Shape Reconstruction by Active Spline Curves

1
Shape Reconstruction by Active Spline Curves
Wenping Wang and Dominic Cheng University of
Hong Kong
2

• Active Splines for Shape Reconstruction
• Shape reconstruction refers to obtaining a
  certain computer representation of the shape of
  input geometric data, called the target shape.
• A target shape can be given by a curve, a
  surface, or a set of data points.

3

B-spline curve fitting Consider the problem of
computing a planar B-spline curve P(t) to
approximate a sequence of ordered data points
Xi, i1, 2, ., N.
4

• The conventional approach requires that the data
  points be assigned parameter values ti,
  i1,2,..., N, and uses the least squares method
  to find the control points Pi of P(t) by
  minimizing the following error
• E ? (P(ti) - Xi)2

5
Problems with data parameterization

• Different methods for data parameterization
• Chord-length,
• centripetal,
• intrinsic,
• etc.

6

• Different ways of data parameterizations yield
  different B-spline curves P(t).
• E.g., in many cases the chord-length
  parameterization does not produce a P(t) that
  passes through the data points with minimum
  Euclidean distance.
• The notion of optimal parameterization is
  elusive.

7
Unorganized point clouds

• B-spline curve interpolation methods based on
  data parameterization cannot be easily used for
  approximating unorganized data points, especially
  point clouds of certain thickness.

8
Squared Distance Formula by Pottmann et al

• Pottmann and co-workers propose the active
  B-spline based the active contour method (i.e.
  snake).
• The originally proposed active B-spline curve
  assumes that the target shape be given by a
  smooth curve whose curvature information can
  readily be obtained.

9

• Given a target shape, an active B-spline curve
  starts with a simple initial shape and is updated
  by minimizing some goal function defined in terms
  of the SD.
• The curve converges to converge to the target
  shape a number of iterations, provided that a
  good initial B-spline is specified.

10
Example 1
11
A different initial spline curve
12
Control point removal
13
A new method

• The goal of this study is to extend the active
  B-spline curve method to shape reconstruction
  from unorganized data points, for which it is
  difficult to derive an SD formula, since it is no
  longer represented by a smooth curve.

14
Attach SD to active curve
We attache the SD formula to the active curve
instead. Let F(Xi) be the squared distance from
data point Xi to curve P(t). We express F(Xi)
as a quadratic function of the control points Pn.
15
Goal Function
Now the task in each iteration is simply to
minimize the following error function E sum_i
F_i(X_i) smoothing term to find the control
points P_i.
16

• The final B-spline curve approximating the target
  shape is obtained by iteration.
• 14 iterations are used in this example.

17
Non-uniform data (8 iterations)
18
Point cloud (20 iterations)
19
Point cloud II (31 iterations)
20
Further Problems

• 1. Add an expansion force to improve the global
  convergence.
• 2. Use a better way of specifying the initial
  spline curve, i.e. with a closer match to the
  target shape.
• 3. The extension to spline curve and surface
  reconstruction from point clouds in 3D space.
• 4. Control points adjustment.

21

• Thank you!