The 3L algorithm for fitting implicit polynomial curves and surfaces to data

Abstract

Of great importance to a wide variety of computer vision and image analysis problems is the ability to represent two-(2D) and three-dimensional (3D) data or objects. Implicit polynomial curves and surfaces are two of the most useful representations available. Their representational power is evidenced by their ability to smooth noisy data and to interpolate through sparse or missing data. Furthermore, their associated Euclidean and a ne invariants are powerful discriminators, making implicit polynomials a computationally attractive technology for recognizing objects in arbitrary positions with respect to cameras or range sensors. In this paper, we introduce a completely new approach to tting implicit polynomials to data. The algorithm represents a signi cant advancement of implicit polynomial technology for three important reasons. First, it is orders of magnitude faster than existing methods. Second, it has signi cantly better repeatability and numerical stability than current methods. Third, it can easily t polynomials of high, such as 14 th to 18 th , degree. In addition, this approach provides a completely new way of thinking about and handling implicit polynomials.

Simultaneous localization and sampled environment mapping

“…The main reason for this phenomenon is that there are not many laser points observed at the two ends, causing the measurements variance computed in (15) to be larger than that in the middle part. Thus in the procedure of observation updation (20), the samples at the two ends get little modification. The nonlinear observation model of the arcs may also contribute negative influence.…”

“…Fitting a set of points with a second order polynomial can be solved by 3L-Fitting method quickly [20], [21].…”

Implicit B-Spline Surface Reconstruction

“…Indeed, it considers the absolute values normalized by the gradient | f |/ ∇ f . The 3L algorithm is another fitting technique that exploits more geometric clues of the point set [12]. In this method the original data is supported by two additional offsets in order to control the function values around the zero set.…”

“…The 3L algorithm, as mentioned in section II-B, is an algebraic IP fitting technique that exploits local geometric clues in the data set [12]. It generates an inner and outer offset denoted by P −δ and P +δ at the distance ±δ from the original data set P 0 .…”

“…These functions are constructed from locally supported functions that enable local control, but unlike the parametric B-splines, they avoid parameterization problems. First, we propose the 3L-IBS algorithm to provide the IBS coefficients through a fast algebraic fitting method [12]. Then, we show how the obtained IBS surface can be further controlled by a proper regularization term and additional constraints.…”