatial location index at resolution l. The gross approximation is represented by f_a,L(m,n) where the subscript a is used instead of k to denote approximation coefficients. In the second stage, modifying selected wavelet coefficients embeds the watermark bit stream.

To embed a binary watermark of length Nw denoted as f(i), i =1,2,...... N_w, a user-defined coefficient selection key ckey(i), i = 1,2, ......N_w, is employed. The particular wavelet coefficient at which ith watermark bit f(i) is to be embedded is given by ckey(i). Each element of ckey(i), is distinct so that two bits are not marked at the same location, causing an ambiguity or error. In addition, selection of the coefficients is random and well spread spatially throughout each resolution level. In the proposed techniue the ckey(i) is generated by randomly selecting a coefficient from the set {f_{h, l} (m,n), f_dl (m,n)_v,l (m,n)} for each l and (m,n). Thus, one detail coefficient at each resolution and spatial location is marked. The binary watermark is also randomly generated using a uniform distribution and is set to be of the same length as ckey(i). The watermark bit f(i) is embedded into the coefficient ckey(i) through an appropriate quantization procedure [18,20]. In the final stage, the corresponding Lth level inverse wavelet
transform of the marked image components is computed to form the tamper-proofed image.

Watermark extraction is performed by taking Lth level discrete wavelet transform (DWT) of the given image and the coefficient selection key ckey(i) is used to determine the marked coefficients. A quantization function Q(.) is applied to each of these coefficients to extract the watermark values f(i) [18]. Thus, for authentication, the authorized user’s public key ckey(i) is applied to the extracted watermark to obtain the identification code f(i). Almost any tampering of the image will cause authentication procedure to fail, as the decryption procedure is highly sensitive to changes in the watermark. Thus, authentication is possible only if the extracted watermark is identical to the embedded. If public key authentication fails, then tamper assessment is employed to determine the credibility of the modified multimedia content.