Analogies and differences in optical and mathematical systems and approaches

Publication in the conference proceedings of SampTA, Bremen, Germany, 201

Mod\'elisations de textures par champ gaussien \`a orientation locale prescrite

This paper presents two new models of oriented texture, based on a new class of Gaussian fields, called locally anisotropic fractional Brownian fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation with the Cholesky method or the turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation.Comment: in Frenc

Quaternionic Representation of the Riesz Pyramid for Video Magnification

Recently, we presented a new image pyramid, called the Riesz pyramid, that uses the Riesz transform to manipulate the phase in non-oriented sub-bands of an image sequence to produce real-time motion-magnified videos. In this report we give a quaternionic formulation of the Riesz pyramid, and show how several seemingly heuristic choices in how to use the Riesz transform for phase-based video magnification fall out of this formulation in a natural and principled way. We intend this report to accompany the original paper on the Riesz pyramid for video magnification

3D steerable wavelets and monogenic analysis for bioimaging

ABSTRACT In this paper we introduce a 3D wavelet frame that has the key property of steerability. The proposed wavelet frame relies on the combination of a 3D isotropic wavelet transform with the 3D Riesz operator which brings steerability to the pyramid. The novel transform enjoys self reversibility and exact steering of the basis functions in any 3D direction by linear combination of the primary coefficients. We exploit the link between the Riesz transform and the directional Hilbert transform to define a multiresolution monogenic signal analysis in 3D which achieves multiscale AM/FM signal decomposition. We give an example of application of the 3D monogenic wavelet frame in biological imaging with the enhancement of anisotropic structures in 3D fluorescence microscopy

2-D Prony-Huang Transform: A New Tool for 2-D Spectral Analysis

This work proposes an extension of the 1-D Hilbert Huang transform for the analysis of images. The proposed method consists in (i) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure, and (ii) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2-D mode decompositions based on non-smooth convex optimization: a "Genuine 2-D" approach, that constrains the local extrema of the IMFs, and a "Pseudo 2-D" approach, which constrains separately the extrema of lines, columns, and diagonals. The spectral analysis step is based on Prony annihilation property that is applied on small square patches of the IMFs. The resulting 2-D Prony-Huang transform is validated on simulated and real data.Comment: 24 pages, 7 figure

Riesz pyramids for fast phase-based video magnification

We present a new compact image pyramid representation, the Riesz pyramid, that can be used for real-time phase-based motion magnification. Our new representation is less overcomplete than even the smallest two orientation, octave-bandwidth complex steerable pyramid, and can be implemented using compact, efficient linear filters in the spatial domain. Motion-magnified videos produced with this new representation are of comparable quality to those produced with the complex steerable pyramid. When used with phase-based video magnification, the Riesz pyramid phase-shifts image features along only their dominant orientation rather than every orientation like the complex steerable pyramid.Quanta Computer (Firm)Shell ResearchNational Science Foundation (U.S.) (CGV-1111415)Microsoft Research (PhD Fellowship)Massachusetts Institute of Technology. Department of MathematicsNational Science Foundation (U.S.). Graduate Research Fellowship (Grant 1122374