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 distanceweighted
shape vector image diffusion.
Algorithms
 Firstly, our framework conformally maps a
tobeanalyzed 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 vectorvalued
diffusion space through our novel geodesic
distanceweighted 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 scaleaware 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 scaleaware 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 realworld applications. For
example, it is very useful for processing
intersubject 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
realworld 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.
Results
 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 distanceweighted diffusion.
 The following subfigures (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(ce), (gi) and (km) 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 distanceweighted 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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