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# Selecting the Best B-spline Fit to a Closed Curve

IP.com Disclosure Number: IPCOM000109370D
Original Publication Date: 1992-Aug-01
Included in the Prior Art Database: 2005-Mar-24
Document File: 2 page(s) / 71K

IBM

## Related People

Flickner, M: AUTHOR [+4]

## Abstract

Two new methods for obtaining a shift invariant uniform B-spline fitting are presented. The goal is to select the shift of the data that minimizes the fitting error. This can be decided without computing the different approximations.

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Selecting the Best B-spline Fit to a Closed Curve

Two new methods for obtaining a shift invariant uniform
B-spline fitting are presented.  The goal is to select the shift of
the data that minimizes the fitting error.  This can be decided
without computing the different approximations.

If a continuous curve or a set of data points is to be
approximated by splines, the use of the so called "B-spline" basis
leads to an efficient solution.  However, a number of decisions have
to be made: degree of the polynomial pieces, number and location of
the knots, etc.

A reasonable requirement for an approximation algorithm is to
be invariant with regard to certain simple transformations of the
data.  Uniform B-splines (where the discontinuity points are
uniformly sampled in the parametric domain) provide an invariant
scheme when affine mappings of the data are taken into account.  This
method is not invariant to the selection of the starting point of the
data (shift invariance).  However, such invariance may be obtained by
selecting the shift that provides the minimum error of fit.  We
present two efficient methods for choosing such shift before fitting.

When a set of data points {Yj}mj=1 is given as the input, it
will be assumed that these are the values of an (unknown) function f
at a increasing sequence of points {uj}mj=1, belonging to the
parametric domain.  These data point parameters uj should satisfy
uj+1 = uj + Wu with p, where m is the number of data points,
k is the number of pieces, and p is a positive integer.  This is
known as "uniform parameterization".  The first method reduces the
domain of search of the be...