Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 25

Wavelet transform is a signal processing tool used for the analysis of optical and digital signals. It has good local optimization features as well as the multi-resolution analysis features, which makes it suitable for information processing applications [28]. Discrete wavelet transform (DWT) is any wavelet transform that uses a discrete set of wavelet scales and transformation to process signals. WT has been used in analyzing and processing optical signals due to its scaling and shift parameters. Its application in image compression is well established. Therefore, there is an inherent capability of combining WT not only in securing information but in compressing the secured data. This helps communicate the secured data through conventional communication channels. The optical implementation of the WT for 2D objects has been reported. Considering 1D representation, a mother wavelet h(x) is a finite-duration window function that can generate a family of daughter wavelets by varying scale a and shift b.

ha,b(x)=1ahx−ba(2.32)

The mother wavelet should satisfy the admissibility conditions that it must be oscillatory, have fast decay to zero, and integrate to zero. WT is defined as an inner product between a signal and a set of wavelets as

wf(a,b)=∫−∞∞ha,b*(x)f(x)dx(2.33)

where * denotes the complex conjugate and w_f(a,b) is considered as a function of spatial shift b for each fixed scale a that displays the information f(x) at various resolution levels.

Further, the scaling factor of WT and fractional orders of FRT has been combined, which is called fractional WT (FWT). This scheme has been used for image encryption. The FWT is a linear transformation without introducing any cross-term interference [29, 30]. It combines the virtues of the WT as well as the FRT and possesses multi-resolution features. The concept behind this transform is extracting the fractional spectrum of the signal via FRT operation followed by WT operation of the fractional spectrum. This concept of FWT has also been applied to optical information security applications. The fractional orders of FRT and scaling factors of wavelets serve as the additional keys to the cryptosystem. If these keys are chosen randomly then the unauthorized people cannot retrieve the actual input data. So the information can be well protected.

The use of WT in information security is more suitable while particularly securing multispectral data [31, 32]. In which, images collected through different sensors at different wavelengths are received. Such multi-spectral images need to be fused (merging of different frequency sub-bands) first and then secured. The multispectral data received from satellites and airborne sensors are becoming increasingly available for analysis for various applications such as remote sensing. Fusion techniques play an important role in processing all such multispectral data. This technique maximizes the information of the fused image and thus improves the image content. If such fusion schemes can be applied for information security application, then the degrees of freedom may be greatly enhanced. Combining the concept of wavelets with GT, gyrator wavelet transform (GWT) has also been reported, which finds applications in image encryption and compression [33, 34].