Curve Registration

1
Curve Registration

• The rigid metric of physical time may not be
  directly relevant to the internal dynamics of
  many real-life systems.
• Rather, there can be a sort of biological or
  meteorological time scale that can be nonlinearly
  related to physical time

2

• Amplitude Variation vs Phase Variation
• Shift Registration
• Landmark Registration
• More General Registration Techniques

3
Shift Registration

• The goal is to estimate a d for each curve such
  that the criteria (REGSSE) is minimized
• The Procrustes Method involves estimating a delta
  for each curve. Then, one reestimates with the
  newly registered curves the mean curve and
  repeats the process.

4
Numerical method for a simple shift

• How do we estimate each delta?
• Newton-Raphson
• 1. Choose an initial delta, perhaps using visual
  landmarks.
• 2. Modify delta by
• 3. Update the mean and continue Procrustes
  Method.
• In the text, R S use this method to improve
  visual alignment of the pinch force data.

5
Landmark Registration

• Mark important structural events (characteristic
  points).
• Sometimes this is done visually. Other times,
  events in the derivative (minima and maxima) may
  be useful.
• How do we use these events to align the curve?
• Time Warping Functiona strictly increasing
  transformation of time.
• h(t) where h(0)0 and h(T)T for our time in
  0,T
• Must use a reference function (often the mean)
  and h(t0f)tif for each feature f.
• Gasser et al use average times to do this. So,
  there is no target function.
• Our registered sample functions will be
  x(h(t))x(t).
• These together will yield the structural average.

6
Interpolation

• In RS, the example uses linear extrapolation to
  fit h(t) through each of the adjusted values for
  the handwriting data.
• This is data from Ramsays A guide to curve
  registration. Here a smooth interpolation is
  used.
• Another example is Gasser et all using piecewise
  cubic polynomials to piece together human growth
  curve data.
• One problem with curve registration is the
  regularity of landmark features across the
  sample. Sometimes there is ambiguity in maxima
  and minima and visual or intuitive methods must
  be used.

7
General Method

• The problem is obtaining a strictly increasing
  function that aligns h(0)0 and h(T)T. We
  restrict h to be in this family and to be twice
  differentiable.
• Thus,
• Which has the solution,

8
Modeling warping using SDEs

• Ramsey and Wang mention that an alternative model
  would be
• Where everything is as before except z is a
  stochastic process
• If we take B to be Brownian motion (w.o. drift)
  there is a solution
• This may be envisaged as a clock that is
  running fast or slow from instant to instant,
  constantly undergoing a percentage change in rate
  in a memoryless chaotic manner.

9
Finding w(t)

• Now, we look for a w(t) that will minimize
• Lambda is a penalty on the second derivative of
  h(t), which ensures a smooth h.
• Ramsay and Li recommend an order 1 B-spline to
  approximate w.
• This allows for a closed form solution of
  h(t)which, while not absolutely necessary, is
  desirable.

10
Height Acceleration
This was done with lambda 0.01 and breakvalues
4,7,10,12,14,16,18.
11
An Extension

• Alternatively, you could minimize
• This allows for differing weights over the
  intervals, across derivatives, and over vectors
  if the function is vector-valued.

12
An alternate minimizing criteria

• What if the sample functions are multiples of the
  target function (the mean function)?
• Minimize the log of the smallest eigenvalue of
• If these really are multiples, the smallest
  eigenvalue should be zero.

13
Other Sources

• Silverman includes the warping function as part
  of his principal component analysis. He uses a
  parametric model for the warping function.
• Sakoe and Chiba did a much earlier version by
  minimizing a weighted distance between curves
  (with appropriate restrictions on h) using
  dynamic programming
• Kneip and Gasser (1992) went through a detailed
  analysis of the statistical properties of using
  warping functions.