Some properties of the spinor fourier transform

In this paper, the theory of the spinor Fourier transform introduced in [Batard T, Berthier M, Saint-Jean C, Clifford-Fourier Transform for Color Image Processing, Geometric Algebra Computing for Engineering and Computer Science (E. Bayro-Corrochano and G. Scheuermann Eds.), Springer, London, 2010, pp. 135–161] is further developed. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra. Next, two bases are determined under which this Fourier transform is diagonalizable. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. This problem will be tackled in the final section of this paper

Spinor Fourier Transform for Image Processing

We propose in this paper to introduce a new spinor Fourier transform for both grey-level and color image processing. Our approach relies on the three following considerations: mathematically speaking, defining a Fourier transform requires to deal with group actions; vectors of the acquisition space can be considered as generalized numbers when embedded in a Clifford algebra; the tangent space of the image surface appears to be a natural parameter of the transform we define by means of so-called spin characters. The resulting spinor Fourier transform may be used to perform frequency filtering that takes into account the Riemannian geometry of the image. We give examples of low-pass filtering interpreted as diffusion process. When applied to color images, the entire color information is involved in a really non marginal process

Hyperparameter-free losses for model-based monocular reconstruction

This work proposes novel hyperparameter-free losses for single view 3D reconstruction with morphable models (3DMM). We dispense with the hyperparameters used in other works by exploiting geometry, so that the shape of the object and the camera pose are jointly optimized in a sole term expression. This simplification reduces the optimization time and its complexity. Moreover, we propose a novel implicit regularization technique based on random virtual projections that does not require additional 2D or 3D annotations. Our experiments suggest that minimizing a shape reprojection error together with the proposed implicit regularization is especially suitable for applications that require precise alignment between geometry and image spaces, such as augmented reality. We evaluate our losses on a large scale dataset with 3D ground truth and publish our implementations to facilitate reproducibility and public benchmarking in this field.Peer ReviewedPostprint (author's final draft

Variational models for color image correction inspired by visual perception and neuroscience

Reproducing the perception of a real-world scene on a display device is a very challenging task which requires the understanding of the camera processing pipeline, the display process, and the way the human visual system processes the light it captures. Mathematical models based on psychophysical and physiological laws on color vision, named Retinex, provide efficient tools to handle degradations produced during the camera processing pipeline like the reduction of the contrast. In particular, Batard and Bertalmío [J Math. Imag. Vis. 60(6), 849-881 (2018)] described some psy-chophysical laws on brightness perception as covariant derivatives, included them into a variational model, and observed that the quality of the color image correction is correlated with the accuracy of the vision model it includes. Based on this observation, we postulate that this model can be improved by including more accurate data on vision with a special attention on visual neuro-science here. Then, inspired by the presence of neurons responding to different visual attributes in the area V1 of the visual cortex as orientation, color or movement, to name a few, and horizontal connections modeling the interactions between those neurons, we construct two variational models to process both local (edges, textures) and global (contrast) features. This is an improvement with respect to the model of Batard and Bertalmío as the latter can not process local and global features independently and simultaneously. Finally, we conduct experiments on color images which corroborate the improvement provided by the new models

Derivatives and Inverse of a Linear-Nonlinear Multi-Layer Spatial Vision Model

Analyzing the mathematical properties of perceptually meaningful linear-nonlinear transforms is interesting because this computation is at the core of many vision models. Here we make such analysis in detail using a specific model [Malo & Simoncelli, SPIE Human Vision Electr. Imag. 2015] which is illustrative because it consists of a cascade of standard linear-nonlinear modules. The interest of the analytic results and the numerical methods involved transcend the particular model because of the ubiquity of the linear-nonlinear structure. Here we extend [Malo&Simoncelli 15] by considering 4 layers: (1) linear spectral integration and nonlinear brightness response, (2) definition of local contrast by using linear filters and divisive normalization, (3) linear CSF filter and nonlinear local con- trast masking, and (4) linear wavelet-like decomposition and nonlinear divisive normalization to account for orientation and scale-dependent masking. The extra layers were measured using Maximum Differentiation [Malo et al. VSS 2016]. First, we describe the general architecture using a unified notation in which every module is composed by isomorphic linear and nonlinear transforms. The chain-rule is interesting to simplify the analysis of systems with this modular architecture, and invertibility is related to the non-singularity of the Jacobian matrices. Second, we consider the details of the four layers in our particular model, and how they improve the original version of the model. Third, we explicitly list the derivatives of every module, which are relevant for the definition of perceptual distances, perceptual gradient descent, and characterization of the deformation of space. Fourth, we address the inverse, and we find different analytical and numerical problems in each specific module. Solutions are proposed for all of them. Finally, we describe through examples how to use the toolbox to apply and check the above theory. In summary, the formulation and toolbox are ready to explore the geometric and perceptual issues addressed in the introductory section (giving all the technical information that was missing in [Malo&Simoncelli 15])

The minor house dust mite allergen Der p 13 is a fatty acid binding protein and an activator of a TLR2-mediated innate immune response

Background: The house dust mite (HDM) allergen Der p 13 could be a lipid-binding protein able to activate key innate signaling pathways in the initiation of the allergic response. We investigated the IgE reactivity of recombinant Der p 13 (rDer p 13), its lipid binding activities and its capacity to stimulate airway epithelium cells. Methods: Purified rDer p 13 was characterized by mass spectrometry, circular dichroism, fluorescence-based lipid binding assays and in-silico structural prediction. IgE binding activity and allergenic potential of Der p 13 were examined by ELISA, basophil degranulation assays and in-vitro airway epithelial cell activation assays. Results: Protein modeling and biophysical analysis indicated that Der p 13 adopts a β barrel structure with a predominately apolar pocket representing a potential binding site for hydrophobic ligands. Fluorescent lipid binding assays confirmed that the protein is highly selective for ligands and that it binds a fatty acid with a dissociation constant typical of lipid transporter proteins. The low IgE binding frequency (7%, n= 224) in Thai HDM-allergic patients as well as the limited propensity to activate basophil degranulation classifies Der p 13 as a minor HDM allergen. Nevertheless, the protein with its presumptively associated lipid(s) triggered the production of IL-8 and GM-CSF in respiratory epithelial cells through a TLR2-, MyD88-, NF-kB- and MAPK-dependent signaling pathway. Conclusions: Although a minor allergen, Der p 13 may, through its lipid binding capacity, play a role in the initiation of the HDM allergic response through TLR2 activation

The house dust mite allergen Der p 5 binds lipid ligands and stimulates airway epithelial cells through a TLR2‐dependent pathway

Background: Protein crystallographic studies suggest that the house dust mite (HDM) allergen Der p 5 potentially interacts with hydrophobic ligands. Der p 5, in association with its ligand(s), might therefore trigger innate immune signalling pathways in the airway epithelium and influence the initiation of the HDM‐allergic response. Objective: We investigated the lipid binding propensities of recombinant (r)Der p 5 and characterized the signalling pathways triggered by the allergen in airway epithelial cells. Methods: rDer p 5 was produced in Pichia pastoris and characterized by mass spectrometry, multi‐angle light scattering and circular dichroism. Its interactions with hydrophobic ligands were investigated in fluorescence‐based lipid binding assays and in‐silico docking simulations. Innate immune signalling pathways triggered by rDer p 5 were investigated in airway epithelial cell activation assays in vitro. Results: Biophysical analysis showed that rDer p 5 was monomeric and adopted a similar α‐helix‐rich fold at both physiological and acidic pH. Spectrofluorimetry experiments showed that rDer p 5 is able to selectively bind lipid ligands, but only under mild acidic pH conditions. Computer‐based docking simulations identified potential binding sites for these ligands. This allergen, with putatively associated lipid(s), triggered the production of IL‐8 in respiratory epithelial cells through a TLR2‐, NF‐kB‐ and MAPK‐dependent signalling pathway. Conclusions and Clinical Relevance: Despite the fact that Der p 5 represents a HDM allergen of intermediate prevalence, our findings regarding its lipid binding and activation of TLR2 indicate that it could participate in the initiation of the HDM‐allergic state

Géométrie différentielle des fibrés vectoriels et algèbres de Clifford appliquées au traitement d'images multicanaux

This thesis is devoted to supply applications of Clifford algebras to multichannel image processing. Moreover, we introduce the use of vector bundles framework in image processing. Part 1 is devoted to multichannel image segmentation. We generalize Di Zenzo's approach to edge detection by constructing metric tensors related to the choice of the segmentation. Using the framewok of Clifford algebras bundles, we show that the choice of a segmentation of an image is related to the choice of a connection and a section on such a bundle. Part 2 is devoted to regularization. We make use of heat equations associated to generalized Laplacians on vector bundles. The main result of this part is the following. Considering the heat equation associated to the Hodge operator on the Clifford bundle of a well-chosen Riemannian manifold, we obtain a common framework for anisotropic regularization of images (videos), and related fields such as vector fields and orthonormal frame fields. At last, in Part 3, we deal with spectral analysis via the definition of a Fourier transform of a multichannel image. This definition is related to an abstract theory of Fourier transform based on the notion of group representation. From this point of view, the usual Fourier transform of grey level images is related with irreducible representations of the translations of the plane. We extend this Fourier transform to multichannel images by considering reducible representations of this group.Le sujet de cette thèse est l'apport d'applications du formalisme des algèbres de Clifford au traitement d'images multicanaux. Nous y introduisons également l'utilisation du cadre des fibrés vectoriels en traitement d'image. La Partie 1 est consacrée à la segmentation d'images multicanaux. Nous généralisons l'approche de Di Zenzo pour la détection de contours en construisant des tenseurs métriques adaptés au choix de la segmentation. En utilisant le cadre des fibrés en algèbres de Clifford, nous montrons que le choix d'une segmentation d'une image est directement lié au choix d'une métrique, d'une connexion et d'une section sur un tel fibré. La Partie 2 est consacrée à la régularisation. Nous utilisons le cadre des équations de la chaleur associées à des Laplaciens généralisés sur des fibrés vectoriels. Le résultat principal que nous obtenons est qu'en considérant l'équation de la chaleur associée à l'opérateur de Hodge sur le fibré de Clifford d'une variété Riemannienne bien choisie, nous obtenons un cadre global pour régulariser de manière anisotrope des images (vidéos) multicanaux, et des champs s'y rapportant tels des champs de vecteurs ou des champs de repères orthonormés. Enfin, dans la Partie 3, nous nous intéressons à l'analyse spectrale via la définition d'une transformée de Fourier d'une image multicanaux. Cette définition repose sur une théorie abstraite de la transformée de Fourier basée sur la notion de représentation de groupe. De ce point de vue, la transformée de Fourier usuelle pour les images en niveau de gris est basée sur les représentations irréductibles du groupe des translations du plan. Nous l'étendons aux images multicanaux en lui associant les représentations réductibles de ce groupe