Linear canonical transformation

Summary

In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of the transform

Official source

https://en.wikipedia.org/wiki/Linear_canonical_transformation

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Fresnelets: New Multiresolution Wavelet Bases for Digital Holography

Thierry Blu, Michaël Unser

We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of

L _{ 2 }

to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberg-like uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor functions, bringing together two separate aspects of the holography inventor's work. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their near-optimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several user-defined, wavelength-independent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zero-order and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms.

IEEE2003

We present a new class of wavelet bases—Fresnelets—which is obtained by applying the Fresnel transform operator to a wavelet basis of

L _{ 2 }

. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays. We first investigate the multiresolution properties (translation, dilation) of the Fresnel transform that are needed to construct our new wavelet. We derive a Heisenberg-like uncertainty relation that links the localization of the Fresnelets with that of the original wavelet basis. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. We conclude that the Fresnel B-splines are particularly well suited for processing holograms because they tend to be well localized in both domains.

SPIE2001

Complex-Wave Retrieval from a Single Off-Axis Hologram

Thierry Blu, Michaël Unser

We present a new digital two-step reconstruction method for off-axis holograms recorded on a CCD camera. First, we retrieve the complex object wave in the acquisition plane from the hologram's samples. In a second step, if required, we propagate the wave front by using a digital Fresnel transform to achieve proper focus. This algorithm is sufficiently general to be applied to sophisticated optical setups that include a microscope objective. We characterize and evaluate the algorithm by using simulated data sets and demonstrate its applicability to real-world experimental conditions by reconstructing optically acquired holograms.

OSA2004

We present a new class of wavelet bases—Fresnelets—which is obtained by applying the Fresnel transform operator to a wavelet basis of

L _{ 2 }

SPIE2001

Fresnelets: New Multiresolution Wavelet Bases for Digital Holography

Thierry Blu, Michaël Unser

L _{ 2 }

IEEE2003