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Geodesic Distance-Weighted Shape Vector Image Diffusion

Problem Definition

  • To date, 3D surface matching and data visualization is still a very challenging research problem in many visual data processing and analysis fields. This is partly because surfaces may have highly flexible freeform shape characteristics which are difficult to be captured and used for matching and registration purposes, especially under noisy conditions. One of the fundamental issues in surface matching is the shape representation scheme. According to the conformal geometry theory, each 3D shape can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism. These maps are stable, insensitive to resolution changes and robust to occlusion and noises. Therefore, accurate and efficient 3D shape analysis may be achieved through 2D image analysis. Along this direction, we proposed a novel and efficient surface matching and visualization framework based on the geodesic distance-weighted shape vector image diffusion.


  • Firstly, our framework conformally maps a to-be-analyzed surface to a canonical 2D domain. The surface curvatures and conformal factors are then interpolated and encoded into the rectangular 2D domain, which we call shape vector image in this work. As the surface curvatures and conformal factors can uniquely define the surface, the vector image composed by curvature and conformal factors can serve as the shape signature. In the shape vector image, the distances between sampling pixels are the actual geodesic distances on the manifold. Since the mapping is independent of the mesh resolution, the resulting shape vector image is robust to different samplings of the surface. In order to extract the most robust and salient features to abstract the shape vector image, we propose to create a multiscale vector-valued diffusion space through our novel geodesic distance-weighted shape vector image diffusion. As a result, analysis of the shape vector image in its diffusion space is similar to the direct diffusion analysis of the 3D model. A valuable point here is that our computation is executed in a regular 2D domain, which is much simpler than in the 3D domain.

  • In the diffusion space, we can then extract distinctive features used for matching and analysis. A rich set of scale-aware features can be extracted from the diffusion space representation. Similar to the feature extraction technique in [10], our approach detects the extrema across the scales as keypoints. We then calculate the orientation histograms around the keypoints as feature descriptors, which provide distinctive bases for representing the 3D geometry of the original shape. These scale-aware geometric features can directly be used for robust matching and registration against the noises and distortions. Therefore, statistical analysis and visualization of surface properties across subjects become readily available. This is important for many real-world applications. For example, it is very useful for processing inter-subject brain surfaces from medical scans of different subjects since these surfaces exhibit the inherited physiological variances among subjects. We have conducted extensive experiments on scanned real-world surface models and real 3D human neocortical surfaces, through which we demonstrate the excellent performance of our approach in surface matching and registration, statistical analysis, and integrated visualization of the multimodality volumetric data over the shape vector image.


  • The following figure shows shape vector image. Figure (a) shows the Igea (5002 vertices) surface and mesh; figures (b) and (c) show the mean curvature channel and conformal factor channel of the shape vector image representation of the Igea model; and figure (d) is the composite shape vector image including both channels.
  • The following figure shows the diffused shape vector images, consisting both curvature and conformal factor channels, of the Igea model at different diffusion scales, t, computed by the geodesic distance-weighted diffusion.
  • The following sub-figures (a)-(m) are keypoint detection in the diffusion space. Figure(a): The Igea model with all the detected keypoints at different scales indicated by the points of different colors and sizes. Figure(b): All the detected keypoints shown on the curvature channel of the shape vector image. Figures(c-e), (g-i) and (k-m) show the intermediate curvature channel images of the DoDs across scales t and the detected extrema (shown by points) on the rresponding DoDs at different scales. Figures(f), (j) and (n) show again the extrema detected at figures(e), (i) and (m), respectively, with the different sizes of circles indicating the sizes of scales at which these extrema are detected.
  • The following figure illustrates repeatability of keypoint features when the Igea model is under different Gaussian noise levels. The left panel shows the Igea models (with the computed curvature colormaps) with 4% and 10% additive Gaussian noise and their corresponding shape vector images. The detected keypoints are shown in the shape vector images. The right panel shows the repeatability of the feature points extracted by our geodesic distance-weighted shape vector image diffusion method. The comparison to the conventional anisotropic and isotropic diffusion methods is demonstrated.
  • Matching of face models with different expressions from the same subject is showed in the following figure. The left panel shows all the matched keypoints between the two surfaces. The right panel shows the scales of the matched keypoints.
  • The multimodality image analysis pipeline is showed in the following figure. The referenced brain is used as the template SVI (TSVI), and then all other brain SVIs are registered based on this TSVI. Based on the registered shape vector images, multimodality data such as the PET and DTI, can be integrated over the SVI images to perform the multimodality analysis.

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    Updated by 09/28/2011