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Random Phase Textures: Theory and Synthesis

Random Phase Textures: Theory and Synthesis

Random Phase Textures: Theory and Synthesis B. Galerne ∗ , Y. Gousseau † and J.-M. Morel ∗ Abstract This paper explores the mathematical and algorithmic properties of two sample-based micro- texture models: random phase noise (RPN ) and asymptotic discrete spot noise (ADSN ). These models permit to synthesize random phase textures. They arguably derive from linearized ver- sions of two early Julesz texture discrimination theories. The ensuing mathematical analysis shows that, contrarily to some statements in the literature, RPN and ADSN are different stochastic processes. Nevertheless, numerous experiments also suggest that the textures ob- tained by these algorithms from identical samples are perceptually similar. The relevance of this study is enhanced by three technical contributions to micro-texture synthesis from samples. A solution is proposed to three obstacles that prevented the use of RPN or ADSN to emulate micro-textures. First, RPN and ADSN algorithms are adapted to color images. Second, a pre- processing is proposed to avoid artifacts due to the non-periodicity of real-world texture samples. Finally, the method is extended to synthesize textures with arbitrary size from a given sample. Keywords: texture synthesis, random phase, shot noise, spot noise
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Regularity and localized representations of random-phase textures

Regularity and localized representations of random-phase textures

1.1. Textures, algorithms and models 11 dimensions of the texture (Chapter 2) and zooming in within a continuous model (Chapters 4 and 5). The continuous model is connected to the study of random Fourier series, which has been first developped by Zygmund along with Paley [108], [109], [110] and Salem [119], and widely discussed still in the one-dimensional case by Kahane [69] and in more general- ity by Marcus and Pisier [96]. In Chapter 4 and 5, we focus on extending to the general finite-dimensional case some results which are well known in the one-dimensional case with respect to convergence ([15] and [69]), continuity ([15], [69], [45] and [32]) and regularity ([69] and [32]). Recent papers have been working on a similar path of generalizations from the one-dimensional to the finite-dimensional case with a focus on continuity, let us cite e.g. [27]. In the remaining of this section, we propose a non-exhaustive tour of texture algorithms and models. As discussed supra, randomness plays a very important role within texture al- gorithms/models. As we shall see, they can indeed be characterized by a varying degree of randomness. Indeed, a continuum can be drawn from deterministic tilings to filtered white noise algorithms, which can be translated in terms of information theory by an increasing entropy from deterministic textures (strongly structured) to noises (strongly unstructured). In between these two extremes lie two important families of texture algorithms: examplar- based synthesis, which can be characterized as “weakly structured”, and noise synthesis based on statistical analysis qualified as “weakly unstructured”. The former roughly consists in adding randomness to deterministic algorithms supported by rigid structures, and the latter in adding control to noisy images through constraints based on statistical analysis of some inputs.
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Motion clouds: model-based stimulus synthesis of natural-like random textures for the study of motion perception

Motion clouds: model-based stimulus synthesis of natural-like random textures for the study of motion perception

5 Discussion In this article we described the mathematical framework and provide the computer implementation of a set of complex stimuli that we call Motion Clouds. Those are an instantiation of a more generic class of stimuli called Random Phase Textures. These stimuli, presented herein in the context of visual motion perception, represent an attempt to fill the gap between simple stimuli (such as spots of light or sinusoidal gratings), stimulus ensembles consisting of simple stimuli (for instance, white noise patterns) and natural stimuli [ Felsen and Dan , 2005 , Rust and Movshon , 2005 ]. Similar approaches have been used before in the case of motion detection [ Schrater et al. , 2000 ] but stimuli have been described in a somewhat incomplete and non accessible way. Here, our goal was to provide a complete and rigorous mathematical description of those stimuli, as well as tools for generating them. We have also given a few examples of different subset of Motion Clouds that could be used for probing detection, integration and segmentation stages at both psychophysical and neurophysiological levels. To conclude, we indicate a few future extensions and possible uses.
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Interactive Quadrangulation with Reeb Atlases and Connectivity Textures Interactive Quadrangulation with Reeb Atlases and Connectivity Textures

Interactive Quadrangulation with Reeb Atlases and Connectivity Textures Interactive Quadrangulation with Reeb Atlases and Connectivity Textures

Fig. 12. Various examples of user designed quad meshes generated with our framework accompanied by quality statistics related to the vertices (vertex count, extraordinary vertex count and the max difference from the ideal valence), the average mesh angle and scaled Jacobian. (corresponding to critical points of f ) will correspond to extraordinary vertices in the final mesh. The user can edit the valence and location of those extraordinary vertices with a set of curve editing operations applied on the chart boundaries as demonstrated in the accompanying video. Second, when the user is satisfied with the Reeb atlas segmentation, local views of each Reeb chart are opened to design connectivity textures via subdivisions, deletions and element movements. Typically, our experiments showed that connectivity texturing was often achieved through a sequence of global subdivisions, possibly with intermediate cube subdivisions (Fig. 9). Finally, an automated stitching algorithm generates the final output mesh by composing the connectivity textures.
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A recursive non-linear pre-encryption for opto-digital double random phase encoding

A recursive non-linear pre-encryption for opto-digital double random phase encoding

Although the above optical and opto-digital DRPE versions are efficient, they may not resist to some attacks [ 14–16 ]. This is mainly due to the fact that the transforms employed are linear operators and the associated scrambling schemes can also be considered as linear transformations, which make the entire resulting DRPE a linear image encryption technique and hence fragile to some attacks. In order to overcome this linearity problem, a non-linear pre-processing has recently been proposed in [ 17 ]. It consists of obtaining a pre-processed image by performing the XOR operation in the spatial domain between each pixel of the input image and its corresponding pixel in the same position of a randomly created image before applying a DRPE. It has been shown in [ 17 ] that this non-linear pre-processing coupled with an opto-digital MPDFrFT-DRPE leads to a new opto-digital DRPE that outperforms the other existing DRPE versions, specifically in terms of the secret key sensitivity. Even though the pre-processing proposed in [ 17 ] is very efficient and attractive, the non-linearity offered by the XOR operation has been introduced therein for each pixel separately and independently. However, from the encryption point of view, it is highly desirable to construct a more complex non-linear pre-processing. This is achieved in this paper by introducing a new recursive non-linear pre-encryption to be applied prior to any DRPE. It consists of (1) resizing the input image into a vector, (2) chaotically scrambling the resulting vector using the piecewise linear chaotic map (PLCM), (3) performing the bit-wise XOR operation recursively between two adjacent elements of the resulting scrambled vector, where the first element is bit-XORed with a random eight-bit integer, and (4) reshaping the resulting XORed vector into a matrix to obtain the pre-encrypted image. The recursive property of the proposed pre-encryption introduces some dependency between the pixels of the pre-processed image and hence ensures the accumulation and propagation of the error to all pixels in the case of any wrongness in the decryption key. Therefore, any opto-digital cryptosystem obtained by coupling the proposed pre-encryption with an optical DRPE cryptosystem including the FT-DRPE would resist to decryption attacks. As mentioned above, in order to further increase the security of an opto-digital cryptosystem, it is highly desirable to use transforms having optical implementation and increased number of independent parameters such as FrFT and MPDFrFT.
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Synthèse de textures multifractales directement sur des surfaces 3D

Synthèse de textures multifractales directement sur des surfaces 3D

1 Introduction Les textures jouent un rôle crucial dans notre perception des objets dans une scène. La géométrie des objets dans l’espace est aussi essentielle et nous percevons globalement une surface courbe texturée. Il existe de nombreuses méthodes pour simu- ler un objet texturé en infographie. La texture peut être utili- sée pour modifier différentes propriétés de l’objet telles que sa réflectivité, sa couleur, son relief ou sa transparence. Les approches se répartissent essentiellement en deux grandes ca- tégories : la synthèse à partir d’un échantillon et la synthèse procédurale. Si l’on souhaite reproduire exactement une tex- ture donnée sur une surface 3D, la méthode la plus naturelle
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Exact asymptotics for phase retrieval and compressed sensing with random generative priors

Exact asymptotics for phase retrieval and compressed sensing with random generative priors

Figure 6: Algorithmic gap ∆ alg = α alg − α IT for small ρ and linear activation, as a function of (left) the compression β ≡ β l for fixed depth L = 4 and of (right) depth for a fixed compression β = 2. 4 Conclusion and perspectives In this manuscript we analysed how generative priors from an ensemble of random multi-layer neural networks impact signal reconstruction in the high-dimensional limit of two important inverse problems: real-valued phase retrieval and linear estimation. More specifically, we characterised the phase diagrams describing the interplay between number of measurements needed at a given signal compression ρ, for a range of shallow and multi-layer architectures for the generative prior. We observed that although present, the algorithmic gap significantly decreases with depth in the studied architectures. This is particularly striking when compared with sparse priors at ρ  1, for which the algorithmic gap is considerably wider. In practice, this means generative models given by random multi-layer neural networks allow for efficient compressive sensing in these problems.
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Random matrix study of the phase structure of QCD with two colors

Random matrix study of the phase structure of QCD with two colors

unphysical results including negative baryon densities for small µ and a chiral condensate which varies with µ. These pathologies can be eliminated by the inclusion of all Matsubara frequencies. This extension of the random matrix model also correctly produces an essential singularity in the second-order line at T = 0. Neither variant of the random matrix model considered here can support diquark condensation at arbitrarily high baryon density since neither contains a true Fermi sea of states. A microscopic model capable of supporting diquark condensation at all densities does so at the cost of introducing significant model dependence. Near the condensation edge, a three dimensional model also induces power law corrections to the temperature dependence of the onset chemical potential. This result is robust.
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By-example Synthesis of Architectural Textures

By-example Synthesis of Architectural Textures

Content synthesis from example Example-based texture syn- thesis methods [Wei et al. 2009] generate new textures of any size resembling a small input image. Most of these methods focus on textures of unstructured, homogeneous aspect and require to pre- compute and store the resulting images. The approach of Sun et al. [2005] fills holes in structured images by pasting overlapping patches at discrete locations along user defined curves, minimizing for color differences. Sch¨odl et al. [2000] synthesize infinite video loops by cutting and re-ordering pieces of an example sequence. Loops are formed by jumping backward in time to matching frames. This could be applied to images, jumping from one column to an- other. Compared to this scheme our graph formulation allows for arbitrarily shaped cuts and is not limited to backward jumps.
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Descripteurs de Textures pour la Segmentation d'Images Echographiques 3D

Descripteurs de Textures pour la Segmentation d'Images Echographiques 3D

Pour calculer ces attributs de texture, différents modèles sont décrits dans les sous-sections suivantes. 3.1 Étude géométrique des textures 3D La structure géométrique des textures est descriptible à l’aide de composantes connexes qui sont une représenta- tion des motifs d’une texture. Pour le calcul des compo- santes connexes, il est nécessaire de réaliser une binarisa- tion de l’image originale. Dans notre méthode, nous uti- lisons une binarisation globale qui nécessite la définition d’un seuil choisi par l’utilisateur du système. La figure 2 présente une image binarisée avec des exemples de com- posantes connexes. Soit A et B deux points d’un sous- ensemble S de l’image I. A et B sont dit connectés dans S si et seulement si il existe un chemin connexe dans S reliant A et B. La relation "être connecté" dans S est une rela- tion d’équivalence. Les composantes connexes de l’image sont égales aux classes d’équivalence de cette relation [12]. Calculer les composantes connexes d’une image binaire re- vient à associer à chacune d’elles une même étiquette. Pour cela, il existe plusieurs algorithmes 2D dont les principaux sont expliqués par Chassery et Montanvert dans [3]. Parmi ces méthodes, nous avons choisi d’adapter à la 3D un al- gorithme ne nécessitant que deux passages sur une image pour calculer les composantes connexes. Ici la complexité est uniquement dépendante de la taille de l’image alors que dans un algorithme purement séquentiel, le nombre d’ité- rations dépend de la complexité des objets. Par exemple, avec un algorithme séquentiel, le nombre d’itérations pour traiter une spirale sera très important.
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Compact Representations of Stationary Dynamic Textures

Compact Representations of Stationary Dynamic Textures

6. NUMERICAL RESULTS In order to test the proposed algorithms, we compiled a dataset of stationary dynamic textures 1 containing 27 differ- ent color dynamic textures. It includes dynamic sequences of boiling water, clouds, fire, fog, fountain, waterfall, snow, ocean waves, ponds, and steam. Figure 1 presents the re- sults of analysis and synthesis on the exemplar texture mov- ing goldenlines and waterfall. Figure 1 (b) shows that the learned dynTextons decay very fast in space and time. Fig- ure 1 (c)-(d) and (e)-(f) compare the synthesized results of those using full-size dynTextons and truncated dynTextons. It demonstrates that thresholding the dynTextons does not affect the synthesized results, due to their compactness. Moreover, we observe that the synthesized results of these two Gaus- sian models are visually comparable. In particular, compared with the LDS model [6], in which case the a matrix is of size N × N , the proposed AR-dyntextions are much more com- pact. More results and videos can be found in the link http: //www.enst.fr/ ∼ xia/dynTextures.html .
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Phase diagram of the random frequency oscillator: The case of Ornstein-Uhlenbeck noise

Phase diagram of the random frequency oscillator: The case of Ornstein-Uhlenbeck noise

The long time behavior of the stochastic oscillator with random frequency is controlled by the sign of the Lyapunov exponent. In the case of a white noise perturbation, this exponent can be calculated exactly and the phase diagram of the stochastic oscillator can then be rigorously determined. The aim of this work is to study the effects of time correlations on the noise-induced bifurcation of the stochastic oscillator. In the case of an Ornstein-Uhlenbeck noise, an exact calculation seems to be out of reach and therefore we use various approximation schemes in order to derive analytical expressions for the Lyapunov exponent and to draw the phase diagram. Since the different approximations have distinct regions of validity, their study has allowed us to derive a global picture of the behavior of the system in the parameter space. In particular, we have derived the scaling behavior of the phase boundary in regions where the amplitude of the noise is small or large. Our results agree fairly well with numerical simulations and with exact perturbative expansions. These comparisons allow us to test the validity of the different approximation schemes.
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Motifs locaux binaires pour la classification d'images de textures multispectrales

Motifs locaux binaires pour la classification d'images de textures multispectrales

Résumé – Les motifs locaux binaires (LBP) permettent d’extraire des descripteurs de texture spatiaux pour discriminer les images de textures en niveaux de gris. Dans cet article, nous proposons un opérateur LBP qui extrait conjointement les informations de texture spatiales et spectrales à partir de l’image brute fournie par une caméra équipée d’une matrice de filtres multispectraux. Nous montrons expérimentalement que le descripteur proposé présente à la fois un faible coût de calcul et un haut pouvoir discriminant par rapport aux descripteurs LBP classiques appliqués aux images multispectrales.
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Méthodologie de conception de textures pour les interfaces tactiles à frottement programmable

Méthodologie de conception de textures pour les interfaces tactiles à frottement programmable

et leurs opportunités pour la conception et le dimensionne- ment d’environnements tactiles numériques, ainsi que leurs différences fondamentales avec les autres systèmes à retour tactile. Les méthodes psychophysiques permettent d’évaluer précisément la relation entre la valeur du signal et sa détec- tion ou sa discrimination avec un autre signal. À partir de ces règles de conception de base, il est possible de construire une grande variété d’éléments, composés de variations de frottement dans l’espace et dans le temps. Ces textures or- ganisées dans l’espace peuvent alors avoir de nouvelles caractéristiques de frottement, mais aussi des propriétés d’orientation ou de variation de densité. Elles forment des icônes tactiles pouvant s’organiser en un ensemble cohérent pour une situation d’usage donnée. Ce travail de concep- tion doit être mené de manière spécifique pour s’adapter aux spécificités des dispositifs à frottement programmable, et prendre ainsi en compte les paramètres d’action de l’utilisa- teur. Comme proposé par Beaudouin-Lafon [5], cette tâche consiste en la conception d’interactions, en tant que phéno- mènes sensorimoteurs relatifs à une activité avec un produit, et non uniquement du design d’interface.
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Random optimization on random sets

Random optimization on random sets

γ P Essmin w F pXq Ă L 0 pD, F q, i.e. such that cpγq ě cpˆ γq. We then claim that any ˆ γ P Essmin w F pXq Ă L 0 pD, F q satisfies min xPDpωq cpω, xq “ cpˆ γq. Theorem 6.1. Consider a random F -measurable set D. Let c be a (random) F -normal integrand and consider the random order x " y if cpxq ě cpyq. Suppose that the associated essential minimum (see Definition 3.7) Essmin w F pL 0 pD, F qq is not empty. Then,

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Synthèse de textures dynamiques pour l'étude de la vision en psychophysique et électrophysiologie

Synthèse de textures dynamiques pour l'étude de la vision en psychophysique et électrophysiologie

1.3 Contributions In this chapter, we attempt to engender a better understanding of hu- man perception by improving generative models for dynamic texture synthesis. From that perspective, we motivate the generation of optimal visual stimula- tion within a stationary Gaussian dynamic texture model. We develop our current model by extending, mathematically detailing and robustly testing previously introduced dynamic noise textures [169, 178, 194] coined “Motion Clouds” (MC or MCs). Our first contribution is a complete axiomatic deriva- tion of the model, seen as a shot noise aggregation of dynamically warped “textons”. Within our generative model, the parameters correspond to av- erage spatial and temporal transformations (ie zoom, orientation and trans- lation speed) and associated standard deviations of random fluctuations, as illustrated in Figure 2.1, with respect to external (objects) and internal (ob- servers) movements. A second contribution is the explicit demonstration of the equivalence between this model and a class of linear sPDEs. This shows that our model is a generalization of the well-known luminance conservation equation 4.2. This sPDE formulation has two chief advantages: it allows for a real-time synthesis using an AR recurrence and allows one to recast the log- likelihood of the model as a generalization of the classical motion energy model, which in turn is crucial to allow for Bayesian modeling of perceptual biases. Finally, we provide the source code 1 of this model of dynamic textures as an
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2014 — Discrimination de textures et quantification de rugosité par algorithme d'apprentissage

2014 — Discrimination de textures et quantification de rugosité par algorithme d'apprentissage

connaissance à 97% sur 144 images de 64x64 pixels. Picard et al. (1993) remettent en question les études trop conditionnées à des images découpées sur peu de photos car elles ne permettent pas une bonne globalisation. Ils proposent donc une étude comparant deux algorithmes : un SAR et un algorithme de reconnaissance des composantes principales. Il est important de men- tionner que ces algorithmes sont rendus peu sensibles à la rotation et aux variations d’échelles. Cette étude a été réalisée sur plus de 1 000 images venant de plus de 100 photos de la base d’image Brodatz, une des bases de photos de textures les plus utilisées. En conclusion l’algo- rithme SAR est beaucoup plus performant (91% contre 78%). De nombreuses études se sont aussi intéressées au filtre de Gabor. Ces filtres visent à être peu sensibles à l’orientation et l’échelle de l’image. Néanmoins leur théorie est peu expliquée et assez complexe. Les résul- tats sont par contre intéressants : Jain et Healey (1998) montrent l’intérêt d’utiliser une image multi-bandes. Il propose un algorithme utilisant différents descripteurs issu d’une image po- lychromatique filtrée avec un filtre de Gabor. En testant cet algorithme sur une base de plus de 2500 photos, il montre que cet algorithme a de meilleures performances que si on travaille uniquement avec une image en nuances de gris. Shi et Healey (2003) poursuivent ce travail en cherchant les descripteurs les plus efficaces. Il est intéressant de voir que ceux-ci ont beaucoup d’influence, les résultats varient de 35% à plus de 90% suivant le nombre de caractéristiques sélectionnées et du nombre de bandes spectrales. Néanmoins, ces méthodes utilisent des al- gorithmes lourds. Paschos (2000) propose une méthode beaucoup plus légère en utilisant les moments chromatiques. Cette approche consiste à calculer un indice pour chaque pixel dépen- dant des valeurs RGB de celui-ci. En faisant varier les moments, les résultats varient de 56 à 99% sur des sets de photos de marbres et granits.
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Percolation by cumulative merging and phase transition for the contact process on random graphs

Percolation by cumulative merging and phase transition for the contact process on random graphs

The motivation for introducing this partition arises from a connection with the contact process as it roughly describes the geometry of the sets where the process survives for a long time. We give a sufficient condition on a graph to ensure that the contact process has a non trivial phase transition in terms of the existence of an infinite cluster. As an application, we prove that the contact process admits a sub-critical phase on d-dimensional random geometric graphs and on random Delaunay triangulations. To the best of our knowledge, these are the first examples of graphs with unbounded degrees where the critical parameter is shown to be strictly positive.
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Partial mu-tau textures and leptogenesis

Partial mu-tau textures and leptogenesis

η M jV e3 j ðη M < 0Þ: (31) These approximations appear in Fig. 2 in the form of dotted curves, and it is apparent that they fit the numerical results extremely well. This tight prediction of the Dirac phase δ D as a function of jV e3 j (along with the prediction of an inverted spectrum) is a most important element of the ansatz as it can be easily falsified as new neutrino data and global fits further tighten the bounds on leptonic CP violation.

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A random matrix analysis of random Fourier features: beyond the Gaussian kernel, a precise phase transition, and the corresponding double descent

A random matrix analysis of random Fourier features: beyond the Gaussian kernel, a precise phase transition, and the corresponding double descent

1.1 Our Main Contributions We consider the RFF model in the more realistic large n, p, N limit. While, in this setting, the RFF empirical Gram matrix does not converge to the Gaussian kernel matrix, we can characterize its behavior as n, p, N → ∞ and provide asymptotic performance guarantees for RFF on large- scale problems. We also identify a phase transition as a function of the ratio N/n, including the corresponding double descent phenomenon. In more detail, our contributions are the following. 1.We provide a precise characterization of the asymptotics of the RFF empirical Gram matrix, in the large n, p, N limit (Theorem 1). This is accomplished by constructing a deterministic equivalent for the resolvent of the RFF Gram matrix. Based on this, the behavior of the RFF model is (asymptot- ically) accessible through a fixed-point equation, that can be interpreted in terms of an angle-like correction induced by the non-trivial large n, p, N limit (relative to the N → ∞ alone limit). 2.We derive the asymptotic training and test mean squared errors (MSEs) of RFF ridge regression, as a function of the ratio N/n, regularization penalty λ, training as well as test sets (Theorem 2 and 3, respectively). We identify precisely the under- to over-parameterization phase transition, as a function
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