Yang Mingqiang

A survey of shape feature extraction techniques

1 Introduction "A picture is worth one thousand words". This proverb comes from Confucius a Chinese philosopher before about 2500 years ago. Now, the essence of these words is universally understood. A picture can be magical in its ability to quickly communicate a complex story or a set of ideas that can be recalled by the viewer later in time.

Visual information plays an important role in our society, it will play an increasingly pervasive role in our lives, and there will be a growing need to have these sources processed further. The pictures or images are used in many application areas like architectural and engineering design, fashion, journalism, advertising, entertainment, etc. Thus it provides the necessary opportunity for us to use the abundance of images. However, the knowledge will be useless if one can't nd it.

In the face of the substantive and increasing apace images, how to search and to retrieve the images that we interested with facility is a fatal problem: it brings a necessity for image retrieval systems. As we know, visual features of the images provide a description of their content. Content-based image retrieval (CBIR), emerged as a promising mean for retrieving images and browsing large images databases. CBIR has been a topic of intensive research in recent years. It is the process of retrieving images from a collection based on automatically extracted features. This paper focuses on presenting the existing approaches of shape-based feature extraction. Ecient shape features must present some essential properties such as: identiability: shapes which are found perceptually similar by human have the same feature dierent from the others. translation, rotation and scale invariance: the location, rotation and scaling changing of the shape must not aect the extracted features.

ane invariance: the ane transform performs a linear mapping from 2D coordinates to other 2D coordinates that preserves the "straightness" and "parallelism" of lines. Ane transform can be constructed using sequences of translations, scales, ips, rotations and shears. The extracted features must be as invariant as possible with ane transforms.

noise resistance: features must be as robust as possible against noise, i.e., they must be the same whichever be the strength of the noise in a give range that aects the pattern. occultation invariance: when some parts of a shape are occulted by other objects, the feature of the remaining part must not change compared to the original shape. statistically independent: two features must be statistically independent. This represents compactness of the representation. reliable: as long as one deals with the same pattern, the extracted features must remain the same.

In general, shape descriptor is some set of numbers that are produced to describe a given shape feature. A descriptor attempts to quantify shape in ways that agree with human intuition (or task-specic requirements). Good retrieval accuracy requires a shape descriptor to be able to eectively nd perceptually similar shapes from a database. Usually, the descriptors are in the form of a vector. Shape descriptors should meet the following requirements: the descriptors should be as complete as possible to represent the content of the information items. the descriptors should be represented and stored compactly. The size of descriptor vector must not be too large. the computation of distance between descriptors should be simple; otherwise the execution time would be too long.

Shape feature extraction and representation plays an important role in the following categories of applications: shape retrieval: searching for all shapes in a typically large database of shapes that are similar to a query shape.

Usually all shapes within a given distance from the query are determined or the rst few shapes that have the smallest distance. shape recognition and classication: determining whether a given shape matches a model suciently, or which of representative class is the most similar. shape alignment and registration: transforming or translating one shape so that it best matches another shape, in whole or in part. shape approximation and simplication: constructing a shape of fewer elements (points, segments, triangles, etc.), that is still similar to the original.

Many shape description and similarity measurement techniques have been developed in the past. A number of new techniques have been proposed in recent years. There are 3 main dierent classication methods as follows:

Contour-based methods and region-based methods [START_REF]MPEG-7 overview (version 10)[END_REF]. This is the most common and general classication and it is proposed by MPEG-7. It is based on the use of shape boundary points as opposed to shape interior points. Under each class, dierent methods are further divided into structural approaches and global approaches. This sub-class is based on whether the shape is represented as a whole or represented by segments/sections (primitives).

Space domain and transform domain. Methods in space domain match shapes on point (or point feature) basis, while feature domain techniques match shapes on feature (vector) basis.

Information preserving (IP) and non-information preserving (NIP). IP methods allow an accurate reconstruction of a shape from its descriptor, while NIP methods are only capable of partial ambiguous reconstruction. For object recognition purpose, IP is not a requirement.

Unlike the traditional classication, the approaches of shape-based feature extraction and representation are classied according to their processing approaches. The gure 1 shows the hierarchy of the classication of shape feature extraction 2 Shape parameters Basically, shape-based image retrieval consists of the measuring of similarity between shapes represented by their features. Some simple geometric features can be used to describe shapes. Usually, the simple geometric features can only discriminate shapes with large dierences; therefore, they are usually used as lters to eliminate false hits or combined with other shape descriptors to discriminate shapes. They are not suitable to be stand alone shape descriptors. A shape can be described by dierent aspects. These shape parameters are Center of gravity, Axis of least inertia, Digital bending energy, Eccentricity, Circularity ratio, Elliptic variance, Rectangularity, Convexity, Solidity, Euler number, Proles, Hole area ratio. They will be introduced in this section.

Center of gravity

The center of gravity is also called centroid. Its position should be xed in relation to the shape.

If a shape is represented by its region function Eq. ??, its centroid (gx, gy) is:

gx = 1 N N i=1 xi gy = 1 N N i=1 yi (1)
where N is the number of point in the shape, (xi, yi) ∈ {(xi, yi) | f (xi, yi) = 1}.

If a shape is represented by its contour Eq. ??, the position of its centroid is given below:

gx = 1 6A N -1 i=0 (xi + xi+1)(xiyi+1 -xi+1yi) gy = 1 6A N -1 i=0 (yi + yi+1)(xiyi+1 -xi+1yi) (2)
where A is the contour's area and

A = 1 2 N -1 i=0 (xiyi+1 -xi+1yi) (3)
The position of shape centroid is xed with dierent points distribution on a contour. One can notice that the position of the centroid in Figure 2 is xed no matter how the points distribution is.

(a) (b)

The dots are points distributed on contour by uniformly (a) and non-uniformly (b). The star is the centroid of original contour; the inner dot is the centroid of sampled contour.

Figure 2: Centroid of contour So using Eq. 2, we can obtain the genuine centroid of a contour under whatever the contour is normalized.

Axis of least inertia

The axis of least inertia is unique to the shape. It serves as a unique reference line to preserve the orientation of the shape. The axis of least inertia (ALI) of a shape is dened as the line for which the integral of the square of the distances to points on the shape boundary is a minimum.

Since the axis of inertia pass through the centroid of a contour, to nd the ALI, transfer the shape and let the centroid of the shape be the origin of Cartesian coordinate system. Let x sin θ -y cos θ = 0 be the parametric equation of ALI.

The slope angle θ is estimated as follows:

Let α be the angle between the axis of least inertia and the x-axis. The inertia is given by [START_REF] Horn | Robot Vision[END_REF][START_REF] Tsai | Object recognition by a linear weight classier[END_REF]:

I = 1 2 (a + c) - 1 2 (a -c) cos(2α) - 1 2 bsin(2α) where a = N -1 i=0 x 2 i , b = 2 N -1 i=0 xiyi, c = N -1 i=0 y 2 i . Hence, dI dα = (a -c) sin(2α) -b cos(2α) d 2 I dα 2 = 2(a -c) cos(2α) + 2b sin(2α) Let dI/dα = 0, we obtain α = 1 2 arctan b a -c , - π 2 < α < π 2
The slope angle θ is given by

θ = α + π 2 if d 2 I dα 2 < 0 α otherwise

Average bending energy

Average bending energy BE is dened by

BE = 1 N N -1 s=0 K(s) 2
where K(s) is the curvature function, s is the arc length parameter, and N is the number of points on a contour [START_REF] Loncaric | A survey of shape analysis techniques[END_REF].

In order to compute the average bending energy more eciently, Young et. al. [START_REF] Young | An analysis technique for biological shape[END_REF] did the Fourier transform of the boundary and used Fourier coecients and Parseval's relation.

One can prove that the circle is the shape having the minimum average bending energy.

Eccentricity

Eccentricity is the measure of aspect ratio. It is the ratio of the length of major axis to the length of minor axis. It can be calculated by principal axes method or minimum bounding rectangle method.

Principal axes method

Principal axes of a given shape can be uniquely dened as the two segments of lines that cross each other orthogonally in the centroid of the shape and represent the directions with zero cross-correlation [START_REF] Peura | Eciency of simple shape descriptors[END_REF]. This way, a contour is seen as an instance from a statistical distribution. Let us consider the covariance matrix C of a contour:

C = 1 N N -1 i=0 xi -gx yi -gy xi -gx yi -gy T = cxx cxy cyx cyy (4)
where

cxx = 1 N N -1 i=0 (xi -gx) 2 cxy = 1 N N -1 i=0 (xi -gx)(yi -gy) cyx = 1 N N -1 i=0 (yi -gy)(xi -gx) cyy = 1 N N -1 i=0 (yi -gy) 2
G(gx, gy) is the centroid of the shape. Clearly, here cxy = cyx. The lengths of the two principal axes equal the eigenvalues λ1 and λ2 of the covariance matrix C of a contour, respectively.

So the eigenvalues λ1 and λ2 can be calculated by

det(C -λ1,2I) = det cxx -λ1,2 cxy cyx cyy -λ1,2 = (cxx -λ1,2)(cyy -λ1,2) -c 2 xy = 0 So        λ1 = 1 2 cxx + cyy + (cxx + cyy) 2 -4 cxxcyy -c 2 xy λ2 = 1 2 cxx + cyy -(cxx + cyy) 2 -4 cxxcyy -c 2 xy
Then, eccentricity can be calculated:

E = λ2/λ1 (5)

Minimum bounding rectangle

Minimum bounding rectangle is also called minimum bounding box. It is the smallest rectangle that contains every point in the shape. For an arbitrary shape, eccentricity is the ratio of the length L and width W of minimal bounding rectangle of the shape at some set of orientations. Elongation, Elo, is an other concept based on eccentricity (cf. Figure 3):

Elo = 1 -W/L (6)
Figure 3: Minimum bounding rectangle and corresponding parameters for elongation

Elongation is a measure that takes values in the range [0, 1]. A symmetrical shape in all axes such as a circle or square will have an elongation value of 0 whereas shapes with large aspect ratios will have an elongation closer to 1.

Circularity ratio

Circularity ratio represents how a shape is similar to a circle [START_REF] Zhang | Review of shape representation and description techniques[END_REF]. There are 3 denitions: Circularity ratio is the ratio of the area of a shape to the area of a circle having the same perimeter:

C1 =

As Ac [START_REF] Zhang | Review of shape representation and description techniques[END_REF] where As is the area of the shape and Ac is the area of the circle having the same perimeter as the shape. Assume the perimeter is O, so Ac = O 2 /4π. Then C1 = 4π • As/O 2 . As 4π is a constant, so we have the second circularity ratio denition.

Circularity ratio is the ratio of the area of a shape to the shape's perimeter square:

C2 = As O 2 (8)
Circularity ratio is also called circle variance, and dened as:

Cva = σR µR (9)
where µR and σR are the mean and standard deviation of the radial distance from the centroid (gx, gy) of the shape to the boundary points (xi, yi), i ∈ [0, N -1]. They are the following formulae respectively:

µR = 1 N N -1 i=1 di and σR = 1 N N -1 i=1 (di -µR) 2 where di = (xi -gx) 2 + (yi -gy) 2 .
The most compact shape is a circle. See Figure 4.

Ellipse variance

Ellipse variance Eva is a mapping error of a shape to t an ellipse that has an equal covariance matrix as the shape:

C ellipse = C (cf. Eq.4). It is practically eective to apply the inverse approach yielding.

We assume

Vi = xi -gx yi -gy d i = V T i • C -1 ellipse • Vi µ R = 1 N N -1 i=1 d i and σ R = 1 N N -1 i=1 (d i -µ R ) 2 Then Eva = σ R µ R (10)
Comparing with Eq. 9, intuitively, Eva represents a shape more accurately than Cva, cf. Figure 5.

Rectangularity

Rectangularity represents how rectangular a shape is, i.e. how much it lls its minimum bounding rectangle:

Rectangularity = AS/AR
where AS is the area of a shape; AR is the area of the minimum bounding rectangle.

Convexity

Convexity is dened as the ratio of perimeters of the convex hull O Convexhull over that of the original contour O [START_REF] Peura | Eciency of simple shape descriptors[END_REF]:

Convexity = O Convexhull O (11)
Figure 6: Illustration of convex hull

The region R 2 is a convex if and only if for any two points P1, P2 ∈ R 2 , the whole line segment P1P2 is inside the region. The convex hull of a region is the smallest convex region including it. In Figure 6, the outline is the convex hull of the region.

In [START_REF] Peura | Eciency of simple shape descriptors[END_REF], the authors presented the algorithm for constructing a convex hull by traversing the contour and minimizing turn angle in each step.

Solidity

Solidity describes the extent to which the shape is convex or concave [START_REF] Cheng | Image retrieval based on region shape similarity[END_REF] and it is dened by

Solidity = As/H
where, As is the area of the shape region and H is the convex hull area of the shape. The solidity of a convex shape is always 1.

Euler number

Euler number describes the relation between the number of contiguous parts and the number of holes on a shape. Let S be the number of contiguous parts and N be the number of holes on a shape. Then the Euler number is:

Eul = S -N For example
Euler Number equal to 1, -1 and 0, respectively.

Proles

The proles are the projection of the shape to x-axis and y -axis on Cartesian coordinate system. We obtain two onedimension functions:

P rox(i) = jmax j=j min f (i, j) and P roy(j)

= imax i=i min f (i, j)
where f (i, j) represents the region of shape Eq. ??. See Figure 7.

Hole area ratio

Hole area ratio HAR is dened as

HAR = A h As
where As is the area of a shape and A h is the total area of all holes in the shape. Hole area ratio is most eective in discriminating between symbols that have big holes and symbols with small holes [START_REF] Soffer | Negative shape features for image databases consisting of geographic symbols[END_REF].

3 One-dimensional function for shape representation

The one-dimensional function which is derived from shape boundary coordinates is also often called shape signature [START_REF] Zhang | A comparative study of fourier descriptors for shape representation and retrieval[END_REF][START_REF] Kauppinen | An experimental comparison of auto-regressive and fourier-based descriptors in 2-D shape classication[END_REF]. The shape signature usually captures the perceptual feature of the shape [START_REF] Yadava | Retrieval and classication of shape-based objects using fourier, generic fourier, and wavelet-fourier descriptors technique: A comparative study[END_REF]. Complex coordinates, Centroid distance function, Tangent angle (Turning angles), Curvature function, Area function, Triangle-area representation and Chord length function are the commonly used shape signatures.

Shape signature can describe a shape all alone; it is also often used as a preprocessing to other feature extraction algorithms, for example, Fourier descriptors, wavelet description. In this section, the shape signatures are introduced.

Complex coordinates

A complex coordinates function is simply the complex number generated from the coordinates of boundary points,

Pn(x(n), y(n)), n ∈ [1, N ]: z(n) = [x(n) -gx] + i[y(n) -gy]
where (gx, gy) is the centroid of the shape, given by Eq. 2.

Centroid distance function

The centroid distance function is expressed by the distance of the boundary points from the centroid (gx, gy) (Eq. 2) of a shape

r(n) = [(x(n) -gx) 2 + (y(n) -gy) 2 ] 1/2
Due to the subtraction of centroid, which represents the position of the shape, from boundary coordinates, both complex coordinates and centroid distance representation are invariant to translation.

Tangent angle

The tangent angle function at a point Pn(x(n), y(n)) is dened by a tangential direction of a contour [START_REF] Zhang | A comparative study on shape retrieval using fourier descriptors with dierent shape signatures[END_REF]:

θ(n) = θn = arctan y(n) -y(n -w) x(n) -x(n -w)
since every contour is a digital curve; w is a small window to calculate θ(n) more accurately.

Tangent angle function has two problems. One is noise sensitivity. To decrease the eect of noise, a contour is ltered by a low-pass lter with appropriate bandwidth before calculating the tangent angle function. The other is discontinuity, due to the fact that the tangent angle function assumes values in a range of length 2π, usually in the interval of [-π, π] or [0, 2π]. Therefore θn in general contains discontinuities of size 2π. To overcome the discontinuity problem, with an arbitrary starting point, the cumulative angular function ϕn is dened as the angle dierences between the tangent at any point Pn along the curve and the tangent at the starting point P0 [START_REF] Zahn | Fourier descriptors for plane closed curves[END_REF][START_REF] Lu | Compliant mechanism synthesis for shape-change applications: Preliminary results[END_REF]:

ϕ(n) = [θ(n) -θ(0)]
In order to be in accordance with human intuition that a circle is shapeless , assume t = 2πn/N , then ϕ(n) = ϕ(tN/2π). A periodic function is termed as the cumulative angular deviant function ψ(t) and is dened as

ψ(t) = ϕ( N 2π t) -t t ∈ [0, 2π]
N is the total number of contour points.

In [START_REF] Latecki | Shape similarity measure based on correspondence of visual parts[END_REF], the authors proposed a method based on tangent angle. Curvature is a very important boundary feature for human to judge similarity between shapes. It also has salient perceptual characteristics and has proven to be very useful for shape recognition [START_REF] Wang | Multiscale curvature-based shape representation using B-spline wavelets[END_REF]. In order to use K(n) for shape representation, we quote the function of curvature, K(n), from [START_REF] Mokhtarian | A theory of multiscale, curvature-based shape representation for planar curves[END_REF][START_REF] Jalba | Shape representation and recognition through morphological curvature scale spaces[END_REF] as:

K(n) = ẋ(n)ÿ(n) -ẏ(n)ẍ(n) ( ẋ(n) 2 + ẏ(n) 2 ) 3/2 (12)
Therefore, it is possible to compute the curvature of a planar curve from its parametric representation. If n is the normalized arc-length parameter s, then Eq. 12 can be written as:

K(s) = ẋ(s)ÿ(s) -ẏ(s)ẍ(s) (13)
As given in Eq. 13, the curvature function is computed only from parametric derivatives, and, therefore, it is invariant under rotations and translations. However, the curvature measure is scale dependent, i.e., inversely proportional to the scale. A possible way to achieve scale independence is to normalize this measure by the mean absolute curvature, i.e.,

K (s) = K(s) 1 N N s=1 |K(s)|
where N is the number of points on the normalized contour.

When the size of the curve is an important discriminative feature, the curvature should be used without the normalization; otherwise, for the purpose of scale-invariant shape analysis, the normalization should be performed.

An approximate arc-length parametrization based on the centripetal method is given by the following [START_REF] Jalba | Shape representation and recognition through morphological curvature scale spaces[END_REF]:

Let P = N n=1 dn be the perimeter of the curve and L = N n=1 √ dn, where dn is the length of the chord between points pn and pn+1, n=1, 2, . . . , N-1. The approximate arc-length parametrization relations: s1 = 0;

s k = s k-1 + P √ d k-1 L , k = 2, 3, . . . , N .
Starting from an arbitrary point and following the contour clockwise, we compute the curvature at each interpolated point using Eq. 13. Convex and concave vertices will imply negative and positive values, respectively (the opposite is veried for counterclockwise sense). Figure 9 is an example of curvature function. Clearly, as a descriptor, the curvature function can distinguish dierent shapes. The area function is linear under ane transform. However, this linearity only works for shape sampled at its same vertices.

Triangle-area representation

The triangle-area representation (TAR) signature is computed from the area of the triangles formed by the points on the shape boundary [START_REF] Alajlan | Multi-object image retrieval based on shape and topology[END_REF][START_REF] Alajlan | Shape retrieval using triangle-area representation and dynamic space warping[END_REF]. The curvature of the contour point (xn, yn) is measured using the TAR as follows.

For each three consecutive points Pn-t s (xn-t s , yn-t s ), Pn(xn, yn), and Pn+t s (xn+t s , yn+t s ), where n ∈ [1, N ] and ts ∈ [1, N/2 -1], N is even. The signed area of the triangle formed by these points is given by:

T AR(n, ts) = 1 2 xn-t s yn-t s 1 xn yn 1 xn+t s yn+t s 1 (14)
When the contour is traversed in counter clockwise direction, positive, negative and zero values of TAR mean convex, concave and straight-line points, respectively. Figure 11 demonstrates these three types of the triangle areas and the complete TAR signature for the hammer shape. In [START_REF] Alajlan | Shape retrieval using triangle-area representation and dynamic space warping[END_REF], authors show that the multi-scale space TAR is relatively invariant to the ane transform and robust to non-rigid transform. The computation complexity of the TAR stage is O(N 2 ).

Chord length function

The chord length function is derived from shape boundary without using any reference point. For each boundary point p, its chord length function is the shortest distance between p and another boundary point p' such that line pp' is perpendicular to the tangent vector at p [START_REF] Zhang | A comparative study of fourier descriptors for shape representation and retrieval[END_REF].

The chord length function is invariant to translation and it overcomes the biased reference point (which means the centroid is often biased by boundary noise or defections) problems. However, it is very sensitive to noise, there may be drastic burst in the signature of even smoothed shape boundary.

Discussions

A shape signature represents a shape by a 1-D function derived from shape contour. To obtain the translation invariant property, they are usually dened by relative values. To obtain the scale invariant property, normalization is necessary. In order to compensate for orientation changes, shift matching is needed to nd the best matching between two shapes. Having regard to occultation, Tangent angle, Contour curvature and Triangle-area representation have invariance property. In addition, shape signatures are computationally simple.

Shape signatures are sensitive to noise, and slight changes in the boundary can cause large errors in matching.

Therefore, it is undesirable to directly describe shape using a shape signature. Further processing is necessary to increase its robustness and reduce the matching load. For example, a shape signature can be simplied by quantizing the signature into a signature histogram, which is rotationally invariant.

Polygonal approximation

Polygonal approximation can be set to ignore the minor variations along the edge, and instead capture the overall shape. This is useful because it reduces the eects of discrete pixelization of the contour. In general, there are two methods to realize it. One is merging, the other is splitting [START_REF] Han | An invariant feature representation for shape retrieval[END_REF].

Merging methods

Merging methods add successive pixels to a line segment if each new pixel that is added doesn't cause the segment to deviate too much from a straight line.

Distance threshold method

Choose one point as a starting point, on the contour, for each new point that we add, let a line go from the starting point to this new point. Then, we compute the squared error for every point along the segment/line. If the error exceeds some threshold, we keep the line from the start point to the previous point and start a new line.

In practice, the most of practical error measures in use are based on distance between vertices of the input curve and the approximation linear segments. The distance d k (i, j) from curve vertex P k = (x k , y k ) to the corresponding approximation linear segments (Pi, Pj) is dened as follows (cf. Figure 13):

d k (i, j) = |(xj -xi)(yi -y k ) -(xi -x k )(yj -yi)| (xj -xi) 2 + (yj -yi) 2

Tunneling method

If we have thick boundaries rather than single-pixel thick ones, we can still use a similar approach called tunneling.

Imagine that we're trying to lay straight rods along a curved tunnel, and that we want to use as few as possible. We can start at one point and lay as long a straight rod as possible. Eventually, the curvature of the tunnel won't let us go any further, so we lay another rod and another until we reach the end.

Both the distance threshold and tunneling methods can do polygonal approximation eciently. However, the great disadvantage is that the position of starting point will aect greatly the approximate polygon.

Polygon evolution

The basic idea of polygons evolution in [START_REF] Latecki | Convexity rule for shape decomposition based on discrete contour evolution[END_REF] is very simple: in every evolution step, a pair of consecutive line segments (the line segment is the line between two consecutive vertices) s1,s2 is substituted with a single line segment joining the endpoints of s1 and s2.

The key property of this evolution is the order of the substitution. The substitution is done according to a relevance measure K given by

K(s1, s2) = β(s1, s2)l(s1)l(s2) l(s1) + l(s2) ,
where β(s1, s2) is the turn angle at the common vertex of segments s1, s2 and l(α) is the length of α, α = s1 or s2, normalized with respect to the total length of a polygonal curve. The evolution algorithm is assuming that vertices which are surrounded by segments with a high value of K(s1, s2) are important while those with a low value are not. The curve evolution method achieves the task of shape simplication, i.e., the process of evolution compares the signicance of vertices of the contour based on a relevance measure. Since any digital curve can be regarded as a polygon without loss of information (with possibly a large number of vertices), it is sucient to study evolutions of polygonal shapes for shape feature extraction.

Splitting methods

Splitting methods work by rst drawing a line from one point on the boundary to another. Then, we compute the perpendicular distance from each point along the boundary segment to the line. If this exceeds some threshold, we break the line at the point of greatest distance. We then repeat the process recursively for each of the two new lines until we don't need to break any more. See Figure 15 for an example.

Figure 15: Splitting methods for polygonal approximation This is sometimes known as the t and split algorithm. For a closed contour, we can nd the two points that lie farthest apart and t two lines between them, one for one side and one for the other. Then, we can apply the recursive splitting procedure to each side.

Discussions

Polygonal approximation technique can be used as a simple method for contour representation and description. The polygon approximation have some interesting properties:

it leads to simplication of shape complexity with no blurring eects.

it leads to noise elimination.

although irrelevant features vanish after polygonal approximation, there is no dislocation of relevant features.

the remaining vertices on a contour do not change their positions after polygonal approximation.

Polygonal approximation technique can also be used as preprocessing method for further extracting features from a shape.

5 Spatial interrelation feature Spatial interrelation feature describes the region or the contour of shape by the relation of their pixels or curves. In general, the representation is done by using their geometric features: length, curvature, relative orientation and location, area, distance and so on.

Adaptive grid resolution

The adaptive grid resolution (AGR) was proposed by [START_REF] Chakrabarti | Similar shape retrieval in mars[END_REF]. In the AGR, a square grid that is just big enough to cover the entire shape is overlaid on a shape. A resolution of the grid cells varies from one portion to another according to the content of the portion of the shape. On the borders or the detail portion on the shape, the higher resolution, i.e. the smaller grid cells, are applied; on the other hand, in the coarse regions of the shape, lower resolution, i.e. the bigger grid cells, are applied.

To guarantee rotation invariance, it needs to convert an arbitrarily oriented shape into a unique common orientation.

First, nd the major axis of the shape. The major axis is the straight line segment joining the two points P1 and P2 on the boundary farthest away from each other. Then we rotate the shape so that its major axis is parallel to the x-axis. This orientation is still not unique as there are two possibilities: P1 can be on the left or on the right. This problem is solved by computing the centroid of the polygon and making sure that the centroid is below the major axis, thus guaranteeing a unique orientation.

Let us now consider scale and translation invariance. We dene the bounding rectangle (BR) of a shape as the rectangle with sides parallel to the x and y axes just large enough to cover the entire shape (after rotation). Note that the width of the BR is equal to the length of the major axis. To achieve scale invariance, we proportionally scale all shapes so that their BRs have the same xed width (pixels).

The method of computation of the AGR representation of a shape applies quad-tree decomposition on the bitmap representation of the shape. The decomposition is based on successive subdivision of the bitmap into four equal-size quadrants. If a bitmap-quadrant does not consist entirely of part of shape, it is recursively subdivided into smaller and smaller quadrants until we reach bitmap-quadrants, i.e., termination condition of the recursion is that the resolution reaches that one predened. Figure 16(a) is an example of AGR. To represent the AGR image, in [START_REF] Chakrabarti | Similar shape retrieval in mars[END_REF], quad-tree method is applied. Each node in the quad-tree covers a square region of the bitmap. The level of the node in the quad-tree determines the size of the square. The internal nodes (shown by gray circles) represent partially covered regions; the leaf nodes shown by white boxes represent regions with all 0s while the leaf nodes shown by black boxes represent regions with all 1s. The all 1s regions are used to represent the shape, Figure 16(b). Each rectangle can be described by 3 numbers: its center C = (Cx, Cy) and its size (i.e. side length) S. So each shape can be mapped to a point in 3n-dimensional space (n is the number of the rectangles occupied by the shape region).

Due to the fact that the normalization before computing AGR, AGR representation is invariant under rotation, scaling and translation. It is also computationally simple.

Bounding box

Bounding box computes homeomorphisms between 2D lattices and its shapes. Unlike many other methods, this mapping is not restricted to simply connected shapes but applies to arbitrary topologies [START_REF] Bauckhage | Bounding box splitting for robust shape classication[END_REF].

To make bounding box representation invariant to rotation, a shape should be normalized by the same method as for AGR (Subsection 5.1) before further computation. After the normalization, a shape S is a set of L pixels, S = p k ∈ R 2 |k = 1, 2, • • • , L and also write |S| = L. The minimum bounding rectangle or bounding box of S is denoted by B(S); its width and height, are called w and h, respectively.

Figure 17 shows the algorithm owchart based on bounding box that divides a shape S into m (row) × n (column) parts. The output B is a set of bounding boxes.

An illustration of this procedure and its result is shown in Figure 18. To represent each bounding box, one method is that partial points of the set of bounding boxes are sampled. Figure 19 shows an example. x

, µ i(1)j(1) y , • • • , µ i(k)j(k) x , µ i(k)j(k) y
where i(α) < i(β) if α < β and likewise for the index j.

Bounding box representation is a simple computational geometry approach to compute homeomorphisms between shapes and lattices. It is storage and time ecient. It is invariant to rotation, scaling and translation and also robust against noisy shape boundaries.

Convex hull

The approach is that the shape is represented by a serie of convex hulls. The convex region has be dened in Sebsection 1.2.8. The convex hull H of a region is its smallest convex region including it. In other words, for a region S, the convex hull conv(S) is dened as the smallest convex set in R 2 containing S. In order to decrease the eect of noise, common practice is to rst smooth a boundary prior to partitioning.

The representation of the shape may be obtained by a recursive process which results in a concavity tree. See Figure 20. Each concavity can be described by its area, chord (the line connects the cut of the concavity) length, maximum curvature, distance from maximum curvature point to the chord. The matching between shapes becomes a string or a graph matching. Convex hull representation has a high storage eciency. It is invariant to rotation, scaling and translation and also robust against noisy shape boundaries (after ltering). However, extracting the robust convex hulls from the shape is where the shoe pinches. [START_REF] Davies | Machine Vision: Theory, Algorithms, Practicalities[END_REF][START_REF] Gonzalez | Digital image processing[END_REF] and [START_REF] Sonka | Image Processing, Analysis and Machine Vision[END_REF] gave the boundary tracing method and morphological methods to achieve convex hulls respectively.

Chain code

Chain code is a common approach for representing dierent rasterized shapes as line-drawings, planar curves, or contours.

Chain code describes an object by a sequence of unit-size line segments with a given orientation [START_REF] Zhang | Review of shape representation and description techniques[END_REF]. Chain code can be viewed as a connected sequence of straight-line segments with specied lengths and directions [START_REF] Liu | Compressed vertex chain codes[END_REF].

Basic chain code

Freeman [START_REF] Hu | Visual pattern recognition by moment invariants[END_REF] rst introduced a chain code that describes the movement along a digital curve or a sequence of border By encoding relative, rather than absolute position of the contour, the basic chain code is translation invariant. We can match boundaries by comparing their chain codes, but with the two main problems: 1) it is very sensitive to noise;

2) it is not rotationally invariant. To solve these problems, dierential chain codes (DCC) and resampling chain codes (RCC) were proposed.

Dierential chain codes (DCC) is encoding dierences in the successive directions. This can be computed by subtracting each element of the chain code from the previous one and taking the result modulo n, where n is the connectivity. This dierencing allows us to rotate the object in 90-degree increments and still compare the objects, but it doesn't get around the inherent sensitivity of chain codes to rotation on the discrete pixel grid.

Re-sampling chain codes (RCC) consists in re-sampling the boundary onto a coarser grid and then computing the chain codes of this coarser representation. This smoothes out small variations and noise but can help compensate for dierences in chain-code length due to the pixel grid.

Vertex chain code (VCC)

To improve chain code eciency, in [START_REF] Liu | Compressed vertex chain codes[END_REF] the authors proposed a chain code for shape representation according to vertex chain code (VCC). An element of the VCC indicates the number of cell vertices, which are in touch with the bounding contour of the shape in that element's position. Only three elements 1, 2 and 3 can be used to represent the bounding contour of a shape composed of pixels in the rectangular grid. Figure 22 shows the elements of the VCC to represent a shape. Iivarinen and Visa derive a chain code histogram (CCH) for object recognition [START_REF] Iivarinen | Shape recognition of irregular objects[END_REF]. The CCH is computed as hi = {i∈M, M is the range of chain code}, {α} denotes getting the number of the value α.

The CCH reects the probabilities of dierent directions present in a contour.

If the chain code is used for matching it must be independent of the choice of the starting pixel in the sequence. The chain code usually has high dimensions and is sensitive to noise and any distortion. So, except the CCH, the other chain code approaches are often used as contour representations, but is not as contour attributes.

Smooth curve decomposition

In [START_REF] Berretti | Retrieval by shape similarity with perceptual distance and eective indexing[END_REF], the authors proposed smooth curve decomposition as shape descriptor. The segment between the curvature zero-crossing points from a Gaussian smoothed boundary are used to obtain primitives, called tokens. The feature for each token is its maximum curvature and its orientation. In Figure 23, the rst number in the parentheses is its maximum curvature and the second is its orientation.

Symbolic representation based on the axis of least inertia

In [START_REF] Guru | Symbolic representation of two-dimensional shapes[END_REF], a method of representing a shape in terms of multi-interval valued type data is proposed. The proposed shape representation scheme extracts symbolic features with reference to the axis of least inertia, which is unique to the shape.

The axis of least inertia (ALI) of a shape is dened as the line for which the integral of the square of the distances to points on the shape boundary is a minimum (cf. Subsection 1.2.2).

Once the ALI is calculated, each point on the shape curve is projected on to ALI. The two farthest projected points say E1 and E2 on ALI are chosen as the extreme points as shown in Figure 24. The Euclidean distance between these two extreme points denes the length of ALI. The length of ALI is divided uniformly by a xed number n; the equidistant points are called feature points. At every feature point chosen, an imaginary line perpendicular to the ALI is drawn. It is interesting to note that these perpendicular lines may intersect the shape curve at several points. The length of each imaginary line in shape region is computed and the collection of these lengths in an ascending order denes the value of the feature at the respective feature point. The feature vector F representing the shape S is invariant to image transformations viz., uniform scaling, rotation, translation and ipping (reection).

Beam angle statistics

Beam angle statistics (BAS) shape descriptor is based on the beams originated from a boundary point, which are dened as lines connecting that point with the rest of the points on the boundary [START_REF] Arica | BAS: a perceptual shape descriptor based on the beam angle statistics[END_REF].

Let B be the shape boundary. B = {P1, P2, • • • , PN } is represented by a connected sequence of points, Pi = (xi, yi), i = 1, 2, • • • , N, where N is the number of boundary points. For each point Pi, the beam angle between the forward beam vector V i+k = ----→ PiP i+k and backward beam vector V i-k = ----→ PiP i-k in the k th order neighborhood system, is then computed as (see Figure 25, k =5 for example)

C k (i) = (θV i+k -θV i-k )
where θV i+k = arctan For each boundary point Pi of the contour, the beam angle C k (i) can be taken as a random variable with the probability density function P (C k (i)). Therefore, beam angle statistics (BAS), may provide a compact representation for a shape descriptor. For this purpose, m th moment of the random variable C k (i) is dened as follows:

y i+k -yi x i+k -xi , θV i-k = arctan y i-k -yi x i-k -xi
E[C m (i)] = (N/2)-1 k=1 C m k (i) • P k (C k (i)) m = 1, 2, • • •
In the above formula E indicates the expected value. See Figure 26 as an example.

Beam angle statistics shape descriptor captures the perceptual information using the statistical information based on the beams of individual points. It gives globally discriminative features to each boundary point by using all other boundary points. BAS descriptor is also quite stable under distortions and is invariant to translation, rotation and scaling.

Shape matrix

Shape matrix descriptor is an M × N matrix to present a region shape. There are two basic modes of shape matrix:

Square model [START_REF] Flusser | Invariant shape description and measure of object similarity[END_REF] and Polar model [START_REF] Taza | Discrimination of planar shapes using shape matrices[END_REF].

Square model shape matrix

Square model of shape matrix, also called grid descriptor [START_REF] Lu | Region-based shape representation and similarity measure suitable for content based image retrieval[END_REF][START_REF] Flusser | Invariant shape description and measure of object similarity[END_REF], is constructed by the following: for the shape S, construct a square centered on the center of gravity G of S ; the size of each side is equal to 2L, L is the maximum Euclidean distance from G to a point M on the boundary of the shape. Point M lies in the center of one side and GM is perpendicular to this side.

Divide the square into N × N subsquares and denote S kj , k, j = 1, • • • , N , the subsquares of the constructed grid. Dene the shape matrix SM = [B kj ],

B kj = 1 ⇔ µ(S kj ∩ S) ≥ µ(S kj )/2 0 otherwise
where µ(F ) is the area of the planar region F. Figure 27 shows an example of square model of shape matrix. For a shape with more than one maximum radius, it can be described by several shape matrices and the similarity distance is the minimum distance between these matrices. In [START_REF] Flusser | Invariant shape description and measure of object similarity[END_REF], authors gave a method to choose the appropriate shape matrix dimension.

Polar model shape matrix

Polar model of shape matrix is constructed by the following steps. Let G be the center of gravity of the shape, and GA is the maximum radius of the shape. Using G as center, draw n circles with radii equally spaced. Starting from GA, and counterclockwise, draw radii that divide each circle into m equal arcs. The values of the matrix are same as in square model shape matrix. Figure 28 shows an example, where n = 5 and m =12. Its polar model of shape matrix is maximum radii are on the shape. However, since the sampling density is not constant with the polar sampling raster, a weighed shape matrix is necessary. For the detail, refer to [START_REF] Taza | Discrimination of planar shapes using shape matrices[END_REF].

P SM =       1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0      
The shape matrix exists for every compact shape. There is no limit to the scope of the shapes that the shape matrix can represent. It can describe even shapes with holes. Shape matrix is also invariant under translation, rotation and scaling of the object. The shape of the object can be reconstructed from the shape matrix; the accuracy is given by the size of the grid cells.

Shape context

In [START_REF] Belongie | Shape matching and object recognition using shape context[END_REF], the shape context has been shown to be a powerful tool for object recognition tasks. It is used to nd corresponding features between model and image.

Shape contexts analysis begins by taking N samples from the edge elements on the shape. These points can be on internal or external contours. Consider the vectors originating from a point to all other sample points on the shape.

These vectors express the appearance of the entire shape relative to the reference point. This descriptor is the histogram of the relative polar coordinates of all other points: of object recognition problems [START_REF] Belongie | Shape matching and object recognition using shape context[END_REF][START_REF] Mori | Estimating human body congurations using shape context matching[END_REF][START_REF] Zhang | Learning a discriminative classier using shape context distances[END_REF][START_REF] Thayananthan | Shape context and chamfer matching in cluttered scenes[END_REF]. The shape context descriptor has the following invariance properties: translation: the shape context descriptor is inherently translation invariant as it is based on relative point locations.

hi(k) = {Q = Pi : (Q -Pi) ∈ bin(k)}
scaling: for clutter-free images the descriptor can be made scale invariant by normalizing the radial distances by the mean (or median) distance between all point pairs. rotation: it can be made rotation invariant by rotating the coordinate system at each point so that the positive x-axis is aligned with the tangent vector.

shape variation: the shape context is robust against slight shape variations.

few outliers: points with a nal matching cost larger than a threshold value are classied as outliers. Additional `dummy' points are introduced to decrease the eects of outliers.

Chord distribution

The basic idea of chord distribution is to calculate the lengths of all chords in the shape (all pair-wise distances between boundary points) and to build a histogram of their lengths and orientations [START_REF] Smith | Chord distribution for shape matching[END_REF]. The lengths histogram is invariant to rotation and scales linearly with the size of the object. The angles histogram is invariant to object size and shifts relative to object rotation. Figure 30 gives an example of chord distribution. Medial axis is an image processing operation which reduces input shapes to axial stick-like representations. It is as the loci of centers of bi-tangent circles that t entirely within the foreground region being considered. Figure 31 illustrates the medial axis for a rectangular shape. We notice that the radius of each circle is variable. This variable is a function of the loci of points on the medial axis. We call this function as the radius function.

A shock graph is a shape abstraction that decomposes a shape into a set of hierarchically organized primitive parts.

Siddiqi and Kimia dene the concept of a shock graph [START_REF] Siddiqi | A shock grammar for recognition[END_REF] as an abstraction of the medial axis of a shape onto a directed acyclic graph (DAG). Shock segments are curve segments of the medial axis with monotonic ow, and give a more rened partition of the medial axis segments. Figure 32 is for example. To calculate the distance between two shock graphs, in [START_REF] Sebastian | Recognition of shapes by editing their shock graphs[END_REF], the authors employ a polynomial-time edit-distance algorithm. It shows this algorithm has the good performances for boundary perturbations, articulation and deformation of parts, segmentation errors, scale variations, viewpoint variations and partial occultation. However the authors also indicate the computation complexity is very high. The matching shape typically takes about 3-5 minutes on an SGI Indigo II (195 MHz), which limits the number of shapes that can be practically matched.

Discussions

Spacial feature descriptor is a direct method to describe a shape. These descriptors can apply the theory of tree-based (Adaptive grid resolution and Convex hull), statistic (Chain code histogram, Beam angle statistics, Shape context and Chord distribution) or syntactic analysis (Smooth curve decomposition) to extract or represent the feature of a shape.

This description scheme not only compresses the data of a shape, but also provides a compact and meaningful form to facilitate further recognition operations.

Moments

The concept of moment in mathematics evolved from the concept of moment in physics. It is an integrated theory system. For both contour and region of a shape, one can use moment's theory to analyze the object.

Boundary moments

Boundary moments, analysis of a contour, can be used to reduce the dimension of boundary representation [START_REF] Sonka | Image Processing, Analysis and Machine Vision[END_REF]. Assume shape boundary has been represented as a 1-D shape representation z(i) as introduced in Section 3, the r th moment mr and central moment µr can be estimated as

mr = 1 N N i=1 [z(i)] r and µr = 1 N N i=1 [z(i) -m1] r
where N is the number of boundary points. The normalized moments mr = mr/(µ2) r/2 and μr = µr/(µ2) r/2 are invariant to shape translation, rotation and scaling. Less noise-sensitive shape descriptors can be obtained from

F1 = (µ2) 1/2 m1 , F2 = µ3 (µ2) 3/2 and F3 = µ4 (µ2) 2 ,
The other boundary moments method treats the 1-D shape feature function z(i) as a random variable v and creates a K bins histogram p(vi) from z(i). Then, the r th central moment is obtained by

µr = K i=1 (vi -m) r p(vi) and m = K i=1 vip(vi)
The advantage of boundary moment descriptors is that it is easy to implement. However, it is dicult to associate higher order moments with physical interpretation.

Region moments

Among the region-based descriptors, moments are very popular. These include invariant moments, Zernike moments, Radial Chebyshev moments, etc.

The general form of a moment function mpq of order (p + q) of a shape region can be given as:

mpq = x y Ψpq(x, y)f (x, y) p, q = 0, 1, 2 • • •
where Ψpq is known as the moment weighting kernel or the basis set ; f (x, y) is the shape region Eq. ??.

Invariant moments (IM)

Invariant moments (IM) are also called geometric moment invariants. Geometric moments, are the simplest of the moment functions with basis Ψpq = x p y q , while complete, is not orthogonal [START_REF] Hu | Visual pattern recognition by moment invariants[END_REF]. Geometric moment function mpq of order (p + q) is given as: mpq =

x y

x p y q f (x, y) p, q = 0, 1, 2 • • •

The geometric central moments, which are invariant to translation, are dened as: µpq =

x y (x -x) p (y -ȳ) q f (x, y) p, q = 0, 1, 2 • • • where x = m10/m00 and ȳ = m01/m00

A set of 7 invariant moments (IM) are given by [START_REF] Hu | Visual pattern recognition by moment invariants[END_REF]:

φ1 = η20 + η02 φ2 = (η20 -η02) 2 + 4η 2 11 φ3 = (η30 -3η12) 2 + (3η21 -η03) 2 φ4 = (η30 + η12) 2 + (η21 + η03) 2 φ5 = (η30 -3η12)(η30 + η12) (η30 + η12) 2 -3(η21 + η03) 2 + (3η21 -η03)(η21 + η03) • 3(η30 + η12) 2 -(η21 + η03) 2 φ6 = (η20 -η02) (η30 + η12) 2 -(η21 + η03) 2 + 4η 2 11 (η30 + η12)(η21 + η03) φ7 = (3η21 -η03)(η30 + η12) (η30 + η12) 2 -3(η21 + η03) 2 + (3η12 -η03)(η21 + η03) • 3(η30 + η12) 2 -(η21 + η03) 2
where ηpq = µpq/µ γ 00 and γ = 1 + (p + q)/2 for p + q = 2, 3, • • • IM are computationally simple. Moreover, they are invariant to rotation, scaling and translation. However, they have several drawbacks [START_REF] Celebi | A comparative study of three moment-based shape descriptors[END_REF]:

information redundancy: since the basis is not orthogonal, these moments suer from a high degree of information redundancy.

noise sensitivity: higher-order moments are very sensitive to noise. large variation in the dynamic range of values: since the basis involves powers of p and q, the moments computed have large variation in the dynamic range of values for dierent orders. This may cause numerical instability when the image size is large.

Algebraic moment invariants

The algebraic moment invariants are computed from the rst m central moments and are given as the eigenvalues of predened matrices, M [j,k] , whose elements are scaled factors of the central moments [START_REF] Taubin | Recognition and positioning of rigid objects using algebraic moment invariants[END_REF]. The algebraic moment invariants can be constructed up to arbitrary order and are invariant to ane transformations. However, algebraic moment invariants performed either very well or very poorly on the objects with dierent conguration of outlines.

Zernike moments (ZM)

Zernike Moments (ZM) are orthogonal moments [START_REF] Celebi | A comparative study of three moment-based shape descriptors[END_REF]. The complex Zernike moments are derived from orthogonal Zernike polynomials:

Vnm(x, y) = Vnm(r cos θ, sin θ) = Rnm(r) exp(jmθ)

where Rnm(r)is the orthogonal radial polynomial:

Rnm(r) = (n-|m|)/2 s=0 (-1) s (n -s)! s! × n-2s+|m| 2 ! n-2s-|m| 2 ! r n-2s n = 0, 1, 2, • • • ; 0 ≤ |m| ≤ n; and n -|m| is even.
Zernike polynomials are a complete set of complex valued functions orthogonal over the unit disk, i.e., x 2 + y 2 ≤ 1. The Zernike moment of order n with repetition m of shape region f (x, y) (Eq. ??) is given by:

Znm = n + 1 π r θ f (r cos θ, r sin θ) • Rnm(r) • exp(jmθ) r ≤ 1
Zernike moments (ZM) have the following advantages [START_REF] Mukundan | Image analysis by tchebichef moments[END_REF]:

rotation invariance: the magnitudes of Zernike moments are invariant to rotation.

robustness: they are robust to noise and minor variations in shape.

expressiveness: since the basis is orthogonal, they have minimum information redundancy.

However, the computation of ZM (in general, continuous orthogonal moments) pose several problems: coordinate space normalization: the image coordinate space must be transformed to the domain where the orthogonal polynomial is dened (unit circle for the Zernike polynomial).

numerical approximation of continuous integrals: the continuous integrals must be approximated by discrete summations. This approximation not only leads to numerical errors in the computed moments, but also severely aects the analytical properties such as rotational invariance and orthogonality.

computational complexity: computational complexity of the radial Zernike polynomial increases as the order becomes large.

Radial Chebyshev moments (RCM)

The radial Chebyshev moment of order p and repetition q is dened as [START_REF] Mukundan | A new class of rotational invariants using discrete orthogonal moments[END_REF]:

Spq = 1 2πρ(p, m) m-1 r=0 2π θ=0 tp(r) • exp(-jqθ) • f (r, θ)
where tp(r) is the scaled orthogonal Chebyshev polynomials for an image of size N × N : t0

(x) = 1 t1(x) = (2x -N + 1)/N tp(x) = (2p -1)t1(x)tp-1(x) -(p -1) 1 - (p -1) 2 N 2 tp-2(x) p , p > 1 ρ(p, N ) is the squared-norm: ρ(p, N ) = N 1 - 1 N 2 1 - 2 2 N 2 • • • 1 - p 2 N 2 2p + 1 p = 0, 1, • • • , N -1 and m = (N/2) + 1.
The mapping between (r, θ) and image coordinates (x, y) is given by:

x = rN 2(m -1) cos(θ) + N 2 y = rN 2(m -1) sin(θ) + N 2
Compared to Chebyshev moments, radial Chebyshev moments possess rotational invariance property.

Discussions

Besides the previous moments, there are other moments for shape representation, for example, homocentric polar-radius moment [START_REF] Jin | Homocentric polar-radius moment for shape classication[END_REF], orthogonal Fourier-Mellin moments (OFMMs) [START_REF] Kan | Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments[END_REF], pseudo-Zernike Moments [START_REF] Mehtre | Shape measures for content based image retrieval: A comparison[END_REF], etc. The study shows that the moment-based shape descriptors are usually concise, robust and easy to compute. It is also invariant to scaling, rotation and translation of the object. However, because of their global nature, the disadvantage of moment-based methods is that it is dicult to correlate high order moments with a shape's salient features.

Scale space approaches

In scale space theory a curve is embedded into a continuous family {Γσ : σ ≥ 0} of gradually simplied versions. The main idea of scale spaces is that the original curve Γ = Γ0 should get more and more simplied, and so small structures should vanish as parameter σ increases. Thus due to dierent scales (values of σ), it is possible to separate small details from relevant shape properties. The ordered sequence {Γσ : σ ≥ 0} is referred to as evolution of Γ. Scale-spaces nd wide application in computer vision, in particular, due to smoothing and elimination of small details.

A lot of shape features can be analyzed in scale-space theory to get more information about shapes. Here we introduced 2 scale-space approaches: curvature scale-space (CSS) and intersection points map (IPM).

Curvature scale-space

The curvature scale-space (CSS) method, proposed by F. Mokhtarian in 1988, was selected as a contour shape descriptor for MPEG-7. This approach is based on multi-scale representation and curvature to represent planar curves. For convenience, we copy the nature parametrization equation (Eq. ??) as following:

Γ(µ) = (x(µ), y(µ)) (15)
An evolved version of that curve is dened by

Γσ(µ) = (X(µ, σ), Y (µ, σ))
where X(µ, σ) = x(µ) * g(µ, σ) and Y (µ, σ) = y(µ) * g(µ, σ), * is the convolution operator, and g(µ, σ) denotes a Gaussian lter with standard deviation σ dened by

g(µ, σ) = 1 σ √ 2π exp( -µ 2 2σ 2 )
Functions X(µ, σ) and Y (µ, σ) are given explicitly by

X(µ, σ) = ˆ∞ -∞ x(v) 1 σ √ 2π exp( -(µ -v) 2 2σ 2 )dv Y (µ, σ) = ˆ∞ -∞ y(v) 1 σ √ 2π exp( -(µ -v) 2 2σ 2 )dv
The curvature of is given by

k(µ, σ) = Xµ(µ, σ)Yµµ(µ, σ) -Xµµ(µ, σ)Yµ(µ, σ) (Xµ(µ, σ) 2 -Yµ(µ, σ) 2 ) 3/2
where

Xµ(µ, σ) = ∂ ∂µ (x(µ) * g(µ, σ)) = x(µ) * gµ(µ, σ) Xµµ(µ, σ) = ∂ 2 ∂µ 2 (x(µ) * g(µ, σ)) = x(µ) * gµµ(µ, σ) Yµ(µ, σ) = ∂ ∂µ (y(µ) * g(µ, σ)) = y(µ) * gµ(µ, σ) Yµµ(µ, σ) = ∂ 2 ∂µ 2 (y(µ) * g(µ, σ)) = y(µ) * gµµ(µ, σ
) Note that σ is also referred to as a scale parameter. The process of generating evolved versions of Γσ as σ increases from 0 to ∞ is referred to as the evolution of Γσ. This technique is suitable for removing noise and smoothing a planar curve as well as gradual simplication of a shape.

The function dened by k(µ, σ) = 0 is the CSS image of Γ. it captures the main features of a shape, enabling similarity-based retrieval; it is robust to noise, changes in scale and orientation of objects; it is compact, reliable and fast; It retains the local information of a shape. Every concavity or convexity on the shape has its own corresponding contour on the CSS image.

Although CSS has a lot of advantages, it does not always give results in accordance with human vision system. The main drawbacks of this description are due to the problem of shallow concavities/convexities on a shape. It can be shown that the shallow and deep concavities/convexities may create the same large contours on the CSS image. In [START_REF] Abbasi | Enhancing css-based shape retrieval for objects with shallow concavities[END_REF][START_REF] Yang | Scale-controlled area dierence shape descriptor[END_REF], the authors gave some methods to alleviate these eects.

Intersection points map

Similarly to the CSS, many methods also use a Gaussian kernel to progressively smooth the curve relatively to the varying bandwidth. In [START_REF] Kpalma | Multiscale contour description for pattern recognition[END_REF], the authors proposed a new algorithm, intersection points map (IPM), based on this principle, instead of characterizing the curve with its curvature involving 2 nd order derivatives, it uses the intersection points between the smoothed curve and the original. As the standard deviation of the Gaussian kernel increases, the number of the intersection points decreases. By analyzing these remaining points, features for a pattern can be dened.

Since this method deals only with curve smoothing, it needs only the convolution operation in the smoothing process. So this method is faster than the CSS one with equivalent performances. Figure 35 The IPM pattern can be identied regardless of its orientation, translation and scale change. It is also resistant to noise for a range of noise energy. The main weakness of this approach is that it fails to handle occulted contours and those having undergone a non-rigid deformation.

Discussions

As multi-resolution analysis in signal processing, scale-space theory can obtain abundant information about a contour with dierent scales. In scale-space, global pattern information can be interpreted from higher scales, while detailed pattern information can be interpreted from lower scales. Scale-space algorithm benet from the boundary information redundancy in the new image, making it less sensitive to errors in the alignment or contour extraction algorithms. The great advantages are the high robustness to noise and the great coherence with human perception.

Shape transform domains

The transform domain class includes methods which are formed by the transform of the detected object or the transform of the whole image. Transforms can therefore be used to characterize the appearance of images. The shape feature is represented by the all or partial coecients of a transform.

Fourier descriptors

Although, Fourier descriptor (FD) is a 40-year-old technique, it is still considered as a valid description tool. The shape description and classication using FD either in contours or regions are simple to compute, robust to noise and compact.

It has many applications in dierent areas.

One-dimensional Fourier descriptors

In general, Fourier descriptor (FD) is obtained by applying Fourier transform on a shape signature that is a onedimensional function which is derived from shape boundary coordinates (cf. Section 3). The normalized Fourier transformed coecients are called the Fourier descriptor of the shape. FD derived from dierent signatures has signicant dierent performance on shape retrieval. As shown in [START_REF] Zhang | A comparative study of fourier descriptors for shape representation and retrieval[END_REF][START_REF] Zhang | A comparative study of curvature scale space and fourier descriptors for shape-based image retrieval[END_REF], FD derived from centroid distance function r(t) outperforms FD derived from other shape signatures in terms of overall performance. The discrete Fourier transform of r(t) is then given by

an = 1 N N -1 t=0 r(t)exp -j2πnt N , n = 0, 1, • • • , N -1
Since the centroid distance function r(t) is only invariant to rotation and translation, the acquired Fourier coecients have to be further normalized so that they are scaling and start point independent shape descriptors. From Fourier transform theory, the general form of the Fourier coecients of a contour centroid distance function r(t) transformed through scaling and change of start point from the original function r(t) (o) is given by

an = exp(jnτ ) • s • a (o)
n where an and a

(o) n are the Fourier coecients of the transformed shape and the original shape, respectively, τ is the angles incurred by the change of start point; s is the scale factor. Now considering the following expression:

bn = an a1 = exp(jnτ ) • s • a (o) n exp(jτ ) • s • a (o) 1 = a (o) n a (o) 1 exp[j(n -1)τ ] = b (o) n exp[j(n -1)τ ]
where bn and b

(o) n are the normalized Fourier coecients of the transformed shape and the original shape, respectively.

If we ignore the phase information and only use magnitude of the coecients, then |bn| and b

(o) n are the same. In other words, |bn| is invariant to translation, rotation, scaling and change of start point.

The set of magnitudes of the normalized Fourier coecients of the shape {|bn| , 0 < n < N } are used as shape descriptors, denoted as {F Dn, 0 < n < N } .

One-dimensional FD has several nice characteristics such as simple derivation, simple normalization and simple to do matching. As indicated by [START_REF] Zhang | A comparative study of curvature scale space and fourier descriptors for shape-based image retrieval[END_REF], for ecient retrieval, 10 FDs are sucient for shape description.

Region-based Fourier descriptor

The region-based FD is referred to as generic FD (GFD), which can be used for general applications. Basically, GFD is derived by applying a modied polar Fourier transform (MPFT) on shape image [START_REF] Zhang | Enhanced generic fourier descriptors for object-based image retrieval[END_REF][START_REF] Yadava | Retrieval and classication of shape-based objects using fourier, generic fourier, and wavelet-fourier descriptors technique: A comparative study[END_REF]. In order to apply MPFT, the polar shape image is treated as a normal rectangular image. The steps are 1. the approximated normalized image is rotated counter clockwise by an angular step suciently small.

2. the pixel values along positive x-direction starting from the image center are copied and pasted into a new matrix as row elements.

3. the steps 1 and 2 are repeated until the image is rotated by 360°.

The result of these steps is that an image in polar space plots into Cartesian space. The Fourier transform is acquired by applying a discrete 2D Fourier transform on this shape image.

pf (ρ, φ) = r i f (r, θi)exp[j2π( r R ρ + 2πi T φ)]
where 0 ≤ r = [(x -gx) 2 + (y -gy) 2 ] < R, and θi = i(2π/T ); 0 ≤ ρ < R, 0 ≤ φ < T . (gx, gy) is the center of mass of the shape; R and T are the radial and angular resolutions. The acquired Fourier coecients are translation invariant.

Rotation and scaling invariance are achieved by the following:

GF D = |pf (0, 0)| area , |pf (0, 1)| |pf (0, 0)| , • • • , |pf (0, n)| |pf (0, 0)| , • • • , |pf (m, 0)| |pf (0, 0)| , • • • , |pf (m, n)| |pf (0, 0)|
where area is the area of the bounding circle in which the polar image resides. m is the maximum number of the radial frequencies selected and n is the maximum number of angular frequencies selected. m and n can be adjusted to achieve hierarchical coarse to ne representation requirement.

For ecient shape description, in the implementation of [START_REF] Zhang | Enhanced generic fourier descriptors for object-based image retrieval[END_REF], 36 GFD features reecting m = 4 and n = 9 are selected to index the shape. The experimental results have shown GFD as invariant to translation, rotation, and scaling. For obtaining the ane and general minor distortions invariance, in [START_REF] Zhang | Enhanced generic fourier descriptors for object-based image retrieval[END_REF], the authors proposed Enhanced Generic Fourier Descriptor (EGFD) to improve the GFD properties.

Wavelet transform

A hierarchical planar curve descriptor is developed by using the wavelet transform [START_REF] Chuang | Wavelet descriptor of planar curves: Theory and applications[END_REF]. This descriptor decomposes a curve into components of dierent scales so that the coarsest scale components carry the global approximation information while the ner scale components contain the local detailed information. The wavelet descriptor has many desirable properties such as multi-resolution representation, invariance, uniqueness, stability, and spatial localization. In [START_REF] Khalil | A dyadic wavelet ane invariant function for 2D shape recognition[END_REF], the authors use dyadic wavelet transform deriving an ane invariant function. In [START_REF] Chen | Invariant fourier-wavelet descriptor for pattern recognition[END_REF], a descriptor is obtained by applying the Fourier transform along the axis of polar angle and the wavelet transform along the axis of radius. This feature is also invariant to translation, rotation, and scaling. At same time, the matching process of wavelet descriptor can be accomplished cheaply.

Angular radial transformation

The angular radial transformation (ART) is based in a polar coordinate system where the sinusoidal basis functions are dened on a unit disc. Given an image function in polar coordinates, f (ρ, θ), an ART coecient Fnm (radial order n, angular order m ) can be dened as [START_REF] Ricard | Generalizations of angular radial transform for 2D and 3D shape retrieval[END_REF]:

Fnm = ˆ2π 0 ˆ1 0 Vnm(ρ, θ)f (ρ, θ)ρdρdθ
Vnm(ρ, θ) is the ART basis function and is separable in the angular and radial directions so that:

Vnm(ρ, θ) = Am(θ)Rn(ρ)

The angular basis function, Am, is an exponential function used to obtain orientation invariance. This function is dened as:

Am(θ) = 1 2π e jmθ
Rn, the radial basis function, is dened as:

Rn(ρ) = 1 if n = 0 2 cos(πnρ) if n = 0
In MPEG-7, twelve angular and three radial functions are used (n < 3, m < 12). Real parts of the 2-D basis functions are shown in Figure 37. For scale normalization, the ART coecients are divided by the magnitude of ART coecient of order n = 0, m = 0.

MPEG-7 standardization process showed the eciency of 2-D angular radial transformation. This descriptor is robust to translations, scaling, multi-representation (remeshing, weak distortions) and noises.

Shape signature harmonic embedding

A harmonic function is obtained by a convolution between the Poisson kernel PR(r, θ) and a given boundary function u(Re jφ ). Poisson kernel is dened by

PR(r, θ) = R 2 -r 2 R 2 -2Rr cos(θ) + r 2
The boundary function could be any real-or complex-valued function, but here we choose shape signature functions for the purpose of shape representation. For any shape signature s[n], n = 0, 1, • • • , N -1, the boundary values for a unit disk can be set as

u(Re jφ ) = u(Re jω 0 n ) = s[n]
where ω0 = 2π/N , φ = ω0n.

So the harmonic function u can be written as

u(re jθ ) = 1 2π ˆ2π 0 u(Re jφ )PR(r, φ -θ)dφ (16)
The Poisson kernel PR(r, θ) has a low-pass lter characteristic, where the radius r is inversely related to the bandwidth of the lter. The radius r is considered as scale parameter of a multi-scale representation [START_REF] Lee | A shape representation for planar curves by shape signature harmonic embedding[END_REF]. Another important property is PR(0, θ) = 1, indicating u(0) is the mean value of boundary function u(Re jφ ).

In [START_REF] Lee | A shape representation for planar curves by shape signature harmonic embedding[END_REF], the authors proposed a formulation of a discrete closed-form solution for the Poisson's integral formula Eq.

16, so that one can avoid the need for approximation or numerical calculation of the Poisson summation form.

As in Subsection 8.1.2, the harmonic function inside the disk can be mapped to a rectilinear space for a better illustration. Figure 38 shows an example for a star shape. Here, we used curvature as the signature to provide boundary values. The zero-crossing image of the harmonic functions is extracted as shape feature. This shape descriptor is invariant to translation, rotation and scaling. It is also robust to noise. Figure 39 is for example. The original curve is corrupted with dierent noise levels, and the harmonic embeddings show robustness to the noise. Figure 39: centroid distance signature harmonic embedding that is robust to noisy boundaries At same time, it is more ecient than CSS descriptor. However, it is not suitable for similarity retrieval, because it is unconsistent to non-rigid transform.

-Transform

The -Transform to represent a shape is based on the Radon transform. The approach is presented as follow. We assume that the function f is the domain of a shape, cf. Eq. ??. Its Radon transform is dened by: The following transform is dened as -transform:

TR(ρ, θ) = ˆ∞ -∞ ˆ∞ -∞ f (x,
f (θ) = ˆ∞ -∞ T 2 R (ρ, θ)dρ
where TR(ρ, θ) is the Radon transform of the domain function f . In [START_REF] Tabbone | A new shape descriptor dened on the radon transform[END_REF], the authors show the following properties of f (θ):

periodicity: f (θ ± π) = f (θ)
rotation: a rotation of the image by an angle θ0 implies a translation of the -transform of θ0: f (θ + θ0).

translation: the -transform is invariant under a translation of the shape f by a vector -→ u = (x0, y0).

scaling: a change of the scaling of the shape f induces a scaling in the amplitude only of the -transform.

Given a large collection of shapes, one -transform per shape is not ecient to distinguish from the others because the -transform provides a highly compact shape representation. In this perspective, to improve the description, each shape is projected in the Radon space for dierent segmentation levels of the Chamfer distance transform. Chamfer distance transform is introduced in [START_REF] Borgefors | Distance transformations in digital images[END_REF][START_REF] Di Baja | Skeletonization algorithm running on path-based distance maps[END_REF] (See Appendix A for detail).

Given the distance transform of a shape, the distance image is segmented into N equidistant levels to keep the segmentation isotropic. For each distance level, pixels having a distance value superior to that level are selected and at each level of segmentation, an -transform is computed. In this manner, both the internal structure and the boundaries of the shape are captured.

Since a rotation of the shape implies a corresponding shift of the -transform. Therefore, a one-dimensional Fourier transform is applied on this function to obtain the rotation invariance. After the discrete one-dimensional Fourier transform F, -transform descriptor vector is dened as follows:

RT D = (

F 1 ( π M ) F 1 (0) , • • • , F 1 ( iπ M ) F 1 (0) , • • • , F 1 (π) F 1 (0) , • • • , F N ( π M ) F N (0) , • • • , F N ( iπ M ) F N (0) , • • • , F N (π) F N (0) )
where i∈ [1, M ], M is the angular resolution. F α is the magnitude of Fourier transform to -transform. α ∈ [1, N ], is the segmentation level of Chamfer distance transform. Clearly, computing the shape script B is a non-trivial task, since ∆ is over-complete and there will be multiple sets of bases that reconstruct the curve with equal precision. [START_REF] Dubinskiy | A multi-scale generative model for animate shapes and parts[END_REF] gave some pursuit algorithms to use shapelets representing a shape.

Discussions

As a kind of global shape description technique, shape analysis in transform domains takes the whole shape as the shape representation. The description scheme is designed for this representation. Unlike the spacial interrelation feature analysis, shape transform projects a shape contour or region into an other domain to obtain some of its intrinsic features.

For shape description, there is always a trade-o between accuracy and eciency. On one hand, shape should be described as accurate as possible; on the other hand, shape description should be as compact as possible to simplify indexing and retrieval. For a shape transform analysis algorithm, it is very exible to accomplish a shape description with dierent accuracy and eciency by choosing the number of transform coecients.

Summary table

For convenience to compare these shape feature extraction approaches in this chapter, we summarize their properties in Table 1.

Frankly speaking, it is not equitable to arm a property of an approach by rudely speaking good or bad. Because certain approaches have great dierent performances under dierent conditions. For example, the method area function is invariant with ane transform under the condition of the contours sampled at its same vertices; whereas it is not robust to ane transform if the condition can't be contented. In addition, some approaches have good properties for certain type shapes; however it is not for the others. For example, the method shapelets representation is especially suitable for blobby objects, and it has shortcomings in representing elongated objects. So the simple evaluations in this table are only as a reference. These evaluations are drawn by assuming that all the necessary conditions have been contented for each approach.

Conclusion

In this chapter we made a study and a comparison the methods of shape-based feature extraction and representation.

About 40 techniques for extraction of shape features have been shortly described and compared. Unlike the traditional classication, the approaches of shape-based feature extraction and representation were classied by their processing approaches. These processing approaches included shape signatures, polygonal approximation methods, spatial interrelation feature, moments approaches, scale-space methods and shape transform domains: in such way, one can easily select the appropriate processing approach. A synthetic table has been established for a fast and global comparison of the performances of these approaches.

Extracting a shape feature in accordance with human perception is not an easy task. Due to the fact that human vision and perception are an extraordinary complicated system, it is a utopia to hope that the machine vision has super excellent performance with small complexity. In addition, choosing appropriate features for a shape recognition system must consider what kinds of features are suitable for the task. There exists no general feature which would work best for every kind of images.

Figure 4 :

 4 Figure 4: Circle variance

Figure 5 :

 5 Figure 5: Ellipse variance

Figure 7 :

 7 Figure 7: Proles

  It is called tangent space representation. A digital curve C simplied by polygon evolution is represented in the tangent space by the graph of a step function, where the x-axis represents the arc length coordinates of points in C and the y-axis represents the direction of the line segments in the decomposition of C. For example, gure 8 shows a digital curve and its step function representation in the tangent space.

Figure 8 :

 8 Figure 8: Digital curve and its step function representation in the tangent space

  (a) Contours normalized to 128 points; the dots marked star are the starting points on the contours; (b) curvature functions; the curvature is computed clockwise.

Figure 9 :

 9 Figure 9: Curvature function

Figure 10

 10 Figure 10 shows an example. Let S(n) be the area between the successive boundary points Pn, Pn+1 and center of gravity G.

Figure 10 :

 10 Figure 10: Area function

Figure 11 :

 11 Figure 11: Three dierent types of the triangle-area values and the TAR signature for the hammer shape

Figure 12

 12 Figure12shows the multi-scale space TAR function and its dynamic space warping (DSW) matching (cf. Subsection 1.3.2). In (a), the correspondent points on the model contour to these on the query contour are consistent with human perception after DSW matching.

Figure 12 :

 12 Figure 12: Dynamic space warping (DSW) matching

Figure 13 :

 13 Figure 13: Illustration of the distance from a point on the boundary to a linear segment

Figure 14 is

 14 Figure 14 is an example.

Figure 14 :

 14 Figure 14: A few stages of polygon evolution according to a relevant measure

  (a) Adaptive Grid Resolution (AGR) image; (b) quad-tree decomposition of AGR.

Figure 16 :

 16 Figure 16: Adaptive resolution representations

Figure 17 :

 17 Figure 17: Flowchart of shape divided by bounding box

  (a) Compute the bounding box B(S) of a pixel set S ; (b) subdivide S into n vertical slices; (c) compute the bounding box B(Sj) of each resulting pixel set Sj, where j = 1, 2, • • • , n; (d) subdivide each B(Sj) into m horizontal slices; (e) compute the bounding box B(Sij) of each resulting pixel set Sij, where i = 1, 2, • • • , m.

Figure 18 :

 18 Figure 18: The ve steps of bounding box splitting

Figure 19 :

 19 Figure 19: A sample points on lattice and examples of how it is mapped onto dierent shapes

  (a) Convex hull and its concavities; (b) concavity tree representation of convex hull.

Figure 20 :

 20 Figure 20: Illustrates recursive process of convex hull

  scheme {i|i = 0, 1, 2, • • • , 7} or {i|i = 0, 1, 2, 3}denoting a counter-clockwise angle of 45 • × i or 90 • × i regarding the positive x -axis, as shown in Figure 21.

Figure 21 :

 21 Figure 21: Basic chain code direction

Figure 22 :

 22 Figure 22: Vertex chain code

  θ is the orientation of this token; (b) an example of smooth curve decomposition.

Figure 23 :

 23 Figure 23: Smooth curve decomposition

Figure 24 :

 24 Figure 24: Symbolic features based axis of least inertia

Figure 25 :

 25 Figure 25: Beam angle at the neighborhood system 5 for a boundary point

Figure 26 :

 26 Figure 26: The BAS descriptor for original and noisy contour

  (a) Original shape region; (b) square model shape matrix; (c) reconstruction of the shape region.

Figure 27 :

 27 Figure 27: Square model shape matrix

Figure 28 :

 28 Figure 28: Polar model shape

  An example is shown in Figure 29. (c) is the diagram of log-polar histogram that has 5 bins for the polar direction and 12 bins for the angular direction. The histogram of a point Pi is formed by the following: putting the center of the histogram bins diagram on the point Pi, each bin of this histogram contains a count of all other sample points on the shape falling into that bin. Note on this gure, the shape contexts (histograms) for the points marked by • (in (a)), (in (b)) and (in (a)) are shown in (d), (e) and (f ), respectively. It is clear that the shape contexts for the points marked by • and , which are computed for relatively similar points on the two shapes, have visual similarity. By contrast, the shape context for is quite dierent from the others. Obviously, this descriptor is a rich description, since as N gets large, the representation of the shape becomes exact. (a) and (b) Sampled edge points of two shapes; (c) diagram of log-polar histogram bins used in computing the shape contexts; (d), (e) and (f ) shape contexts for reference sample points marked by •, and in (a) and (b), respectively.(Dark=large value).

Figure 29 :

 29 Figure 29: Shape context computation and graph matching

  (a) Original contour; (b) chord lengths histogram; (c) chord angles histogram (each stem covers 3 degrees).

Figure 30 :

 30 Figure 30: Chord distribution

Figure 31 :

 31 Figure 31: Medial axis of a rectangle dened in terms of bi-tangent circles

Figure 32 :

 32 Figure 32: Shock segments

Figure 33 :

 33 Figure 33: Examples of shapes and their shock graphs

Figure 34

 34 is a CSS image examples. Evolution of Africa: from left to right σ = 0(original), σ = 4, σ = 8 and σ = 16, respectively; (b) CSS image of Africa.

Figure 34 :

 34 Figure 34: Curvature scale-space image

  is an example of IPM. (a) An original contour; (b) an IPM image in the (u, σ) plane. The IPM points indicated by (1)(6) refer to the corresponding intersection points in (a).

Figure 35 :

 35 Figure 35: Example of the IPM

Figure 36

 36 Figure36shows the polar shape image turning into normal rectangular image.

Figure 36 :

 36 Figure 36: The polar shape image turns into normal rectangular image.

Figure 37 :

 37 Figure 37: Real parts of the ART basis functions

  (a) Example shape; (b) harmonic function within the unit disk; (c) rectilinear mapping of the function.

Figure 38 :

 38 Figure 38: Harmonic embedding of curvature signature

  (a) Original and noisy shapes; (b) harmonic embedding images for centroid distance signature.

  y)δ(x cos θ + y sin θ -ρ)dxdy where δ(.) is the Dirac delta-function:δ(x) = 1 if x = 0 0 otherwise θ ∈ [0, π] and ρ ∈ (-∞, ∞).In other words, Radon transform TR(ρ, θ) is the integral of f over the line L (ρ,θ) dened by ρ = x cos θ + y sin θ.

Figure 40

 40 Figure 40 is an example of a shape and its Radon transform.

Figure 40 :

 40 Figure 40: A shape and its Radon transform

8. 6

 6 Shapelets descriptorShapelets descriptor was proposed to present a model for animate shapes and extracting meaningful parts of objects.The model assumes that animate shapes (2D simple closed curves) are formed by a linear superposition of a number of shape bases. A basis function ψ(s; µ, σ) is dened in[START_REF] Dubinskiy | A multi-scale generative model for animate shapes and parts[END_REF]: µ ∈ [0, 1] indicates the location of the basis function relative to the domain of the observed curve, and σ is the scale of the function ψ.

Figure 41

 41 shows the shape of the basis function ψ at dierent σ values. It displays variety with dierent parameter and transforms.

Figure 41 :d 2 kFigure 42 :

 41242 Figure 41: Each shape base is a lobe-shaped curve The basis functions are subject to ane transformations by a 2 Ö 2 matrix of basis coecients:A k = a k b k c k d kThe variables for describing a base are denoted by b k = (A , µ k , σ k ) and are termed basis elements. The shapelet is dened by