Related Experiment Videos
Improving the stability of algebraic curves for applications
T Tasdizen1, J P Tarel, D B Cooper
1Division of Engineering, Brown University, Providence, RI 02906, USA. tt@lems.brown.edu
Summary
This study addresses the instability in fitting algebraic curves to data. A new ridge regression method enhances stability for reliable shape recognition and tracking, improving image database indexing.
Area of Science:
- Computer Vision
- Computational Geometry
- Machine Learning
Background:
- Algebraic curves model complex shapes beyond conics.
- Shape representation using algebraic curves faces repeatability challenges in data fitting.
- Previous fitting methods lacked stability, hindering practical applications.
Purpose of the Study:
- Investigate the stability issues in algebraic curve fitting.
- Propose a novel, stable fitting method for algebraic curves.
- Enable robust shape recognition, estimation, and tracking.
Main Methods:
- Developed a fitting method using ridge regression.
- Constrained polynomial representations to well-behaved subsets.
- Investigated the properties of the proposed fitting algorithm.
Main Results:
- The proposed method demonstrates sufficient stability for algebraic curve fitting.
- Achieved fast, position-invariant shape recognition and tracking.
- New representations enable effective shape-based indexing.
Conclusions:
- The ridge regression-based method overcomes previous stability limitations.
- This approach facilitates practical applications in computer vision and image retrieval.
- Enables efficient and reliable shape analysis for complex datasets.
Related Concept Videos
01:24Pole and System Stability
The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.
01:17Curve Equations
Curves are essential geometric elements characterized by tangent distance, chord length, middle ordinate, and total arc length. These measurements are crucial in understanding a curve's geometric and spatial properties and are defined by the relationship between its radius and its central angle.The tangent distance (T) refers to the straight-line measurement from the intersection point of two tangents to either the start or end of the curve. This distance is influenced by the curve's radius (R)...
01:20Elevation of Intermediate Points on Vertical Curves
Vertical curves are essential in roadway design because they provide smooth transitions between varying roadway grades. Designing vertical curves involves calculating intermediate elevations and identifying the curve's highest or lowest point, which is essential for optimal roadway performance.Intermediate elevations on a vertical curve are determined using the tangent offset method. This method considers the initial elevation at the start of the curve, the grades, and the curve's geometry. The...
01:29Application of Nonlinear Inequalities
A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality: can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values are 3...
01:14Stability of structures
In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...
01:22Curve Sketching and Derivatives
Understanding the behavior of a function through its first and second derivatives is essential for analyzing its graph. Derivatives provide insight into where a function increases or decreases, where it attains local maxima or minima, and how its curvature behaves across different intervals.The first derivative of a function reveals the slope of the tangent line at any given point. Points where the derivative is zero or undefined are considered critical, as they often indicate potential extrema...