Carlos Brito-Loeza

Multigrid Algorithm for High Order Denoising

Image denoising has been a research topic deeply investigated within the last two decades. Excellent results have been obtained by using such models as the total variation (TV) minimization by Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259-268], which involves solving a second order PDE. In more recent years some effort has been made [Y.-L. You and M. Kaveh, IEEE Trans. Image Process., 9 (2000), pp. 1723-1730; M. Lysaker, S. Osher, and X.-C. Tai, IEEE Trans. Image Process., 13 (2004), pp. 1345-1357; M. Lysaker, A. Lundervold, and X.-C. Tai, IEEE Trans. Image Process., 12 (2003), pp. 1579-1590; Y. Chen, S. Levine, and M. Rao, SIAM J. Appl. Math., 66 (2006), pp. 1383-1406] in improving these results by using higher order models, particularly to avoid the staircase effect inherent to the solution of the TV model. However, the construction of stable numerical schemes for the resulting PDEs arising from the minimization of such high order models has proved to be very difficult due to high nonlinearity and stiffness. In this paper, we study a curvature-based energy minimizing model [W. Zhu and T. F. Chan, Image Denoising Using Mean Curvature, preprint, http://www.math.nyu.edu/ wzhu/], for which one has to solve a fourth order PDE. For this model we develop two new algorithms: a stabilized fixed point method and, based upon this, an efficient nonlinear multigrid (MG) algorithm. We will show numerical experiments to demonstrate the very good performance of our MG algorithm.

On High-Order Denoising Models and Fast Algorithms for Vector-Valued Images

Variational techniques for gray-scale image denoising have been deeply investigated for many years; however, little research has been done for the vector-valued denoising case and the very few existent works are all based on total-variation regularization. It is known that total-variation models for denoising gray-scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector-valued models. High-order models, on the contrary, do not present staircasing. In this paper, we introduce three high-order and curvature-based denoising models for vector-valued images. Their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided. AMS subject classifications: 68U10, 65F10, 65K10.

A Fourth-Order Variational Image Registration Model and Its Fast Multigrid Algorithm

Several partial differential equations (PDEs) based variational methods can be used for deformable image registration, mainly differing in how regularization for deformation fields is imposed [J. Modersitzki, Numerical Methods for Image Restoration, Oxford University Press, Oxford, 2004]. On one hand for smooth problems, models of elastic-, diffusion-, and fluid-image registration are known to generate globally smooth and satisfactory deformation fields. On the other hand for nonsmooth problems, models based on the total variation (TV) regularization are better for preserving discontinuities of the deformation fields. It is a challenge to design a deformation model suitable for both smooth and nonsmooth deformation problems. One promising model that is based on a curvature type regularizer and appears to deliver excellent results for both problems is proposed and studied in this paper. A related work due to B. Fischer and J. Modersitzki [J. Math. Imaging Vision, 18 (2003), pp. 81–85] and then refined by S...

A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation

Variational image restoration models for both additive and multiplicative noise (MN) removal are rarely encountered in the literature. This paper proposes a new variational model and a fast algorithm for its numerical approximation to remove independent additive and MN from digital images. Two previous works by L. Rudin, S. Osher, and E. Fatemi [Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), pp. 259–268] and Z. Jin and X. Yang [Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl. 362 (2010), pp. 415–426] are used to develop the new model. As a result, developing a fast numerical algorithm is difficult because the associated Euler–Lagrange equation is highly nonlinear and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient nonlinear multigrid approach via a robust fixed-point smoother. Numerical tests using both synthetic and realistic images not only confirm that our new model delivers quality results but also that the proposed numerical algorithm allows a very fast numerical realization of the model.

Total variation regularization cost function for demodulating phase discontinuities

We introduce a method based on the minimization of a total variation regularization cost function for computing discontinuous phase maps from fringe patterns. The performance of the method is demonstrated by numerical experiments with both synthetic and real data.

Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing

A number of successful variational models for processing planar images have recently been generalized to three-dimensional (3D) surface processing. With this new dimensionality, the amount of numerical computations to solve the minimization of such new 3D formulations naturally grows up dramatically. Though the need of computationally fast and efficient numerical algorithms able to process high resolution surfaces is high, much less work has been done in this area. Recently, a two-step algorithm for the fast solution of the total curvature model was introduced in Tasdizen, Whitaker, Burchard and Osher [Geometric surface processing via normal maps, ACM Trans. Graph. 22(4) (2003), pp. 1012–1033]. In this paper, we generalize and modify this algorithm to the solution of analogues of the mean curvature model of Droske and Martin Rumpf [A level set formulation for Willmore flow, Interfaces Free Bound. 6(3) (2004), pp. 361–378] and the Gaussian curvature model of Elsey and Esedoḡlu [Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul. 7(4) (2009), pp. 1549–1573]. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.

Chagas Parasite Detection in Blood Images Using AdaBoost

The Chagas disease is a potentially life-threatening illness caused by the protozoan parasite, Trypanosoma cruzi. Visual detection of such parasite through microscopic inspection is a tedious and time-consuming task. In this paper, we provide an AdaBoost learning solution to the task of Chagas parasite detection in blood images. We give details of the algorithm and our experimental setup. With this method, we get 100% and 93.25% of sensitivity and specificity, respectively. A ROC comparison with the method most commonly used for the detection of malaria parasites based on support vector machines (SVM) is also provided. Our experimental work shows mainly two things: (1) Chagas parasites can be detected automatically using machine learning methods with high accuracy and (2) AdaBoost + SVM provides better overall detection performance than AdaBoost or SVMs alone. Such results are the best ones known so far for the problem of automatic detection of Chagas parasites through the use of machine learning, computer vision, and image processing methods.

An automatic algorithm for the detection of Trypanosoma cruzi parasites in blood sample images

Chagas disease is a tropical parasitic disease caused by the flagellate protozoan Trypanosoma cruzi (T. cruzi) and currently affecting large portions of the Americas. One of the standard laboratory methods to determine the presence of the parasite is by direct visualization in blood smears stained with some colorant. This method is time-consuming, requires trained microscopists and is prone to human mistakes. In this article we propose a novel algorithm for the automatic detection of T. cruzi parasites, in microscope digital images obtained from peripheral blood smears treated with Wrights stain. Our algorithm achieved a sensitivity of 0.98 and specificity of 0.85 when evaluated against a dataset of 120 test images. Experimental results show the versatility of the method for parasitemia determination.

Variational phase recovering without phase unwrapping in phase-shifting interferometry

ABSTRACT We present a variational method for recovering the phase term from the information obtained from phase-shifting methods. First we introduce the new method based on a variational approach and then describe the numerical solution of the proposed cost function, which results in a simple algorithm. Numerical experiments with both synthetic and real fringe patterns show the accuracy and simplicity of the resulting algorithm.