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Normalization with F=1


Another commonly used normalization is to set F=1. If we use the same notations as in the last subsection, the problem becomes to minimize the following function:


where tex2html_wrap_inline2725 is the sixth element of vector tex2html_wrap_inline2527 , i.e., tex2html_wrap_inline2729 .

Indeed, we seek for a least-squares solution to tex2html_wrap_inline2731 under the constraint tex2html_wrap_inline2733 . The equation can be rewritten as


where tex2html_wrap_inline2735 is the matrix formed by the first (n-1) columns of tex2html_wrap_inline2739 , tex2html_wrap_inline2741 is the last column of tex2html_wrap_inline2739 and tex2html_wrap_inline2745 is the vector tex2html_wrap_inline2747 . The problem can now be solved by the technique described in sect:Ax=b.

In the following, we present another technique for solving this kind of problems, i.e.,


based on eigen analysis, where we consider a general formulation, that is tex2html_wrap_inline2739 is a tex2html_wrap_inline2751 matrix, tex2html_wrap_inline2527 is a m-vector, and tex2html_wrap_inline2757 is the last element of vector tex2html_wrap_inline2527 . The function to minimize is


As in the last subsection, the symmetric matrix tex2html_wrap_inline2653 can be decomposed as in (5), i.e., tex2html_wrap_inline2763 . Now if we normalize each eigenvalue and eigenvector by the last element of the eigenvector, i.e.,


where tex2html_wrap_inline2765 is the last element of the eigenvector tex2html_wrap_inline2665 , then the last element of the new eigenvector tex2html_wrap_inline2769 is equal to one. We now have


where tex2html_wrap_inline2771 and tex2html_wrap_inline2773 . The original problem (7) becomes:

Find tex2html_wrap_inline2669 such that tex2html_wrap_inline2671 is minimized with

tex2html_wrap_inline2779 subject to tex2html_wrap_inline2781 .

After some simple algebra, we have


The problem now becomes to minimize the following unconstrained function:


where tex2html_wrap_inline2677 is the Lagrange multiplier. Setting the derivatives of tex2html_wrap_inline2695 with respect to tex2html_wrap_inline2681 through tex2html_wrap_inline2683 and tex2html_wrap_inline2677 yields:


The unique solution to the above equations is given by


where tex2html_wrap_inline2793 . The solution to the problem (7) is given by


Note that this normalization (F=1) has singularities for all conics going through the origin. That is, this method cannot fit such conics because they require to set F=0. This might suggest that the other normalizations are superior to the F=1 normalization with respect to singularities. However, as shown in [20], the singularity problem can be overcome by shifting the data so that they are centered on the origin, and better results by setting F=1 has been obtained than by setting A+C=1.

next up previous contents
Next: Least-Squares Fitting Based on Up: Least-Squares Fitting Based on Previous: Normalization with #tex2html_wrap_inline2635#

Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996