Efficient planar object tracking and parameter estimation usingcompactly represented cubic B-spline curves

In this paper, we consider the problem of matching 2D planar object curves from a database, and tracking moving object curves through an image sequence. The first part of the paper describes a curve data compression method using B-spline curve approximation. We present a new constrained active B-spline curve model based on the minimum mean square error (MMSE) criterion, and an iterative algorithm for selecting the “best” segment border points for each B-spline curve. The second part of the paper describes a method for simultaneous object tracking and affine parameter estimation using the approximate curves and profiles. We propose a novel B-spline point assignment algorithm which incorporates the significant corners for interpolating corresponding points on the two curves to be compared. A gradient-based algorithm is presented for simultaneously tracking object curves, and estimating the associated translation, rotation and scaling parameters. The performance of each proposed method is evaluated using still images and image sequences containing simple objects     

3-D curve matching using splines

A machine vision algorithm to find the longest common subcurve of two 3-D curves is presented. The curves are represented by splines fitted through sequences of sample points extracted from dense range data. The approximated 3-D curves are transformed into 1-D numerical strings of rotation and translation invariant shape signatures, based on a multiresolution representation of the curvature and torsion values of the space curves. The shape signature strings are matched using an efficient hashing technique that finds longest matching substrings. The results of the string matching stage are later verified by a robust, least-squares, 3-D curve matching technique, which also recovers the Euclidean transformation between the curves being matched. This algorithm is of average complexity O(n) where n is the number of the sample points on the two curves. The algorithm has applications in assembly and object recognition tasks. Results of assembly experiments are included.     

Curve fitting and optimal interpolation on CNC machines based on quadratic B-splines

In this paper,curve fitting of 3-D points generated by G01 codes and interpolation based on quadratic B-splines are studied.Feature points of G01 codes are selected using an adaptive method.Next,quadratic B-splines are obtained as the fitting curve by interpolating these feature points.Computations required in implementing the velocity planning algorithm mentioned by Timer et al.are very complicated because of the appearance of high-order curves.An improved time-optimal method for the quadratic B-spline cur...     

数控加工中的二次曲线拟合与最优插补控制算法

本文提出一种基于二次B样条曲线对G01代码的拟合及插补方法,先通过自适应方法选取G01代码所描述的加工轨迹形状特征的各个特征点,再用通过所有特征点的二次B样条曲线拟合待加工路径.由于Timar等人提出的最优速度规划算法对于三次及三次以上的高阶次样条曲线的计算极为复杂,本文提出了一种改进的基于二次B样条曲线的时间最优插补计算方法.最后,将所提出的算法应用到实际的图案加工仿真与实际数控加工中.     

Discrete contours in multiple views: approximation and recognition

Recognition of discrete planar contours under similarity transformations has received a lot of attention but little work has been reported on recognizing them under more general transformations. Planar object boundaries undergo projective or affine transformations across multiple views. We present two methods to recognize discrete curves in this paper. The first method computes a piecewise parametric approximation of the discrete curve that is projectively invariant. A polygon approximation scheme and a piecewise conic approximation scheme are presented here. The second method computes an invariant sequence directly from the sequence of discrete points on the curve in a Fourier transform space. The sequence is shown to be identical up to a scale factor in all affine related views of the curve. We present the theory and demonstrate its applications to several problems including numeral recognition, aircraft recognition, and homography computation.     

Extracting Group Transformations from Image Moments

Image distortion induced by the relative motion between an observer and the scene is an important cue for recovering the motion and the structure of the scene. It is known that the distortion in images can be described by transformation groups, such as Euclidean, affine, and projective groups. In this paper, we investigate how the moments of image curves are changed by group transformations, and we derive a relationship between the change in image moments and the invariant vector fields of the transformation groups. The results are used to formalize a method for extracting invariant vector fields of affine transformations from changes in the moments of orientation of curve segments in images. The method is applied to a realtime robot visual navigation task.     

Homography and Fundamental Matrix Estimation from Region Matches Using an Affine Error Metric

Matching a pair of affine invariant regions between images results in estimation of the affine transformation between the regions. However, the parameters of the affine transformations are rarely used directly for matching images, mainly due to the lack of an appropriate error metric of the distance between them. In this paper we derive a novel metric for measuring the distance between affine transformations: Given an image region, we show that minimization of this metric is equivalent to the minimization of the mean squared distance between affine transformations of a point, sampled uniformly on the image region. Moreover, the metric of the distance between affine transformations is equivalent to the l ₂ norm of a linear transformation of the difference between the six parameters of the affine transformations. We employ the metric for estimating homographies and for estimating the fundamental matrix between images. We show that both homography estimation and fundamental matrix estimation methods, based on the proposed metric, are superior to current linear estimation methods as they provide better accuracy without increasing the computational complexity.     

Smoothing and matching of 3-d space curves

We present a new approach to the problem of matching 3-D curves. The approach has a low algorithmic complexity in the number of models, and can operate in the presence of noise and partial occlusions.Our method builds upon the seminal work of Kishon et al. (1990), where curves are first smoothed using B-splines, with matching based on hashing using curvature and torsion measures. However, we introduce two enhancements:

Object recognition by combining paraperspective images

This paper provides a study on object recognition under paraperspective projection. Discussed is the problem of determining whether or not a given image was obtained from a 3-D object to be recognized. First it is clarified that paraperspective projection is the first-order approximation of perspective projection. Then it is shown that, if we represent an object as a set of its feature points and the object undergoes a rigid transformation or an affine transformation, any paraperspective image can be expressed as a linear combination of several appropriate paraperspective images: we need at least three images for rigid transformations; whereas we need at least two images for affine transformations. Particularly in the case of a rigid transformation, the coefficients of the combination have to satisfy two conditions: orthogonality and norm equality. A simple algorithm to solve the above problem based on these properties is presented: a linear, single-shot algorithm. Some experimental results with synthetic images and real images are also given.This work was done while the author was with ATR Auditory and Visual Perception Research Laboratories.Advanced Research Laboratory Hitachi, Ltd.     

Matching 3-D Models to 2-D Images

We consider the problem of analytically characterizing the set of all 2-D images that a group of 3-D features may produce, and demonstrate that this is a useful thing to do. Our results apply for simple point features and point features with associated orientation vectors when we model projection as a 3-D to 2-D affine transformation. We show how to represent the set of images that a group of 3-D points produces with two lines (1-D subspaces), one in each of two orthogonal, high-dimensional spaces, where a single image group corresponds to one point in each space. The images of groups of oriented point features can be represented by a 2-D hyperbolic surface in a single high-dimensional space. The problem of matching an image to models is essentially reduced to the problem of matching a point to simple geometric structures. Moreover, we show that these are the simplest and lowest dimensional representations possible for these cases.We demonstrate the value of this way of approaching matching by applying our results to a variety of vision problems. In particular, we use this result to build a space-efficient indexing system that performs 3-D to 2-D matching by table lookup. This system is analytically built and accessed, accounts for the effects of sensing error, and is tested on real images. We also derive new results concerning the existence of invariants and non-accidental properties in this domain. Finally, we show that oriented points present unexpected difficulties: indexing requires fundamentally more space with oriented than with simple points, we must use more images in a motion sequence to determine the affine structure of oriented points, and the linear combinations result does not hold for oriented points.     

Least square geometric iterative fitting method for generalized B-spline curves with two different kinds of weights

Generalized B-spline bases are generated by monotone increasing and continuous “core” functions; thus generalized B-spline curves and surfaces not only hold almost the same perfect properties which classical B-splines hold but also show more flexibility in practical applications. Geometric iterative method (also known as progressive iterative approximation method) has good adaptability and stability and is popular due to its straight geometric meaning. However, in classical geometric iterative method, the number of control points is the same as that of data points. It is not suitable when large numbers of data points need to be fitted. In order to combine the advantages of generalized B-splines with those of geometric iterative method, a fresh least square geometric iterative fitting method for generalized B-splines is given, and two different kinds of weights are also introduced. The fitting method develops a series of fitting curves by adjusting control points iteratively, and the limit curve is weighted least square fitting result to the given large data points. Detailed discussion about choosing of core functions and two kinds of weights are also given. Plentiful numerical examples are also presented to show the effectiveness of the method.     

Design and recovery of 2-D and 3-D shapes using rational Gaussian curves and surfaces

A new representation for parametric curves and surfaces is introduced here. It is in rational form and uses rational Gaussian bases. This representation allows design of 2-D and 3-D shapes, and makes recovery of shapes from noisy image data possible. The standard deviations of Gaussians in a curve or surface control the smoothness of a recovered shape. The control points of a surface in this representation are not required to form a regular grid and a scattered set of control points is sufficient to reconstruct a surface. Examples of shape design, shape recovery, and image segmentation using the proposed representation are given.     

Model-based invariants for 3-D vision

Invariance under a group of 3-D transformations seems a desirable component of an efficient 3-D shape representation. We propose representations which are invariant under weak perspective to either rigid or linear 3-D transformations, and we show how they can be computed efficiently from a sequence of images with a linear and incremental algorithm. We show simulated results with perspective projection and noise, and the results of model acquisition from a real sequence of images. The use of linear computation, together with the integration through time of invariant representations, offers improved robustness and stability. Using these invariant representations, we derive model-based projective invariant functions of general 3-D objects. We discuss the use of the model-based invariants with existing recognition strategies: alignment without transformation, and constant time indexing from 2-D images of general 3-D objects.     

带最多独立形状参数的三阶三次均匀B样条曲线

构造了三阶三次等距结点的多项式B样条参数曲线,给出了de Boor控制顶点与分段三次Bézier控制顶点的关系式。该曲线具有一些类似于二次B样条曲线的性质：关于参变量为C¹连续,每个样条区间上的曲线由三个de Boor控制顶点的线性组合表示,具有仿射变换下的不变性,包含了二次均匀B样条曲线等。还具有形状可调性质：调配函数中含有形状参数,具有明显的几何意义,可用于调控曲线的形状或变形。给出了其具有凸包性、对de Boor控制多边形保形性等性质及其条件,讨论了形状参数对曲线形状的影响。     

基于三维网格物体的鲁棒性盲水印方案

提出了一种基于3D网格物体的鲁棒性盲水印方案。先将3D物体模型转换到仿射不变空间,然后把顶点坐标转换到球面坐标,利用球面坐标来嵌入水印。由于在仿射不变空间下,3D物体模型不受平移、旋转和比例变换的影响,故该水印方案对于此类变化具有很强的鲁棒性。试验结果也证明了这一点,且该水印具有良好的不可见性。     

基于B样条多分辨率表示的一种数字水印方法

本文根据B样条多分辨率表示的特点提出了一种盲数字水印方法,把原图像做二次小波分解,在低频子图像上用Harris特征点检测算法选取特征点,在以特征点为圆心的同心圆上选点的灰度值作为控制顶点建立B样条曲线,对B样条曲线做多分辨率分解,在低分辨率系数中嵌入水印。由于小波分解后选特征点以及Harris特征点具有几何变换不变性,试验表明该方法对JPEG压缩、缩放、旋转、平移、剪切、滤波等具有较强的抵抗力。