**3.2 Mathematical expression**

Let us assume that image, denoted by *f xy* <sup>1</sup> ( , ) , is modulated by two sinusoidal phase gratings along the reference abscissa and of the form *<sup>m</sup> i ux* π <sup>1</sup> exp sin2 <sup>2</sup> where *u*1 is the spatial frequency and m is the phase contrast. In such a case, the first modulated object may be expressed as:

$$s\_1(\mathbf{x}, \boldsymbol{\chi}) = f\_1(\mathbf{x}, \boldsymbol{\chi}) \exp\left[i\left(\frac{m}{2}\right) \sin 2\pi\mu\_1 \boldsymbol{\chi}\right] \tag{1}$$

Invoking the well-known convolution theorem, the Fourier transform of this modulated object is given by,

$$S\_1(u,v) = \left[F\_1(u,v)\right] \oplus \left[\sum\_{q=-a}^{q\to+a} I\_q\left(\frac{m}{2}\right) \delta\left(u-qu\_1,v\right)\right] = \left[\sum\_{q=-a}^{q\to+a} I\_q\left(\frac{m}{2}\right) F\_1\left(u-qu\_1,v\right)\right] \tag{2}$$

*<sup>q</sup>* where is the qth order Bessel Function of *J* the 1st kind. The diffraction order is represented by the parameter 'q'.

The diffraction pattern, as given by Eq. (2) is a series of diffraction spots each containing the object spectrum. Considering the zero order and the first two orders of the spectrum, Eq. (2) may be represented by,

$$\mathcal{S}\_1(\boldsymbol{u}, \boldsymbol{v}) = \left[ J\_0\left(\frac{m}{2}\right) F\_1(\boldsymbol{u}, \boldsymbol{v}) + J\_{-1}\left(\frac{m}{2}\right) F\_1(\boldsymbol{u} + \boldsymbol{u}\_1, \boldsymbol{v}) + J\_{+1}\left(\frac{m}{2}\right) F\_1(\boldsymbol{u} - \boldsymbol{u}\_1, \boldsymbol{v}) \right] \tag{3}$$

#### **3.3 Retrieval of images from spectral band**

The reconstruction of images is achieved by decrypting the encrypted plane followed by plotting an intensity graph of the spectrum horizontally. This intensity graph is generated using the intensity values with respect to its location (pixel position). In the intensity graph, peak values indicate the brightest spots, where information of images are maximum. To avoid any human intervention, system automatically finds the intensity level horizontally, vertically and diagonally. In addition, regional Inverse Fourier Transform is applied around the peaks by using a

**Figure 1.** *Sinusoidal phase grating along x axis.*

**Figure 2.** *Spectrum of an image due to modulation.*

fixed cut-off frequency. As only one sideband is enough for image reconstruction, hence we select one side-band. This spectrum is filtering using a cut-off value of 120. Proper operating frequency is selected and there is no overlapping during the band selection, so the extracted images are free from aliasing problem. As center band contains the information of all images, therefore filtering is not applied across center band.
