Nonlinear Multiscale Analysis Models for Filtering of 3D + Time Biomedical Images
We review nonlinear partial differential equations (PDEs) in the processing of 2D and 3D images. At the same time we present recent models introduced for processing of space-time image sequences and apply them to 3D echocardiography. The nonlinear (degene
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DEIS, University of Bologna, Italy Department of Mathematics, Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovakia Department of Mathematics, University of Bologna, Italy
Abstract. We review nonlinear partial differential equations (PDEs) in the processing of 2D and 3D images. At the same time we present recent models introduced for processing of space-time image sequences and apply them to 3D echocardiography. The nonlinear (degenerate) diffusion equations filter the sequence with a keeping of space-time coherent structures. They have been developed using ideas of regularized Perona-Malik anisotropic diffusion and geometrical diffusion of mean curvature flow type, combined with Galilean invariant movie multiscale analysis of Alvarez, Guichard, Lions and Morel. A discretization of space-time filtering equations is discussed. Computational results in processing of 3D echocardiographic sequences obtained by rotational acquisition technique and by Real-Time 3D Echo Volumetrics aquisition technique are presented.
7.1
Introduction
The aim of this contribution is to present mathematical models, numerical methods and computational results in processing of three-dimensional (3D) image sequences. We apply the proposed models and methods to 3D echocardiography. The models which we use for space-time filtering are based on partial differential equations (PDEs) approach, namely PDEs of degenerate diffusion type are applied to initially given image sequence. Since the images are given on discrete grids, the nonlinear PDEs are discretized by semi-implicit finite volume method in order to get fast and stable solution. Two-dimensional (2D) echocardiography is an imaging modality frequently used in cardiology due to its simplicity, lack of ionizing radiation and a relative low cost. However, 2D echocardiography allows visualization of only tomographic planar sections of the heart; thus to obtain a complete evaluation of the heart anatomy and function, the physician must reassemble mentally a 3D model from multiple two-dimensional images. Moreover, 2D echocardiography relies on geometrical assumptions for the determination of heart chamber volumes and thus presents a considerable measurement error. 3D echocardiography may avoid the need for geometrical assumptions, thereby allowing accurate evaluation of chambers size and shape, even in the case of cavities with irregular or distorted geometry. The correct visualization R. Malladi (ed.), Geometric Methods in Bio-Medical Image Processing © Springer-Verlag Berlin Heidelberg 2002
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Sarti, Mikula, Sgallari, Lamberti
and interpretation of 3D echo images is often affected by the high amount of noise intrinsically linked to the acquisition method. It is absolutely necessary to submit the data to pre-processing in order to improve their legibility from a clinical point of view. The pre-processing algorithm should be able to distinguish the noise from the contours of the different cardiac structures by using both spatial and temporal coherence. In this paper we use two t
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