W HAT IS IT ? Point cloud - Set of vertices in a 3D coordinate system. These vertices are usually...
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W HAT IS IT ? Point cloud - Set of vertices in a 3D coordinate system. These vertices are usually defined by X, Y, and Z coordinates, and typically are
W HAT IS IT ? Point cloud - Set of vertices in a 3D coordinate
system. These vertices are usually defined by X, Y, and Z
coordinates, and typically are intended to be representative of the
external surface of an object. It is a point-based rendering, i.e.,
the 3D models are represented by points (point cloud) and not by
triangles, as they are commonly represented.
Slide 3
M OTIVATION Evolution of point acquisitions devices. Best
efficiency when compared with rendering based on triangles or
another primitive that needs information about connectivity of
points. High quality (with efficiency) of rendering when the point
cloud is dense. Hardware implementation is possible.
Slide 4
P IPELINE 3D Acquisition Spherical Flipping Convex Hull Point
Selection Rendering
Slide 5
3D ACQUISITION (S CANNER ) 3D Scanning - A scan is taken at
some viewpoint. View planning - A decision is made about where to
position the scanner relative to the object in order to perform the
next scan. Registration - After all scans have been acquired, a
global registration algorithm may be used to simultaneously
minimize misalignment errors over all pairs of overlapping scans.
Merging - The aligned scans frequently contain signicant regions in
which many scans overlap. Merging these logically separate scans
into a single model both reduces storage and averages away some of
the scanning noise. Post-processing - Depending on the nal use for
the model, the output of the merging step may undergo further
processing, including noise or outlier reduction, automated lling
of small remaining holes, and curvature-adaptive resampling or
decimation.
Slide 6
3D ACQUISITION (S CANNER ) Triangulation-based
Triangulation-based 3D scanners nd the positions of points on a
surface by computing corresponding pixels from two viewpoints. The
correspondence denes a pair of rays in space, and the intersection
of the rays determines a 3D position.
Slide 7
3D ACQUISITION (S CANNER ) Passive Stereo Active Stereo
Structured Light Light Stripe Multiview Triangulation and Structure
from Motion Pulsed and Modulated Time of Flight Shape from Shading
and Photometric Stereo
Slide 8
3D ACQUISITION (P HOTOGRAPHY )
Slide 9
Slide 10
S PHERICAL F LIPPING Consider a D-dimensional sphere with
radius R, centered at the origin (C), and constrained to include
all the points in P. Spherical flipping reflects a point P with
respect to the sphere (spherical mirror) by applying the following
equation: where: = point inside the sphere R = radius of
sphere
Slide 11
S PHERICAL F LIPPING Intuitively, spherical flipping reflects
every point internal to the sphere along the ray from C to to its
image outside the sphere. Figure 2 - Spherical flipping (in red) of
a 2D curve (in blue) using a sphere (in green) centered at the view
point (in magenta). Figure 1Figure 2
Slide 12
C ONVEX H ULL The convex hull of a set Q of points is the
smallest conve x polygon P for which each point Q is either on the
boun dary of P or in its interior.
Slide 13
C ONVEX H ULL (3D) 3 rd Party library
Slide 14
C ONVEX H ULL Denote by the transformed point cloud of P:
Calculate the convex hull of U {C}, i.e., the set that contains the
transformed point cloud AND the center of the sphere.
Slide 15
A point is marked visible from C if its inverted point lies on
the convex hull of U {C}. P OINT S ELECTION
Slide 16
B EFORE
Slide 17
A FTER
Slide 18
R EFERENCES Point-Based Graphics (The Morgan Kaufmann Series in
Computer Graphics) Markus Gross, Hanspeter Pfister Direct
Visibility of Point Sets Sagi Katz, Ayellet Tal, Ronen Basri Mesh
segmentation using feature point and core extraction Sagi Katz,
George Leifman, Ayellet Tal Introduction to Algorithms Thomas H.
Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein
Slide 19
P OINT S ELECTION (P ROVE ) The shape of L for different values
of (in degrees), where = (10;0), R = 30. Result of spherical
flipping
Slide 20
L and the X-axis define the empty region associated with. How
much is visible, depends on the size of the region. For every given
point, there exist two special points on either side of P, and P.
The region bounded between the curves in the L equation, from
through and from through, is the largest possible empty region. P
OINT S ELECTION (P ROVE )
Slide 21
The empty gray region between Lj and Lk, as defined by the
values of j + k. From L Equation, it can be deduced that the
largest region corresponds to the smallest . This means that j and
k that correspond to the largest possible empty region, are the
smallest possible for. Note that j and k can be extracted from L
Equation. For to be visible, the sum of j and k should satisfy j +
k = const (i.e., a large empty region is associated with )