Recent 3D image machinery delivers sequences of large scale 3D images with a considerably small sampling width in time. In medical as well as in engineering applications one is especially interested in underlying deformation, growth or motion phenomena. A robust method is presented to extract motion velocities from such image sequences. To avoid an ill-posedness of the problem one has to restrict to certain motion types, which are related to the concrete application. Differences to the general motion case are discussed. The derived formulas for the motion velocities clearly reflect the geometry of the motion. Robustness of the presented implementation is based on the local projection of the actual discrete data on suitable polynomial spaces in space-time. Required geometric quantities on the image sequences are then evaluated on the local projection. Examples outline the potential of the proposed method in medical applications (3D ultra sound sequences) and experimental fluid dynamics (3D flow in porous media).
Furthermore an effective denoising method based on anisotropic geometric diffusion for 3D data sets is discussed, which respects important features on level sets such as edges and corners and preserves them during the smoothing process. Its application as a pre-processing step turns out to be especially advisable for experimental image sequences with a considerably small signal to noise ratio. It does not destroy the essential motion feature of the data set but allows us to visualize the motion fields on properly regularized level sets in the 3D images.
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From the noisy image sequence set, obtained by echocardiography of one cardiac cycle of the human heart, we extract the motion of the level sets. On the left one noisy frame is shown, whereas on the right the result of the application of the anisotropic geometric smoothing process is depicted. * Movie: cardiac
cycle, noisy data |
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For successive frames of the whole sequence of echocardiographical data the extracted velocities are shown. A color ramp from blue (moving inward) to red (moving outward) indicates the normal component of the velocity. |
For the test case of a deforming ellipsoid, we depict the extracted velocity. In the upper row the normal component (blue=moving inward, red=moving outward) is depicted, whereas the lower row shows the tangential component of the corresponding frame in a color ramp from blue to red. |
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For one frame of the deforming ellipsoid sequence, we depict the splitting of the velocity in tangential (= red arrows) and normal (=blue arrows) component. The color coding of the level set gives the absolute value of the tangential component of the velocity. |
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From experimental fluid dynamics the salt concentration in a 3D box was measured using an MRI device. During the experiment, salt concentration flows into a box, which was originally filled with fresh water. Then an outlet on the top right is opened, such that the liquid flows out. We have extracted the motion of the isosurfaces and again color coded the normal velocity on the surfaces. |
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For one frame of the fluid dynamical experiment, we depict the tangential component of the velocity. The color coding from blue to red indicates the absolute value of the tangential velocity. We emphasize that the values are about 6 orders of magnitude less than the normal velocities. This can be seen as an indication of the underlying physical phenomenon. Here, according to Darcy's law the flow of matter is in direction of the concentration gradient (= normal to the level sets). |
