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Shape FeaturesJamil Ahmad*

ContentsShape FeaturesProperties of Shape FeaturesShape DescriptorDescriptors ClassificationSimple Shape FeaturesOne-dimensional function for shape representationSome other shape featuresConclusionReferences*

Shape FeatureA prominent attribute or aspect of a shape*

Properties of Shape FeaturesEfficient shape features must present some essential properties such as:Identifiability: perceptually similar objects have similar (or the same) features.Translation, rotation and scale invariance: the location, the rotation and the scaling changing of the shape must not affect the extracted features.*

Properties of Shape FeaturesAffine invariance: (preserving parallelism and straightness): shape distortion that preserve shape characteristics should not alter the descriptor.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 affects the pattern.

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Properties of Shape Featuresoccultation invariance: partial occlusion should not change the descriptor.Statistically independent: compact descriptor.Reliability: as long as one deals with the same pattern, the extracted features must remain the same.*

Shape DescriptorShape Descriptor is a set of numbers that are produced to represent a given shape feature. A descriptor attempts to quantify the shape in ways that agree with human intuition.Usually, the descriptors are in the form of a vector.*

Descriptors ClassificationContour-Based Methods:Use shape boundary pointsRegion-Based Methods: Use shape interior points*

Simple Shape FeaturesSome simple geometric features can be used to describe shapes. Usually, the simple geometric features can only discriminate shapes with large differences; therefore, they are usually used as filters to eliminate false hits or combined with other shape descriptors to discriminate shapes.Center of gravityEccentricityCircularity ratioRectangularityConvexitySolidityEuler numberHole area ratio*

1. Center of GravityThe center of gravity is also called centroid. Its position should be fixed in relation to the shape.

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2. Circularity ratioCircularity ratio represents how a shape is similar to a circle.Circularity ratio is the ratio of the area of a shape to the shape's perimeter square:*

3. RectangularityRectangularity represents how rectangular a shape is, i.e. how much it fills its minimum bounding rectangle:

where AS is the area of a shape; AR is the area of the minimum bounding rectangle.*

4. ConvexityConvexity is defined as the ratio of perimeters of the convex hull over that of the original contour.*

5. SoliditySolidity describes the extent to which the shape is convex or concave.

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.*

6. Euler numberEuler 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:*

9. Hole area ratioHole area ratio HAR is defined as

where As is the area of a shape and Ah is the total area of all holes in the shape*

One-dimensional function for shape representationAlso known as Shape SignatureIt is derived from shape boundary coordinatesThe shape signature usually captures the perceptual feature of the shapeCentroid distance functionArea functionChord length function*

1. Centroid distance functionThe centroid distance function is expressed by the distance of the boundary points from the centroid of a shape:

Translation Invariant*

2. Area FunctionWhen the boundary points change along the shape boundary, the area of the triangle formed by two successive boundary points and the center of gravity also changes. This forms an area function which can be exploited as shape representation.*

3. Chord Length FunctionFor each boundary point P, its chord length function is the distance between P and another boundary point P.*

Some other shape featuresBasic Chain CodeDifferential Chain Code Chain Code HistogramShape Matrix*

Chain Codes (Basic + DifferentialNotation to record a list of boundary bounds along a contour.describes the movement along a digital curve or a sequence of border pixels by using so-called 8-connectivity or 4-connectivity.*

Shape NumbersTo make the chain code rotational invariant, we consider all cyclic rotations of the differential chain code and choose among them the lexicographically smallest such code. The resulting code is called Shape Number.Chain Code: 3 0 0 3 0 1 1 2 1 1 2 3 2Differential Chain Code:1 0 3 1 1 0 1 3 1 1 3Shape Number:0 1 3 1 1 3 1 0 3 1 1*

Chain Code HistogramThe CCH reflects the probabilities of different directions present in a contour.*

Shape MatrixShape matrix descriptor is an M N matrix to represent a shape region.Matrix itself is not a featureHistograms in x- and y-directionTexture of shape matrixFrequency analysis*

ConclusionA shape signature represents a shape by a 1-D function derived from shape contour.To obtain the translation invariant property, they are usually defined by relative values.To obtain the scale invariant property, normalization is necessary.Shape signatures are sensitive to noise, and slight changes in the boundary can cause large errors in matching procedure.A shape signature can be simplified by quantizing the signature into a signature histogram, which is rotationally invariant.*

ReferencesYang Mingqiang, Kpalma Kidiyo and Ronsin Joseph, A Survey of Shape Feature Extraction Techniques Pattern Recognition Techniques, Technology and Applications, Book edited by: Peng-Yeng Yin, ISBN 978-953-7619-24-4, pp. 626, November 2008, I-Tech, Vienna, Austria*

Thank you*

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