REDUCED DATA FOR CURVE MODELING – APPLICATIONS IN GRAPHICS, COMPUTER VISION AND PHYSICS Curve Modeling - Applications Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics Reduced Data for Curve Modeling - Applications in Graphics, Computer Vision and Physics

interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . ABSTRACT In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate the missing knots Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } m i =0 for non-parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . t i = i . However, this simplistic method frequently renders surprisingly undesired results. Following discussion from [4] and [6] there exists a strong indication, that method of guessing interpolation , where Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n . Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori speciﬁed interpolation knots . We discuss two approaches to estimate missing knots { t i } mi =0 for non- parametric data (i.e. collection of points { q i } mi =0 , where q i ∈ R n ). The ﬁrst approach ( uniform evaluation ) is based on blind guess in which knots { ˆ t i } mi =0 are chosen uniformly. The second approach ( cumulative chord parameterization ), incorporates the geometry of the distribution of data points. More precisely the diﬀerence ˆ t i +1 − ˆ t i is equal to the Euclidean distance between data points q i +1 and q i . The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for ﬁtting non-parametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n .


INTRODUCTION
In this paper we consider the problem of modeling curves via interpolation based on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), where q i ∈ R n . The term reduced data corresponds to the ordered sequence of m+1 input points in R n stripped from the tabular parameters ve Modeling -Applications ter Vision and Physics Kozera 1,2  , where qi ∈ R n ). The n) is based on blind guess in which knots he second approach (cumulative chord the geometry of the distribution of data encet i+1 −ti is equal to the Euclidean +1 and qi. The second method partially nformation carried by the reduced data. of the above schemes for fitting nonaphics (light-source motion rendering), entation) and in physics (high velocity Though experiments are conducted for ethod is equally applicable in R n . uter vision and graphics, physics. per we consider the problem of modeling curves on without a priori specified interpolation knots. aches to estimate missing knots {ti} m i=0 for noncollection of points {qi} m i=0 , where qi ∈ R n ). The evaluation) is based on blind guess in which knots iformly. The second approach (cumulative chord orporates the geometry of the distribution of data the differencet i+1 −ti is equal to the Euclidean a points qi+1 and qi. The second method partially oss of the information carried by the reduced data. application of the above schemes for fitting nonmputer graphics (light-source motion rendering), mage segmentation) and in physics (high velocity odeling). Though experiments are conducted for he entire method is equally applicable in R n . tion, computer vision and graphics, physics.
(where t 0 = 0 < t 1 <t 2 < ... <t m = T < ∞ ), usually referred in the literature as interpolation knots. To perform any in terpolation scheme we need first to estimate the unknown knots t i . One approach is to choose the parameters act. In this paper we consider the problem of modeling curves via interpolation without a priori specified interpolation knots. cuss two approaches to estimate missing knots {ti} m i=0 for nonetric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The proach (uniform evaluation) is based on blind guess in which knots are chosen uniformly. The second approach (cumulative chord terization), incorporates the geometry of the distribution of data More precisely the differencet i+1 −ti is equal to the Euclidean e between data points qi+1 and qi. The second method partially nsates for the loss of the information carried by the reduced data. o present the application of the above schemes for fitting nonetric data in computer graphics (light-source motion rendering), puter vision (image segmentation) and in physics (high velocity es trajectory modeling). Though experiments are conducted for in R 2 and R 3 the entire method is equally applicable in R n .
ords: interpolation, computer vision and graphics, physics.
uction we consider the problem of modeling curves via interpolation based led discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), n . The term reduced data corresponds to the ordered sequence of oints in R n stripped from the tabular parameters {t i } m i=0 . More preain reduced data by sampling parametric curve γ : [0, T ] → R n with here 0 ≤ i ≤ m) in arbitrary Euclidian space without provision of nding parameters {t i } m i=0 (where t 0 = 0 < t 1 < t 2 < ... < t m = T < referred in the literature as interpolation knots. To perform any incheme we need first to estimate the unknown knots t i . One approach the parameters {t i } m i=0 ∈ [0,T ] m+1 blindly, by assigning them e.g. bers in the uniform manner:t i = i. However, this simplistic method nders surprisingly undesired results. Following discussion from [4] exists a strong indication, that method of guessing interpolation

Introduction
In this paper we consider the problem of modeling curves via interpolation based on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), where q i ∈ R n . The term reduced data corresponds to the ordered sequence of m+1 input points in R n stripped from the tabular parameters {t i } m i=0 . More precisely we obtain reduced data by sampling parametric curve γ : [0, T ] → R n with γ(t i ) = q i (where 0 ≤ i ≤ m) in arbitrary Euclidian space without provision of the corresponding parameters {t i } m i=0 (where t 0 = 0 < t 1 < t 2 < ... < t m = T < ∞), usually referred in the literature as interpolation knots. To perform any interpolation scheme we need first to estimate the unknown knots t i . One approach is to choose the parameters {t i } m i=0 ∈ [0,T ] m+1 blindly, by assigning them e.g. natural numbers in the uniform manner:t i = i. However, this simplistic method frequently renders surprisingly undesired results. Following discussion from [4] and [6] there exists a strong indication, that method of guessing interpolation = i. However, this simplistic method frequently renders surprisingly undesired results. Following discussion from [5] and [8] a strong indication exists that the method of guessing interpolation knots

Reduced Data for Curve Modeling -Applications in Graphics, Computer Vision and Physics
Ma lgorzata Janik 1 , Ryszard Kozera 1,2 , and Przemys law Kozio l 1 Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate missing knots {ti} m i=0 for nonparametric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The first approach (uniform evaluation) is based on blind guess in which knots {t i} m i=0 are chosen uniformly. The second approach (cumulative chord parameterization), incorporates the geometry of the distribution of data points. More precisely the differencet i+1 −ti is equal to the Euclidean distance between data points qi+1 and qi. The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for fitting nonparametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n .
Keywords: interpolation, computer vision and graphics, physics.

Introduction
In this paper we consider the problem of modeling curves via interpolation based on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), where q i ∈ R n . The term reduced data corresponds to the ordered sequence of m+1 input points in R n stripped from the tabular parameters {t i } m i=0 . More precisely we obtain reduced data by sampling parametric curve γ : should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [5] and [8], and is later referred to in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calcu lating the distance between consecutive different points {q i , q i+1 } and use the cumulative distance as respective values for the unknown knots: i.e.

Reduced Data for Curve Modeling -Applications
should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is . The problem of fitting non-parametric data is not only an abstract mathemati cal concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate missing knots {ti} m i=0 for nonparametric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The first approach (uniform evaluation) is based on blind guess in which knots {t i} Introduction this paper we consider the problem of modeling curves via interpolation based the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), ere q i ∈ R n . The term reduced data corresponds to the ordered sequence of 1 input points in R n stripped from the tabular parameters {t i } m i=0 . More preely we obtain reduced data by sampling parametric curve γ : Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate missing knots {ti} m i=0 for nonparametric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The first approach (uniform evaluation) is based on blind guess in which knots {t i} m i=0 are chosen uniformly. The second approach (cumulative chord parameterization), incorporates the geometry of the distribution of data points. More precisely the differencet i+1 −ti is equal to the Euclidean distance between data points qi+1 and qi. The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for fitting nonparametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n .
Keywords: interpolation, computer vision and graphics, physics.

Introduction
In this paper we consider the problem of modeling curves via interpolation b on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m where q i ∈ R n . The term reduced data corresponds to the ordered sequenc m+1 input points in R n stripped from the tabular parameters {t i } m i=0 . More cisely we obtain reduced data by sampling parametric curve γ : In this paper we consider the problem of modeling curves nterpolation without a priori specified interpolation knots. two approaches to estimate missing knots {ti} m i=0 for nondata (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The h (uniform evaluation) is based on blind guess in which knots chosen uniformly. The second approach (cumulative chord ation), incorporates the geometry of the distribution of data e precisely the differencet i+1 −ti is equal to the Euclidean ween data points qi+1 and qi. The second method partially s for the loss of the information carried by the reduced data. sent the application of the above schemes for fitting nondata in computer graphics (light-source motion rendering), vision (image segmentation) and in physics (high velocity jectory modeling). Though experiments are conducted for and R 3 the entire method is equally applicable in R n .
interpolation, computer vision and graphics, physics.
ion onsider the problem of modeling curves via interpolation based iscrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), he term reduced data corresponds to the ordered sequence of in R n stripped from the tabular parameters {t i } m i=0 . More preeduced data by sampling parametric curve γ : .. < t m = T < ed in the literature as interpolation knots. To perform any ine we need first to estimate the unknown knots t i . One approach arameters {t i } m i=0 ∈ [0,T ] m+1 blindly, by assigning them e.g. n the uniform manner:t i = i. However, this simplistic method s surprisingly undesired results. Following discussion from [4] ts a strong indication, that method of guessing interpolation Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate missing knots {ti} m i=0 for nonparametric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The first approach (uniform evaluation) is based on blind guess in which knots {t i} m i=0 are chosen uniformly. The second approach (cumulative chord parameterization), incorporates the geometry of the distribution of data points. More precisely the differencet i+1 −ti is equal to the Euclidean distance between data points qi+1 and qi. The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for fitting nonparametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n .
Keywords: interpolation, computer vision and graphics, physics.

Introduction
In this paper we consider the problem of modeling curves via interpolation based on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), where q i ∈ R n . The term reduced data corresponds to the ordered sequence of m+1 input points in R n stripped from the tabular parameters {t i } m i=0 . More precisely we obtain reduced data by sampling parametric curve γ : , usually referred in the literature as interpolation knots. To perform any interpolation scheme we need first to estimate the unknown knots t i . One approach is to choose the parameters {t i } m i=0 ∈ [0,T ] m+1 blindly, by assigning them e.g. natural numbers in the uniform manner:t i = i. However, this simplistic method frequently renders surprisingly undesired results. Following discussion from [4] and [6] there exists a strong indication, that method of guessing interpolation are chosen uniformly. The second approach (cumulative chord parameterization) incorporates the geometry of the distribution of data points. More precisely, the difference Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate missing knots {ti} m i=0 for nonparametric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The first approach (uniform evaluation) is based on blind guess in which knots {t i} m i=0 are chosen uniformly. The second approach (cumulative chord parameterization), incorporates the geometry of the distribution of data points. More precisely the differencet i+1 −ti is equal to the Euclidean distance between data points qi+1 and qi. The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for fitting nonparametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n .
Keywords: interpolation, computer vision and graphics, physics.

Introduction
In this paper we consider the problem of modeling curves via interpolation based on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), where q i ∈ R n . The term reduced data corresponds to the ordered sequence of m+1 input points in R n stripped from the tabular parameters {t i } m i=0 . More precisely we obtain reduced data by sampling parametric curve γ : , usually referred in the literature as interpolation knots. To perform any interpolation scheme we need first to estimate the unknown knots t i . One approach is to choose the parameters {t i } m i=0 ∈ [0,T ] m+1 blindly, by assigning them e.g. natural numbers in the uniform manner:t i = i. However, this simplistic method frequently renders surprisingly undesired results. Following discussion from [4] and [6] there exists a strong indication, that method of guessing interpolation such as medical image processing or high-velocity particle trajec tory modeling. Such examples are implemented here. The presented method can also be applied in modeling of different technical processes, i.e. [6] or [7,9].

Concepts
Spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [2]) of class C 2 . The essential idea is to fit the data γ(t 0 ), γ(t 2 ), ..., γ(t m ) with a piecewise cubic S : [0, T] → R n of the form: 2 Reduced Data for Curve Modeling -Applications knots {t i } m i=0 should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is to fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the form: Again by [1] (see Chapt. 4) the latter coefficients (with the aid of Newton's divided differences) read as: where s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). We consider here the second case. In doing so, we recall that values of s i for i = 1, ..., m − 1 can be derived from: P i (t i+1 ) = P i+1 (t i+1 ) (see also [1]). If s 0 and s m are given then we deal with the so-called complete spline.
On the other hand, if s 0 and s m are also unknown, we can add constraints γ(t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines with P 0 (t 0 ) = P i−1 (t m ) = 0. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [1]). (1) Reduced Data for Curve Modeling -Applications should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is to fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the form: Again by [1] (see Chapt. 4) the latter coefficients (with the aid of Newton's divided differences) read as: where s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). We consider here the second case. In doing so, we recall that values of s i for i = 1, ..., m − 1 can be derived from: P i (t i+1 ) = P i+1 (t i+1 ) (see also [1]). If s 0 and s m are given then we deal with the so-called complete spline.
On the other hand, if s 0 and s m are also unknown, we can add constraints γ(t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines with P 0 (t 0 ) = P i−1 (t m ) = 0. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [1]).
Again by [2] the latter coefficients (with the aid of Newton's divided differences) read as: should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is to fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the form: Again by [1] (see Chapt. 4) the latter coefficients (with the aid of Newton's divided differences) read as: where s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). We consider here the second case. In doing so, we recall that values of s i for i = 1, ..., m − 1 can be derived from: P i (t i+1 ) = P i+1 (t i+1 ) (see also [1]). If s 0 and s m are given then we deal with the so-called complete spline.
On the other hand, if s 0 and s m are also unknown, we can add constraints γ(t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines with P 0 (t 0 ) = P i−1 (t m ) = 0. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [1] should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is to fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the form: Again by [1] (see Chapt. 4) the latter coefficients (with the aid of Newton's divided differences) read as: where s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). We consider here the second case. In doing so, we recall that values of s i for i = 1, ..., m − 1 can be derived from: P i (t i+1 ) = P i+1 (t i+1 ) (see also [1]). If s 0 and s m are given then we deal with the so-called complete spline.
On the other hand, if s 0 and s m are also unknown, we can add constraints γ(t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines with P 0 (t 0 ) = P i−1 (t m ) = 0. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [1]).
and Δt i = t i+1 -t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). The latter case is considered here. In doing so, we recall that values of s i for i = 1, ..., m -1 can be derived from: Reduced Data for Curve Modeling -Applications should incorporate the geometry of the distribution of sampling oints Q m . Such possible method is analyzed in [4] and [6], and later referred our paper as cumulative chord knot evaluation method. In this approach we ompensate for the loss of the information carried by the reduced data by calcuting the distance between consecutive points {q i , q i+1 } and use the cumulative istance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . he problem of fitting non-parametric data is not only an abstract mathematial concept, but can be applied in real life. The latter happens e.g. in computer raphics (motion rendering), computer vision (image segmentation) and other pplications such as medical image processing or high-velocity particle trajecory modeling. Such examples are implemented here. Presented method can be lso applied in modeling of differet technical processes, i.e. [8] or [9,10].

.1 Concepts
pline interpolation is a form of interpolation, where the interpolant is a special ype of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a iecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is o fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the rm: Again by [1] (see Chapt. 4) the latter oefficients (with the aid of Newton's divided differences) read as: here s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e.
i are known (Hermite interpolation) and s i are unknown (a common case in ractice). We consider here the second case. In doing so, we recall that values f s i for i = 1, ..., m − 1 can be derived from: ). If s 0 and s m are given then we deal with the so-called complete spline. n the other hand, if s 0 and s m are also unknown, we can add constraints (t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines The natural spline determines the smoothest of all ossible interpolating curves in the sense that it minimizes the integral of the quare of the second derivative (see [1]).
(see also [2]). If s 0 and s m are given then we deal with the so-called complete spline. On the other hand, if s 0 and s m are also unknown, we can add constraints 2 Reduced Data for Curve Modeling -Applications should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is to fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the form: with constant vectors a i , b i , c i , d i ∈ R n . Again by [1] (see Chapt. 4) the latter coefficients (with the aid of Newton's divided differences) read as: where s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). We consider here the second case. In doing so, we recall that values of s i for i = 1, ..., m − 1 can be derived from: P i (t i+1 ) = P i+1 (t i+1 ) (see also [1]). If s 0 and s m are given then we deal with the so-called complete spline.
On the other hand, if s 0 and s m are also unknown, we can add constraints γ(t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines with P 0 (t 0 ) = P i−1 (t m ) = 0. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [1]).
. Such boundary conditions render the so-called natural splines with 2 Reduced Data for Curve Modeling -Applications should incorporate the geometry of the distribution of sampling points Q m . Such possible method is analyzed in [4] and [6], and later referred in our paper as cumulative chord knot evaluation method. In this approach we compensate for the loss of the information carried by the reduced data by calculating the distance between consecutive points {q i , q i+1 } and use the cumulative distance as values for the unknown knots: i.e.t 0 = 0 andt i+1 = q i+1 − q i +t i . The problem of fitting non-parametric data is not only an abstract mathematical concept, but can be applied in real life. The latter happens e.g. in computer graphics (motion rendering), computer vision (image segmentation) and other applications such as medical image processing or high-velocity particle trajectory modeling. Such examples are implemented here. Presented method can be also applied in modeling of differet technical processes, i.e. [8] or [9,10].

Concepts
Spline interpolation is a form of interpolation, where the interpolant is a special type of piecewise polynomial called a spline (see e.g. [11]). A cubic spline is a piecewise cubic polynomial (see [1]; Chapt. 4) of class C 2 . The essential idea is to fit the data γ(t 0 ),γ(t 2 ),...,γ(t m ) with a piecewise cubic S : [0, T ] → R n of the form: with constant vectors a i , b i , c i , d i ∈ R n . Again by [1] (see Chapt. 4) the latter coefficients (with the aid of Newton's divided differences) read as: where s i =γ(t i ) and ∆t i = t i+1 − t i . There are two possible cases here: i.e. s i are known (Hermite interpolation) and s i are unknown (a common case in practice). We consider here the second case. In doing so, we recall that values of s i for i = 1, ..., m − 1 can be derived from: P i (t i+1 ) = P i+1 (t i+1 ) (see also [1]). If s 0 and s m are given then we deal with the so-called complete spline.
On the other hand, if s 0 and s m are also unknown, we can add constraints γ(t 0 ) =γ(t m ) = 0. Such boundary conditions render the so-called natural splines with P 0 (t 0 ) = P i−1 (t m ) = 0. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [1]).
. The natural spline determines the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative (see [2]).

NON-PARAMETRIC INTERPOLATION AND KNOT EVALUATION METHODS
Some practical problems exist while dealing with the incomplete data set. We can consider many problems where the sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters pkoziol@student.mini.pw.edu.pl 2 Faculty of Applied Informatics and Mathematics Warsaw University of Life Sciences -SGGW 02-776 Nowoursynowska 159, Warsaw Poland ryszard_kozera@sggw.edu.pl Abstract. In this paper we consider the problem of modeling curves in R n via interpolation without a priori specified interpolation knots. We discuss two approaches to estimate missing knots {ti} m i=0 for nonparametric data (i.e. collection of points {qi} m i=0 , where qi ∈ R n ). The first approach (uniform evaluation) is based on blind guess in which knots {t i} m i=0 are chosen uniformly. The second approach (cumulative chord parameterization), incorporates the geometry of the distribution of data points. More precisely the differencet i+1 −ti is equal to the Euclidean distance between data points qi+1 and qi. The second method partially compensates for the loss of the information carried by the reduced data. We also present the application of the above schemes for fitting nonparametric data in computer graphics (light-source motion rendering), in computer vision (image segmentation) and in physics (high velocity particles trajectory modeling). Though experiments are conducted for points in R 2 and R 3 the entire method is equally applicable in R n .
Keywords: interpolation, computer vision and graphics, physics.

Introduction
In this paper we consider the problem of modeling curves via interpolation based on the so-called discrete reduced data Q m = (q 0 , q 1 , ..., q m ) (for i ∈ {0, 1, ..., m}), where q i ∈ R n . The term reduced data corresponds to the ordered sequence of m+1 input points in R n stripped from the tabular parameters {t i } m i=0 . More precisely we obtain reduced data by sampling parametric curve γ : , usually referred in the literature as interpolation knots. To perform any interpolation scheme we need first to estimate the unknown knots t i . One approach is to choose the parameters {t i } m i=0 ∈ [0,T ] m+1 blindly, by assigning them e.g. natural numbers in the uniform manner:t i = i. However, this simplistic method frequently renders surprisingly undesired results. Following discussion from [4] and [6] there exists a strong indication, that method of guessing interpolation . Such a task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply a scheme based on non-parametric interpolation, careful guessing of the knots Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 3

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2).

Non-parametric Methods
There exist some practica set. We can consider many the unknown curve γ with coined as fitting the reduce data is called non-parame on non-parametric interp [0,T ] m+1 needs to be mad Eq. (1)) yields the best p the analysis of C 0 piecewi C 1 or C 2 piecewise-cubics

Uniform Knot Ev
The simplest and the mos mate the unknown {t i } m i=0 withT = m. The potentia in Fig. 1 and Fig. 2. We pr reproducing the sector of the case, when the points uniform evaluation metho very well (see Fig. 1). But i along the circle, strong de Fig. 2).

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2). (here Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 3

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2). = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [5] and [8] for the analysis of C° piecewise-cubics and piecewise-quadratics or see [4] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simple stand the most natural fashion of choosing the knots is to approximate the unknown Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 3

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2). in the uniform manner: Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 3

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2).

Non-parametric Interpolation and Knot Eval Methods
There exist some practical problems, while dealing with the inc set. We can consider many problems, where sequence of points Q m the unknown curve γ with no provision of knot parameters {t i } m i=0 coined as fitting the reduced data Q m and any interpolation scheme data is called non-parametric interpolation. In order to apply any on non-parametric interpolation, the careful guessing of the kn [0,T ] m+1 needs to be made so that the resulting interpolant γ (h Eq. (1)) yields the best possible orders of convergence -see e.g. the analysis of C 0 piecewise-cubics and piecewise-quadratics or se C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots mate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: The potential problems in selecting {t i } m i=0 blindly in Fig. 1 and Fig. 2. We present here interpolation problems, that c reproducing the sector of the circle. We specify two different set o the case, when the points are distributed in the regular, uniform uniform evaluation method, not surprisingly, is able to reproduc very well (see Fig. 1). But in the case, when points are placed in irre along the circle, strong deviations from the original curve can be Fig. 2).

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2). blindly are illustrated in Figure 1 and Figure 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of point q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Figure 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Figure 2).

Non-parametric Interpolation and Knot Evaluation Methods
There exist some practical problems, while dealing with the incomplete data set. We can consider many problems, where sequence of points Q m interpolates the unknown curve γ with no provision of knot parameters {t i } m i=0 . Such task is coined as fitting the reduced data Q m and any interpolation scheme based on such data is called non-parametric interpolation. In order to apply any scheme based on non-parametric interpolation, the careful guessing of the knots {t i } m i=0 ∈ [0,T ] m+1 needs to be made so that the resulting interpolant γ (here γ = S, see Eq. (1)) yields the best possible orders of convergence -see e.g. [4] and [6] for the analysis of C 0 piecewise-cubics and piecewise-quadratics or see [5] or [3] for C 1 or C 2 piecewise-cubics, respectively.

Uniform Knot Evaluation Method
The simplest and the most natural fashion of choosing the knots is to approximate the unknown {t i } m i=0 ∈ [0, T ] m+1 in the uniform manner: withT = m. The potential problems in selecting {t i } m i=0 blindly are illustrated in Fig. 1 and Fig. 2. We present here interpolation problems, that can arise while reproducing the sector of the circle. We specify two different set of points q i . In the case, when the points are distributed in the regular, uniform manner the uniform evaluation method, not surprisingly, is able to reproduce the curve γ very well (see Fig. 1). But in the case, when points are placed in irregular intervals along the circle, strong deviations from the original curve can be observed (see Fig. 2).

Cumulative Chord Knot Evaluation Method
Following [4] or [6] instead of choosing the knots blindly (e.g. as by (3)) we can assign to them the values of the cumulative distance between the interpolated

Cumulative Chord Knot Evaluation Method
Following [5] or [8] instead of choosing the knots blindly (e.g. as by (3)) we can assign to them the values of the cumulative distance between the interpolated points:

Cumulative Chord Knot Evaluation Method
Following [4] or [6] instead of choosing the knots blindly (e.g. as by (3)) we can assign to them the values of the cumulative distance between the interpolated points:t where · denotes a standard Euclidean norm in R n . Formula (4) for estimating knots t i takes into account the geometrical distribution of the points Q m for an arbitrary dimensions, which makes our procedure usable for any non-parametric interpolation problem. The results of the interpolation of the points placed on the sector of the circle can be compared in Fig. 1 (for uniformly distributed points) and in Fig. 2 (for data distributed in irregular manner).

Comparison of Knot Evaluation Methods -Examples
Following experiments performed here (see Fig. 3) certain facts should be emphasized: 1. If the number of interpolation points Q m is small and the data are distributed in highly irregular manner the uniform method creates irregularities in trajectory estimation, while the curve obtained by chord evaluation method maintains plain and smooth shape. 2. If the data are distributed in the uniform manner then both methods work equally well, since uniform distribution of knots reflects uniform distribution of the data. 3. If the number of points Q m is large then the results from both methods appear to be very similar, but in fact the convergence order of the approximation to the trajectory is not fast for uniform knot evaluation method and would give big errors while estimating the length of the curve [4] or [6]. This does not happen with item 1 from above.

Cumulative Chord Knot Evaluation Method
Following [4] or [6] instead of choosing the knots blindly (e.g. as by (3)) we can assign to them the values of the cumulative distance between the interpolated points:t where · denotes a standard Euclidean norm in R n . Formula (4) for estimating knots t i takes into account the geometrical distribution of the points Q m for an arbitrary dimensions, which makes our procedure usable for any non-parametric interpolation problem. The results of the interpolation of the points placed on the sector of the circle can be compared in Fig. 1 (for uniformly distributed points) and in Fig. 2 (for data distributed in irregular manner).

Comparison of Knot Evaluation Methods -Examples
Following experiments performed here (see Fig. 3) certain facts should be emphasized: 1. If the number of interpolation points Q m is small and the data are distributed in highly irregular manner the uniform method creates irregularities in trajectory estimation, while the curve obtained by chord evaluation method maintains plain and smooth shape. 2. If the data are distributed in the uniform manner then both methods work equally well, since uniform distribution of knots reflects uniform distribution of the data. 3. If the number of points Q m is large then the results from both methods appear to be very similar, but in fact the convergence order of the approximation to the trajectory is not fast for uniform knot evaluation method and would give big errors while estimating the length of the curve [4] or [6]. This does not happen with item 1 from above.
, where ith (a) uniform knot evaluation (red line) and (b) r points distributed in irregular fashion.

t Evaluation Method
osing the knots blindly (e.g. as by (3)) we can cumulative distance between the interpolated where · denotes a standard (4) for estimating knots t i takes into account e points Q m for an arbitrary dimensions, which ny non-parametric interpolation problem. The e points placed on the sector of the circle can mly distributed points) and in Fig. 2 (for data aluation Methods -Examples here (see Fig. 3) certain facts should be empoints Q m is small and the data are distributed e uniform method creates irregularities in tracurve obtained by chord evaluation method hape. the uniform manner then both methods work tribution of knots reflects uniform distribution is large then the results from both methods t in fact the convergence order of the approxit fast for uniform knot evaluation method and timating the length of the curve [4] or [6]. This from above.
denotes a standard Euclidean norm in R n . Formula (4) for estimating knots t i takes into account the geometrical distribution of the points Q m for an arbitrary dimensions, what makes our procedure usable for any non-parametric interpolation problem. The results of the interpolation of the points placed on the sector of the circle can be compared in Figure 1 (for uniformly distributed points) and in Figure 2 (for data distributed in irregular manner).

Comparison of Knot Evaluation Methods -Examples
Following experiments performed here (see Figure 3) certain facts should be emphasized: 1. If the number of interpolation points Q m is small and the data are distributed in highly irregular manner the uniform method creates irregularities in trajectory estimation, while the curve obtained by chord evaluation method maintains plain and smooth shape. 2. If the data are distributed in the uniform manner then both methods work equally well, since uniform distribution of knots reflects uniform distribution of the data. 3. If the number of points Q m is large then the results from both methods appear to be very similar, but in fact the convergence order of the approximation to the trajectory is not fast for uniform knot evaluation method and would give big errors while estimating the length of the curve [5] or [8]. This does not happen with item 1 from above.
For data distributed in the uniform manner even for simple guess (a) (b) (c) Fig. 3. Cubic spline interpolation using both knot evaluation methods: uniform (red line) and cumulative chord (green line). Example scenarios: (a) number of interpolation points is small and the data are distributed in highly irregular manner, (b) data are distributed in the uniform manner, (c) number of points is large.
For data distributed in the uniform manner even for simple guesst i = i we obtain desired results. However, there are some problems for which we do not have control over specifying interpolation points, or even if we have, we want to specify only small collection of points. In the latter case to correctly reproduce the curve we need to choose more points in the area where the curve is changing rapidly, than in places where it remain steady. Such procedure would result in increasing density of points in some regions, yielding in non-uniformly distributed data.

Sphere Illumination (Computer Graphics)
The main goal of the sphere illumination module is to present the estimation of the trajectory of the light-source movement on the basis of a sparse sequence of observed frames, which are defined on the basis of the position of the lightsource. Each frame is created by illuminating the same three dimensional object in the same place in space by light-source. Frames differ from each other only by the assigned a place in sequence and the position of a source of light in 3D space. The sphere illumination module estimates the position of a source of light in an exact number of frames placed between each frame of the input data. Therefore the resulting sequence of frames consists of the initial set of frames and the set of estimated frames forming altogether the estimation of the movement of the source of light. For sphere illumination, Phong reflection model [7] is used. To calculate the intensity of each pixel we apply: where I a is the intensity of ambient colour of the pixel, the I d is the intensity of colour for diffuse reflection of light at the pixel and I s is the intensity of colour for specular reflection of light at the pixel. The ambient colour parameters are constant for a particular object and does not depend on the position of observer we obtain desired results. However, there are some problems for which we do not have control over specifying interpolation points, or even if we have, we want to specify only small collection of points. In the latter case to correctly reproduce the curve we need to choose more points in the area where the curve is changing rapidly, than in places where it remain steady. Such procedure would result in increasing density of points in some regions, yielding in non-uniformly distributed data.

SPHERE ILLUMINATION (COMPUTER GRAPHICS)
The main goal of the sphere illumination module is to present the estimation of the trajectory of the light-source movement on the basis of a sparse sequence of observed frames, which are defined on the basis of the position of the lightsource. Each frame is created by illuminating the same three dimensional object in the same place in space by light-source. Frames differ from each other only by the assigned a place in sequence and the position of a source of light in 3D space. The sphere illumination module estimates the position of a source of light in an exact number of frames placed between each frame of the input data. Therefore the resulting sequence of frames consists of the initial set of frames and the set of estimated frames forming altogether the estimation of the movement of the source flight. For sphere illumination, Phong reflection model [10] is used. To calculate the intensity of each pixel we apply: where I a is the intensity of ambient colour of the pixel, the I d is the intensity of colour for diffuse reflection of light at the pixel and I s is the inten- sity of colour for specular reflection of light at the pixel. The ambient colour parameters are constant for a particular object and do not depend on the position of the observer and the position of lightsource. Therefore, the equation for the ambient property is of a form I a = k a , where k a is a constant value of colour intensity. The I d is the diffuse property of the material. The basic form of an equation for the I d intensity of diffuse compound of colour for a given pixel is I a = k a · cosϑ , where 6 Reduced Data for Curve Modeling -Applications and the position of light-source. Therefore the equation for the ambient property is of a form where k a is a constant value of colour intensity. The I d is the diffuse property of the material. The basic form of an equation for the I d intensity of diffuse compound of colour for a given pixel is where k R d is a constant value of the diffuse property and ϑ is the angle between the surface normal and the vector pointing from the surface point to the light source. The I s is the specular property of the material. The basic form of an equation for the I s intensity of specular compound of colour for a given pixel is where k s is a constant value of specular property of a material, which is illuminated by the white light, p determines the size of the highlight spot and ϕ is an angle between the vector pointing from the specified point to the position of observer and the ideal reflection vector.

Experimental Concept
In the sphere illumination model we implemented two different knot evaluation methods for determining the trajectory of the light-source, namely uniform and cumulative chord. The trajectory is obtained by interpolating the curve through specified points in the three dimensional space (see Fig. 4). The experimental task was to study the differences between methods simulating the sphere illumination by the moving light-source, where the light-source travels with constant velocity.

Example
We prepared a set of input data consisting of points shown in Tab. 1. Those input data points define the position of the light-source, which illuminated the object in each of frames. For this set of coordinates we simulated the movement of the light-source applying both knots evaluation methods (see Eqs. (3) and (4)). The trajectories of the light-source for both methods are shown in Fig.  4. More precisely, Fig. 4 (a) and 4 (b) present the same set of frames, which were an input for interpolation task. However, the images do not exactly match, as the scale on those picture differs. This difference originates from significant differences in coordinates of estimated points on trajectories. Algorithms for Phong illumination model and spline interpolation are applied in exactly the same fashion. As a result we obtained two different sequences of images for the same frame sequences within the whole resulting set of frames. Fig. 5 presents frames between 8 and 13 (row ordered) of the set obtained for uniform evaluation of knots. Fig. 6 presents the same set of frames obtained for evaluation of knots based on the length of chord.
is a constant value of the diffuse property and ϑ is the angle between the surface normal and the vector pointing from the surface point to the light source. The I s is the specular property of the material. The basic form of the equation for the I s intensity of specular compound of colour for a given pixel is I a = k a · (cosφ) p , where k s is a constant value of specular property of a material, which is illumi nated by the white light, p determines the size of the highlight spot and φ is an angle between the vector pointing from the specified point to the position of the observer and the ideal reflection vector.

Experimental Concept
In the sphere illumination model we implemented two different knot evaluation methods for determining the trajectory of the light-source, namely uniform and cumulative chord. The trajectory is obtained by interpolating the curve through specified points in the three dimensional space (see Figure 4). The experimental task was to study the differences between methods simulating the sphere illumi nation by the moving light-source, where the light-source travels with constant velocity.

Example
We prepared a set of input data consisting of points shown in Table 1. Those input data points Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 7 Table 1. Input data for sphere illumination module.
Frame number X Y Z 1 120 120 120 2 120 220 120 3 120 220 320 4 820 620 320 6 220 120 20   define the position of the light-source, which illuminated the object in each of the frames. For this set of coordinates we simulated the movement of the light-source applying both knot evaluation methods (see Eqs (3) and (4)). The trajectories of the light-source for both methods are shown in Figure 4. More precisely, Figure 4a and 4b present the same set of frames, which were an input for the interpolation task. However, the images do not exactly match, as the scales on these picture differ. This difference originates from significant differences in coordinates of the estimated points on trajectories. Algorithms for Phong illumination model and spline interpolation are applied in exactly the same fashion. As a result we obtained two different sequences of images for the same frame sequences within the whole resulting set of frames. Figue 5 presents frames between 8 and 13 (row ordered) of the set obtained for a uniform evaluation of knots. Figure 6 presents the same set of frames obtained for the evaluation of knots based on the length of chord.

IMAGE SEGMENTATION (COMPUTER VISION)
The main goal of the image segmentation module is to present the border line surrounding a certain area in the picture on the basis of a sequence of points marked by the user as interpolation points. Each point that is marked by the user is drawn on the picture in real time and the current shape of the curve is plotted onto the image. As all of the significant points are marked user closes the curve by splitting the image into two regions. The user can calculate the number of pixels within or outside of the region closed by the curve, which is realized by the Flood Fill Algorithm [1], which counts all points of the area until it recognizes reaching the border. The border curve (see Eq. (2)) may be calculated by applying two different knot evaluation modules discussed herein.

Experiment Concept
In the image segmentation model two different knot eval uation methods are implemented for determining the shape of the curve (see Eqs (3) and (4)). The experimental task is to study the impact of the evaluation methods on curve's shape and the area of a region bounded by this curve.

Example
We prepared two input images. Over the first one, we marked points as shown in Table 2 and 3. Over the second one, we marked points as indicated in Table 4. For this set of coordinates we evaluated the shape of the curve applying both knots evaluation methods. The coordinates for the first and the last points are identical, as the curve is closed. For both methods we also calculated the area within the selected region. Algorithms for the calculation of the area based on the Flood Fill Algorithm [1] with pixel count and spline interpolation are applied in exactly the same way. As a result we obtained two different shapes of unknown curve and consecutively two different sizes of a region bordered by the curve. Figure  7 presents the curve obtained for selected points with the uniform evaluation of knots applied. The computed size of the area within the curve is 10220 pixels and 1117 pixels for left and right canal respectively. Figure 7 presents the curve obtained for selected points with chord evaluation of the knots applied. The resulting size of the area within the curve was 10540 pixels (left canal) and 1366 pixels (right canal). Visibly the chord method outperforms the uniform one. The same observations originate from a comparison of curves bounding the cell, which is presented at Figure 8. The computed size of a cell within the curve was 44925 pixels using the uniform knot

TRAJECTORY MODELING (PHYSICS)
The main goal of the trajectory modeling module is to present the most accu rate estimation of the shape of the trajectory obtained as an image of observed physical process and to provide analytical formula for estimated curve. The user is expected to mark points over the trajectory. Each point that is marked by the user is drawn on the picture in real time and the current shape of the curve is plotted onto the image. Therefore, the user can decide in which moment the whole trajectory is covered by the interpolating curve and perform the analysis of curve equations. The curve can be calculated by applying two different knot evaluation modules (i.e. uniform and cumulative chord).

Experiment Concept
As in the trajectory modeling, two different knot evalua tion methods are implemented for determining the shape of the curve by interpolating the knots' values from the sequence of two dimensional points. The experimental task is to study the differences between the two methods to evaluate their impact on the analytical formulas obtained for both interpolants (serving as the boundary seg menting the image).

Example
We prepared an input image over which points as listed in Table 5 are marked. For this set of coordinates we evaluated the shape of the curve applying both knot evaluation methods. For both methods we also calculated the curvature at points {(443, 395), (611, 318)}. The calculation is performed as presented below. Curvature K(t) for curve γ(t) = (x(t), y(t)) ∈ R 2 is defined as: Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 11

Experiment Concept
As in the trajectory modeling, there are implemented two different knot evaluation methods for determining the shape of the curve by interpolating the knots values from the sequence of two dimensional points. The experimental task is to study the differences between two methods to evaluate their impact on the analytical formulas obtained for both interpolants (serving as the boundary segmenting the image).

Example
We prepared an input image over which there are marked points as listed in Tab. 5. For this set of coordinates we evaluated the shape of the curve applying both knots evaluation methods. For both methods we also calculated the curvature in points {(443, 395), (611, 318)}. The calculation is performed as presented below. Curvature K(t) for curve γ(t) = (x(t), y(t)) ∈ R 2 is defined as: x (t)y (t) − x (t)y (t) ((x (t)) 2 + (y (t)) 2 ) 3/2 .
Momentum p of the particle of charge q moving within the magnetic field B reads as (see [12]; Chapt. 5): where the circle radius r can be estimated by the curvature K: The analytical formula for S(t) = (S 1 (t), S 2 (t)) obtained from spline computation (see Eq. (1)) by (5) yields K. Since the charge q can be +1 or −1 the latter does not change the value of the momentum. Hence (with aid of Eqs. (6) and (7)) we obtain: p = B/K. The final unit of the momentum is kg · pixel s (if the input value of the magnetic field B was given in T (Tesla)). As a result we obtained two different shapes of resulting curve and consecutively two different values of curvature. Fig. 9 (a) presents the curve obtained for selected points with uniform evaluation of knots applied. The resulting curvature in the point (443, 395) amounted Momentum p of the particle of charge q moving within the magnetic field B reads as (see [12]): where the circle radius r can be estimated by the curvature K: Ma lgorzata Janik, Ryszard Kozera, Przemys law Kozio l 11

Experiment Concept
As in the trajectory modeling, there are implemented two different knot evaluation methods for determining the shape of the curve by interpolating the knots values from the sequence of two dimensional points. The experimental task is to study the differences between two methods to evaluate their impact on the analytical formulas obtained for both interpolants (serving as the boundary segmenting the image).

Example
We prepared an input image over which there are marked points as listed in Tab. 5. For this set of coordinates we evaluated the shape of the curve applying both knots evaluation methods. For both methods we also calculated the curvature in points {(443, 395), (611, 318)}. The calculation is performed as presented below. Curvature K(t) for curve γ(t) = (x(t), y(t)) ∈ R 2 is defined as: x (t)y (t) − x (t)y (t) ((x (t)) 2 + (y (t)) 2 ) 3/2 .
Momentum p of the particle of charge q moving within the magnetic field B reads as (see [12]; Chapt. 5): where the circle radius r can be estimated by the curvature K: The analytical formula for S(t) = (S 1 (t), S 2 (t)) obtained from spline computation (see Eq. (1)) by (5) yields K. Since the charge q can be +1 or −1 the latter does not change the value of the momentum. Hence (with aid of Eqs. (6) and (7)) we obtain: p = B/K. The final unit of the momentum is kg · pixel s (if the input value of the magnetic field B was given in T (Tesla)). As a result we obtained two different shapes of resulting curve and consecutively two different values of curvature. Fig. 9 (a) presents the curve obtained for selected points with uniform evaluation of knots applied. The resulting curvature in the point (443, 395) amounted The analytical formula for S(t) = (S 1 (t), S 2 (t)) obtained from spline computa tion (see Eq. (1)) by (5) yields K. Since the charge q can be +1 or -1 the latter does not change the value of the momentum. Hence, (with aid of Eqs (6) and (7)) we obtain: p = B/K. The final unit of the momentum is kg · pixel/s (if the input value of the magnetic field B was given in T (Tesla)). As a result we ob-

Conclusions
Our experiments show that one needs to be very careful while fitting nonparametric data. A proper knot parameterization, taking into account the geometrical distribution of data points must be selected. The experiments confirm the flexibility of cumulative chord knot parameterization. The latter is not preserved by the naïve blind guess of uniform parameterization.
tained two different shapes of the resulting curve and consecutively two different values of curvature. Figure 9a presents the curve obtained for selected points with a uniform eval uation of knots applied. The resulting curvature in point (443, 395) amounted to -0.0015 1/pixel and in point (611, 318) amounted to -0.0018 1/pixel. Figure  9b presents the curve obtained for selected points with ap plied chord evaluation of knots. The resulting curvature in the point (443, 395) amounted to -0.0003 1/pixel and in point (611, 318) amounted to -0,0011 1/pixel.

CONCLUSIONS
Our experiments show that one needs to be very careful while fitting non-parametric data. A proper knot parameterization must be selected with consideration for the geo metrical distribution of data points. The experiments confirm the flexibility of cumulative chord knot parameterization. The latter is not pre served by a naive blind guess of the uniform parameterization.