The problem of tracking curves in dense visual clutter is challenging. Kalman filtering is inadequate because it is based on Gaussian densities which, being unimo dal, cannot represent simultaneous alternative hypotheses. The Condensation algorithm uses “factored sampling”, previously applied to the interpretation of static images, in which the probability distribution of possible interpretations is represented by a randomly generated set. Condensation uses learned dynamical models, together with visual observations, to propagate the random set over time. The result is highly robust tracking of agile motion. Notwithstanding the use of stochastic methods, the algorithm runs in near real-time.

/pdf/c-ondensation-conditional-density-propagation-forvisual-2zlkn4lnac.pdf

C ONDENSATION —Conditional Density Propagation forVisual Tracking

One Curve and Surface Basics.- 1.1 Implicit and Parametric Forms.- 1.2 Power Basis Form of a Curve.- 1.3 Bezier Curves.- 1.4 Rational Bezier Curves.- 1.5 Tensor Product Surfaces.- Exercises.- Two B-Spline Basis Functions.- 2.1 Introduction.- 2.2 Definition and Properties of B-spline Basis Functions.- 2.3 Derivatives of B-spline Basis Functions.- 2.4 Further Properties of the Basis Functions.- 2.5 Computational Algorithms.- Exercises.- Three B-spline Curves and Surfaces.- 3.1 Introduction.- 3.2 The Definition and Properties of B-spline Curves.- 3.3 The Derivatives of a B-spline Curve.- 3.4 Definition and Properties of B-spline Surfaces.- 3.5 Derivatives of a B-spline Surface.- Exercises.- Four Rational B-spline Curves and Surfaces.- 4.1 Introduction.- 4.2 Definition and Properties of NURBS Curves.- 4.3 Derivatives of a NURBS Curve.- 4.4 Definition and Properties of NURBS Surfaces.- 4.5 Derivatives of a NURBS Surface.- Exercises.- Five Fundamental Geometric Algorithms.- 5.1 Introduction.- 5.2 Knot Insertion.- 5.3 Knot Refinement.- 5.4 Knot Removal.- 5.5 Degree Elevation.- 5.6 Degree Reduction.- Exercises.- Six Advanced Geometric Algorithms.- 6.1 Point Inversion and Projection for Curves and Surfaces.- 6.2 Surface Tangent Vector Inversion.- 6.3 Transformations and Projections of Curves and Surfaces.- 6.4 Reparameterization of NURBS Curves and Surfaces.- 6.5 Curve and Surface Reversal.- 6.6 Conversion Between B-spline and Piecewise Power Basis Forms.- Exercises.- Seven Conics and Circles.- 7.1 Introduction.- 7.2 Various Forms for Representing Conics.- 7.3 The Quadratic Rational Bezier Arc.- 7.4 Infinite Control Points.- 7.5 Construction of Circles.- 7.6 Construction of Conies.- 7.7 Conic Type Classification and Form Conversion.- 7.8 Higher Order Circles.- Exercises.- Eight Construction of Common Surfaces.- 8.1 Introduction.- 8.2 Bilinear Surfaces.- 8.3 The General Cylinder.- 8.4 The Ruled Surface.- 8.5 The Surface of Revolution.- 8.6 Nonuniform Scaling of Surfaces.- 8.7 A Three-sided Spherical Surface.- Nine Curve and Surface Fitting.- 9.1 Introduction.- 9.2 Global Interpolation.- 9.2.1 Global Curve Interpolation to Point Data.- 9.2.2 Global Curve Interpolation with End Derivatives Specified.- 9.2.3 Cubic Spline Curve Interpolation.- 9.2.4 Global Curve Interpolation with First Derivatives Specified.- 9.2.5 Global Surface Interpolation.- 9.3 Local Interpolation.- 9.3.1 Local Curve Interpolation Preliminaries.- 9.3.2 Local Parabolic Curve Interpolation.- 9.3.3 Local Rational Quadratic Curve Interpolation.- 9.3.4 Local Cubic Curve Interpolation.- 9.3.5 Local Bicubic Surface Interpolation.- 9.4 Global Approximation.- 9.4.1 Least Squares Curve Approximation.- 9.4.2 Weighted and Constrained Least Squares Curve Fitting.- 9.4.3 Least Squares Surface Approximation.- 9.4.4 Approximation to Within a Specified Accuracy.- 9.5 Local Approximation.- 9.5.1 Local Rational Quadratic Curve Approximation.- 9.5.2 Local Nonrational Cubic Curve Approximation.- Exercises.- Ten Advanced Surface Construction Techniques.- 10.1 Introduction.- 10.2 Swung Surfaces.- 10.3 Skinned Surfaces.- 10.4 Swept Surfaces.- 10.5 Interpolation of a Bidirectional Curve Network.- 10.6 Coons Surfaces.- Eleven Shape Modification Tools.- 11.1 Introduction.- 11.2 Control Point Repositioning.- 11.3 Weight Modification.- 11.3.1 Modification of One Curve Weight.- 11.3.2 Modification of Two Neighboring Curve Weights.- 11.3.3 Modification of One Surface Weight.- 11.4 Shape Operators.- 11.4.1 Warping.- 11.4.2 Flattening.- 11.4.3 Bending.- 11.5 Constraint-based Curve and Surface Shaping.- 11.5.1 Constraint-based Curve Modification.- 11.5.2 Constraint-based Surface Modification.- Twelve Standards and Data Exchange.- 12.1 Introduction.- 12.2 Knot Vectors.- 12.3 Nurbs Within the Standards.- 12.3.1 IGES.- 12.3.2 STEP.- 12.3.3 PHIGS.- 12.4 Data Exchange to and from a NURBS System.- Thirteen B-spline Programming Concepts.- 13.1 Introduction.- 13.2 Data Types and Portability.- 13.3 Data Structures.- 13.4 Memory Allocation.- 13.5 Error Control.- 13.6 Utility Routines.- 13.7 Arithmetic Routines.- 13.8 Example Programs.- 13.9 Additional Structures.- 13.10 System Structure.- References.

https://www.springer.com/productFlyer_978-3-540-61545-3.pdf?SGWID=0-0-1297-1491853-0

The NURBS Book

A major new professional reference work on fingerprint security systems and technology from leading international researchers in the field Handbook provides authoritative and comprehensive coverage of all major topics, concepts, and methods for fingerprint security systems This unique reference work is an absolutely essential resource for all biometric security professionals, researchers, and systems administrators

/pdf/handbook-of-fingerprint-recognition-16mh887j24.pdf

Handbook of Fingerprint Recognition

The problem of tracking curves in dense visual clutter is a challenging one. Trackers based on Kalman filters are of limited use; because they are based on Gaussian densities which are unimodal, they cannot represent simultaneous alternative hypotheses. Extensions to the Kalman filter to handle multiple data associations work satisfactorily in the simple case of point targets, but do not extend naturally to continuous curves. A new, stochastic algorithm is proposed here, the Condensation algorithm — Conditional Density Propagation over time. It uses ‘factored sampling’, a method previously applied to interpretation of static images, in which the distribution of possible interpretations is represented by a randomly generated set of representatives. The Condensation algorithm combines factored sampling with learned dynamical models to propagate an entire probability distribution for object position and shape, over time. The result is highly robust tracking of agile motion in clutter, markedly superior to what has previously been attainable from Kalman filtering. Notwithstanding the use of stochastic methods, the algorithm runs in near real-time.

https://link.springer.com/content/pdf/10.1007%2FBFb0015549.pdf

Contour Tracking by Stochastic Propagation of Conditional Density

A survey of shape analysis techniques

1 Introduction 2 Preliminaries 3 Hermite and Cubic Spline Interpolation 4 A Simple Approximation Technique - Uniform Cubic B-splines 5 Splines in a More General Setting 6 The One-Sided Basis 7 Divided Differences 8 General B-splines 9 B-spline Properties 10 Bezier Curves 11. Knot Insertion 12 The Oslo Algorithm 13 Parametric vs. Geometric Continuity 14 Uniformly-Shaped Beta-spline Surfaces 15 Geometric Continuity, Reparametrization, and the Chain Rule 16 Continuously-Shaped Beta-splines 17 An Explicity Formulation for Cubic Beta-splines 18 Discretely-Shaped Beta-splines 19 B-spline Representations for Beta-splines 20 Rendering and Evaluation 21 Selected Applications

An Introduction to Splines for Use in Computer Graphics and Geometric Modeling

The increasing availability of affordable color raster graphics displays has made it important to develop a better understanding of how color can be used effectively in an interactive environment. Most contemporary graphics displays offer a choice of some 16 million colors; the user's problem is to find the right color.Folklore has it that the RGB color space arising naturally from color display hardware is user-hostile and that other color models such as the HSV scheme are preferable. Until now there has been virtually no experimental evidence addressing this point.We describe a color matching experiment in which subjects used one of two tablet-based input techniques, interfaced through one of five color models, to interactively match target colors displayed on a CRT.The data collected show small but significant differences between models in the ability of subjects to match the five target colors used in this experiment. Subjects using the RGB color model matched quickly but inaccurately compared with those using the other models. The largest speed difference occurred during the early convergence phase of matching. Users of the HSV color model were the slowest in this experiment, both during the convergence phase and in total time to match, but were relatively accurate. There was less variation in performance during the second refinement phase of a match than during the convergence phase.Two-dimensional use of the tablet resulted in faster but less accurate performance than did strictly one-dimensional usage.Significant learning occurred for users of the Opponent, YIQ, LAB, and HSV color models, and not for users of the RGB color model.

An experimental comparison of RGB, YIQ, LAB, HSV, and opponent color models

Principles and techniques useful for calibrated color reproduction are defined. These results are derived from a project to take digital images designed on a variety of different color monitors and accurately reproduce them in a journal using digital offset printing. Most of the images printed were reproduced without access to the image as viewed in its original form; the color specification was derived entirely from calorimetric specification. The techniques described here are not specific to offset printing and can be applied equally well to other digital color devices.The reproduction system described is calibrated using CIE tristimulus values. An image is represented as a set of three-dimensional points, and the color output device as a three-dimensional solid surrounding the set of all reproducible colors for that device, called its gamut. The shapes of the monitor and the printer gamuts are very different, so it is necessary to transform the image points to fit into the destination gamut, a process we call gamut mapping. This paper describes the principles that control gamut mapping. Included also are some details on monitor and printer calibration, and a brief description of how digital halftone screens for offset printing are prepared.

Color gamut mapping and the printing of digital color images

The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated. Using a restricted form of quintic Hermite interpolation, it is possible to allow distinct bias and tension parameters at each joint without destroying geometric continuity. This provides a new means of obtaining local control of bias and tension in piecewise polynomial curves and surfaces.

John C. Beatty

Papers

An Introduction to Splines for Use in Computer Graphics and Geometric Modeling

An experimental comparison of RGB, YIQ, LAB, HSV, and opponent color models

Color gamut mapping and the printing of digital color images

Local Control of Bias and Tension in Beta-splines

An introduction to the use of splines in computer graphics