**Authors: **[tex2html_wrap4106]*F. Leymarie, M. D. Levine*

**Investigator username:** levine

**Category: ** perception

**Subcategory:** computer vision

We have explored a new representation for surfaces based on the notion
of differential geometry and topography. The curvature indicatrix, as
it is usually defined with respect to differentiable surfaces, can be
modified when applied to surfaces corresponding to graphs of 2D functions
(i.e., which can be parametrized with a single coordinate chart). In this
context, one can take advantage of the special vertical dimension and define a
new indicatrix in terms of sets rather than curves. The concept of a curvature
indicatrix is thus amenable to the description of quantized and discretized 2D
signals. We call this new indicatrix a *structural gauge figure*.

Using this gauge figure idea we propose a multiscale representation of the
class of surfaces corresponding to graphs of some 2D discrete functions. This
representation is regional and permits us to distinguish and characterize
surface patches by one of the four classes: *valley, ridge, flat-slanted*
and *plateau*. Furthermore, the representation is both a quantitative and
a qualitative one. Directional labels are attached to each surface point, with
associated quantitative measures pertaining to curvature and scale.

Once a topographic representation has been obtained through the use of the structural gauge figure, a relaxation labelling process is applied in order to retrieve consistent labellings. Further steps of building graphs made of regional patches, of reconstructing the 3D shape from such graphs, and of extracting valley-lines and ridge-lines (i.e., watersheds) remain to be explored.

TRACES is being applied to four types of 2D signals at the moment: Laser range data, intensity images and their smoothed derivatives, distance transforms of binary shapes, and potential fields for robot motion planning. Refer to Figure 1 for a comparison of TRACES with edge extraction.

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baron@cim.mcgill.ca