Self-fractional Fourier Functions Research Articles

Hybrid image fusion scheme using self-fractional Fourier functions and multivariate empirical mode decomposition

Image fusion has emerged as a promising area of research and a bivariate empirical mode decomposition based fusion scheme has recently been proposed in the literature. In this paper, a hybrid fusion scheme combining self-fractional Fourier function (SFFF) decomposition and multivariate empirical mode decomposition is proposed. In the proposed image fusion technique, images to be fused are decomposed into SFFF images. The SFFF images are further decomposed into intrinsic mode functions (IMFs) using multivariate empirical mode decomposition (MEMD). Corresponding IMFs of same decomposition level of SFFF images are fused using local variance based adaptive weight fusion rule to obtain fused IMF images. The fused image is obtained by applying inverse transformation on fused IMF images. The proposed technique provides flexibility in the number of functions in the SFFF decomposition, transform before SFFF decomposition, and the types of source images (real and complex) to be fused. Simulations are performed for fusion of test images with different SFFF decomposition levels and the results are compared with other existing methods. It is seen that the simulation results are comparable to the existing schemes.

Image fusion based on image decomposition using self-fractional Fourier functions

Image fusion has been receiving increasing attention in the research community in a wide spectrum of applications. Several algorithms in spatial and frequency domains have been developed for this purpose. In this paper we propose a novel algorithm which involves the use of fractional Fourier domains which are intermediate between spatial and frequency domains. The proposed image fusion scheme is based on decomposition of source images (or its transformed version) into self-fractional Fourier functions. The decomposed images are then fused by maximum absolute value selection rule. The selected images are combined and inverse transformation is taken to obtain the final fused image. The proposed decomposition scheme and the use of some transformation before the decomposition step offer additional degrees of freedom in the image fusion scheme. Simulation results of the proposed scheme for different transformation of the source images for two different sets of images are also presented. It is observed through the simulation results that the use of taking the transformation before the decomposition step improves the quality of fused image. In particular the results of using the fractional Fourier transform and discrete cosine transform before the decomposition step are encouraging.

Uniqueness of self-fractional Fourier transform with irrational order and a supplement to Caola–Mendlovic, Ozaktas and Lohmann's rule

In this communication we first prove uniqueness of self fractional Fourier function (SFFF) for irrational order, the SFFF uniquely determined is given by f( x)= cexp{− x 2/2}, where c can be any constant. As a supplement to Caola–Mendlovic, Ozaktas and Lohmann's rule, a simple rule for constructing self fractional Fourier transform with order N/(2 mM 1) (m⩾0 , N and M 1 odd) is presented.

Random fractional Fourier transform: stochastic perturbations along the axis of propagation

The fractional Fourier transform (FRT) is known to be optically implementable with use of a medium with a perfect radial quadratic-index profile. Using the quantum-mechanical operator formalism, we examine the effects on the FRT action of such a medium that are due to small random inhomogeneities in the longitudinal direction, the direction of propagation, and we formulate the random fractional Fourier transform (RFRT). Applying the RFRT to a self-fractional Fourier function, a Gaussian function, we discuss both the total power and the variance. The random Fourier transform is examined as a special limiting case.

Self-imaging in fractional Fourier transform systems

The self-imaging phenomenon in a fractional Fourier transform optical system is described in the framework of self-fractional Fourier functions. The main properties of these functions are investigated. The possible application of fractional Fourier transform optical systems in optical signal processing is discussed.

Self-fractional Fourier functions and selection of modes

It is shown that any function from can be represented as a linear combination of M self-fractional Fourier functions of order M, which are orthogonal to each other. Each of them contains a selection of Hermite - Gauss modes of the generator function. The physical meaning of the synthesis of self-fractional Fourier functions is discussed.

Imaging in first-order optical systems

A general analysis of imaging phenomena based on the properties of the canonical integral transform has been carried out for first-order (ABCD) optical systems. It is shown that under some conditions the effect of the canonical integral transform on an input complex amplitude, being the eigenfunction of another canonical operator, generates its scaled replica with an additional quadratic phase shift. As particular examples, the imaging of a period icinput complex amplitude and a self-fractional Fourier function are considered. For imaging of a transparency the additional quadratic phase shift vanishes with use of a wave with a spherocylindrical phase front for illumination

On the self-fractional Fourier functions

It is shown that a self-fractional Fourier function for an angle where N and M are indivisible integers (N < M), is also a self-fractional Fourier function for angles (j = 1, 2, ...). This allows us to define a self-fractional Fourier function of order M. An eigenvalue of a fractional Fourier operator for angle is equal to , where L is an integer.