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# Computer Vision, Imaging and Computer Graphics

## Team and Interlocutors

P. C. Carvalho, L. Velho, J. P. Zubelli, R. Beauclair.

A. Grunbaum (University of California at Berkeley), D. Mumford, (Harvard & Brown Universities), G. Papanicolaou (Stanford University).

## Vision

The main goal in Computer Vision is determining characteristics of the objects present in a given image. A large variety of problems is obtained according to the nature of the images and of the information to be extracted. Classically, an image is a map from region of the plane into a color space, corresponding to a mathematical model of the light that reaches the retina. However, more general models are considered, in such a way as to include results produced by a variety of sensors (for instance, the three-dimensional images representing tissue densities produced by X-ray tomography).

The development of algorithms capable of extracting information from images is strongly influenced by the understanding of how images are acquired and perceived in the visual system of men and other animals. The classical model of Marr, for instance, includes "vision modules" that range from "low-level processing" (image acquisition by the eye, filtering, edge detection, clustering) to "shape detection" (using information from texture, motion, shading and stereo vision) and, finally, to "high-level processing" (forming objects and recognizing them by comparison with prototypes).

One of such discoveries is the fact that animals process visual information in several scales simultaneously, even at the low processing levels ? in other words, natural vision is capable of obtaining characteristics of large structures in a image ignoring distractions produced by its details. This property is essential to the robustness of any visual process (this is what allows us to understand noisy images, mosaics and even television images). A mathematical model that deals with these features is provided by the theory of Scale Spaces. Actually, Scale Spaces are the foundation for an entire researc line of Partial Differential Equations and Harmonic Analysis applied to Image Processing, which is extremely active and includes topics such as snakes, wavelets and level set methods. In this project, this line of research is represented by Luiz Velho and his collaborators Ralph Teixeira e Marcos Craizer.

On the other hand, at the higher visual levels we have the process of visual perception, whereby the acquired images are compared to images and models already stored in the memory, in order to recognize patterns observed in the past. A fundamental aspect of this field is constructing probabilistic models for the vision variables (the images observed; the shape, positioning, texture and illumination of objects; etc) and obtaining samples and estimates from such models. This line of research is represented in this project by David Mumford, responsible for important contributions in the fields Algebraic Geometry and Computer Vision and recipient of the Fields Medal.

## Medical Imaging and Inverse Problems

A few projects that are currently being developed in this general area are the following:

Tomography in the presence of scattering and diffusion: In this project, Jorge P. Zubelli, in collaboration with F. A. Grunbaum at Berkeley, is developing methods to tackle the extremely difficult and computationally intensive problem of reconstructing the interior of objects and bodies from scattered and diffused radiation. A key application of such techniques concerns the possibility of using near infra-red radiation as a diagnosing tool. Grunbaum and Zubelli already hold a patent (jointly with J. R. Singer) on the subject and currently many groups are working on the problem.

The development of fast algorithms for handling the multitude of important inverse problems that appear in petrol exploration, including seismic imaging is crucial for the state of the art of oil prospection. Zubelli is currently studying inverse problems related to wave propagation in stratified media. Some of these problems have direct connections with optical diffuse tomography problems, due to the fact that in some approximations the inverse problem for wave propagation naturally leads to the linear Boltzmann transport equation. In this area Zubelli has been discussing with G. Papanicolaou (Stanford) in the recent possibilities that were unveiled by the latter's on wave propagation in stochastic media.

The massive data analysis necessary for handling and extracting features from different biological experiments can only be dealt with by means of effective computational and mathematical techniques. One area, which has received recent attention is that of extracting features from DNA arrays through the use singular value decomposition (SVD) techniques. More precisely in genome-wide expression data processing and modeling. The possibilities here are multi-fold, especially due to the possibility of using wavelet techniques as well as recent generalizations of the SVD, such as the ones put forward by D. Donoho (Stanford).

Finally, another aspect considered in this project is the extraction of anatomic information from three-dimensional medical data (CT, MRI, ultra-sound, etc). More especifically, in collaboration with the Medical School at the State University of Rio de Janeiro, we tackle the problem of reconstructing lung structures from CT images, in order to detect, measure and analyze nodules. In particular, we want to determine the growth rate of the tumor and whether surgery is indicated. Preliminary results regarding this subproject, under the responsibility of P. C. P. Carvalho.