unified approaches to time series and shape analysis and some
TRANSCRIPT
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Unified approaches to time series and shape analysis and some fuzzy extensions
VASILE GEORGESCU
University of Craiova, Faculty of Economics and Business Administration
13 A.I.Cuza, Craiova, Romania
e-mail: [email protected], [email protected]
Abstract. Despite their differences in nature, time series analysis and statistical shape analysis have much in common because they can share a unified methodological framework. Basically, it is based on
transforming a shape closed contour to a time series. Statistical Shape Analysis involves methods for the
geometrical study of random objects where location, rotation and scale information can be removed.
Time series analysis is a widely spread technique that takes into consideration the temporal nature of
data. However, it has been proved to be well adapted to represent or describe two-dimensional closed
contours of shapes. The aim of this paper is to extend the methods of transforming a shape closed contour
to a time series in a fuzzy context, based on recent advances in fuzzy digital geometry. The main
ingredient in this generalization is to consider geodesic distances between two points in a fuzzy object
(for example, between its centroid and its fuzzy boundaries). A geodesic distance is taken along a
geodesic path (i.e., a path of minimum length). Unlike in crisp (binary) convex objects, the shortest path
(when it exists) between to points in a fuzzy convex object is not necessarily a straight line segment. The
notions of fuzzy geodesic distance and fuzzy distance transform are foundational for developing a unitary
framework allowing to deal with both fuzzy shapes and fuzzy time series as equivalent objects.
1. TRANSFORMING A CRISP SHAPE CLOSED CONTOUR TO A TIME SERIES
The contour of a shape can be described by a function. There are two basic ideas to introduce such a
function:
i) the contour of a figure can be symmetric with respect to a line; than the orthogonal distance of the
contour point from the symmetry line as a function of position on the symmetry line can be considered
as a contour function;
ii) the contour function may be periodic, the contour itself can be considered as a periodic function;
assuming that the contour has some desirable properties such as star-shapedness (i.e., for each point
Ay ∈ , the line segment connecting y with the centroid is contained in A ) or convexity, a relatively
simple contour functions, such as the radius-vector or support functions, can be introduced, otherwise
more complex contour functions, such as the tangent-angle function, can be considered as contour
functions.
The radius-vector function rX (ϕ) is the distance from the reference point O (usually the center of
gravity) to the contour in the direction of the ϕ-ray where πϕ 20 ≤≤ (Fig. 1 a). In the general case,
however, description by the radius-vector function is not suitable for non-star-shaped figures (Fig. 1
b).
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ϕ
Xr X
(ϕ)
O
ϕ
Xr X
(ϕ)
O
a b
Figure 1. a) Radius-vector function; b) Problems with the radius-vector function occur if the figure is
not star-shaped.
An example of a star-shaped figure and its radius-vector function is given in Fig. 2.
ϕ
O
Xr X
(ϕ)
a
Figure 2. a) A start-shaped figure X ; b) Radius-vector function rX (ϕ) of the figure X . The gravity center of the figure was used as the origin to generate the radius-vector function.
When the radius-vector function rX (ϕ) of a star-shaped figure X is available some geometrical figure
parameters can be obtained. Integrating of rX (ϕ) yields the perimeter P(X), area A(X) and mean
radius-vector length Xr :
∫∫∫ ==′+=
πππ
ϕϕπ
ϕϕϕϕϕ2
0
2
0
2
2
0
22 )(2
1,)(
2
1)(,)()()( drrdrXAdrrXP XXXXX
The tangent angle at different points of the contour can also be used for the description of a shape
contour. It is assumed that the contour of the considered figure is piecewise-smooth so that a tangent
may not exist at a finite number of points. Let the perimeter of the figure X be L . Every point pl of the
contour of X can thus be identified with a number l, with 0≤ l≤ L, run through anti-clockwise. A
pointer is places at p0 so that its zero position coincides with the tangent direction at p0 . If the pointer
moves on the contour then it changes its direction in such a way that it is always in the direction of the
tangent, where its orientation is given by the direction of the movement. The angle given by the
pointer direction at pl is denoted )(lXφ where )0(Xφ = 0 and )(LXφ = π2 (Fig. 3). The function )(lXφ
is called the tangent-angle function.
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Figure 3. a, b) Definition of the tangent-angle function )(lXφ of figure X .
All these functions can be interpreted as signatures of a shape. Furthermore, they can be regarded as
parameterized functions of time, thus acting as converters from shape contours to time series.
2. REPRESENTATION OF CRISP AND FUZZY SHAPES IN A CONTINUOUS SPACE
2.1. CRISP SHAPES
Crisp shapes represent objects with crisp borders. Furthermore, if a texture is associated with the
object, it has to be uniformly represented (e.g. a digitized image, where all pixels are classified as
object pixels, or as background pixels).
The contour functions introduced in the previous section are good candidates for a shape descriptor. It
designates a signature of the shape and is based on a one-dimensional functional representation of the
two-dimensional shape boundary. The simplest way to generate a signature is to use the radial function
(also called centroid distance function), which express the radial distance from the centroid to the
boundary, as a function of the angle. Thus, for crisp objects, the shape signature function corresponds
to the Euclidean distance between each boundary point ( ))(),()( tytxtA = and the centroid
( )ccc yxA ,= of the shape:
( ) ( )22)()()( cc ytyxtxtCD −+−=
2.2. CONTINUOUS FUZZY SHAPES
In the case of a fuzzy object, boundary points are not strictly defined; there is a progressive transition
of the membership values from the support outline to the core outline. Fuzziness of an image
representation can arise from various reasons, such as limited acquisition conditions (scanning
resolution in digital images), but also as intrinsic property of the image, which may have imprecise
borders. In such cases, pixels close to the border of the object have assigned to them a fuzzy
membership value according to the extent of their belongingness to the object.
Continuous fuzzy shapes can be described as fuzzy geometric objects. A continuous fuzzy geometric
object A in pℜ is defined as a set of pairs ( ) pA xxx ℜ∈|)(, µ where [ ]1,0: →ℜ pAµ is the
membership function of A in nℜ . For any value θ ∈ [0, 1], θ-support of S, denoted by Θθ(S), is the
hard subset x | nx ℜ∈ and )(xSµ ≥ θ of nℜ . In other words, the θ-support of S is the set of all
points in nℜ with membership values greater than or equal to θ. 0-support will often be referred to as
support and be denoted by Θ(S). A fuzzy subset with a bounded support is called bounded. S is said to
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be convex if, for every three collinear points x, y, and z in nℜ such that y lies between x and z, µS(y) ≥
min[µS(x), µS(z)]. A fuzzy subset is called smooth if it is differentiable at every location nx ℜ∈ .
An alternative representation of fuzzy geometric objects is given by a set of −α cuts:
[ ] 1,0|)( ∈= ααAAC , where αµα ≥ℜ∈= )(| xxA Ap is a crisp object, whose −α level contour is
obtained for αµ =)(xA .
In (Chaudhuri, 1991), basic fuzzy geometric shapes, like point, line, circle, ellipse, and polygon, are
defined on continuous 2D support space. It is assumed that the fuzzy set has a bounded support, is
piecewise constant, and has a finite number n of distinct membership values.
A fuzzy geometric point is a fuzzy set with nonzero membership only at one point of the support
space.
A fuzzy geometric straight (curved) line is a fuzzy set for which any −α cut, ∈α (0, 1], is either
empty, or a straight (curved) line in a support space. A fuzzy line is a connected fuzzy set. A fuzzy
straight line is a convex fuzzy set.
A fuzzy geometric circle is a fuzzy set whose −α cuts, for ∈α (0, 1], are all concentric circles (figure
5(a)).
A fuzzy subset S is a ring if )(xSµ = )(~ rµ , where =r ||x −x0|| for some nx ℜ∈0 and ]1,0[:~ →ℜµ is a
membership function. A convex ring is called a fuzzy disk.
As opposite to a fuzzy geometric circle, the membership function of a fuzzy geometric disk is non-
increasing away from the interior of the object.
Figure 4. Fuzzy point, fuzzy curved line and fuzzy region
For example, in figure 5(b) is shown a fuzzy disk. Its core is a crisp disk defined by
( ) 122
21
21 | rxxxA ≤+ℜ∈= and its contour is the circle defined by ( ) 1
22
21
21 | rxxxA C =+ℜ∈= , where
1r is the length of the corresponding radix at the level 1=α . In general, for any ]1,0[∈α , the −α cut
(or equivalently, −α support) is defined by ( ) αα rxxxA ≤+ℜ∈= 22
21
2| , and the −α level contour is
defined by ( ) αα rxxxA C =+ℜ∈= 22
21
2 | .
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(a) (b)
Figure 5: (a) A fuzzy circle; (b) A fuzzy disk: centroid, core, support, α-level contours, radial distance
Two possible ways of generalizing the shape signature for a continuous fuzzy shape was proposed by
Chanussot et al. (2005):
• as s radial integral of the membership function:
( )∫=−
)(
1 )(),()(
tA
A
Afuzzy
c
dyxtCD ρρρµ
where )(tρρ = is a parameterization of the straight path between a boundary point and the centroid.
• as an average signature obtained from the −α cuts:
∫=−
1
0
2 )()( αα dtCDtCD fuzzy
where fuzzy star-shaped objects are considered, with all the boundaries of there −α cuts jointly
indexed by the same parameter t .
3. REPRESENTATION OF CRISP AND FUZZY SHAPES IN A DIGITAL SPACE
3.1. DIGITAL GEOMETRY
Discrete objects can arise from the digitization of scanned images, which involves sampling the
picture and quantizing the sampled values. A digital picture consists of a finite number of pixels (short
for “picture elements”) or voxels (short for “volume elements”), each of which is defined by a location
and a value at that location.
Digital geometry is the study of geometric or topologic properties of sets of pixels or voxels. It often
attempts to obtain quantitative information and topologic characterizations of pictures or to transform
pictures into “simpler” topologically equivalent pictures.
In the context of digital geometry, pixels and voxels refer to the elements of the medium on which
pictures reside, which is usually defined by a regular orthogonal grid. Thus, they are locations defined
by grid coordinates, and have values defined by a picture.
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In general, a digital grid is a set of points in nℜ . However, under a proper coordinate system, these
points represent the points in nZ where Z is the set of all integers. Moreover, most imaging systems
acquire images in regular orthogonal grids.
Objects in digital geometry often do not behave like Euclidean objects. For example, we can define
adjacency relations between pixels, whereas distinct points cannot be “adjacent” in Euclidean
geometry. Grid points are isolated points in the (real) plane, but, in the grid, adjacency relations
between grid points can be defined.
A digital space D is an ordered pair (G, α) where G is the underlying digital grid and α is a binary
relation on G that indicates the adjacency relationship between every two points in G. The notion of
adjacency is useful to define a path in a digital space and the boundary of a digital object. Only hard
adjacencies will be considered, because the interpretation of fuzziness of adjacencies in the context of
a path is not clear. In other words, α : Zn × Z
n →0, 1. Two points p, q ∈ Zn
are called adjacent if and
only if α(p, q) = 1. We can chose α to be standard 4- or 8-adjacency in 2D, 6-, 18-, or 26-adjacency in
3D, and their higher dimensional analogs.
Two adjacent points are often referred to as neighboring points to each other. For p = (x, y) ∈ Z2, we
define the neighborhoods
N4(p) = (x,y), (x+1,y), (x−1,y), (x,y+1), (x,y−1)
and
N8(p) = N4(p) ∪(x+1,y+1), (x+1,y−1), (x−1,y+1), (x−1,y−1) .
Two grid points p,q ∈ Z2 are called 4-adjacent or proper 4-neighbors (8-adjacent or proper 8-
neighbors) iff p ≠ q and p ∈ N4(q) (p ∈ N8(q)).
A set M (e.g., of grid points) is called connected if for all p,q ∈M there exists a sequence npp ,,0 … of
elements of M such that p0 = p, pn = q, and pi is adjacent to pi−1 (1 ≤ i ≤ n); such a sequence is called a
path and is said to join p and q in M. Maximal connected subsets of M are called (connected)
components of M.
(a) (b) (c) (d)
Figure 6. (a) Cell 1-adjacency and pixel 4-adjacency (left); (b) Neighborhoods (right)
(c) Cell 0-adjacency and pixel 8-adjacency (left); (d) Neighborhoods (right)
Figure 7. Cell- and voxel-adjacency for 3D digital objects
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3.2. FUZZY DIGITAL GEOMETRY
When an object in a scene is represented by a set of pixels in a picture, it may not always be obvious
which pixels belong to the object and which pixels belong to the background or another object. This
suggests that an object can be viewed as a fuzzy set which is specified by some membership function
defined on all picture points.
The concept of fuzzy digital geometry has been introduced by Rosenfeld (1984) and plays a key role
in many image processing applications. The application areas of fuzzy geometry are image
representation, enhancement and segmentation. The process of converting the input image into a fuzzy
set by indicating, for each pixel, the degree of membership to the object, is referred to as “fuzzy
segmentation”. The most straightforward way to perform fuzzy segmentation is to scale grey-levels of
an image to be between 0 and 1. Such grey levels reflect the area coverage of a pixel by the object, and
can be naturally used as membership values. However, in most cases, more advanced segmentation
methods are required, especially since it is rarely sufficient to use only the brightness of pixels to
calculate fuzzy membership values. For example, fully segmented image can be generated by
combining the optimum automatic thresholding procedure with edge detection to produce
continuously connected object border.
The object of interest is represented as a discrete spatial fuzzy subset of a grid. It should be noted,
however, that the discrete fuzzy objects obtained from the digitization of scanned images (say, using a
grey-level scale) are affected by multiple distortions, due to limited representation resolution.
Consequently, their properties are significantly different with respect to those of corresponding
continuous fuzzy objects. Figure 8 shows a discrete fuzzy disk (a) versus a discrete fuzzy hole (c). The
crisp counterpart of a fuzzy disk (b) can be obtained by thresholding a fuzzy (gray-level) image: pixels
with a membership degree below the threshold are lost.
a) (b) (c)
Figure 8: (a) discrete fuzzy disk; (b) crisp (binary) disk; (c) discrete fuzzy hole
For any picture P with pixel values in the range 0, . . . ,Gmax, if we divide the pixel values by Gmax,
we obtain a picture P′ in which the pixel values are in the range [0,1], so that they can be regarded as membership values of the pixels in a fuzzy subset µ of P.
3.3. Fuzzy distances and distance transforms
There are two main approaches in measuring distances when considering fuzzy objects: the first one
basically compares only the membership functions representing the concerned fuzzy object, while the
other one combines spatial distance between objects and membership functions, thus taking into
account both spatial information and information related to the imprecision attached to the image
object.
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Distances between two points in a fuzzy set are typically addressed in order to find the best path in the
geodesic sense in a spatial fuzzy set.
Distances from a point to a set are used when computing distance from a point to a complement of a
fuzzy set, i.e., performing distance transform.
The distances between sets are used in shape matching.
3.3.1. Fuzzy geodesic distance
A geodesic distance between points in a fuzzy set was introduced by Bloch (2000), being defined
conditionally to a reference set X. Thus, a geodesic distance dX(x, y) from x to y is the length of a
shortest path from x to y, completely included in X. Let µ be a fuzzy set on the space S . The
definition of the geodesic distance relies on the degree of connectivity in µ between two points x and
y of S, as defined by Rosenfeld (1984),
=
∈)(minmax),(
),(),(tyxc
yxLtyxLµµ
where L(x, y) denotes a path from x to y, consisting of a sequence of points in S according to the
discrete connectivity defined on S. Let ),(* yxL denote the shortest path between x and y on which µc
is reached; this path is not necessarily unique and can be interpreted as a geodesic path descending as
little as possible in terms of membership degrees. Let l( ),(* yxL ) denote its length (the number of
points along the path). Then the geodesic distance in µ between x and y is defined as
( )),(
),(),(
*
yxc
yxLlyxd
µµ =
If ),( yxcµ = 0, then ∞=),( yxdµ , which corresponds to the result obtained for the classical geodesic
distance in the case where x and y belong to different connected components. The definition
corresponds to the classical geodesic distance computed at the −α cut of µ at level α = ),( yxcµ . In
this −α cut, x and y belong to the same connected component. The definition satisfies the following
set of properties:
• the distance between any two points is non-negative;
• the distance between x and y is the same as the distance between y and x;
• the distance equals zero only between two spatially identical points;
• the distance is defined by the shortest path between x and y that “goes out” of µ “as little
as possible”, and tends to infinity if it is not possible to find a path between x and y
without going through a point t such that µ (t) = 0;
• the distance decreases when µ (x) and µ (y) increase;
• the distance decreases when ),( yxcµ increases;
• the distance is equal to the classical geodesic distance if µ is crisp.
The triangular inequality is not satisfied, but from the given definition it is possible to derive a true
distance, satisfying triangular inequality, while keeping all other properties:
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( ) ( )
+=′
∈ ),(
),(
),(
),(min),(
**
ytc
ytLl
txc
txLlyxd
Stµµ
µ
where S is the whole image space.
Obviously, the geodesic distance between the centroid and fuzzy boundaries of a fuzzy shape
is a good candidate for generalizing the shape signature of discrete fuzzy shapes.
Chanussot et al. (2005) proposed two alternative generalizations of shaoe signature fo a discrete fuzzy
shape:
(1) by using the distance between the boundary points and the centroid, which consists of
the following steps:
• Compute the centroid coordinates.
• Detect the inner boundary of the lowest a-cut using 8-connectivity.
• Compute the signature using length estimation based on local steps for the corresponding
discrete straight line.
The signature of a discrete fuzzy shape, calculated using the (pseudo-)distance between the boundary
points and the centroid, is given by:
( )
( )∑=
−
⋅+
+=
kN
j
kkAk
ccAfuzzydiscrete
jyjxj
yxkCD
1
1_
)(),()(
,)(
µδ
µ
(2) as an average signature obtained from the −α cuts, which consists of the following
steps:
• Compute the centroid coordinates.
• For each α (if the data are quantized using 8 bits per pixel, the total number of α –cuts
is α total = 255).
– Compute the corresponding α –cut.
– Detect the inner boundary of the α –cut.
– Compute the shape signature of the α –cut.
• Resample all the signatures to the number of pixels of the longest obtained signature;
this does not necessary correspond to the signature of the lowest a-cut.
• Average the resampled signatures.
∑=
−− =total
kCDkCD resampledtotal
fuzzydiscrete
α
α
αα
1
2_ )(1
)(
where )(kCD resampled−α is the k th sample of the resampled signature obtained for one −α cut.
3.3.2. Distance transform
For a binary object, distance transform (DT) is a process that assigns a value at each location within
the object that is simply the shortest distance between that location and the complement of the object.
Most DT methods approximate the global Euclidean distance by propagating local distances between
neighboring pixels.
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Let S be a crisp subset of nℜ .We shall use S to denote its complement and Interior (S) to denote its
interior, which is the largest open set contained in S. The distance transform (DT) of S may be
represented as an image (x, DS(x)) | nx ℜ∈ on nℜ where DS is the DT value at x that is defined as
follows.
SyyxxDS ∈−= inf)( ,
where, inf gives the infimum of a set of positive numbers and . is the Euclidean norm. In digital
images, we always deal with bounded objects so that S is always nonempty.
3.3.3. Fuzzy distance transform in continuous spaces
However, this notion of hard DT cannot be applied on fuzzy objects in a meaningful way. The notion
of DT for fuzzy objects, called fuzzy distance transform (FDT), was introduced by Saha et al. (2002).
It is very important in many imaging applications because we often deal with situations with data
inaccuracies, graded object compositions, or limited image resolution..
Similar to ordinary DT, FDT of a fuzzy subset S in nℜ is an image on nℜ , which is denoted by a set
of pairs (x, ΩS(x)) | nx ℜ∈ ; ΩS(x) is the FDT value at x and is defined in the following way.
A path π in nℜ from a point nx ℜ∈ to another point (not necessarily distinct) ny ℜ∈ is a continuous
function π : [0, 1]→ nℜ such that π(0)=x and π(1)= y. The length of a path π in S, denoted by ΠS(π), is
the value of the following integration
( )∫=Π
1
0
)()()( dt
dt
tdtSS
ππµπ
i.e., )(πSΠ is the integral of membership values (in S) along π.
When a path passes through a low density (low membership) region, its length increases slowly and
the portion of the path in the complement of the support of S contributes no length. This approach is
useful to measure regional object depth, object thickness distribution, etc.
Let ζS(x, y) denote a subset of positive real numbers defined as
),( yxSζ = ),(|)( yxPS ∈Π ππ ,
i.e., ),( yxSζ is the set of all possible path lengths in S between x and y. The fuzzy distance from
nx ℜ∈ to ny ℜ∈ in S, denoted as ),( yxSω , is the infimum of ),( yxSζ ; i.e.,
),(inf),( yxyx SS ζω =
For any fuzzy subset S of nℜ and for any nyx ℜ∈, , the fuzzy distance Sω satisfies the metric
properties:
1. ,,0),( yxifyxS ==ω
2. ),,(),( xyyx SS ωω =
3. nSSS zanyforyzzyyx ℜ∈+≤ ),,(),(),( ωωω
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Furthermore, for any nonzero positive number θ ≤1 and for any nyx ℜ∈, , such that either x or y is in
Interior( ))(SθΘ ,
4. ,,0),( yxifyxS ≠>ω
The FDT value )(xSΩ at a point nx ℜ∈ is equal to the fuzzy distance between x and the closest (with
respect to ωS) point in )(SΘ . In other words, the value of )(xSΩ is defined as follows
)(|),(inf)( Syyxx SS Θ∈=Ω ω
Actually, the fuzzy distance Sω is a geodesic distance, which means that the shortest paths (when they
exist) in a fuzzy subset S between two points nyx ℜ∈, are not necessarily a straight line segment even
when S is convex. For example, consider a fuzzy disk where there are two possible membership values
within its support )(SΘ : points with high membership value are shown as dark gray and those with
low membership value (within the outer annular region) are shown as light gray. We also consider two
points x and y as illustrated in the figure below. Assuming the high membership value sufficiently
close to one and the low one close to zero, the shortest path between x, y should be contained within
the light gray region and therefore is not a straight line segment.
(a) Continuous space (b) Digital space
Figure 9. The shortest paths (when they exist) between two points in a convex fuzzy subset are not
necessarily a straight line segment.
3.3.4. Fuzzy distance transform in digital spaces
A fuzzy digital object O is a fuzzy subset (p, µO(p)) | p ∈ Zn, where µO :
nZ →[0, 1] specifies the
membership value at each point in the object.
The support )(OΘ of a fuzzy digital object O is the set of all points in nZ each having a nonzero
object membership value, i.e., )(OΘ = p | p ∈ nZ and µO(p) ≠ 0. A path π in a set S of points from a
point p∈S to another (not necessarily distinct) point q∈S is a sequence of points
q pp pp m == ,,, 21 … such that Spi ∈ for all mi ≤≤1 and jp is adjacent to 1+jp for all 1 ≤ j <
m. The length of the path is m.
A set of points S will be called path connected if and only if, for every two points p, q ∈ S, there is a
path in S from p to q. P(p, q) will denote the set of all paths in nZ from a point p ∈ nZ to another
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point q ∈ nZ . For the purpose of defining the length of a path, we use the notion of a link and its
length. A link is a path consisting of two points. The length of a link qp, in O may be defined in
different ways, e.g.,
(1) qpqp OO −×)(),(max µµ ,
(2) ( ) qpqp OO −×+ )()(2
1µµ , etc.
It may be noted that in both examples, the length of a link has two components—one coming from the
membership values at p and at q and the other from the distance between the two points.
The length )(πOΠ of a path q pp pp m === ,,, 21 …π is the sum of the lengths of all links on the
path, i.e.,
( )∑−
=
++ −×+=Π1
1
11)()(2
1)(
m
i
iiiOiOO pppp µµπ
πp,q ∈P(p, q) is one shortest path from p ∈ nZ to q ∈ nZ in O if )()( , ππ OqpO Π≤Π , ∀ ∈P(p, q).
Unlike the case of the continuous case, one shortest path always exists between two points in a
bounded digital fuzzy object.
Although, the existence of one shortest path is guaranteed, there may be multiple shortest paths
between two points in a fuzzy digital object between two points. We consider fuzzy digital objects
with bounded supports. The fuzzy distance, from p ∈ nZ to q ∈ nZ in O, denoted as ),( qpOω , is the
length of any shortest path in O from p to q. Therefore,
),( qpOω = )(min),(
ππ
OqpP
Π∈
.
For any digital cubic space ( )α,nZD = , for any digital object O , the fuzzy distance Oω satisfies the
metric properties for any nZqp ∈, :
1. ,,0),( qpifqpO ==ω
2. ),,(),( pqqp OO ωω =
3. nOOO Zranyforqrrpqp ∈+≤ ),,(),(),( ωωω
Furthermore, for any nZqp ∈, , such that either p or q is in )(OΘ ,
4. ,,0),( qpifqpO ≠≠ω
Consequently, the fuzzy distance Oω is a metric for )(OΘ .
In conclusion, the FDT value at a point p ∈ nZ in a fuzzy object O over a digital space, denoted by
)(pOΩ , is equal to the fuzzy distance between p and the nearest point in )(OΘ . In other words, the
value of )(pOΩ is defined as follows
),(min)()(
qpp OOq
O ωΘ∈
=Ω
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4. PROCRUSTEAN SHAPE ANALYSIS
The coordinates of selected landmarks for a crisp shape can be arranged in a pn× configuration
matrix A , or equivalently on a np×1 configuration vector )(Aveca = .
A configuration matrix for a discrete p-dimensional fuzzy shape A can be represented by vertical
concatenation of its −α level contours into a pnk × block matrix. Each one of the k sub-matrices
defined at each level α collects pn× landmarks: ( ))()()( 21 ααααAp
AA xxxA …= , where
10| 1 =<<<<=∈ kii ααααα ⋯⋯ .
A similar pnk × -dimensional configuration matrix can be defined for a crisp shape with multiple (say
k ) contours.
A configuration matrix A is not a proper shape descriptor, because it is not pose invariant. For any
similarity transformation, i.e. +ℜ∈s , )(pSOR∈ (the special orthogonal group, i.e. R is ( pp× )
matrix, s.t. IRR =′ ) and pt ℜ∈ , the configuration given by tRAs p′+1 describes the same shape as
A, where p1 is the 1×p vector )111( ′… . To obtain a true shape representation, location, scale and
rotational effects need to be filtered out. This is carried out by shape alignment, i.e. by establishing a
coordinate reference, commonly known as pose. A very popular alignment procedure is Procrustes
shape analysis, which provides a measure, Procrustes distance, that quantifies the dissimilarity of two
configurations, and which is invariant with respect to translation, scaling, and rotation. Procrustes
shape analysis also provides a way to define the average shape, the Procrustes mean shape, which can
be viewed as a representative class template.
The Extended Orthogonal Procrustes (EOP) problem is a least squares method for fitting a given
configuration matrix A to another given matrix B . It is based on the functional model
BtsARE p −′+= 1 and consists of minimizing the Procrustes distance between A and B (i.e. 2
FE ),
under choice of unknown similarity transformation parameters R , t and s . This leads to solving the
problem EEstR
′,,
min , subject to the orthogonality restriction IRR =′ .
Generalized Orthogonal Procrustes (GOP) analysis is a technique that provides least-squares
correspondence of more than two model points. The solution of the problem can be thought as the
search of the unknown optimal matrix W (also named consensus matrix), defined as follows:
mitRAsAEM ipiiiii ,,1;1ˆ…=′+==+
( )( )pni QQNEvec ⊗=Σ 2,0~)( σ
where iE is the random error matrix in normal distribution, Σ is the covariance matrix, nQ is the
cofactor matrix of the n points, pQ is the cofactor matrix of the p coordinates of each point, ⊗
stands for the Kronecker product, and 2σ is the variance factor.
Let mACm
i i∑ ==
1
ˆ be the geometrical centroid of the transformed matrices. Therefore, Generalized
Orthogonal Procrustes problem can be solved minimizing
14
( ) ( )∑∑ ==
−′
−=−m
i ii
m
i i CACAtrmCAm1
2
1
ˆˆˆ
Crosilla and Beinat (2002) proved that the shape mean (centroid) C corresponds to the least squares
estimation M of the true value M : ∑ ===
m
i iAMC1
ˆˆ .
In the case of multiple-contour crisp shapes we can benefit from the Extended Orthogonal Procrustes
method in order to find mutual distances between shape pairs and from the Generalized Orthogonal
Procrustes technique in order to estimate the Procrustes mean shape of a collection of shapes. This is
illustrated in figures 10 and 11.
On the other hand, dealing with the case of fuzzy shapes needs more advanced Procrustean techniques,
which allow us to consider weighted distances between points placed on −α level contours with
different membership degrees. This leads to solve a Weighted Orthogonal Procrustes (WOP) problem.
0 100 200 300 400 500 6000
100
200
300
400
500
1
23
4
5
6
7
8
9
10
Figure 10: Ten double-contour star-shaped 2D objects with 3, 4 and 5 “lobes”
-0.1 -0.05 0 0.05 0.1-0.1
-0.05
0
0.05
0.1
Figure 11: Procrustes mean shape (shape centroid)
A weighting matrix W of the residual E (defined above) is now introduced and the minimization
problem becomes:
2min
FEW
15
subject to orthogonality restriction IRR =′ , 1)det( =R .
Typically, an iterative method is needed to derive a solution to WOP.
5. A GENERALIZATION OF K-MEANS ALGORITHM FOR CLUSTERING FUZZY
SHAPES
K-means is a commonly used data clustering for partitioning data points into disjoint groups such that
data points belonging to same cluster are similar, while data points belonging to different clusters are
dissimilar. The main idea is to define k centroids, one for each cluster, and to take each point
belonging to a given data set and associate it to the nearest centroid. When no point is pending, an
early groupage is done. Next, we need to re-calculate k new centroids of the clusters resulting from the
previous step. After we have these k new centroids, a new binding has to be done between the same
data set points and the nearest new centroid. We continue this loop until no more changes are done.
Clustering of objects or images of objects, according to the shapes of their boundaries is of a key
importance in computer vision and pattern recognition. This paper was intended to pay attention to this
reason by proposing a generalization of K-means algorithm in order to integrate Procrustean metrics
and full mean shape estimation, in a way making it able of clustering objects with either multiple or
fuzzy contours.
We present the algorithm in pseudo-code, as follows:
• Make initial guesses for the mean shapes 1v , 2v , …, kv , by choosing the first k shapes
from a random permutation.
• While any change still exists in any mean shape
o Calculate all pair-wise Procrustes distances between shapes using the Extended
Orthogonal Procrustes algorithm
o Use the estimated mean shapes to assign the shape samples into clusters
o For i from 1 to k
Replace iv with the mean shape of all of the samples for cluster i , using the
Generalized Orthogonal Procrustes algorithm
o end_for
• end_while
The resulting mean shapes for each one of the 3 clusters are shown below.
-0.1 -0.05 0 0.05 0.1-0.1
-0.05
0
0.05
0.1
Figure 12: First cluster: 2, 3, 8. Mean shape and cluster members
16
-0.1 -0.05 0 0.05 0.1-0.1
-0.05
0
0.05
0.1
Figure 13: Second cluster: 1, 6, 7, 9. Mean shape and cluster members
-0.1 -0.05 0 0.05 0.1-0.1
-0.05
0
0.05
0.1
Figure 14: Third cluster: 4, 5, 10. Mean shape and cluster members
Our method is thus graphically validated.
Similar results can be obtained by transforming each shape contour into a time series and applying a
time series clustering techniques (see section 6.3).
6. TIME SERIES KNOWLEDGE MINING
In a recent paper we have published in Fuzzy Economic Review, we proposed a time series knowledge
mining framework, designed to favor the synergy between subsequence time series clustering and
predictive tools such as Hidden Markov Models. Many tasks for temporal data mining rely heavily on
the choice of the representation scheme and the dissimilarity measure. We first presented a detailed
representation taxonomy for numeric and symbolic time series and comprehensive categorization of
distance measures. Subsequence time series clustering methods with a sliding window were addressed
next and a generalization of Fuzzy C-Means algorithm based on the dynamic time warping distance
was proposed as a very effective solution. This involves a shape-based distance tolerant to phase shifts
in time or accelerations/decelerations along the time axis. It also allows to determine the degree to
which set-defined objects, such as subsequence time series and their cluster centroids (similar in
nature) differ from each other. Finally we discussed the integration of clustering algorithms with
probabilistic predictive tools, such as discrete Markov chains or hidden Markov models. We applied
these techniques to clustering of non-overlapping sequences extracted from Standard and Poor’s 500
stock index historical data and we suggested different integrations with markovian models to improve
the predictive power.
17
6.1. TIME SERIES REPRESENTATIONS
Time series data mining involves techniques and methods adjusted in a way that they take into
consideration the temporal nature of data. Actually, it is concerned with data mining using numeric
and symbolic time series and sequences. Preprocessing, noise filtering, outlier detection, scaling,
representation methods, (dis)similarity measures, subsequence matching, indexing, clustering,
classification, segmentation, anomaly detection, motif discovery, rule discovery and prediction are
typical tasks for time series data mining.
The selection of a suitable time series representation is interconnected with the selection of a distance
measure. On the other hand, many tasks, such as clustering and classification of time series, rely
heavily on the similarity measure and the representation scheme selected.
Most of the time series are presented using numeric representations. However, transforming the
original time series in some form other than using the actual values is mainly a way of reducing their
intrinsic high dimensionality. Therefore, many methods have been proposed in order to convert
numeric time series to symbolic representations.
6.2. TIME SERIES DISTANCES
For short time series, shape-based distances are commonly used to compare their overall appearance.
The Euclidean distance is the most widely used shape-based distance for the comparison of numeric
time series. More generally, other pL norms, i.e., Manhattan for 1=p , Euclidean for 2=p ,
Maximum for ∞=p , can be used as well, putting different emphasis on large deviations.
The usage of the Euclidean distance is subject to the constraint that both time subsequences are of the
same length w. Let 11 11,, −+= wmm yyS … and 12 22
,, −+= wmm yyS … be two subsequences with length w
of time series nyyY ,,1 …= , where 2,1,11 =+−≤≤ jwnm j . Each subsequence will be represented
as a vector in a w-dimensional vector space. Thus we can define the dissimilarity between
subsequences 1S and 2S as the Euclidean distance between the two −w dimensional vectors measured
by the 2L norm:
( )∑=
−+−+ −=w
i
imim yySSL1
2
11212 21),(
Figure 15. The Euclidean distance
There are several pitfalls when using pL distances on time series: it does not allow for different
baselines in the time sequences; it is very sensitive to phase shifts in time; it does not allow for
acceleration and deceleration along the time axis (time warping). Another problem with pL distances
of time series is when scaling and translation of the amplitudes or the time axis are considered, or
S1
S2
18
when outliers and noisy regions are present. Care has to be taken in choosing the transformations to
obtain a time series distance measure that is meaningful to the application. A common example where
a translation or scaling is not desired is the comparison of stock prices. The absolute value of a stock is
usually not as interesting as the shape of up and down movements. A linear transformation or
normalization to a fixed or zero mean and unit variance can be a satisfactory solution for this case.
However, a number of non-metric distance measures have been defined to overcome some of these
problems. Distance measures that are robust to extremely noisy data will typically violate the
triangular inequality, i.e., are non-metric. This happens because such measures do not consider equally
all parts of the time series. Examples of such non-metric distance measures are presented in what
follows.
Small distortions of the time axis are commonly addressed with non-uniform time warping, more
precisely with Dynamic Time Warping (DTW). The DTW distance is an extensively used technique in
speech recognition and allows warping of the time axes (acceleration–deceleration of signals along the
time dimension) in order to align the shapes of the two times series better. The two series can also be
of different lengths. The optimal alignment is found by calculating the shortest warping path in the
matrix of distances between all pairs of time points under several constraints. The point-wise distance
is usually the Euclidean or Manhattan distance.
Let us consider two sequences (of possibly different lengths) ,, 1 nqqQ …= and ,,, 1 mccC …= .
To align two sequences using DTW, we construct an n-by-m matrix where the (ith, j
th) element of the
matrix contains the distance d(qi , cj) between the two points qi and cj (i.e. 2)(),( jiji cqcqd −= ). Each
matrix element ),( ji corresponds to the alignment between the points qi and cj . A warping path W is
a contiguous set of matrix elements that defines a mapping between Q and C. The kth element of W is
defined as kk jiw ),(= , so we have:
,,,,,, 21 Kk wwwwW ……= 1),max( −+<≤ nmKnm
The warping path is typically subject to several constraints.
• Boundary conditions: w1 = (1, 1) and wK = (m, n).
• Continuity: Given wk = (a, b), then ),(1 bawk′′=− , where 1≤′− aa and 1≤′− bb . This restricts the
allowable steps in the warping path to adjacent cells.
• Monotonicity: Given wk = (a, b), then ),(1 bawk′′=− , where 0≥′− aa and 0≥′− bb . This forces
the points in W to be monotonically spaced in time.
There are exponentially many warping paths that satisfy the above conditions, however we are
interested only in the path which minimizes the warping cost:
KwCQDTWK
i
k
= ∑
=1
min),(
The K in the denominator is used to compensate for the fact that warping paths may have different
lengths.
19
This path can be found efficiently using dynamic programming to evaluate the following recurrence,
which defines the cumulative distance ),( jiγ as the distance d(i, j) found in the current cell and the
minimum of the cumulative distances of the adjacent elements:
),( jiγ = d(qi, cj) + min γ (i −1, j −1), γ (i −1, j), γ (i, j −1)
The Euclidean distance between two sequences can be seen as a special case of DTW where the kth
element of W is constrained such that kjijiw kk === ,),( .
The warping path is also constrained in a global sense by limiting how far it may stray from the
diagonal. The subset of the matrix that the warping path is allowed to visit is called the warping
window. The two most common constraints in the literature are the Sakoe-Chiba band and the Itakura
parallelogram. We can view a global or local constraint as constraining the indices of the warping path
kk jiw ),(= , such that j − r ≤ i ≤ j + r, where r is a term defining the allowed range of warping, for a
given point in a sequence. In the case of the Sakoe-Chiba band, r is independent of i; for the Itakura
parallelogram, r is a function of i.
Figure 16. Optimal warping path with the Sakoe-Chiba band as global constraints
Figure 17. Aligning two time sequences using DTW
6.3. TIME SERIES CLUSTERING
The idea in subsequence time series clustering is as follows. Just a single long time series is given at
the start of the clustering process, from which we extract short series with a sliding window. The
resulting set of subsequences are then clustered, such that each time series is allowed to belong to each
cluster to a certain degree, because of the fuzzy nature of the fuzzy c-means algorithm we use. The
window width and the time delay between consecutive windows are two key choices. The window
width depends on the application; it could be some larger time unit (e.g., 24 days for time series
sampled as daily S&P 500 stock index, in our application). Overlapping or non-overlapping windows
can be used. If the delay is equal to the window width, the problem is essentially converted to non-
overlapping subsequence time series clustering. We will follow this approach, being motivated by the
Keogh’s criticism, where using overlapping windows has been shown to produce meaningless results.
When using a time delay of one, almost all time points contribute equally to each position within the
sliding window. Experiments with k-Means showed that clusters of sine waves are produced, the
prototypes of which add up to the constant function. Using larger time delays for placing the windows
20
does not really solve the problem as long as there is some overlap. Also, the less overlap, the more
problematic the choice of the offsets becomes.
In a recent paper we have generalized the fuzzy c-means algorithm to subsequence time series
clustering. Since clustering relies strongly on a good choice of the dissimilarity measure, this leads to
adopting an appropriate distance, depending on the very nature of the subsequence time series: type of
representation (numeric/symbolic), presence of time axis shifts (uniform/non-uniform time warping),
etc.
Three shape-based distances between numeric time series have been used as alternative choices in our
implementation of fuzzy c-means algorithm: the 2L (Euclidian) distance, the LB-Keogh’s distance
and the DTW distance.
DTW is a much more robust distance measure for time series than 2L , allowing similar shapes to
match even if they are out of phase in the time axis.
Figure 18. Fuzzy c-means clustering of time sequences using the DTW distance
7. FUZZY TIME SERIES
Typically, in conventional time series analysis, we assume that the generating mechanism is
probabilistic and that the observed values …… ,,,, 21 txxx are realizations of stochastic processes
…… ,,,, 21 tXXX . In contrast to the conventional time series, the observations of fuzzy time series
are fuzzy sets (the observations of conventional time series are real numbers).
Fuzzy randomness arises when random variables – e.g. as a result of changing boundary conditions –
cannot be observed with exactness. Fuzzy random variables may also be interpreted as fuzzified
21
random variables, as the random event can only be observed in an uncertain manner. If the fuzzy
random function is solely dependent on time, a fuzzy random process is obtained.
Fuzzy random processes can be used for modeling time series with fuzzy data. In other words,
imprecise data at equally spaced discrete time points are modeled as fuzzy variables. The set of this
discrete fuzzy data forms a fuzzy time series.
Fuzzy time series are regarded as realizations of fuzzy random processes. Fig. 20 gives illustration of
such a realization. For example, it can be obtained by a fuzzification of numerical data based on the
histograms of each weekday. The time series thus obtained is assumed to be stationary.
Song and Chissom (1993) give a thorough treatment of time series models. They define a fuzzy time
series as follows.
Let Y(t) (t=…, 0, 1, 2,…), a subset of ℜ , be the universe of discourse on which fuzzy sets if (t) (i =
1,2,…) are defined and let F(t) be a collection of if (t). Then, F(t) is called a fuzzy time series on Y(t)
(t = …, 0, 1, 2,…).
Figure 19. Fuzzy functions as realizations of a fuzzy random function
Figure 20. Time series with fuzzy data
22
In the case of fuzzy time series, the fuzzy relational equations can be employed as models.
Analogously to the conventional time series models, it is assumed that the observation at the time t
accumulates the information of the observation at the previous times.
Let F(t) and F(t−1) be fuzzy time series on Y(t) and Y(t−1) (t = …, 0, 1, 2,…), for any jf (t)∈F(t),
where j∈J, there exists an if (t−1)∈F(t−1) where i∈I , I and J are indices sets for F(t−1) and F(t),
respectively, such that there exists a first-order fuzzy relation )1,( −ttR and
)1,()1()( −−= ttRtftf ijij , where “ ° ” is the composition, then F(t) is said to be caused by F(t-1)
only. Denote this as if (t−1) → jf (t) or equivalently F(t−1) → F(t).
8. CONCLUSIONS
The main area of interest for this paper was to extend the methods of transforming a shape closed
contour to a time series in a fuzzy context, based on recent advances in fuzzy digital geometry. For
developing this unitary methodological framework, allowing to deal with both fuzzy shapes and fuzzy
time series as equivalent objects, the concept of fuzzy geodesic distance is foundational.
However, apart from this opportunity, robust advances in fuzzy time series area of research provide us
with powerful mathematical machinery for modeling and predicting fuzzy time series generated in
terms of imprecise data as realizations of fuzzy random processes.
REFERENCES
[1] Berndt D. J. and J. Clifford. “Finding patterns in time series: A dynamic programming approach”.
In U. M. Fayyad, G. Piatetsky-Shapiro, P. Smyth, and R. Uthurusamy, editors, Advances in
Knowledge Discovery and Data Mining, pages 229-248. AAAI Press, 1996.
[2] Bloch I. “On fuzzy distances and their use in image processing under imprecision“. Pattern
Recognition, 32:1873–1895, 1999.
[3] Bloch I. “Geodesic balls in a fuzzy set and fuzzy geodesic mathematical morphology“. Pattern
Recognition, 33:897–905, 2000.
[4] Bloch I. and H. Maître. “Fuzzy mathematical morphologies: A comparative study“. Pattern
Recognition, 28(9):1341–1387, 1995.
[5] Borgefors G. “Distance transformations in digital images“. Computer Vision, Graphics, and
Image Processing, 34:344–371, 1986.
[6] Buckley J. and E. Eslami. “Fuzzy plane geometry I: Points and lines“. Fuzzy Sets and Systems,
86:179–187, 1997.
[7] Bicego M., V. Murino, and M. Figueiredo. “Similarity-based classification of sequences using
Hidden Markov Models”. Pattern Recognition, 37(12):22812291, 2004.
[8] Chanussot J., Nyström I., Sladoje S. - “Shape signatures of fuzzy star-shaped sets based on
distance from the centroid“, Pattern Recognition Letters 26 (2005) 735–746.
[9] Chaudhuri B. “Some shape definitions“, Pattern Recognition Letters, 12:531–535, 1991.
[10] Chen, S.-M. & Chung, N.-Y. “Forecasting enrollments using high-order fuzzy time series and
genetic algorithms“, International Journal of Intelligent Systems, 21, 2006, 485–501.
[11] Chen, S.-M. “Forecasting enrollments based on high-order fuzzy time series“, Cybernetics and
Systems: An International Journal, 33, 2002, 1-16.
[12] Chiu B., E. Keogh and S. Lonardi. “Probabilistic discovery of time series motifs”. Proceedings of
the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining,
2003
23
[13] Crosilla F., Beinat A. (2002) “Use of Generalised Procrustes Analysis for Photogrammetric
Block Adjustment by Independent Models", ISPRS Journal of Photogrammetry and Remote
Sensing, Elsevier, 56(3), 195-209
[14] Dryden I. L., Mardia K. W. (1998) Statistical shape analysis. John Wiley & Sons, Chichester,
England, 83-107.
[15] Fujimaki R., S. Hirose and T. Nakata. “Theoretical analysis of subsequence time-series clustering
from a frequency-analysis viewpoint”. SIAM International Conference on Data Mining
(SDM2008).
[16] Ge X. and P. Smyth. “Deformable Markov model templates for time-series pattern matching”. In
R. Ramakrishnan, S. Stolfo, R. Bayardo, and I. Parsa, editors, Proceedings of the 6th ACM
SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD'00), pages
81-90. ACM Press, 2000.
[17] Georgescu V. “A time series knowledge mining framework exploiting the synergy between
subsequence clustering and predictive Markovian models”, Fuzzy Economic Review, vol. XIV,
No.1, 2009, pp.41-66
[18] Georgescu V. “Clustering of Fuzzy Shapes by Integrating Procrustean Metrics and Full Mean
Shape Estimation into K-Means Algorithm”, Proceedings of the 13th IFSA World Congress and
the 6th Conference of EUSFLAT, 20-24 July 2009, Lisbon, Portugal
[19] D. Gunopulos and G. Das. “Time series similarity measures and time series indexing”. ACM
SIGMOD Conference, Santa Barbara, CA., 2001
[20] Huarng, K. “Effective lengths of intervals to improve forecasting in fuzzy time series“, Fuzzy
Sets and Systems, 123, 2001, 387-394.
[21] Huarng, K. & Yu, Tiffany H.-K. “Ratio-based lengths of intervals to improve fuzzy time series
forecasting“, IEEE Transactions on Systems, Man, and Cybernetics-Part B : Cybernetics, 36(2),
April, 2006, 328-340.
[22] Klette, R., Rosenfeld A. Digital Geometry. Geometric Methods for Digital Picture Analysis,
Morgan Kaufmann Publishers, 2004
[23] Keogh E. and M. Pazzani. “An enhanced representation of time series which allows fast and
accurate classification, clustering, and relevance feedback”. In R. Agrawal, P. E. Stolorz, and G.
Piatetsky-Shapiro, editors, Proceedings of the 4th International Conference on Knowledge
Discovery and Data Mining (KDD'98), pages 239-241. AAAI Press, 1998.
[24] Keogh E. and M. J. Pazzani. “Scaling up dynamic time warping to massive datasets”. In J. M.
Zytkow and J. Rauch, editors, Proceedings of the 3rd European Conference on Principles of
Data Mining and Knowledge Discovery (PKDD'99), pages 1-11. Springer, 1999.
[25] Keogh E., J. Lin and W. Truppel. “Clustering of time series subsequences is meaningless:
implications for previous and future research”. Proceedings of the 3rd IEEE International
Conference on Data Mining, 2003 pp. 115–122.
[26] Keogh E. and C. A. Ratanamahatana. “Exact indexing of dynamic time warping”. Knowledge
and Information Systems, 7, 2005, pp. 358-386
[27] Keogh E., S. Chu, D. Hart, and M. Pazzani. “Segmenting time series: A survey and novel
approach”. In M. Last, A. Kandel, and H. Bunke, editors, Data Mining In Time Series Databases,
pages 1-22. World Scientific, 2004
[28] Lee, L.-W., Wang, L.-H., Chen, S.-M. & Leu, Y.-H. “Handling forecasting problems based on
two-factors high-order fuzzy time series“, IEE Transactions on Fuzzy Systems, 2006.
[29] Lin J., E. Keogh, S. Lonardi, and B. Chiu. “A symbolic representation of time series, with
implications for streaming algorithms”. In Proceedings of the 2003 ACM SIGMOD Workshop on
Research Issues in Data Mining and Knowledge Discovery, pages 2-11. ACM Press, 2003a.
[30] Lin J., M. Vlachos, E. Keogh, and D. Gunopulos. “A MPAA-based iterative clustering algorithm
augmented by nearest neighbors search for time-series data streams”. In T. B. Ho, D. Cheung,
and H. Liu, editors, Proceedings of the 9th Paci_c-Asia Conference on Knowledge Discovery and
Data Mining (PAKDD'05), pages 333-342. Springer, 2005.
[31] Lin W., M. Orgun, and G. Williams. “Temporal data mining using Hidden Markov-local
polynomial models”. In D. Cheung, G. Williams, and Q. Li, editors, Proceedings of the 5th
Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD'01), pages 324-
335. Springer, 2001.
24
[32] Mallat S. G. A Wavelet Tour of Signal Processing. Academic Press, 1999.
[33] Mörchen F. “Time series feature extraction for data mining using DWT and DFT“. Technical
Report 33, Department of Mathematics and Computer Science, Philipps-University Marburg,
Germany, 2003.
[34] Mörchen F. and A. Ultsch. “Discovering temporal knowledge in multivariate time series“. In C.
Weihs and W. Gaul, editors, Proceedings of the 28th Annual Conference of the German
Classification Society (GfKl'04), pages 272-279. Springer, 2005a.
[35] Pal S. “Fuzzy skeletonization of an image“. Pattern Recognition Letters, 10:17–23, 1989.
[36] Rabiner L. R. “A tutorial on Hidden Markov Models and selected applications in speech
recognition”. Proceedings of IEEE, 77(2):257-286, 1989.
[37] Rosenfeld A. “The fuzzy geometry of image subsets“. Pattern Recognition Letters, 2 (1984) 311–
317.
[38] Saha P. K., F. W. Wehrli, and B. R. Gomberg. “Fuzzy distance transform: Theory, algorithms,
and applications. Computer Vision and Image Understanding“, 86:171–190, 2002.
[39] Song, Q. & Chissom, B. S. “Forecasting enrollments with fuzzy time series-part I“, Fuzzy Sets
and Systems, 54, 1993, 1-9.
[40] Song, Q. & Chissom, B. S. “Fuzzy time series and its models“, Fuzzy Sets and Systems, 54, 1993,
269-277.
[41] Song, Q. & Chissom, B. S. “Forecasting enrollments with fuzzy time series-part“, Fuzzy Sets and
Systems, 62, 1994, 1-8.