THEORETICAL MODELS

In this section, the mathematical descriptions of wavelet transform are briefly explained. The DWT –based algorithm to embed watermarks and the corresponding detection procedures are presented. In addition, the block diagram of the watermarking algorithm is included.

The Wavelet Transform

The continuous-time wavelet transform (CTWT) of a function x(t) is defined as (Daubechies, 1992; Young, 1993):

(Equ. 1)

where g(t) is a window function known as the analyzing wavelet, the dilation parameter “a” is known as the scale factor, and “b” is a translation factor. Unlike the Fourier Transform where infinite duration sinusoids are used as basis functions, the basis function in Equ. 1 decay rapidly to zero and are zero mean. The wavelet is dilated or compressed by the scale factor. Thus, at low scales high frequency behavior is localized, while at high scales (when the wavelet is stretched out) low frequency features are better resolved. This is of significant benefit when one is dealing with signals containing features with various frequency characteristics. Another advantage of the wavelet transform is that the analyzing wavelet can be chosen based on the application.

DWT Watermarking Technique

We applied the wavelet-based watermarking technique based on the method that was proposed by Dugad, Ratakonda, and Ahuja, in 1998 [4]. The new watermarking method does not require the original image for watermark detection and adds the watermark in selected coefficients with significant image energy in the transform domain in order to ensure non-erasability of the watermark. Moreover, this technique improves resistance to such attacks as lousy compression, linear or non-linear filtering, scaling, or cropping on the watermark.

The watermark is embedded into the image according to the following procedures:

Compute the DWT coefficients with a Daubechies filter.
Add watermark to those coefficients that are above a given threshold (T1) in the sub-bands other than the low pass sub-band.
Compute the inverse DWT to reconstruct the watermarked image.

The equation used for watermark casting and detection is the following:

(Equ. 2)

where N represents the number of wavelet coefficients greater than T1 (barring the low pass component). Vi denotes the corresponding DWT coefficients of the original image and Vi’ denotes the DWT coefficients of the watermarked image. Xi is generated from a uniform distribution of zero mean and unit variance and a is a constant number.

On the other hand, the watermark can be detected according to the same procedure as the embedded technique. However, only the wavelet coefficients > T2 > T1 are selected in order to correlate with the watermark. The correlation Z between the DWT coefficients V’’ of the corrupted watermarked image and a possible different watermark Y is computed as

(Equ. 3)

where i runs over all coefficients > T2 > T1 and M is the number of such coefficients.

The threshold S is defined according to the following formula:

(Equ. 4)

In addition, Figure 2 shows the block diagram of the embedding technique.

Figure 2: Block Diagram of the DWT Watermarking Technique

RESULTS AND DISCUSSIONS

In this section, the DWT-based watermarked images are constructed and the corresponding detector responses are generated.

Discrete Wavelet Transform Watermarking Technique

Figure 3: Original Lena, Watermarked Image, and the Watermark

Original Lena Watermarked Lena The Watermark

Figure 3 shows the original Lena image, the watermarked image, and the watermark that was embedded into the image. We can observe that watermark is only added to the image edges because adding the watermark to significant coefficients in the high frequency bands is equivalent to adding the watermark to only the edge areas of the image.

Figure 4 Attacks and detector response of watermarked images

Cropped Image	Add Noise	Wiener Filtering	Median Filtering

Figure 4 shows the watermarked images under several attacks and the corresponding detector responses. We can confirm that the watermark is robust since only one sharp peak in the cross-correlation coefficient is detected after 1000 sets of different random numbers are tested. In all cases, the detector response is well above the threshold. Moreover, Figure 5 shows the watermarked images that are compressed with different JPEG quality values. The detector response is well above the threshold for JPEG compression quality greater than 20% if we set a = 0.8. Although the detector response is below the threshold for 10% and 0% compression quality, the detector response can be well above the threshold if a is set to be much greater than 1.

Figure 5: JPEG Compression of the Watermarked Image

0% Compression Quality	10% Compression Quality	20% Compression Quality	30% Compression Quality


40% Compression Quality	50% Compression Quality	60% Compression Quality	70% Compression Quality


80% Compression Quality	90% Compression Quality	100% Compression Quality

Furthermore, we attacked the watermarked image by first adding random noise to the image and then filtering the noised image with a wiener filter. Third, we compressed the filtered image with 50% JPEG compression quality and finally filtered the compressed image with a 3 x 3 median filter. After each attack, we detected the corrupted image with the original watermark that was embedded into the image. The results show that the watermark is robust after each attack. Figure 6 shows the Mandrill watermarked image under multiple attacks and the corresponding detector response.

Figure 6: Watermarked Image after Multiple Attacks

Original Mandrill	1. Watermarked Mandrill	2. Add Noise	3. Wiener Filtering	4. 50% JPEG Compression	5. Median Filtering

Based on our investigations, we found that the DWT-based watermarking technique can be used to embed information into the image and does not require the original image for watermarking detection. The watermark is still robust under several attacks such as filtering, rotating, compressing, and cropping.