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10Principal Curve•••••Data PointProjection Point•••••••Figure 2.2: Principal curve example.background noise is uniformly distributed over the region of the image (this isequivalent to Poisson background noise). We assume that the feature points aredistributed uniformly along the true underlying feature; that is, their projectionsonto the feature’s principal curve are drawn randomly from a uniform distributionU(0, ν j ), where ν jis the length of the j-th curve. We assume that the featurepoints are distributed normally about the true underlying feature, with mean zeroand variance σj 2 . Distance about the curve is the orthogonal distance from a pointto the curve; if the point projects to an endpoint of the curve, it is simply thedistance from the point to the curve endpoint. The (K + 1) clusters are combinedin a mixture model, and we denote the unconditional probability of belonging tothe j-th feature by π j (j = 0, 1 . . . K).Let θ denote the entire set of parameters, {ν j , σ 2 j , π j: j = 1, . . . , K}, notincluding the curves themselves. Then the likelihood is L(X|θ) = ∏ Ni=1 π j L(x i |θ),