Theoretical primer
The goal of the PFH formulation is to encode a point’s k-neighborhood
geometrical properties by generalizing the mean curvature around the point
using a multi-dimensional histogram of values. This highly dimensional
hyperspace provides an informative signature for the feature representation, is
invariant to the 6D pose of the underlying surface, and copes very well with
different sampling densities or noise levels present in the neighborhood.
A Point Feature Histogram representation is based on the relationships between
the points in the k-neighborhood and their estimated surface normals. Simply
put, it attempts to capture as best as possible the sampled surface variations
by taking into account all the interactions between the directions of the
estimated normals. The resultant hyperspace is thus dependent on the quality of
the surface normal estimations at each point.
The figure below presents an influence region diagram of the PFH computation
for a query point (
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), marked with red and placed in the middle of a
circle (sphere in 3D) with radius
r, and all its
k neighbors (points
with distances smaller than the radius
r) are fully interconnected in a
mesh. The final PFH descriptor is computed as a histogram of relationships
between all pairs of points in the neighborhood, and thus has a computational
complexity of
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.
To compute the relative difference between two points
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and their associated normals
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, we
define a fixed coordinate frame at one of the points (see the figure below).
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Using the above uvw frame, the difference between the two normals
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greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la
tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
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can be expressed as a set of angular features as
follows:
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where d is the Euclidean distance between the two points
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. The quadruplet
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Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar
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is computed for each pair of two
points in k-neighborhood, therefore reducing the 12 values (xyz and normal
information) of the two points and their normals to 4.
To estimate a PFH quadruplet for a pair of points, use:
computePairFeatures (const Eigen::Vector4f &p1, const Eigen::Vector4f &n1,
const Eigen::Vector4f &p2, const Eigen::Vector4f &n2,
float &f1, float &f2, float &f3, float &f4);
See the API documentation for additional details.
To create the final PFH representation for the query point, the set of all
quadruplets is binned into a histogram. The binning process divides each
features’s value range into b subdivisions, and counts the number of
occurrences in each subinterval. Since three out of the four features presented
above are measure of the angles between normals, their values can easily be
normalized to the same interval on the trigonometric circle. A binning example
is to divide each feature interval into the same number of equal parts, and
therefore create a histogram with
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bins in a fully correlated space.
In this space, a histogram bin increment corresponds to a point having certain
values for all its 4 features. The figure below presents examples of Point
Feature Histograms representations for different points in a cloud.
In some cases, the fourth feature, d, does not present an extreme
significance for 2.5D datasets, usually acquired in robotics, as the distance
between neighboring points increases from the viewpoint. Therefore, omitting
d for scans where the local point density influences this feature dimension
has proved to be beneficial.
Note
For more information and mathematical derivations, including an analysis of PFH signatures for different surface geometries please see [RusuDissertation].