## 4 Optical Implementation  
Inspired from Equations (1)-(3), we intend to produce the ring-to-point transformation illustrated in Figure 1 (do not consider dashed lines). The objective is to optically implement the integral of Equation (3). All information over each ring of the radius \(r\) is summed and the result is projected onto a point shifted by \(r\) from the optical axis in a response plane located at a distance \(z\) behind the object. The result of the transformation is, thus, a line segment which can be freely oriented in the response plane. For a given radius \(r\), the rays covering the optical paths \(s(r,\theta)\) must arrive to the collecting point with the same phase, that is, the phase corresponding to \(s(r,0)\). To satisfy this constraint, we have to introduce an additional phase distribution \(p(r,\theta)\) in the plane of the object fulfilling the following condition:  
\[p(r,\theta)+\frac{2\pi}{\lambda}s(r,\theta)=p(r,0)+\frac{2\pi}{\lambda}s(r,0)+ 2k\pi \tag{4}\]  
where \(k\) is an integer and \(s(r,0)=z\). By choosing, for instance \(p(r,0)=0\), the required phase distribution is expressed as follows (see Figure 1):  
\[p(r,\theta)=\frac{2\pi}{\lambda}z-\frac{2\pi}{\lambda}\sqrt{z^{2}+4r^{2}sin^{2 }\left(\frac{\theta}{2}\right)}+2k\pi \tag{5}\]In the framework of the paraxial assumption, we can use the following approximation (\(\sqrt{1+x^{2}}\cong 1+\frac{x}{2}\) if x \(\ll 1\). This approximation is used in the Fresnel transform [11; 12]):  
\[\sqrt{1+\frac{4r^{2}}{z^{2}}sin^{2}\bigg{(}\frac{\theta}{2}\bigg{)}}\cong 1+ \frac{2r^{2}}{z^{2}}sin^{2}\bigg{(}\frac{\theta}{2}\bigg{)}\]  
and the required transmittance \(t(r,\theta)\) is obtained (for \(k=0\)):  
\[t(r,\theta)=exp(ip(r,\theta))=exp\bigg{(}-4i\pi\frac{r^{2}}{\lambda z}sin^{2} \bigg{(}\frac{\theta}{2}\bigg{)}\bigg{)} \tag{6}\]  
We notice that if the implementation of the m-order CHC is required, then we need only to add the term exp(-i \(m\theta\)) in the expression (6). For instance, one can use the second-order CHF (\(m=2\)) which is often a good compromise for both an acceptable SNR and the peak sharpness of the correlation peak [10]. It is also possible to combine several CHCs. This goes, however, beyond the scope of this work. We need a diffractive phase element with the transmittance expressed by relation (6), which we refer to as the \"diffractive compressing element (DCE)\". This Fresnel diffractive element has a continuous phase profile (kinoform) [13]. Figure 2 shows a quantized version of such an element, where only four phase levels are used. We projected the phase profile onto the closest phase level in the set {0,\(\pi\)/2, \(\pi\), 3\(\pi\)/2}. We see in the neighborhood of the bottom half of the vertical axis that the profile is similar to that of a one-dimensional Fresnel lens. Over each ring \(r\) of the diffractive element, the phase corresponds to a two-dimensional off-axis Fresnel lens, where the off-axis translation corresponds to \(r\). In order to decrease the resolution needed by the Fresnel diffractive element, we can put a spherical lens beside the diffractive element [14; 15]. In fact, the distribution of DCE becomes the difference between Figure 2 and the distribution of the inserted lens. It is worth noting that the phase pattern becomes more and more dense in the boundary areas. The smallest feature should be large compared to the wavelength (let us say, more than five times) so that we could remain in the scope of scalar diffraction. Figure 1: Rotation invariant image compression: Ring-to-point transformation. Figure 3 shows the setup of the correlator providing \(\widetilde{g}\otimes\widetilde{f}\). The implementation of the projection of the input scene image is provided by a diffractive phase element placed just behind the input image. The result of the ring-to-point projection is observed at a distance \(z\) in the plane \(\mathrm{P_{C}}\). The Fourier plane \(\mathrm{P_{F}}\) presents, as usual, the filter plane. Using the same diffractive phase element, the filter can be optically implemented.