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##
## Copyright (c) 2006-2019 of Toni Giorgino
##
## This file is part of the DTW package.
##
## DTW is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## DTW is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
## License for more details.
##
## You should have received a copy of the GNU General Public License
## along with DTW. If not, see <http://www.gnu.org/licenses/>.
##
"""Step Pattern handling
See documentation for the StepPattern class.
"""
import numpy
class StepPattern:
# IMPORT_RDOCSTRING stepPattern
"""Step patterns for DTW
A ``stepPattern`` object lists the transitions allowed while searching
for the minimum-distance path. DTW variants are implemented by passing
one of the objects described in this page to the ``stepPattern``
argument of the [dtw()] call.
**Details**
A step pattern characterizes the matching model and slope constraint
specific of a DTW variant. They also known as local- or
slope-constraints, transition types, production or recursion rules
(GiorginoJSS).
**Pre-defined step patterns**
::
## Well-known step patterns
symmetric1
symmetric2
asymmetric
## Step patterns classified according to Rabiner-Juang (Rabiner1993)
rabinerJuangStepPattern(type,slope_weighting="d",smoothed=False)
## Slope-constrained step patterns from Sakoe-Chiba (Sakoe1978)
symmetricP0; asymmetricP0
symmetricP05; asymmetricP05
symmetricP1; asymmetricP1
symmetricP2; asymmetricP2
## Step patterns classified according to Rabiner-Myers (Myers1980)
typeIa; typeIb; typeIc; typeId;
typeIas; typeIbs; typeIcs; typeIds; # smoothed
typeIIa; typeIIb; typeIIc; typeIId;
typeIIIc; typeIVc;
## Miscellaneous
mori2006;
rigid;
A variety of classification schemes have been proposed for step
patterns, including Sakoe-Chiba (Sakoe1978); Rabiner-Juang
(Rabiner1993); and Rabiner-Myers (Myers1980). The ``dtw`` package
implements all of the transition types found in those papers, with the
exception of Itakura’s and Velichko-Zagoruyko’s steps, which require
subtly different algorithms (this may be rectified in the future).
Itakura recursion is almost, but not quite, equivalent to ``typeIIIc``.
For convenience, we shall review pre-defined step patterns grouped by
classification. Note that the same pattern may be listed under different
names. Refer to paper (GiorginoJSS) for full details.
**1. Well-known step patterns**
Common DTW implementations are based on one of the following transition
types.
``symmetric2`` is the normalizable, symmetric, with no local slope
constraints. Since one diagonal step costs as much as the two equivalent
steps along the sides, it can be normalized dividing by ``N+M``
(query+reference lengths). It is widely used and the default.
``asymmetric`` is asymmetric, slope constrained between 0 and 2. Matches
each element of the query time series exactly once, so the warping path
``index2~index1`` is guaranteed to be single-valued. Normalized by ``N``
(length of query).
``symmetric1`` (or White-Neely) is quasi-symmetric, no local constraint,
non-normalizable. It is biased in favor of oblique steps.
**2. The Rabiner-Juang set**
A comprehensive table of step patterns is proposed in Rabiner-Juang’s
book (Rabiner1993), tab. 4.5. All of them can be constructed through the
``rabinerJuangStepPattern(type,slope_weighting,smoothed)`` function.
The classification foresees seven families, labelled with Roman numerals
I-VII; here, they are selected through the integer argument ``type``.
Each family has four slope weighting sub-types, named in sec. 4.7.2.5 as
“Type (a)” to “Type (d)”; they are selected passing a character argument
``slope_weighting``, as in the table below. Furthermore, each subtype
can be either plain or smoothed (figure 4.44); smoothing is enabled
setting the logical argument ``smoothed``. (Not all combinations of
arguments make sense.)
::
Subtype | Rule | Norm | Unbiased
--------|------------|------|---------
a | min step | -- | NO
b | max step | -- | NO
c | Di step | N | YES
d | Di+Dj step | N+M | YES
**3. The Sakoe-Chiba set**
Sakoe-Chiba (Sakoe1978) discuss a family of slope-constrained patterns;
they are implemented as shown in page 47, table I. Here, they are called
``symmetricP<x>`` and ``asymmetricP<x>``, where ``<x>`` corresponds to
Sakoe’s integer slope parameter *P*. Values available are accordingly:
``0`` (no constraint), ``1``, ``05`` (one half) and ``2``. See
(Sakoe1978) for details.
**4. The Rabiner-Myers set**
The ``type<XX><y>`` step patterns follow the older Rabiner-Myers’
classification proposed in (Myers1980) and (MRR1980). Note that this is
a subset of the Rabiner-Juang set (Rabiner1993), and the latter should
be preferred in order to avoid confusion. ``<XX>`` is a Roman numeral
specifying the shape of the transitions; ``<y>`` is a letter in the
range ``a-d`` specifying the weighting used per step, as above;
``typeIIx`` patterns also have a version ending in ``s``, meaning the
smoothing is used (which does not permit skipping points). The
``typeId, typeIId`` and ``typeIIds`` are unbiased and symmetric.
**5. Others**
The ``rigid`` pattern enforces a fixed unitary slope. It only makes
sense in combination with ``open_begin=True``, ``open_end=True`` to find
gapless subsequences. It may be seen as the ``P->inf`` limiting case in
Sakoe’s classification.
``mori2006`` is Mori’s asymmetric step-constrained pattern (Mori2006).
It is normalized by the matched reference length.
[mvmStepPattern()] implements Latecki’s Minimum Variance Matching
algorithm, and it is described in its own page.
**Methods**
``print_stepPattern`` prints an user-readable description of the
recurrence equation defined by the given pattern.
``plot_stepPattern`` graphically displays the step patterns productions
which can lead to element (0,0). Weights are shown along the step
leading to the corresponding element.
``t_stepPattern`` transposes the productions and normalization hint so
that roles of query and reference become reversed.
Parameters
----------
x :
a step pattern object
type :
path specification, integer 1..7 (see (Rabiner1993), table 4.5)
slope_weighting :
slope weighting rule: character `"a"` to `"d"` (see (Rabiner1993), sec. 4.7.2.5)
smoothed :
logical, whether to use smoothing (see (Rabiner1993), fig. 4.44)
... :
additional arguments to [print()].
Notes
-----
Constructing ``stepPattern`` objects is tricky and thus undocumented.
For a commented example please see source code for ``symmetricP1``.
References
----------
- (GiorginoJSS) Toni Giorgino. *Computing and Visualizing Dynamic Time
Warping Alignments in R: The dtw Package.* Journal of Statistical
Software, 31(7), 1-24.
`doi:10.18637/jss_v031.i07 <https://doi.org/10.18637/jss_v031.i07>`__
- (Itakura1975) Itakura, F., *Minimum prediction residual principle
applied to speech recognition,* Acoustics, Speech, and Signal
Processing, IEEE Transactions on , vol.23, no.1, pp. 67-72, Feb 1975.
`doi:10.1109/TASSP.1975.1162641 <https://doi.org/10.1109/TASSP.1975.1162641>`__
- (MRR1980) Myers, C.; Rabiner, L. & Rosenberg, A. *Performance
tradeoffs in dynamic time warping algorithms for isolated word
recognition*, IEEE Trans. Acoust., Speech, Signal Process., 1980, 28,
623-635.
`doi:10.1109/TASSP.1980.1163491 <https://doi.org/10.1109/TASSP.1980.1163491>`__
- (Mori2006) Mori, A.; Uchida, S.; Kurazume, R.; Taniguchi, R.;
Hasegawa, T. & Sakoe, H. Early Recognition and Prediction of Gestures
Proc. 18th International Conference on Pattern Recognition ICPR 2006,
2006, 3, 560-563.
`doi:10.1109/ICPR.2006.467 <https://doi.org/10.1109/ICPR.2006.467>`__
- (Myers1980) Myers, Cory S. *A Comparative Study Of Several Dynamic
Time Warping Algorithms For Speech Recognition*, MS and BS thesis,
Dept. of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, archived Jun 20 1980,
https://hdl_handle_net/1721.1/27909
- (Rabiner1993) Rabiner, L. R., & Juang, B.-H. (1993). *Fundamentals of
speech recognition.* Englewood Cliffs, NJ: Prentice Hall.
- (Sakoe1978) Sakoe, H.; Chiba, S., *Dynamic programming algorithm
optimization for spoken word recognition,* Acoustics, Speech, and
Signal Processing, IEEE Transactions on , vol.26, no.1, pp. 43-49,
Feb 1978
`doi:10.1109/TASSP.1978.1163055 <https://doi.org/10.1109/TASSP.1978.1163055>`__
Examples
--------
>>> from dtw import *
>>> import numpy as np
The usual (normalizable) symmetric step pattern
Step pattern recursion, defined as:
g[i,j] = min(
g[i,j-1] + d[i,j] ,
g[i-1,j-1] + 2 * d[i,j] ,
g[i-1,j] + d[i,j] ,
)
>>> print(symmetric2) #doctest: +NORMALIZE_WHITESPACE
Step pattern recursion:
g[i,j] = min(
g[i-1,j-1] + 2 * d[i ,j ] ,
g[i ,j-1] + d[i ,j ] ,
g[i-1,j ] + d[i ,j ] ,
)
<BLANKLINE>
Normalization hint: N+M
<BLANKLINE>
The well-known plotting style for step patterns
>>> import matplotlib.pyplot as plt; # doctest: +SKIP
... symmetricP2.plot().set_title("Sakoe's Symmetric P=2 recursion")
Same example seen in ?dtw , now with asymmetric step pattern
>>> (query, reference) = dtw_test_data.sin_cos()
Do the computation
>>> asy = dtw(query, reference, keep_internals=True,
... step_pattern=asymmetric);
>>> dtwPlot(asy,type="density" # doctest: +SKIP
... ).set_title("Sine and cosine, asymmetric step")
Hand-checkable example given in [Myers1980] p 61 - see JSS paper
>>> tm = numpy.reshape( [1, 3, 4, 4, 5, 2, 2, 3, 3, 4, 3, 1, 1, 1, 3, 4, 2,
... 3, 3, 2, 5, 3, 4, 4, 1], (5,5) )
"""
# ENDIMPORT
def __init__(self, mx, hint="NA"):
self.mx = numpy.array(mx, dtype=numpy.double)
self.hint = hint
def get_n_rows(self):
"""Total number of steps in the recursion."""
return self.mx.shape[0]
def get_n_patterns(self):
"""Number of rules in the recursion."""
return int(numpy.max(self.mx[:, 0]))
def T(self):
"""Transpose a step pattern."""
tmx = self.mx.copy()
tmx = tmx[:, [0, 2, 1, 3]]
th = self.hint
if th == "N":
th = "M"
elif th == "M":
th = "N"
tsp = StepPattern(tmx, th)
return tsp
def __str__(self):
np = self.get_n_patterns()
head = " g[i,j] = min(\n"
body = ""
for p in range(1, np + 1):
steps = self._extractpattern(p)
ns = steps.shape[0]
steps = numpy.flip(steps, 0)
for s in range(ns):
di, dj, cc = steps[s, :]
dis = "" if di == 0 else f"{-int(di)}"
djs = "" if dj == 0 else f"{-int(dj)}"
dijs = f"i{dis:2},j{djs:2}"
if cc == -1:
gs = f" g[{dijs}]"
body = body + " " + gs
else:
ccs = " " if cc == 1 else f"{cc:2.2g} *"
ds = f"+{ccs} d[{dijs}]"
body = body + " " + ds
body = body + " ,\n"
tail = " ) \n\n"
ntxt = f"Normalization hint: {self.hint}\n"
return "Step pattern recursion:\n" + head + body + tail + ntxt
def plot(self):
"""Provide a visual description of a StepPattern object"""
import matplotlib.pyplot as plt
x = self.mx
pats = numpy.arange(1, 1 + self.get_n_patterns())
alpha = .5
fudge = [0, 0]
fig, ax = plt.subplots(figsize=(6, 6))
for i in pats:
ss = x[:, 0] == i
ax.plot(-x[ss, 1], -x[ss, 2], lw=2, color="tab:blue")
ax.plot(-x[ss, 1], -x[ss, 2], 'o', color="black", marker="o", fillstyle="none")
if numpy.sum(ss) == 1: continue
xss = x[ss, :]
xh = alpha * xss[:-1, 1] + (1 - alpha) * xss[1:, 1]
yh = alpha * xss[:-1, 2] + (1 - alpha) * xss[1:, 2]
for xx, yy, tt in zip(xh, yh, xss[1:, 3]):
ax.annotate("{:.2g}".format(tt), (-xx - fudge[0],
-yy - fudge[1]))
endpts = x[:, 3] == -1
ax.plot(-x[endpts, 1], -x[endpts, 2], 'o', color="black")
ax.set_xlabel("Query index")
ax.set_ylabel("Reference index")
ax.set_xticks(numpy.unique(-x[:, 1]))
ax.set_yticks(numpy.unique(-x[:, 2]))
return ax
def _extractpattern(self, sn):
sp = self.mx
sbs = sp[:, 0] == sn
spl = sp[sbs, 1:]
return numpy.flip(spl, 0)
def _mkDirDeltas(self):
out = numpy.array(self.mx, dtype=numpy.int32)
out = out[out[:, 3] == -1, :]
out = out[:, [1, 2]]
return out
def _get_p(self):
# Dimensions are reversed wrt R
s = self.mx[:, [0, 2, 1, 3]]
return s.T.reshape(-1)
# Alternate constructor for ease of R import
def _c(*v):
va = numpy.array([*v])
if len(va) % 4 != 0:
_error("Internal error in _c constructor")
va = va.reshape((-1, 4))
return (va)
# Kludge because lambda: raise doesn't work
def _error(s):
raise ValueError(s)
##################################################
##################################################
# Reimplementation of the building process
class _P:
def __init__(self, pid, subtype, smoothing):
self.subtype = subtype
self.smoothing = smoothing
self.pid = pid
self.i = [0]
self.j = [0]
def step(self, di, dj): # equivalent to .Pstep
self.i.append(di)
self.j.append(dj)
return self
def get(self): # eqv to .Pend
ia = numpy.array(self.i, dtype=numpy.double)
ja = numpy.array(self.j, dtype=numpy.double)
si = numpy.cumsum(ia)
sj = numpy.cumsum(ja)
ni = numpy.max(si) - si # ?
nj = numpy.max(sj) - sj
if self.subtype == "a":
w = numpy.minimum(ia, ja)
elif self.subtype == "b":
w = numpy.maximum(ia, ja)
elif self.subtype == "c":
w = ia
elif self.subtype == "d":
w = ia + ja
else:
_error("Unsupported subtype")
if self.smoothing:
# if self.pid==3: import ipdb; ipdb.set_trace()
w[1:] = numpy.mean(w[1:])
w[0] = -1.0
nr = len(w)
mx = numpy.zeros((nr, 4))
mx[:, 0] = self.pid
mx[:, 1] = ni
mx[:, 2] = nj
mx[:, 3] = w
return mx
def rabinerJuangStepPattern(ptype, slope_weighting="d", smoothed=False):
"""Construct a pattern classified according to the Rabiner-Juang scheme (Rabiner1993)
See documentation for the StepPattern class.
"""
f = {
1: _RJtypeI,
2: _RJtypeII,
3: _RJtypeIII,
4: _RJtypeIV,
5: _RJtypeV,
6: _RJtypeVI,
7: _RJtypeVII
}.get(ptype, lambda: _error("Invalid type"))
r = f(slope_weighting, smoothed)
norm = "NA"
if slope_weighting == "c":
norm = "N"
elif slope_weighting == "d":
norm = "N+M"
return StepPattern(r, norm)
def _RJtypeI(s, m):
return numpy.vstack([
_P(1, s, m).step(1, 0).get(),
_P(2, s, m).step(1, 1).get(),
_P(3, s, m).step(0, 1).get()])
def _RJtypeII(s, m):
return numpy.vstack([
_P(1, s, m).step(1, 1).step(1, 0).get(),
_P(2, s, m).step(1, 1).get(),
_P(3, s, m).step(1, 1).step(0, 1).get()])
def _RJtypeIII(s, m):
return numpy.vstack([
_P(1, s, m).step(2, 1).get(),
_P(2, s, m).step(1, 1).get(),
_P(3, s, m).step(1, 2).get()])
def _RJtypeIV(s, m):
return numpy.vstack([
_P(1, s, m).step(1, 1).step(1, 0).get(),
_P(2, s, m).step(1, 2).step(1, 0).get(),
_P(3, s, m).step(1, 1).get(),
_P(4, s, m).step(1, 2).get(),
])
def _RJtypeV(s, m):
return numpy.vstack([
_P(1, s, m).step(1, 1).step(1, 0).step(1, 0).get(),
_P(2, s, m).step(1, 1).step(1, 0).get(),
_P(3, s, m).step(1, 1).get(),
_P(4, s, m).step(1, 1).step(0, 1).get(),
_P(5, s, m).step(1, 1).step(0, 1).step(0, 1).get(),
])
def _RJtypeVI(s, m):
return numpy.vstack([
_P(1, s, m).step(1, 1).step(1, 1).step(1, 0).get(),
_P(2, s, m).step(1, 1).get(),
_P(3, s, m).step(1, 1).step(1, 1).step(0, 1).get()
])
def _RJtypeVII(s, m):
return numpy.vstack([
_P(1, s, m).step(1, 1).step(1, 0).step(1, 0).get(),
_P(2, s, m).step(1, 2).step(1, 0).step(1, 0).get(),
_P(3, s, m).step(1, 3).step(1, 0).step(1, 0).get(),
_P(4, s, m).step(1, 1).step(1, 0).get(),
_P(5, s, m).step(1, 2).step(1, 0).get(),
_P(6, s, m).step(1, 3).step(1, 0).get(),
_P(7, s, m).step(1, 1).get(),
_P(8, s, m).step(1, 2).get(),
_P(9, s, m).step(1, 3).get(),
])
##########################################################################################
##########################################################################################
## Everything here is semi auto-generated from the R source. Don't
## edit!
##################################################
##################################################
##
## Various step patterns, defined as internal variables
##
## First column: enumerates step patterns.
## Second step in query index
## Third step in reference index
## Fourth weight if positive, or -1 if starting point
##
## For \cite{} see dtw.bib in the package
##
## Widely-known variants
## White-Neely symmetric (default)
## aka Quasi-symmetric \cite{White1976}
## normalization: no (N+M?)
symmetric1 = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 0, 1, -1,
2, 0, 0, 1,
3, 1, 0, -1,
3, 0, 0, 1
))
## Normal symmetric
## normalization: N+M
symmetric2 = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 2,
2, 0, 1, -1,
2, 0, 0, 1,
3, 1, 0, -1,
3, 0, 0, 1
), "N+M")
## classic asymmetric pattern: max slope 2, min slope 0
## normalization: N
asymmetric = StepPattern(_c(
1, 1, 0, -1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 0, 1
), "N")
# % \item{\code{symmetricVelichkoZagoruyko}}{symmetric, reproduced from %
# [Sakoe1978]. Use distance matrix \code{1-d}}
#
## normalization: max[N,M]
## note: local distance matrix is 1-d
## \cite{Velichko}
_symmetricVelichkoZagoruyko = StepPattern(_c(
1, 0, 1, -1,
2, 1, 1, -1,
2, 0, 0, -1.001,
3, 1, 0, -1))
# % \item{\code{asymmetricItakura}}{asymmetric, slope contrained 0.5 -- 2
# from reference [Itakura1975]. This is the recursive definition % that
# generates the Itakura parallelogram; }
#
## Itakura slope-limited asymmetric \cite{Itakura1975}
## Max slope: 2; min slope: 1/2
## normalization: N
_asymmetricItakura = StepPattern(_c(
1, 1, 2, -1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 1, 0, 1,
3, 0, 0, 1,
4, 2, 2, -1,
4, 1, 0, 1,
4, 0, 0, 1
))
#############################
## Slope-limited versions
##
## Taken from Table I, page 47 of "Dynamic programming algorithm
## optimization for spoken word recognition," Acoustics, Speech, and
## Signal Processing, vol.26, no.1, pp. 43-49, Feb 1978 URL:
## http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1163055
##
## Mostly unchecked
## Row P=0
symmetricP0 = symmetric2
## normalization: N ?
asymmetricP0 = StepPattern(_c(
1, 0, 1, -1,
1, 0, 0, 0,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 0, -1,
3, 0, 0, 1
), "N")
## alternative implementation
_asymmetricP0b = StepPattern(_c(
1, 0, 1, -1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 0, -1,
3, 0, 0, 1
), "N")
## Row P=1/2
symmetricP05 = StepPattern(_c(
1, 1, 3, -1,
1, 0, 2, 2,
1, 0, 1, 1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 1, 2,
2, 0, 0, 1,
3, 1, 1, -1,
3, 0, 0, 2,
4, 2, 1, -1,
4, 1, 0, 2,
4, 0, 0, 1,
5, 3, 1, -1,
5, 2, 0, 2,
5, 1, 0, 1,
5, 0, 0, 1
), "N+M")
asymmetricP05 = StepPattern(_c(
1, 1, 3, -1,
1, 0, 2, 1 / 3,
1, 0, 1, 1 / 3,
1, 0, 0, 1 / 3,
2, 1, 2, -1,
2, 0, 1, .5,
2, 0, 0, .5,
3, 1, 1, -1,
3, 0, 0, 1,
4, 2, 1, -1,
4, 1, 0, 1,
4, 0, 0, 1,
5, 3, 1, -1,
5, 2, 0, 1,
5, 1, 0, 1,
5, 0, 0, 1
), "N")
## Row P=1
## Implementation of Sakoe's P=1, Symmetric algorithm
symmetricP1 = StepPattern(_c(
1, 1, 2, -1, # First branch: g(i-1,j-2)+
1, 0, 1, 2, # + 2d(i ,j-1)
1, 0, 0, 1, # + d(i ,j)
2, 1, 1, -1, # Second branch: g(i-1,j-1)+
2, 0, 0, 2, # +2d(i, j)
3, 2, 1, -1, # Third branch: g(i-2,j-1)+
3, 1, 0, 2, # + 2d(i-1,j)
3, 0, 0, 1 # + d( i,j)
), "N+M")
asymmetricP1 = StepPattern(_c(
1, 1, 2, -1,
1, 0, 1, .5,
1, 0, 0, .5,
2, 1, 1, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 1, 0, 1,
3, 0, 0, 1
), "N")
## Row P=2
symmetricP2 = StepPattern(_c(
1, 2, 3, -1,
1, 1, 2, 2,
1, 0, 1, 2,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 2,
3, 3, 2, -1,
3, 2, 1, 2,
3, 1, 0, 2,
3, 0, 0, 1
), "N+M")
asymmetricP2 = StepPattern(_c(
1, 2, 3, -1,
1, 1, 2, 2 / 3,
1, 0, 1, 2 / 3,
1, 0, 0, 2 / 3,
2, 1, 1, -1,
2, 0, 0, 1,
3, 3, 2, -1,
3, 2, 1, 1,
3, 1, 0, 1,
3, 0, 0, 1
), "N")
################################
## Taken from Table III, page 49.
## Four varieties of DP-algorithm compared
## 1st row: asymmetric
## 2nd row: symmetricVelichkoZagoruyko
## 3rd row: symmetric1
## 4th row: asymmetricItakura
#############################
## Classified according to Rabiner
##
## Taken from chapter 2, Myers' thesis [4]. Letter is
## the weighting function:
##
## rule norm unbiased
## a min step ~N NO
## b max step ~N NO
## c x step N YES
## d x+y step N+M YES
##
## Mostly unchecked
# R-Myers R-Juang
# type I type II
# type II type III
# type III type IV
# type IV type VII
typeIa = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 0,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 0
))
typeIb = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 1
))
typeIc = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 0
), "N")
typeId = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 2,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 2,
3, 1, 2, -1,
3, 0, 1, 2,
3, 0, 0, 1
), "N+M")
## ----------
## smoothed variants of above
typeIas = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, .5,
1, 0, 0, .5,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, .5,
3, 0, 0, .5
))
typeIbs = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 1
))
typeIcs = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, .5,
3, 0, 0, .5
), "N")
typeIds = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 1.5,
1, 0, 0, 1.5,
2, 1, 1, -1,
2, 0, 0, 2,
3, 1, 2, -1,
3, 0, 1, 1.5,
3, 0, 0, 1.5
), "N+M")
## ----------
typeIIa = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 0, 0, 1
))
typeIIb = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 2,
3, 2, 1, -1,
3, 0, 0, 2
))
typeIIc = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 0, 0, 2
), "N")
typeIId = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 2,
2, 1, 2, -1,
2, 0, 0, 3,
3, 2, 1, -1,
3, 0, 0, 3
), "N+M")
## ----------
## Rabiner [3] discusses why this is not equivalent to Itakura's
typeIIIc = StepPattern(_c(
1, 1, 2, -1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 1, 0, 1,
3, 0, 0, 1,
4, 2, 2, -1,
4, 1, 0, 1,
4, 0, 0, 1
), "N")
## ----------
## numbers follow as production rules in fig 2.16
typeIVc = StepPattern(_c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 1,
3, 1, 3, -1,
3, 0, 0, 1,
4, 2, 1, -1,
4, 1, 0, 1,
4, 0, 0, 1,
5, 2, 2, -1,
5, 1, 0, 1,
5, 0, 0, 1,
6, 2, 3, -1,
6, 1, 0, 1,
6, 0, 0, 1,
7, 3, 1, -1,
7, 2, 0, 1,
7, 1, 0, 1,
7, 0, 0, 1,
8, 3, 2, -1,
8, 2, 0, 1,
8, 1, 0, 1,
8, 0, 0, 1,
9, 3, 3, -1,
9, 2, 0, 1,
9, 1, 0, 1,
9, 0, 0, 1
), "N")
#############################
##
## Mori's asymmetric step-constrained pattern. Normalized in the
## reference length.
##
## Mori, A.; Uchida, S.; Kurazume, R.; Taniguchi, R.; Hasegawa, T. &
## Sakoe, H. Early Recognition and Prediction of Gestures Proc. 18th
## International Conference on Pattern Recognition ICPR 2006, 2006, 3,
## 560-563
##
mori2006 = StepPattern(_c(
1, 2, 1, -1,
1, 1, 0, 2,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 3,
3, 1, 2, -1,
3, 0, 1, 3,
3, 0, 0, 3
), "M")
## Completely unflexible: fixed slope 1. Only makes sense with
## open.begin and open.end
rigid = StepPattern(_c(1, 1, 1, -1,
1, 0, 0, 1), "N")