System and method for registration and modeling of deformable shapes by direct factorization

A method of registering and modeling a deformable shape in a digitized image, comprising the steps of:

• providing a measurement matrix W of N measurements of P points of a D-dimensional deformable shape;
• determining a basis number K for the N measurements, wherein K<N, and selecting K measurements of said measurement matrix W as a basis set;
• decomposing said measurement matrix W into a matrix product M×B, wherein M is a proposed scaled rotation matrix and B is a proposed basis matrix
• computing a matrix Qk defined by
• M
• i
• ⁢
• Q
• k
• ⁢
• M
• j
• T
• =
• {
• I
• D
• ×
• D
• ,
• i
• =
• j
• =
• k
• 0
• D
• ×
• D
• ,
• (
• i
• ,
• j
• )
• ∈
• Φ
• ,
• where Φ stands for {(i,j)|i=1, . . . , K;
• j=1, . . . , N;
• i≠k}, which represents an ambiguity matrix transforming the proposed scaled rotation matrix and the proposed basis matrix into a true scaled rotation matrix and a true basis matrix;
• decomposing the matrix Qk into gkgkT, for k=1, . . . , K, wherein gk is a column of a DK×DK matrix G;
• and
• recovering the true scaled rotation matrix from M×G and the true basis matrix from G−1×B;
• and
• constructing a registered shape from said true scaled rotation matrix and said true basis matrix.

A program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for registering and modeling a deformable shape in a digitized image comprising the steps of:

• providing a set of measurements of a deformable shape, wherein said set of measurements forms a measurement matrix;
• and
• simultaneously solving for a rigid registering transformation and a linear shape model, by forming an average of said set of measurements and subtracting said average from said set of measurements;
• decomposing said measurement matrix into a product of a proposed scaled rotation matrix and a proposed basis matrix;
• calculating a product matrix of an ambiguity transformation from said proposed scaled rotation matrix and said proposed basis matrix;
• constraining and decomposing said product matrix to determine an ambiguity transformation matrix;
• using said ambiguity transformation matrix to determine a true scaled rotation matrix and a true basis matrix;
• and
• constructing a registered shape from said true scaled rotation matrix and said true basis matrix.