Abstract
The gyrator transform (GT) is used for images processing in applications of light propagation. We propose new image processing operators based on the GT, these operators are: Generalized shift, convolution and correlation. The generalized shift is given by a simultaneous application of a spatial shift and a modulation by a pure linear phase term. The new operators of convolution and correlation are defined using the GT. All these image processing operators can be used in order to design and implement new optical image processing systems based on the GT. The sampling theorem for images whose resulting GT has finite support is developed and presented using the previously defined operators. Finally, we describe and show the results for an optical image encryption system using a nonlinear joint transform correlator and the proposed image processing operators based on the GT.
Highlights
The usual convolution and correlation operations have been used in some optical systems for image processing, such as filtering, encryption, decryption, comparison, authentication, pattern recognition and classification of images [1,2,3,4,5,6]
The generalized shift is the simultaneous application of a spatial shift and a modulation by a pure linear phase term that does not introduce a shift in the gyrator domain (GD)
We have proposed new image processing operators based on the gyrator transform (GT); these new operators are: Generalized shift and the convolution and correlation operations in the GD
Summary
The usual convolution and correlation operations have been used in some optical systems for image processing, such as filtering, encryption, decryption, comparison, authentication, pattern recognition and classification of images [1,2,3,4,5,6]. We propose to define a new generalized shift, convolution and correlation operators based on the gyrator transform (GT). We apply the proposed new operators with the purpose of developing and defining the sampling theorem for images whose resulting GT has finite support This result is important when it is necessary to sample properly the spatial domain or the GD of an optical system based on the GT. We can observe in Equations (4) and (5) that a shift or a modulation by a pure linear phase term applied to the function f ( x, y) produces a shift of the GT f α (u, v), which is proportional to the parameters α, x0 , y0 , u0 and v0. Other properties of the GT are described in [14]
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