Mathematical Modeling of Textures: Application to Color Image Decomposition with a Projected Gradient Algorithm

International audienceIn this paper, we are interested in color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling

Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise

Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising

Functional form of motion priors in human motion perception

It has been speculated that the human motion system combines noisy measurements with prior expectations in an optimal, or rational, manner. The basic goal of our work is to discover experimentally which prior distribution is used. More specifically, we seek to infer the functional form of the motion prior from the performance of human subjects on motion estimation tasks. We restricted ourselves to priors which combine three terms for motion slowness, first-order smoothness, and second-order smoothness. We focused on two functional forms for prior distributions: L2-norm and L1-norm regularization corresponding to the Gaussian and Laplace distributions respectively. In our first experimental session we estimate the weights of the three terms for each functional form to maximize the fit to human performance. We then measured human performance for motion tasks and found that we obtained better fit for the L1-norm (Laplace) than for the L2-norm (Gaussian). We note that the L1-norm is also a better fit to the statistics of motion in natural environments. In addition, we found large weights for the second-order smoothness term, indicating the importance of high-order smoothness compared to slowness and lower-order smoothness. To validate our results further, we used the best fit models using the L1-norm to predict human performance in a second session with different experimental setups. Our results showed excellent agreement between human performance and model prediction – ranging from 3% to 8% for five human subjects over ten experimental conditions – and give further support that the human visual system uses an L1-norm (Laplace) prior

Some variational problems from image processing

"Vegeu el resum a l'inici del document del fitxer adjunt"

Image Restoration Using One-Dimensional Sobolev Norm Profiles of Noise and Texture

This work is devoted to image restoration (denoising and deblurring) by variational models. As in our prior work [Inverse Probl. Imaging, 3 (2009), pp. 43-68], the image (f) over tilde to be restored is assumed to be the sum of a cartoon component u (a function of bounded variation) and a texture component v (an oscillatory function in a Sobolev space with negative degree of differentiability). In order to separate noise from texture in a blurred noisy textured image, we need to collect some information that helps distinguish noise, especially Gaussian noise, from texture. We know that homogeneous Sobolev spaces of negative differentiability help capture oscillations in images very well; however, these spaces do not directly provide clear distinction between texture and noise, which is also highly oscillatory, especially when the blurring effect is noticeable. Here, we propose a new method for distinguishing noise from texture by considering a family of Sobolev norms corresponding to noise and texture. It turns out that the two Sobolev norm profiles for texture and noise are different, and this enables us to better separate noise from texture during the deblurring process.open0

Multiphase Object Detection and Image

This paper presents a class of techniques for object detection and image segmentation, using variational models formulated in a level set approach