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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.

Comunicació presentada a: the 7th International Conference on Scale Space and Variational Methods in Computer Vision, celebrada del 30 de juny al 4 de juliol de 2019 a Hofgeismar, Alemanya. In this paper, we establish a connection between image processing, visual perception, and deep learning by introducing a mathematical model inspired by visual perception from which neural network layers and image processing models for color correction can be derived. Our model is inspired by the geometry of visual perception and couples a geometric model for the organization of some neurons in the visual cortex with a geometric model of color perception. More precisely, the model is a combination of a Wilson-Cowan equation describing the activity of neurons responding to edges and textures in the area V1 of the visual cortex and a Retinex model of color vision. For some particular activation functions, this yields a color correction model which processes simultaneously edges/textures, encoded into a Riemannian metric, and the color contrast, encoded into a nonlocal covariant derivative. Then, we show that the proposed model can be assimilated to a residual layer provided that the activation function is nonlinear and to a convolutional layer for a linear activation function. Finally, we show the accuracy of the model for deep learning by testing it on the MNIST dataset for digit classiffication.

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.

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.

Image denoising has been a topic extensively investigated over the last three decades and, as repeatedly shown in this book, denoising algorithms have become incredibly good, so much so that many researchers have started questioning the need to further pursue this line of research. In this chapter, we argue that there is indeed room for improvement of denoising results, and we propose three different avenues to explore, none of which requires the development of new denoising methods. First, we describe how it can be better to denoise a transform of the noisy image rather than denoise the noisy image directly. We mention several possible transforms, and an open problem is to find a transform that is optimal for denoising, according to a proper image quality metric. Next, we point out the importance of having a proper noise model for JPEG pictures, so that a variance stabilization transform can be developed that transforms noise in JPEG images into additive white Gaussian noise, enabling existing denoising methods to be properly applied to the JPEG case. Finally, we highlight the fact that while virtually all denoising methods are optimized and validated in terms of the PSNR or SSIM measures, these metrics are not well correlated with perceived image quality, and therefore, it could be best to optimize the parameter values of denoising methods according to subjective testing. A remaining challenge is to develop perceptually based image quality metrics that match observer preference.

_ Human perception involves many features like contours, shapes, textures, and colors to name a few. Whereas several geometric models for contours, shapes and textures perception have been proposed, the geometry of color perception has received very little attention, possibly due to the fact that our perception of colors is still not fully understood. Nonetheless, there exists a class of mathematical models, gathered under the name Retinex, that aim at modeling the color perception of an image, that are inspired by psychophysical/physiological knowledge about color perception, and that can geometrically be viewed as the averaging of perceptual distances between image pixels. Some of the Retinex models turn out to be associated to an efficient image processing technique for the correction of camera output images. The aim of this paper is to show that this image processing technique can be improved by including more properties of the human visual system. To that purpose, we first present a generalization of the perceptual distance between image pixels by considering the parallel transport map associated to a covariant derivative on a vector bundle, and from which can be derived a new image processing model for color images correction. Then, we show that the family of covariant derivatives constructed in [T. Batard and N. Sochen, J. Math. Imaging Vision, 48(3) (2014), pp. 517-543] can model some color appearance phenomena related to brightness perception. Finally, we conduct experiments in which we show that the image processing techniques induced by these covariant derivatives outperform the original approach. This work was supported by the European Research Council, Starting Grant ref. 306337, by the Spanish government and FEDER Fund, grant ref. TIN2015-71537-P (MINECO/FEDER,UE), and by the Icrea Academia Award.

In vision science, cascades of Linear+Nonlinear transforms are very successful in modeling a number of perceptual experiences. However, the conventional literature is usually too focused on only describing the forward input-output transform. Instead, in this work we present the mathematics of such cascades beyond the forward transform, namely the Jacobian matrices and the inverse. The fundamental reason for this analytical treatment is that it offers useful analytical insight into the psychophysics, the physiology, and the function of the visual system. For instance, we show how the trends of the sensitivity (volume of the discrimination regions) and the adaptation of the receptive fields can be identified in the expression of the Jacobian w.r.t. the stimulus. This matrix also tells us which regions of the stimulus space are encoded more efficiently in multi-information terms. The Jacobian w.r.t. the parameters shows which aspects of the model have bigger impact in the response, and hence their relative relevance. The analytic inverse implies conditions for the response and model parameters to ensure appropriate decoding. From the experimental and applied perspective, (a) the Jacobian w.r.t. the stimulus is necessary in new experimental methods based on the synthesis of visual stimuli with interesting geometrical properties, (b) the Jacobian matrices w.r.t. the parameters are convenient to learn the model from classical experiments or alternative goal optimization, and (c) the inverse is a promising model-based alternative to blind machine-learning methods for neural decoding that do not include meaningful biological information. The theory is checked by building and testing a vision model that actually follows a modular Linear+Nonlinear program. Our illustrative derivable and invertible model consists of a cascade of modules that account for brightness, contrast, energy masking, and wavelet masking. To stress the generality of this modular setting we show examples where some of the canonical Divisive Normalization modules are substituted by equivalent modules such as the Wilson-Cowan interaction model (at the V1 cortex) or a tone-mapping model (at the retina)., This work was partially funded by the Spanish Ministerio de Economia y Competitividad projects CICYT TEC2013-50520-EXP and CICYT BFU2014-59776-R, by the European Research Council, Starting Grant ref. 306337, by the Spanish government and FEDER Fund, grant ref. TIN2015-71537-P(MINECO/FEDER,UE), 1021, and by the ICREA Academia Award.

In the context of fibre bundles theory, there exist some differential operators of order 2, called generalized Laplacians, acting on sections of vector bundles over Riemannian manifolds, and generalizing the Laplace-Beltrami operator. Such operators are determined by covariant derivatives on vector bundles. In this paper, we construct a class of generalized Laplacians, devoted to multi-channel image processing, from the construction of optimal covariant derivatives. The key idea is to consider an image as a section of an associate bundle, that is a vector bundle related to a principal bundle through a group representation. In this context, covariant derivatives are determined by connection 1-forms on principal bundles. We construct optimal connection 1-forms by the minimization of a variational problem on principal bundles. From the heat equations of the generalized Laplacians induced by the corresponding optimal covariant derivatives, we obtain diffusions whose behaviors depend of the choice of the group representation. We provide experiments on color images.

We use the framework of heat equations associated to generalized Laplacians on vector bundles over Riemannian manifolds in order to regularize color images. We show that most methods devoted to image regularization may be considered in this framework. From a geometric viewpoint, they differ by the metric of the base manifold and the connection of the vector bundle involved. By the regularization operator we propose in this paper, the diffusion process is completely determined by the geometry of the vector bundle. More precisely, the metric of the base manifold determines the anisotropy of the diffusion through the computation of geodesic distances whereas the connection determines the data regularized by the diffusion process through the computation of the parallel transport maps. This regularization operator generalizes the ones based on short-time Beltrami kernel and oriented Gaussian kernels. Then we construct particular connections and metrics involving color information such as luminance and chrominance in order to perform new kinds of regularization.

We present a new theoretical framework for multidimensional image processing using Clifford algebras. The aim of the paper is to detect edges by computing the first fundamental form of a surface associated to an image. We propose to construct this metric in the Clifford bundles setting. A nD image, i.e. an image of dimension n, is considered as a section of a trivial Clifford bundle $(CT(D),\widetilde{\pi},D)$ over the domain $D$ of the image and with fiber $Cl(\mathbb{R}^n,\|\|_2)$. Due to the triviality, any connection $\nabla_1$ on the given bundle is the sum of the trivial connection $\widetilde{\nabla}_0$ with $\omega$, a 1-form on $D$ with values in $End(CT(D))$. We show that varying $\omega$ and derivating well-chosen sections with respect to $\nabla_1$ provides all the information needed to perform various kind of segmentation. We present several illustrations of our results, dealing with color (n=3) and color/infrared (n=4) images. As an example, let us mention the problem of detecting regions of a given color with constraints on temperature; the segmentation results from the computation of $\nabla_1(I)=\widetilde{\nabla}_0(I)+ (dx + dy) \otimes \mu \, I$, where $I$ is the image section and $\mu$ is a vector section coding the given color.

The aim of this paper is to present a new framework for regularization by diffusion. The methods we develop in what follows can be used to smooth multichannel images, multichannel image sequences (videos), vector fields, and orthonormal frame fields in any dimension. From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and so-called generalized Laplacians. Sections are regularized from heat equations associated with generalized Laplacians, the solutions being approximated by convolutions with kernels. Then, the behavior of the diffusion is determined by the geometry of the vector bundle, i.e., by the metric of the base manifold and by a connection on the vector bundle. For instance, the heat equation associated with the Laplace-Beltrami operator can be considered from this point of view for applications to images and video regularization. The main topic of this paper is to show that this approach can be extended in several ways to vector fields and orthonormal frame fields by considering the context of Clifford algebras. We introduce Clifford-Beltrami and Clifford-Hodge operators as generalized Laplacians on Clifford bundles over Riemannian manifolds. Laplace-Beltrami diffusion appears as a particular case of diffusion for degree 0 sections (functions). Dealing with base manifolds of dimension 2, applications to multichannel image, two-dimensional vector field, and orientation field regularization are presented.

We extend two existing variational models from the Euclidean space to a vector bundle over a Riemannian manifold. The Euclidean models, dedicated to regularize or enhance some color image features, are based on the concept of nonlocal gradient operator acting on a function of the Euclidean space. We then extend these models by generalizing this operator to a vector bundle over a Riemannian manifold with the help of the parallel transport map associated to some class of covariant derivatives. Through the dual formulations of the proposed models, we obtain the expressions of their solutions, which exhibit the functional spaces that describe the image features. Finally, for a well-chosen covariant derivative and its nonlocal extension, the proposed models perform local contrast modification (reduction or enhancement) and experiments show that they preserve more the aspect of the original image than the Euclidean models do while modifying equally its contrast. This work was supported by the European Research Council, Starting Grant ref. 306337, by the Spanish government, grant ref. TIN2012-38112, and by the Icrea Academia Award.

Comunicació presentada a: IEEE International Conference on Image Processing 2016, celebrada del 25 l 28 de setembre de 2016 a Phoenix, Estats Units d'Amèrica. We propose a fast, local denoising method where the Euclidean curvature of the noisy image is approximated in a regularizing manner and a clean image is reconstructed from this smoothed curvature. User preference tests show that when denoising real photographs with actual noise our method produces results with the same visual quality as the more sophisticated, nonlocal algorithms Non-local Means and BM3D, but at a fraction of their computational cost. These tests also highlight the limitations of objective image quality metrics like PSNR and SSIM, which correlate poorly with user preference. This work was supported by the European Research Council, Starting Grant ref. 306337, by the Spanish government, grant ref. TIN2015-71537-P, and by the Icrea Academia Award.

Comunicació presentada al IS&T International Symposium on Electronic Imaging, celebrat del 14 al 18 de febrer de 2016 a San Francisco (CA, USA) i organitzat per la Society for Imaging Science and Technology. State-of-the-art denoising methods achieve impressive results, even for large noise levels. However, they can not be implemented in camera hardware, mainly due to the fact that they are computationally too intensive. The aim of this paper is then to show that we can obtain comparable denoising results to the ones obtained with state-of-art methods by inserting a well-chosen fast denoising method at the right location in the camera processing pipeline. We evaluate our results visually and with respect to objective measures. This work was supported by the European Research Council, Starting Grant ref. 306337, by the Spanish government, grant ref. TIN2012-38112, and by the Icrea Academia Award.

Comunicació presentada a la 5th International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2015), celebrada a Lège-Cap Ferret (França) del dia 31 de maig al 4 de juny de 2015. In this paper, we show that the anisotropic nonlocal total variation involved in the image regularization model of Gilboa and Osher as well as in the perceptual color correction model of Bertalmío et al. possesses a dual formulation. We then obtain novel formulations of their solutions, which provide new insights on these models. In particular, we show that the model of Bertalmío et al. can be split into two steps: first, it performs global color constancy, then local contrast enhancement. We also extend these two channel-wise variational models in a vectorial way by extending the anisotropic nonlocal total variation to vector-valued functions. This work was supported by the European Research Council, Starting Grant ref. 306337, by the Spanish government, grant ref. TIN2012-38112, and by the Icrea Academia Award.

Due to technical limitations, common display devices can only reproduce images having a low range of intensity values (dynamic range). As a consequence, the dynamic range of images encoding real world scenes, which is large, has to be compressed in order for them to be reproduced on a common display, and this technique is called tone mapping. Because there is no ground truth to compare with, evaluation of a tone mapped image has to be done by comparing with the original high dynamic range image. As standard metrics based on pixel-wise comparisons are not suitable for comparing images of different dynamic range, non local perceptual based metrics are commonly used. We propose a general method for optimizing tone mapped images with respect to a given non local metric. In particular, if the metric is perceptual, i.e. it involves perceptual concepts, we provide an adequate minimization strategy. Experiments on a particular perceptual metric tested with different tone mapped images provided by several tone mapping operators validate our approach. This work was supported by European Research Council, Starting Grant ref. 306337, and by Spanish grants ref. TIN2011-15954-E and ref. TIN2012-38112.

Comunicació presentada a: 6th Pacific-Rim Symposium on Image and Video Technology, celebrat del 28 d'octubre a 1 de novembre de 2013 a Guanajuato, Mèxic. Given any metric that compares images of di erent dynamic range, we propose a method to reduce their distance with respect to this metric. The key idea is to consider the metric as a non local operator. Then, we transform the problem of distance reduction into a non local variational problem. In this context, the low dynamic range image having the smallest distance with a given high dynamic range is the minimum of a suitable energy, and can be reached through a gradient descent algorithm. Dealing with an appropriate metric, we present an application to Tone Mapping Operator (TMO) optimization. We apply our gradient descent algorithm, where the initial conditions are Tone Mapped (TM) images. Experiments show that our algorithm does reduce the distance of the TM images with the high dynamic range source images, meaning that our method improves the corresponding TMOs. This work was supported by the European Research Council, Starting Grant ref. 306337, and by Spanish grants ref. TIN2011-15954-E and ref. TIN2012-38112.

Comunicació presentada a: 6th Pacific-Rim Symposium on Image and Video Technology Workshop, celebrat del 28-29 d'octubre de 2013 a Guanajuato, Mèxic. We present a method to perform histogram matching be- tween two color images based on the concept of harmonic mapping be- tween Riemannian manifolds. The key idea is to associate the histogram of a color image to a Riemannian manifold. In this context, the energy of the matching between the two images is measured by the Dirichlet energy of the mapping between the Riemannian manifolds. Then, we assimilate optimal matchings to critical points of the Dirichlet energy. Such points are called harmonic maps. As there is no explicit expression for harmonic maps in general, we use a gradient descent ow with boundary condition to reach them, that we call harmonic ow. We present an application to color transfer, however many others applications can be envisaged using this general framework. This work was supported by European Research Council, Starting Grant ref. 306337. The second author acknowledges partial support by Spanish grants AACC, ref. TIN2011-15954-E, and Plan Nacional, ref. TIN2012-38112.

Comunicació presentada a: the 6th International Conference, (ICISP 2014), celebrada del 30. de juny al 2 de juliol de 2014 a Cherbourg, França. In this paper, we provide a new non-local method for image denoising. The key idea we develop is to denoise the components of the image in a well-chosen moving frame instead of the image itself.We prove the relevance of our approach by showing that the PSNR of a grayscale noisy image is lower than the PSNR of its components. Experiments show that applying the Non Local Means algorithm of Buades et al. [5] on the components provides better results than applying it directly on the image. This work was supported by European Research Council, Starting Grant ref. 306337, and by Spanish grants AACC, ref. TIN2011-15954-E, and Plan Na- cional, ref. TIN2012-38112. S. Levine acknowledges partial support by NSF-DMS #0915219.

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.

We propose in this chapter to introduce a spinor representation for images based on the work of T. Friedrich. This spinor representation generalizes the usual Weierstrass representation of minimal surfaces (i.e., surfaces with constant mean curvature equal to zero) to arbitrary surfaces (immersed in \(\mathbb{R}^3\) ). We investigate applications to image processing focusing on segmentation and Clifford–Fourier analysis. All these applications involve sections of the spinor bundle of image graphs, that is spinor fields, satisfying the so-called Dirac equation.

We introduce a new Riemannian Fourier transform for color image processing. The construction involves spin characters and spin representations of complex Clifford algebras. Examples of applications to low-pass filtering are presented.

Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamiltons quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts. This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp. It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

Comunicació presentada a: 4th International Conference, (SSVM 2013, celebrada del 2 al 6 de juny de 2013 a Leibnitz, Àustria. We introduce a gradient operator that generalizes the Eu- clidean and Riemannian gradients. This operator acts on sections of vector bundles and is determined by three geometric data: a Rieman- nian metric on the base manifold, a Riemannian metric and a covariant derivative on the vector bundle. Under the assumption that the covariant derivative is compatible with the metric of the vector bundle, we consider the problems of minimizing the L2 and L1 norms of the gradient. In the L2 case, the gradient descent for reaching the solutions is a heat equation of a differential operator of order two called connection Laplacian. We present an application to color image denoising by replacing the regular- izing term in the Rudin-Osher-Fatemi (ROF) denoising model by the L1 norm of a generalized gradient associated with a well-chosen covariant derivative. Experiments are validated by computations of the PSNR and Q-index.

We propose a new mathematical model for color images taking into account that color pixels change under transformation of the light source. For this, we deal with (ρ,G)-equivariant functions on principal bundles, where ρ is a representation of a Lie group G on the color space RGB. We present an application to image regularization, by minimization of the Polyakov action associated to the graph of such functions. We test the groups ${\rm I\!R}^{+\ast}$ , DC(3) of contractions and dilatations of ${\rm I\!R}^3$ and SO(3) with their natural matrix representations, as well as ${\rm I\!R}^{+\ast}$ with its trivial representation. We show that the regularization has denoising properties if the representation is unitary and segmentation properties otherwise.

28 pages, chapter 8.; The aim of this paper is to define a Clifford Fourier transform that is suitable for color image spectral analysis. There have been many attempts to define such a transformation using quaternions or Clifford algebras. We focus here on a geometric approach using group actions. The idea is to generalize the usual definition based on the characters of abelian groups by considering group morphisms from $R^2$ to spinor groups Spin(3) and Spin(4). The transformation we propose is parameterized by a bivector and a quadratic form, the choice of which is related to the application to be treated. A general definition for 4D signal defined on the plane is also given; for particular choices of spinors it coincides with the definitions of S.Sangwine and T. Bülow.

The aim of this paper is to define a Clifford–Fourier transform that is suitable for color image spectral analysis. There have been many attempts to define such a transformation using quaternions or Clifford algebras. We focus here on a geometric approach using group actions. The idea is to generalize the usual definition based on the characters of abelian groups by considering group morphisms from ℝ2 to spinor groups Spin(3) and Spin(4). The transformation we propose is parameterized by a bivector and a quadratic form, the choice of which is related to the application to be treated. A general definition for 4D signal defined on the plane is also given; for particular choices of spinors, it coincides with the definitions of S. Sangwine and T. Bulow.

17 pages; International audience; The aim of this paper is to perform edge detection in color-infrared images from the point of view of Clifford algebras. The main idea is that such an image can be seen as a section of a Clifford bundle associated to the RGBT-space (Red, Green, Blue, Temperature) of acquisition. Dealing with geometric calculus and covariant derivatives of appropriate sections with respect to well-chosen connections allows to get various color and temperature information needed for the segmentation. We show in particular how to recover the first fundamental form of the image embedded in a LSHT-space (Luminance, Saturation , Hue, Temperature) equipped with a metric tensor. We propose applications to color edge detection with respect to a given hue interval and to edge detection in color-infrared images with constraints on temperature. Others applications related to different choices of connections, sections and embedding spaces for nD images may be considered from this general theorical framework.

In this paper, we make use of the theory of Clifford algebras for anisotropic smoothing of vector-valued data. It provides a common framework to smooth functions, tangent vector fields and mappings taking values in $\mathfrak{so}(m)$, the Lie algebra of SO(m), defined on surfaces and more generally on Riemannian manifolds. Smoothing process arises from a convolution with a kernel associated to a second order differential operator: the Hodge Laplacian. It generalizes the Beltrami flow in the sense that the Laplace-Beltrami operator is the restriction to functions of minus the Hodge operator. We obtain a common framework for anisotropic smoothing of images, vector fields and oriented orthonormal frame fields defined on the charts.

6 pages; International audience; In many fields, e.g. decision-making, numerical values in [0,1] are available and one is often interested in detecting which are similar. In this paper, we propose an operator which is able to detect whether some values can be gathered by blocks with respect to their similarity or not. It combines the values and a kernel function using triangular norms and Sugeno integrals. This operator allows to estimate this blockwise similarity at different levels. For illustration purpose, we use it to define an index suitable for the cluster validity problem in pattern recognition

Computational design is one of the most common tasks of immersive computer graphics projects, such as games, virtual reality and special effects. Layout planning is a challenging phase of architectural design, which requires optimization across several conflicting criteria. We present an interactive layout solver that assists designers in layout planning by recommending personalized space arrangements based on architectural guidelines and user preferences. Initialized by the designers high-level requirements, an interactive evolutionary algorithm is used to converge on an ideal layout by exploring the space of potential solutions. The major contributions of our proposed approach are addressing subjective aspects of the design to generate personalized layouts; and the development of a genetic algorithm with a multi-parental recombination method that improves the chance of generating higher quality offspring. We demonstrate the ability of our method to generate feasible floor plans which are satisfactory, based on spatial quality metrics and designers taste. The results show that the presented framework can measurably decrease planning complexity by producing layouts which exhibit characteristics of human-made design.

In this paper, we consider an image decomposition model that provides a novel framework for image denoising. The model computes the components of the image to be processed in a moving frame that encodes its local geometry (directions of gradients and level lines). Then, the strategy we develop is to denoise the components of the image in the moving frame in order to preserve its local geometry, which would have been more affected if processing the image directly. Experiments on a whole image database tested with several denoising methods show that this framework can provide better results than denoising the image directly, both in terms of Peak signal-to-noise ratio and Structural similarity index metrics. The work of G. Ghimpeteanu, T. Batard, and/nM. Bertalmío was supported in part by the Spanish Government under/nGrant TIN2012-38112, in part by the Icrea Academia Award, and in/npart by the European Research Council under Grant 306337. The work/nof S. Levine was supported by the Na/ntional Science Foundation under/nGrant NSF-DMS 1320829.

We present a generalization of the Euclidean and Riemannian gradient operators to a vector bundle, a geometric structure generalizing the concept of a manifold. One of the key ideas is to replace the standard differentiation of a function by the covariant differentiation of a section. Dealing with covariant derivatives satisfying the property of compatibility with vector bundle metrics, we construct generalizations of existing mathematical models for image regularization that involve the Euclidean gradient operator, namely, the linear scale-space and the Rudin--Osher--Fatemi denoising models. For well-chosen covariant derivatives, we show that our denoising model outperforms state-of-the-art variational denoising methods of the same type both in terms of peak signal-to-noise ratio (PSNR) and Q-index [Z. Wang and A. Bovik, IEEE Signal Process. Lett., 9 (2002), pp. 81--84].