Fig 1.
Computation process for Franklin polynomials.
Fig 2.
Pixel information loss during the conversion of an image from Cartesian coordinates to polar coordinates.
Fig 3.
The red region represents the appropriate parameter space for the asymmetric tent mapping.
Fig 4.
Generating mixed low-Order moments feature.
Fig 5.
Procedure of zero-watermarking generation.
Fig 6.
Procedure of zero-watermarking verification.
Fig 7.
The time complexity of recursive method, non-recursive method, and the proposed method.
Fig 8.
Reconstructed image of ‘Baboon’ of size using FFT-based method, Gaussian numerical integration and wavelet numerical integration method.
Fig 9.
Comparative study of different QTFM-based methods with Gaussian integration, wavelet integration, direct computation and FFT-based methods.
Fig 10.
The processing time of different QTFM methods based on Gaussian integration, wavelet integration, direct computation, and FFT.
Fig 11.
Original 7 images and Logo image.
a) ‘Dragonice’ (b) ‘Tambourin’ (c) ‘Klavierstucke’ (d) ‘Fest’ (e) ‘Snow child’ (f) ‘Dvorak’ (g) ‘Saxophone’ (h) Watermark image.
Fig 12.
Attacked images and the verified watermark image under: Blurring with PSNR = 16.8789, Gaussian noise 0.5 with PSNR = 13.8986, Cutting with PSNR = 17.9834, Rotation with PSNR = 8.1672, JPEG compression 0.5 with PSNR = 39.3218.
Table 1.
SSIM values of 7 images under common attacks with different parameters.
Table 2.
SSIM values of 7 images and the mean values of 50 images under mixed attacks.
Table 3.
Comparative study of average SSIM values between the proposed method and similar zero-watermarking algorithms.