Thomas Batard

A Decomposition Framework for Image Denoising Algorithms

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.

On Covariant Derivatives and Their Applications to Image Regularization

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].

Generalized Gradient on Vector Bundle – Application to Image Denoising

We introduce a gradient operator that generalizes the Euclidean and Riemannian gradients. This operator acts on sections of vector bundles and is determined by three geometric data: a Riemannian 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 regularizing 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.

Clifford Bundles: A Common Framework for Image, Vector Field, and Orthonormal Frame Field Regularization

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.

Optimizing layout using spatial quality metrics and user preferences

Abstract 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 designer’s 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 designer’s taste. The results show that the presented framework can measurably decrease planning complexity by producing layouts which exhibit characteristics of human-made design.

Local denoising based on curvature smoothing can visually outperform non-local methods on photographs with actual noise

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.

Denoising an Image by Denoising Its Components in a Moving Frame

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.

Derivatives and inverse of cascaded linear+nonlinear neural models

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).

The Wilson-Cowan model describes Contrast Response and Subjective Distortion

The Wilson-Cowan equations were originally proposed to describe the low-level dynamics of neural populations (Wilson&Cowan 1972). These equations have been extensively used in modelling the oscillations of cortical activity (Cowan et al. 2016). However, due to their low-level nature, very few works have attempted connections to higher level psychophysics (Herzog et al. 2003, Hermens et al. 2005) and, to the best of our knowledge, they have not been used to predict contrast response curves or subjective image quality. Interestingly (Bertalmío&Cowan 2009) showed that Wilson-Cowan models may lead to (high level) color constancy. Moreover, these models may have positive statistical effects similarly to Divisive Normalization, which is the canonical choice to understand contrast response (Watson&Solomon 1997, Carandini&Heeger 2012): while Divisive Normalization reduces redundancy due to predictive coding (Malo&Laparra 2010), Wilson-Cowan leads to local histogram equalization (Bertalmío 2014), another route to increase channel capacity.

Duality Principle for Image Regularization and Perceptual Color Correction Models

In this paper, we show that the anisotropic nonlocal total variation involved in the image regularization model of Gilboa and Osher [15] as well as in the perceptual color correction model of Bertalmio et al. [4] 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 Bertalmio 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.