Metrics of curves in shape optimization and analysis - Andrea Carlo ...

In the case of planar curves c 1 , c 2 , we can apply to above idea to define2.1.2 Shape averagesd(c 1 , c 2 ) = ∣ ∣˚c1 ∆˚c 2∣ ∣ .Many definitions have also been proposed for the average ¯c of a finite set ofshapes c 1 , . . . , c N . There are methods based on the arithmetic mean (that arerepresentation dependent).• One such case is when the shapes are defined by a finite family of Nparameters; so we can define the parametric or landmark averaging¯c(p) = 1 NN∑c n (p)n=1where p is a common parameter/landmark.• More in general, we can define the signed distance level set averagingby using as a representative of a shape its signed distance function, andcomputing the average shape by¯c = { x | ¯b(x) = 0 } , where ¯b(x) = 1 NN∑b cn (x)where b cn is the signed distance function of c n ; or in case of planar curves,of the part of plane enclosed by c n .Then there are non parametric, representation independent, methods. The(possibly) most famous one is theDefinition 2.4 The distance-based average. 2 Given a distance d M betweenshapes, an average shape ¯c may be found by minimizing the sum of its squareddistances.N∑¯c = arg min d M (c, c n ) 2cn=1It is interesting to note that in Euclidean spaces there is an unique minimum,that coincides with the arithmetic mean; while in general there may be nominimum, or more than one.2.1.3 Principal component analysis (PCA)Definition 2.5 Suppose that X is a random vector taking values in lR n ; letX = E(X) be the mean of X. The principal component analysis is therepresentation of X asn∑X = X + Y i S ii=12 Due to Fréchet, 1948; but also attributed to Karcher.n=19