Curvature is a property of the local surface.It is usually regarded as a tool of describing the curve’s degree.On a curved surface,the features we think all fall on the bending places which may be ridges,valleys or peaks.It is feasible to use curvature to characterize the features on a curved surface.This section will introduce some commen shape descriptors based on curvature.
Suppose p be an arbitrary point on a surface in 3D Eucldean space.If p is on a curve,we
…rst de…ne the curvature on a general curve as the reciprocal of the radius of a osculation circle at p.And if there is a plane contained p and the curve,we can see the the unit vector emanating from the point p and perpendicular to the surface is called the unit normal vector. A normal plane at p is a plane that contains the normal vector. Intersection of normal plane and the surface is a curve,The curvature of the planar curve is called normal curvature at p in the speci…c direction.The maximum and minimum normal curvatures at a point de…ne the principle curvature denoted by max and min.
In [7, 39],Gorden provides the de…nition of ridge lines and valley lines.Ridge lines is the local maxima in maxalong the line of maximum curvature and valley lines is similarly the local minima min in along the line of minimum curvature.After computing the principle curvatures at each point,Gorden thresholded the maximum and minimum curvature maps by setting appropriate thresholding value of extreme curvature to …nd the lines what people are really interested in.Fig.2 shows ridge and valley lines for a human face. Note how clearly the characteristic features of the face are displayed by these extrema.
Figure 2:(a)Ridge line:local maxima of in direction of maximum curvature (max ) (b)valley line:local minima of in direction of minimum curvature (min ) [39].
From the principle curvature ,we can derive two curvature measures from the princi-ple curvature, the mean curvature (H) and the Gaussian curvature (K).Gaussian curva-ture,K,is the product of principle curvatures.Formally,it is de…ned as
= max£ min (2.1)
where max and min are maximum and minimum principle curvature as previous men-tioned.Gaussian curvature represents the total bending degree at p on the curved sur-face.Mean curvature,H,is the arithmetic average of principle curvature and is de…ned as
= max+ min
2 (2.2)
where max and min are maximum and minimum principle curvature as previous men-tioned.Mean curvature represents the average bending degree at p on the curved sur-face.These two curvature measures show the characteristics of the local surface around a point.For instance,Points with positive Gaussian curvature are called elliptic,points with negative Gaussian curvature are called hyperbolic and points with zero Gaussian cuvature are at planar.In 1986,Besl introduced the HK segmentation dependimg on the sign of the Gaussian and Mean curvature [13],which is calculated from the two priciple curvatures
max and min.The classi…cation of surface types based on the sign of Gaussian curvature and mean curvature is shown in Table1.
Table1:HK classi…cation based on the sign of the Mean and Gaussian curvatures [13].
Table1 shows there are four kinds of regions are classi…ed:(+)(+) are convex,(+)(¡) are concaves,(¡)(+) are saddle with min + min 0,(¡)(¡) are saddle with
min+ min 0[7].If we apply HK segmentation method on human face analysis,it also mainly segmentation the human face into four kinds of regions.We can see an example in Fig.3.The red zones are elliptical concave regions,green zones are elliptical convex re-gion,yellow zones are hyperbolic concave regions and blue zones are hyperbolic convex region.
Figure 3:Facial segmentation base on sign HK classi…cation [9].
Figure 4:The HK classi…cation map and its threshold version on human face [9].
According to the prior knowledge of human face,we can know most facial feature points ususally have the property of high curvature.Colombo [9] de…ned and as the thresh-old values for isolating the candidate nose tips and eye corners.Fig.4 shows the threshthresh-old- threshold-ing result on human face.Although we can see the good classi…cation ability on thecom-bination of Gaussian curvature and mean curvature from …g.4,the Gaussian curvature is sensitive to the scale.It is not a good shape descriptor which should be irrelated to the scale,translation and rotation and only related to the pure shape.In the following,we will introduce shape index which is irrelated to the scale and more related to the pure shape.
Koenderink and van Doorn proposed an alternative curvature representation,shape index [34].His approach decouples the shape and the magnitude of the curvedness.They de…ned the shape index,S,as a quantitative measure of the shape.The formula is as fol-lowing
= 2
£ tan¡1(max() + min()
max()¡ min()) (2.3) where max and min are the principle curvatures of the surface.The shape index ranges from -1 to 1.A convex surface point with equal principle curvatures has a shape index 1.A concave surface point with equal principal curvatures has a shape index of -1.A saddle surface point with principal curvatures of equal magnitude and opposite sign has
a shape index of 0. The index covers all shape except for the planar shape which has
max = min = 0causing an inderminate value of shape index.
But in 1997,Dorai and Jain found out the local information about each shape category is not maintained distinctly with Keonderink and van Doorn’s de…ntion .Hence Dorai and Jain proposed an extension de…nition of Koenderink and van Doorn’s original de…nition [15, 16] and let the shape index range from 0 to 1.The extension formulation is de…ned as follow
() = 1 2 ¡ 1
tan¡1 max() + min()
max()¡ min() (2.4) where max and min are the principle curvatures of the surface.Every didstinct surface shape corresponds to a unique value of except the planar shape.Nine well-known shape categories and their corresponding shape index are shown in Fig. 5.
Figure 5:Nine well known shape categories and their correspomding shape index [15].
Figure 6:Nine representative shapes on the scale [15].
And the representative shapes from each category are graphically illustrated in Fig. 6.All basic shape types based on the signs of Gaussian and mean curvatures that was adopted by Besl are included in Dorai and Jain’s framework.
The shape index of a point on curved surface is not only independent of its position and orientation in space,but also independent of its scale.In order to obtain the scale di¤erences between objects,Koenderink and van Doorn introduced the curvedness to measure that how highly or gently a curved surface is.The curvedness [15, 16] is de…ned as
() =
r2max() + min2 ()
2 (2.5)
It equals to zero only at a point that has no curvedness,i.e. the point is on a planar patch.The combination classi…cation of the shape index and the curvedness,the SC clas-si…cation,often used to classify the curved surface.It can be applied on human face to isolate the salient feature regions by setting a appropiate threshold value.In [35],Nair and Cavallaro use shape index and curvedness index to describe the facail features as shown
respectively in Fig. 7(a) and Fig. 7(b).
Figure 7(a):Facial features are mapped by shape index [35].
Cantzler and Fisher [13] compared the di¤erence between HK and SC curvature de-scription method by checking the classi…cation ability under the classi…cation threshold varying and the noise increasing.Their conclusion indicates that the SC classi…cation is more stable at low thresholds and can deal better with image noise in image.Therefore SC classi…cation scheme has a slight advantage when dealing with real scenes containing multiple surfaces and moderate noise.