Implicit Polynomial Curves Research Articles

Increasing the Fitting Precision of Implicit Polynomial Curves in 2D Modeling through Signomial Powers

Increasing the Fitting Precision of Implicit Polynomial Curves in 2D Modeling through Signomial Powers

Polynomial Fitting Algorithm Based on Neural Network

As a method of function approximation, polynomial fitting has always been the main research hotspot in mathematical modeling. In many disciplines such as computer, physics, biology, neural networks have been widely used, and most of the applications have been transformed into fitting problems using neural networks. One of the main reasons that neural networks can be widely used is that it has a certain sense of universal approximation. In order to fit the polynomial, this paper constructs a three-layer feedforward neural network, uses Taylor series as the activation function, and determines the number of hidden layer neurons according to the order of the polynomial and the dimensions of the input variables. For explicit polynomial fitting, this paper uses non-linear functions as the objective function, and compares the fitting effects under different orders of polynomials. For the fitting of implicit polynomial curves, the current popular polynomial fitting algorithms are compared and analyzed. Experiments have proved that the algorithm used in this paper is suitable for both explicit polynomial fitting and implicit polynomial fitting. The algorithm is relatively simple, practical, easy to calculate, and can efficiently achieve the fitting goal. At the same time, the computational complexity is relatively low, which has certain application value.

HUMAN-INSPIRED ALGEBRAIC CURVES FOR WEARABLE ROBOT CONTROL.

Having unified representations of human walking gait data is of paramount importance for wearable robot control. In the rehabilitation robotics literature, control approaches that unify the gait cycle of wearable robots are more appealing than the conventional approaches that rely on dividing the gait cycle into several periods, each with their own distinct controllers. In this article we propose employing algebraic curves to represent human walking data for wearable robot controller design. In order to generate algebraic curves from human walking data, we employ the 3L fitting algorithm, a tool developed in the pattern recognition literature for fitting implicit polynomial curves to given datasets. For an impedance model of the knee joint motion driven by the hip angle signal, we provide conditions by which the generated algebraic curves satisfy a robust relative degree condition throughout the entire walking gait cycle. The robust relative degree property makes the algebraic curve representation of walking gaits amenable to various nonlinear output tracking controller design techniques.

An Adaptive and Stable Method for Fitting Implicit Polynomial Curves and Surfaces

Representing 2D and 3D data sets with implicit polynomials (IPs) has been attractive because of its applicability to various computer vision issues. Therefore, many IP fitting methods have already been proposed. However, the existing fitting methods can be and need to be improved with respect to computational cost for deciding on the appropriate degree of the IP representation and to fitting accuracy, while still maintaining the stability of the fit. We propose a stable method for accurate fitting that automatically determines the moderate degree required. Our method increases the degree of IP until a satisfactory fitting result is obtained. The incrementability of QR decomposition with Gram-Schmidt orthogonalization gives our method computational efficiency. Furthermore, since the decomposition detects the instability element precisely, our method can selectively apply ridge regression-based constraints to that element only. As a result, our method achieves computational stability while maintaining fitting accuracy. Experimental results demonstrate the effectiveness of our method compared with prior methods.

Rotations, translations and symmetry detection for complexified curves

A plane algebraic curve can be represented as the zero-set of a polynomial in two—or if one takes homogeneous coordinates: three—variables. The coefficients of the polynomial determine the curve uniquely. Thus features of the curve, like for instance rotation symmetry, must find their correspondence in the algebraic structure of the coefficients of the polynomial. In this article we will investigate how one can extract geometric curve features from the algebraic description of the curve. In particular, we will use a certain complex representation of polynomials introduced by (Tarel, J.-P., Cooper, D.B., 1998. A new complex basis for implicit polynomial curves and its simple exploitation for pose estimation and invariant recognition. In: Conference on Computer Vision and Pattern Recognition (CVPR'98), pp. 111–117; Unel, M., Wolovich, W.A., 1998. Complex representations of algebraic curves. In: International Conference on Image Processing ICIP 1998, pp. 272–276), which is very appropriate for the task of feature detection. In this complex representation actions on the curve parameters induced by geometric rotations or translations of the plane become very simple. Invariant expressions in the complexified parameters and also normal forms are easily accessible. Furthermore this representation allows the detection of rotation symmetry simply by looking at the indices of all non-vanishing complex parameters.

Research on Degree of Fitting Implicit Polynomial Curves and Surfaces

Research on Degree of Fitting Implicit Polynomial Curves and Surfaces

A technique for finding the symmetry axes of implicit polynomial curves under perspective projection

This paper presents an algebraic technique for detecting the symmetry axes of a perspectively projected plane curve. The procedure applies to implicit polynomials which has been fitted to 2D image data acquired by means of a photo camera or a TV set. The effectiveness of the proposed method has been demonstrated experimentally by using both synthetic and real images.

The complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition

Representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast, accurate pose-independent shape recognition under Euclidean transformations with a complete set of invariants, and fast accurate pose-estimation based on all the polynomial coefficients. The latter is accomplished by a centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two-dimensional subspaces or identity transformations within one-dimensional subspaces result from rotations in x, y measured-data space. Angles of these rotations in the two-dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x, y data space. By recasting this approach in terms of a complex variable, i.e., x+iy=z, and complex polynomial-coefficients, further conceptual and computational simplification results. Application to shape-based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.

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

We introduce a completely new approach to fitting implicit polynomial geometric shape models to data and to studying these polynomials. The power of these models is in their ability to represent nonstar complex shapes in two(2D) and three-dimensional (3D) data to permit fast, repeatable fitting to unorganized data which may not be uniformly sampled and which may contain gaps, to permit position-invariant shape recognition based on new complete sets of Euclidean and affine invariants and to permit fast, stable single-computation pose estimation. The algorithm represents a significant advancement of implicit polynomial technology for four important reasons. First, it is orders of magnitude taster than existing fitting methods for implicit polynomial 2D curves and 3D surfaces, and the algorithms for 2D and 3D are essentially the same. Second, it has significantly better repeatability, numerical stability, and robustness than current methods in dealing with noisy, deformed, or missing data. Third, it can easily fit polynomials of high, such as 14th or 16th, degree. Fourth, additional linear constraints can be easily incorporated into the fitting process, and general linear vector space concepts apply.

Pose estimation and object identification using complex algebraic representations

The comparison and alignment of two similar objects is a fundamental problem in pattern recognition and computer vision that has been considered using various approaches. In this work, we employ a complex representation for an algebraic curve, and illustrate how the algebraic transformation which relates two Euclidean equivalent curves can be determined using this representation. The idea is based on a complex representation of 2D points expressed in terms of the orthogonalx andy variables, with rotations of the complex numbers described using Euler's identity. We develop a simple formula for integer multiples of the rotation angle of the Euclidean transformation in terms of the real coefficients of implicit polynomial equations that are used to model 2D free-form objects. When there is a translation, it can be determined using some new results on the conic-line factors of implicit polynomial curves. Experimental results are presented for data sets characterised by both noisy and missing data points to illustrate and validate our procedures.

On Interactive Object Shape Modeling Using Algebraic Curves

Query By Sketch for indexing into an image database involves presenting the machine with a sketch of the object to be found in the database. The sketch can be of the object shape or distinct contours on the image of the object. This sketch can be made from memory, or can be refined interactively in response to what the database search returns at each iteration. Or the sketch can be made by generating curves of an object boundary or object-surface image-discontinuities from an example image. This paper introduces and describes a family of 2D curves (implicit polynomial curves) for this purpose, and an algorithm for generating a representation which passes within e of a set of control points specified by the user. Control points can be placed at arbitrary locations and in arbitrary order, and can be erased by the user at will, in order to arrive at the desired shape representation. Level sets of the object potential field have been used to facilitate the interaction process. The fitting algorithm is formulated in the efficient Linear Programming (LP) framework. We illustrate the use of this method in the application of content-based image retrieval.

Describing complicated objects by implicit polynomials

This paper introduces and focuses on two problems. First is the representation power of closed implicit polynomials of modest degree for curves in 2-D images and surfaces in 3-D range data. Super quadrics are a small subset of object boundaries that are well fitted by these polynomials. The second problem is the stable computationally efficient fitting of noisy data by closed implicit polynomial curves and surfaces. The attractive features of these polynomials for Vision is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>