## 5 Extensions of the Architecture  
### Parallel Pattern Recognition  
Scale considerations can be taken into account. For instance, we can sum information over each ring of radius \(r\) and project the result in a point shifted by the distance \(\alpha\)\(r\) (instead of \(r\)) from the optical axis, yielding:  
\[t(r,\theta)=exp\left(-i\pi\frac{r^{2}}{\lambda z}\left(1+\alpha^{2}-2\alpha \cos(\theta)\right)\right) \tag{7}\]  
Here we used the Law of Cosines, also known as [PERSON]'s theorem (see dashed lines in Figure 1). This technique can be used, for instance, to adapt the compressed data to the features of the spatial light modulator placed in the Fourier plane. Owing to the image information compression, it is also possible to provide parallel pattern recognition. For each input scene image \(g_{m}\) we add in the response plane of the diffractive phase element and in the plane of the compressed data \(\mathrm{P_{C}}\), a linear phase distribution with a certain slope \(exp\left(-i2\pi\frac{x_{m}r}{\lambda f_{1}}\right);f_{l}\) is the focal length of the lens used for the correlation. The Fourier transform of the resulting line segment, containing compressed image data, is laterally shifted by the amount \(x_{m}\) in the filter (Fourier) plane \(\mathrm{P_{F}}\). The different slopes can be generated by an array of mini prisms placed in the plane \(\mathrm{P_{C}}\). An alternative consists of integrating the slopes in the diffractive element. Therefore, for each input image \(g_{m}\), we need a diffractive phase element with the transmittance:  
\[t_{m}(r,\theta)=exp\left(-i\pi\frac{r}{\lambda}\left(\frac{r}{z}\left(1+\alpha ^{2}-2\alpha\cos(\theta)\right)+\frac{2x_{m}}{f_{l}}\right)\right) \tag{8}\]  
We need a bank of references where each elementary reference \(f_{m}\) is compared to one input image \(g_{m}\) (Figure 4a). The two-dimensional input images are arranged in a matrix form. After compression by means of the diffractive compressing element DCE, we obtain, in the plane \(\mathrm{P_{C}}\), an array of line segments \(\widetilde{g}_{m}\) (\(m\)=1,..., \(M\)). In the Fourier plane, each one-dimensional structure \(\widetilde{G}_{m}\), i.e., the Fourier transform of the compressed image \(\widetilde{g}_{m}\) is multiplied by the conjugate of the Fourier transform \(\widetilde{F}_{m}^{*}\) and of the compressed pattern \(\widetilde{f}_{m}\) of a two-dimensional reference \(f_{m}\). The Fourier transform, as well as the compression of each reference \(f_{m}\), are performed numerically. Instead of the output lens \(\mathrm{L_{2}}\), we can use the subsystem of Figure 4b, which provides a one-dimensional Fourier transform, noted by \(\mathrm{FT_{y}}\). The focal length of the lens placed in the middle of the subsystem of Figure 4b is twice as big as the focal length \(f_{l}\) of the two other identical lenses. The incident wavefront is imaged with respect to one dimension and is Fourier-transformed with respect to the other. We note that the various input scene images can be illuminated by spatially incoherent monochromatic sources, such as a matrix of vertical-cavity surface-emitting lasers (VCSELs). The mutual spatial coherence of the sources is not necessary because the correlation products are performed independently. Using the same technique of Figure 4, the scale and rotation invariance can be combined. In this case, several replicas of the input image \(g\) are entered simultaneously, and for each replica, we attribute a scale factor \(\alpha_{m}\) and a translation factor \(x_{m}\). Therefore, for each replica, we need a diffractive phase element:  
\[t_{m}(r,\theta)=exp\left(-i\pi\frac{r}{\lambda}\left(\frac{r}{z}\left(1+ \alpha_{m}^{2}-2\alpha_{m}\cos(\theta)\right)+\frac{2x_{m}}{f_{l}}\right)\right) \tag{9}\]The filter bank, which is the rotation invariant, possesses a two-dimensional structure (Figure 4) where each line segment \(\widetilde{F}_{m}^{*}\), a conjugate of the Fourier transform of \(\widetilde{f}_{m}\), corresponds to a certain scale \(\alpha_{m}\).