The complex wave propagation is generated using the Fresnel transform diffraction through the propagation phenomena (Zhou et al. 2014; Nazeer and Kim 2013). When the Fresnel transform is applied to a multi-resolution bases of the wavelet, it produces the Fresnelet transform basis. These basis have been used to reconstruct the digital hologram with varying sets of parameters. These parameters are composed on value of the resolution scale, the wavelength, and the distance between the propagating objects and the observing plane. Fresnelet transform has been presented to simulate the approximation model of the monochromatic waves propagation. In this regard, one-dimensional data propagation is shown with the Fresnel transform model for a function \( f \in L_{2} \left( {\mathbb{R}} \right) \) that can be represented as the convolution integral:

$$ \tilde{f}_{\tau } \left( x \right) = \left( {f \times k_{\tau } } \right)\left( x \right)\,{\text{with }}\,k_{\tau } \left( x \right) = \frac{1}{\tau }exp\left( {i\pi \frac{{x^{2} }}{{\tau^{2} }}} \right) $$

(1)

where \( k_{\tau } \left( x \right) \) is the one-dimensional kernel. And the normalizing parameter \( \tau > 0 \) depending on the distance \( d \) and on the wavelength \( \lambda \) as follow:

$$ \tau = \sqrt {\lambda d} $$

In addition, two-dimensional data propagation is shown by using the tensor product of the \( k_{\tau } \left( x \right) \), for \( f \in L_{2} \left( {{\mathbb{R}}^{2} } \right) \),

$$ \tilde{f}_{\tau } \left( {x,y} \right) = \left( {f \times K_{\tau } } \right)\left( {x,y} \right)\,{\text{with }}\,K_{\tau } \left( {x,y} \right) = k_{\tau } \left( x \right)k_{\tau } \left( y \right). $$

where \( K_{\tau } \left( {x,y} \right) \) is the separable kernel used to extend the Fresnel transform’s one- dimensional case readily to two-dimensional case (Zhou et al. 2014). Among the various useful properties of the Fresnel transform, the unitary property is the prominent one, so that the given data can be facilitated to obtain the perfect reconstruction.

The separable extension of the one-dimensional wavelet into two-dimensional wavelet can also be obtained. The Riesz basis for \( L_{2} \left( {\mathbb{R}} \right) \), can be defined in terms of a two parameter family \( \left\{ {\psi_{j,l} } \right\}_{{j, l \in {\mathbb{Z}}}} \) on \( L_{2} \left( {\mathbb{R}} \right) \) using the wavelet transform as convolution integrals, where

$$ \left\{ {\psi_{j,l} \left( x \right) = 2^{j/2} \psi \left( {2^{j} x - l} \right)} \right\}_{{j, l \in {\mathbb{Z}}}} . $$

(2)

An orthonormal basis for the Haar wavelet can also be generated for \( L_{2} \left( {\mathbb{R}} \right) \). It is also known as the simplest form of a wavelet that can attain multiresolution decomposition and the perfect reconstruction of the given data as well (Kang and Aoki 2005). Furthermore, the Fresnelet basis is obtained using the Fresnel transform to the Haar wavelet as follows:

$$ \left\{ {\left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} } \right\}_{{j, l \in {\mathbb{Z}}}} \,\,{\text{with}}\,\,\left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} \left( x \right) = 2^{j/2} \tilde{\psi }_{{2^{j} \tau }} \left( {2^{j} x - l} \right). $$

(3)

An orthonormal Fresnelet basis can be attained for fixed \( \tau \), by letting \( \varTheta_{j,l} \left( x \right) = \left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} \left( x \right) \), as follows:

$$ f = \mathop \sum \limits_{j,l} c_{j,l} \varTheta_{j,l} \,{\text{with}}\,c_{j,l} = \left\langle {f,\varTheta_{j,l} } \right\rangle . $$

(4)

The Fresnelet coefficients are represented by \( c_{j,l} \) in (4). Since the separable nature can be used to extend the Fresnelet transform’s one-dimensional domain readily to two-dimensional domain. Following this, we may obtain four possible combinations of the tensor product \( \gamma_{\tau }^{{\left( {ll} \right)}} ,\, \gamma_{\tau }^{{\left( {lh} \right)}} ,\, \gamma_{\tau }^{{\left( {hl} \right)}} , \,{\text{and}} \,\gamma_{\tau }^{{\left( {hh} \right)}} , \) or generating the lower–lower subband, lower-high detail, high-lower detail, and high–high detail subbands, respectively, as follow:

$$ \gamma_{\tau }^{{\left( {ll} \right)}} = \left( {\phi_{j,l} } \right)_{{\tilde{\tau }}} \left( x \right)\left( {\phi_{j,l} } \right)_{{\tilde{\tau }}} \left( y \right), $$

(5)

$$ \gamma_{\tau }^{{\left( {lh} \right)}} = \left( {\phi_{j,l} } \right)_{{\tilde{\tau }}} \left( x \right)\left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} \left( y \right), $$

(6)

$$ \gamma_{\tau }^{{\left( {hl} \right)}} = \left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} \left( x \right)\left( {\phi_{j,l} } \right)_{{\tilde{\tau }}} \left( y \right), $$

(7)

$$ \gamma_{\tau }^{{\left( {hh} \right)}} = \left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} \left( x \right)\left( {\psi_{j,l} } \right)_{{\tilde{\tau }}} \left( y \right), $$

(8)

where the \( \phi \) is representing the scaling function and \( \psi \) is representing the wavelet functions, respectively. Following these functions, the (5) is establishing a low-pass filter and the (6), (7), (8) are establishing high-pass filters. By using the above basis functions to data *f*, the four Fresnelet coefficients are as follows:

$$ f_{\tau ,d}^{{\left( {ll} \right)}} = \left\langle {f,\gamma_{\tau }^{{\left( {ll} \right)}} } \right\rangle ,\,\,f_{\tau ,d}^{{\left( {lh} \right)}} = \left\langle {f, \gamma_{\tau }^{{\left( {lh} \right)}} } \right\rangle , $$

$$ f_{\tau ,d}^{{\left( {hl} \right)}} = \left\langle {f,\gamma_{\tau }^{{\left( {hl} \right)}} } \right\rangle ,\,\,f_{\tau ,d}^{{\left( {hh} \right)}} = \left\langle {f,\gamma_{\tau }^{{\left( {hh} \right)}} } \right\rangle , $$

Where the low-passed data is represented by the coefficient \( f_{\tau ,d}^{{\left( {ll} \right)}} \) and the high-passed detail data are represented by the coefficients \( f_{\tau ,d}^{{\left( {lh} \right)}} \), \( f_{\tau ,d}^{{\left( {hl} \right)}} \) and \( f_{\tau ,d}^{{\left( {hh} \right)}} \), respectively. Figure 1 indicates the Fresnelet coefficients that are acquired from the information data as shown in Fig. 2. It is worth mentioning in Fig. 1 that instead of using the Fresnel transform simple application (Liebling et al. 2003), the information data can be processed by the forward Fresnelet transform so that meaningful information are totally encrypted into dummy image data with four different bands in the form of complex data (Liebling et al. 2003). The magnitudes of the Fresnelet coefficients of the USAF information image are shown in Fig. 1.

Note that by unitary property of the Fresnelet transform, a reconstruction of an information data can be obtained by applying the conjugate transpose of the forward Fresnelet transform. In this case, the reconstruction has a complex valued data form. The first row of Fig. 1 is the decomposition stage of information data (USAF image shown in Fig. 2) into 4 subbands on employing the application of Fresnelet transform using the distance parameter *d*
_{1} and considered as Forward Fresnelet transform propogation. The second row show the central position of information data diffusion with zoomed-in position to show the complete deformation of information data into dummy data (scrambled data). The third row is the inverse propagation of the respective subbands of row one, using the distance parameter *d*
_{2}, and considered as inverse Fresnelet transform. Figure 3a in the proposed method shows the reconstruction (merging) of four subbands of dummy data from Fig. 1 into single complex data image. Complex property of the Fresnelet transform is the multiresolution property as described in Liebling and Unser (2004). For transmitting, the complex data image (e.g. *a* + *ib*) is separated into two parts; where \( a \) and \( b \) are real numbers and \( i \) (imaginary unit) \( i^{2} = \left[ { - 1} \right] \). For transmitting the information data in digital form, we need to separate the complex data as shown in first image of Fig. 3 into real part \( a \) (second image of Fig. 3) and magnitude of imaginary part \( b \) (last image of Fig. 3). On extraction stage, just by multiplying the magnitude of imaginary part \( b \) with \( i \), we can reconstruct our required complex data, for getting the extraction image using inverse process.

To communicate the information data with high privacy and improved secrecy, the proposed algorithm uses the Fresnelet transform that takes into account the wavelength and the distance parameters as keys, which are essential for reconstructing the accurate information data. Moreover, the original form of the information data is attained in the reconstruction phase with the exact keys using the inverse Fresnelet transform processes.