Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

Özaktaş, Haldun M.; Barshan, Billur; Mendlovic, David; Onural, Levent

doi:10.1364/josaa.11.000547

Cited by 452 publications

(255 citation statements)

References 15 publications

(22 reference statements)

Supporting

Mentioning

251

Contrasting

Unclassified

Order By: Relevance

“…First, consider the most general linear estimate of the form (2) Our design criteria is the mean square error (MSE), which is defined as (3) where denotes the expectation operator, and denotes the norm:…”

Section: A Problem Statementmentioning

confidence: 99%

“…However, application of this estimation operator [cf. (2)] on a given distorted and noisy signal would require time, where is the time-bandwidth product of the signals. In this paper, we restrict our estimate so that it corresponds to a multiplication with a filter function in the th fractional Fourier domain.…”

Section: A Problem Statementmentioning

confidence: 99%

“…Several applications of the fractional Fourier transform have been suggested or explored to varying degrees. These include optical diffraction and beam propagation, and optical signal processing [2], [7]- [12], [22]- [25], quantum optics [26]- [30], phase retrieval, signal detection, pattern recognition, noise representation, time-variant filtering and multiplexing, data compression, study of space/TF distributions, and sweptfrequency filters [2], [4], [31], [32]. The fractional Fourier transform is also related to a number of recently developed signal processing tools [33]- [35].…”

Section: Fractional Fourier Transformmentioning

confidence: 99%

“…Recently, we have discussed how various time-variant operations can be performed by multiplying with a filter function in a fractional Fourier domain [2], [3]. A related concept ("swept-frequency filters") has been discussed by Almeida [4].…”

Section: Introductionmentioning

confidence: 99%

“…We observe that they overlap in both the zeroth and first domains (consider the projections onto the and axes), but they do not overlap in the 0.5th domain. (The concept of fractional Fourier domains was developed in [2] and [3] and will be reviewed in Section III). Thus, we can eliminate undesired signal components by using a simple unit amplitude mask in the 0.5th domain.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Optimal filtering in fractional Fourier domains

Kutay

Özaktaş

Ankan³

et al. 1997

IEEE Trans. Signal Process.

260

View full text Add to dashboard Cite

Section: A Problem Statementmentioning

confidence: 99%

Section: A Problem Statementmentioning

confidence: 99%

Section: Fractional Fourier Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal filtering in fractional Fourier domains

Kutay

Özaktaş

Ankan³

et al. 1997

IEEE Trans. Signal Process.

260

View full text Add to dashboard Cite

On fractional paradigm and intermediate zones in electromagnetism: I?Planar observation

Engheta¹

1999

Microw. Opt. Technol. Lett.

View full text Add to dashboard Cite

In this Letter the kernel of the integral transform that relates the field quantities over an observation flat plane to the corresponding quantities on another observation plane parallel with the first one is fractionalized for the two-dimensional (2-D) monochromatic wave propagation. It is shown that such fractionalized kernels, with fractionalization parameter ν between zero and unity, are the kernels of the integral transforms that provide the field quantities over the parallel planes between the two original planes. With proper choice of the first two planes, these fractional kernels can provide us with a natural way of interpreting the fields in the intermediate zones (i.e., the region between the near and the far zones) in certain electromagnetic problems. The evolution of these fractional kernels into the Fresnel and Fraunhofer diffraction kernels is addressed. The limit of these fractional kernels for the static case is also mentioned.

show abstract

Application of the fractional Fourier transform to image reconstruction in MRI

Parot

Sing‐Long

Lizama

et al. 2011

Magnetic Resonance in Med

View full text Add to dashboard Cite

The classic paradigm for MRI requires a homogeneous B 0 field in combination with linear encoding gradients. Distortions are produced when the B 0 is not homogeneous, and several postprocessing techniques have been developed to correct them. Field homogeneity is difficult to achieve, particularly for shortbore magnets and higher B 0 fields. Nonlinear magnetic components can also arise from concomitant fields, particularly in low-field imaging, or intentionally used for nonlinear encoding. In any of these situations, the second-order component is key, because it constitutes the first step to approximate higher-order fields. We propose to use the fractional Fourier transform for analyzing and reconstructing the object's magnetization under the presence of quadratic fields. The fractional fourier transform provides a precise theoretical framework for this. We show how it can be used for reconstruction and for gaining a better understanding of the quadratic field-induced distortions The standard paradigm for MRI requires a strong magnetic field with uniform intensity and time-varying linear encoding gradients across the entire field of view. However, deviations in the main field are common as uniform fields are physically difficult to achieve and also because of off-resonance effects from susceptibility changes. Such frequency variations introduce an accumulating phase over time, which cannot be demodulated easily as it varies spatially. This problem is worse for stronger fields and for sequences with long acquisition times. Great efforts are put in building the system with the highest possible homogeneity, for example, with passive or active shimming which partially correct first-and second-order field variations.Additionally, there has been an increasing interest in spatial encoding by nonhomogeneous, nonbijective spatial encoding magnetic fields (SEMs; Ref. 1). Starting from a general nonlinear field concept, novel techniques including PatLoc (1), O-Space (2), Null Space (3), and PhaseScrambled imaging (4-6) have all introduced the use of second-order SEMs as the first and simplest approach in simulations, custom-built hardware and experiments (7-12). One of the challenges of these approaches is to have an appropriate reconstruction technique.Higher-order fields also appear as a natural component of linear gradients. These concomitant fields can be well approximated by quadratic functions (13,14). In several applications, the magnitude of these fields is not negligible and artifacts appear in image reconstructions (15,16), most notably in very low-field or microtesla imaging (17,18).Several image reconstruction methods have been proposed to correct distortions, or to reconstruct an image produced by nonhomogeneous fields, being an active field of research (19-28). There is a well-known theoretical background for the linear correction approaches, in which an exact analytical solution is provided (19,23). For secondorder and arbitrary field maps, there are several correction techniques that typically trade-off corr...

show abstract

Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

Cited by 452 publications

References 15 publications

Optimal filtering in fractional Fourier domains

Optimal filtering in fractional Fourier domains

On fractional paradigm and intermediate zones in electromagnetism: I?Planar observation

Application of the fractional Fourier transform to image reconstruction in MRI

Contact Info

Product

Resources

About