隐含多项式曲线曲面拟合次数的确定研究

选用合适次数的隐含多项式曲线曲面描述目标物体是处理和识别目标物体的关键,因而需要在理论上解决隐含多项式曲线或者曲面的次数确定问题.根据目标物体本身的特征,从理论上得出隐含多项式曲线描述物体的次数确定定理,并给出了具体计算公式.该方法首先由给定物体边界的轮廓检测出其驻点数,然后根据驻点数得到拟合隐含多项式曲线方程次数的下界,进而推广到三维物体的隐含多项式曲面拟合次数的确定.最后给出的应用实例进一步验证了算法的有效性与可操作性.     

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     

Using symbolic computation to find algebraic invariants

Implicit polynomials have proved themselves as having excellent representation power for complicated objects, and there is growing use of them in computer vision, graphics, and CAD. A must for every system that tries to recognize objects based on their representation by implicit polynomials are invariants, which are quantities assigned to polynomials that do not change under coordinate transformations. In the recognition system developed at the Laboratory for Engineering Man-Machine Studies in Brown University (LEMS), it became necessary to use invariants which are explicit and simple functions of the polynomial coefficients. A method to find such invariants is described and the new invariants presented. This work addresses only the problem of finding the invariants; their stability is studied in another paper     

Computational limitations of model-based recognition

Reliable object recognition is an essential part of most visual systems. Model-based approaches to object recognition use a database (a library) of modeled objects; for a given set of sensed data, the problem of model-based recognition is to identify and locate the objects from the library that are present in the data. We show that the complexity of model-based recognition depends very heavily on the number of object models in the library even if each object is modeled by a small number of discrete features. Specifically, deciding whether a discrete set of sensed data can be interpreted as transformed object models from a given library is NP-complete if the transformation is any combination of translation, rotation, scaling, and perspective projection. This suggests that efficient algorithms for model-based recognition must use additional structure to avoid the inherent computational difficulties. © 1998 John Wiley & Sons, Inc.     

隐式多项式曲线的信息建模研究进展

用隐式多项式曲线来描述数据点集合轮廓具有天然的优势,尤其是在数据点集合轮廓的拟合过程中体现得更为明显。概括了基于隐式多项式曲线的信息建模研究现状,侧重于目前国内外各种隐式多项式曲线拟合算法的分析以及优劣比较。以多个图像物体数据点集合轮廓为例,使用各种拟合算法对其进行拟合,并给出了拟合的效果,分析了算法的优劣和改进措施以及以后的研究方向.     

隐含多项式曲线对称性几何特征及其检测研究

隐舍多项式曲线在物体描述和识别中具有许多优点并得到实际应用,因而物体的对称性检测问题可以转换成对隐含多项式曲线的对称性检测来研究。对隐含多项式曲线对称几何结构性质进行了探讨,提出隐含多项式曲线如果是对称的,则其充分必要条件是首二次因子积组成的椭圆图形是对称的,同时指出椭圆图形对称轴就是隐含多项式曲线的对称轴。算法较为简单和直观．实验结果证明算法的有效性和可操作性。     

Toward a model-based Bayesian theory for estimating and recognizingparameterized 3-D objects using two or more images taken from differentpositions

A parametric modeling and statistical estimation approach is proposed and simulation data are shown for estimating 3-D object surfaces from images taken by calibrated cameras in two positions. The parameter estimation suggested is gradient descent, though other search strategies are also possible. Processing image data in blocks (windows) is central to the approach. After objects are modeled as patches of spheres, cylinders, planes and general quadrics-primitive objects, the estimation proceeds by searching in parameter space to simultaneously determine and use the appropriate pair of image regions, one from each image, and to use these for estimating a 3-D surface patch. The expression for the joint likelihood of the two images is derived and it is shown that the algorithm is a maximum-likelihood parameter estimator. A concept arising in the maximum likelihood estimation of 3-D surfaces is modeled and estimated. Cramer-Rao lower bounds are derived for the covariance matrices for the errors in estimating the a priori unknown object surface shape parameters     

Sparse representation for coarse and fine object recognition

This paper offers a sparse, multiscale representation of objects. It captures the object appearance by selection from a very large dictionary of Gaussian differential basis functions. The learning procedure results from the matching pursuit algorithm, while the recognition is based on polynomial approximation to the bases, turning image matching into a problem of polynomial evaluation. The method is suited for coarse recognition between objects and, by adding more bases, also for fine recognition of the object pose. The advantages over the common representation using PCA include storing sampled points for recognition is not required, adding new objects to an existing data set is trivial because retraining other object models is not needed, and significantly in the important case where one has to scan an image over multiple locations in search for an object, the new representation is readily available as opposed to PCA projection at each location. The experimental result on the COIL-100 data set demonstrates high recognition accuracy with real-time performance.     

Computing stable poses of piecewise smooth objects

When a three dimensional object is known to be lying on a planar surface, its pose is restricted from six to three degrees of freedom. Computer vision algorithms can exploit the few stable poses of modeled objects to simplify scene interpretation and more accurately determine object location. This paper presents necessary and sufficient conditions for the pose of a piecewise smooth curved three-dimensional object to be stable. For objects whose surfaces are represented by implicit algebraic equations, these conditions can be expressed as systems of polynomial equations that are readily solved by homotopy continuation. Examples from the implemented algorithm are presented.     

On using CAD models to compute the pose of curved 3D objects

A new approach is presented for explicitly relating image observables to models of curved three-dimensional objects. This relationship is used for object recognition and positioning. Object models consist of collections of parametric surface patches. The image observables considered are raw range data, surface normal and Gaussian curvature, raw image intensity and intensity gradient, raw image contours, and contour orientation and curvature. Elimination theory provides a method for constructing an implicit equation that relates these observables to the three-dimensional position and orientation of object models. Determining the unknown pose parameters is reduced to a fitting problem between the implicit equation and the observed data points. By considering translation-independent observables such as surface normal and curvature, this process is further decomposed into first determining orientation and then determining translation. Applications to object recognition are described, and an implementation is presented.     

Stable fitting of 2D curves and 3D surfaces by implicit polynomials

This work deals with fitting 2D and 3D implicit polynomials (IPs) to 2D curves and 3D surfaces, respectively. The zero-set of the polynomial is determined by the IP coefficients and describes the data. The polynomial fitting algorithms proposed in this paper aim at reducing the sensitivity of the polynomial to coefficient errors. Errors in coefficient values may be the result of numerical calculations, when solving the fitting problem or due to coefficient quantization. It is demonstrated that the effect of reducing this sensitivity also improves the fitting tightness and stability of the proposed two algorithms in fitting noisy data, as compared to existing algorithms like the well-known 3L and gradient-one algorithms. The development of the proposed algorithms is based on an analysis of the sensitivity of the zero-set to small coefficient changes and on minimizing a bound on the maximal error for one algorithm and minimizing the error variance for the second. Simulation results show that the proposed algorithms provide a significant reduction in fitting errors, particularly when fitting noisy data of complex shapes with high order polynomials, as compared to the performance obtained by the above mentioned existing algorithms.     

The 3L algorithm for fitting implicit polynomial curves andsurfaces 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     

The Classical Theory of Invariants and Object Recognition Using Algebraic Curve and Surfaces

Combining implicit polynomials and algebraic invariants for representing and recognizing complicated objects proves to be a powerful technique. In this paper, we explore the findings of the classical theory of invariants for the calculation of algebraic invariants of implicit curves and surfaces, a theory largely disregarded in the computer vision community by a shadow of skepticism. Here, the symbolic method of the classical theory is described, and its results are extended and implemented as an algorithm for computing algebraic invariants of projective, affine, and Euclidean transformations. A list of some affine invariants of 4th degree implicit polynomials generated by the proposed algorithm is presented along with the corresponding symbolic representations, and their use in recognizing objects represented by implicit polynomials is illustrated through experiments. An affine invariant fitting algorithm is also proposed and the performance is studied.     

基于BP神经网络的隐式曲线构造方法

隐式曲线与曲面是当前计算机图形学研究的热点之一。通过把BP神经网络与隐式曲线构造原理相结合，提出了一种构造隐式曲线的新方法，即首先由约束点构造神经网络的输入与输出，把描述物体边界曲线的隐式函数转化为显式函数；然后用BP神经网络对此显式函数进行逼近；最后由仿真曲面得到物体边界的拟合曲线。该新方法不同于传统的对显式函数的逼近方法，因为传统方法无法描述封闭的曲线；也不同于基于优化的拟合隐式曲线方法，因为它无须考虑函数的形式或多项式的次数。实验表明，该新方法有很强的物体边界描述能力和缺损修复能力，因而在物体边界重建、缺损图像复原等领域有一定的应用前景。     

A complete and extendable approach to visual recognition

A framework for 3D object recognition is presented. Its flexibility and extensibility are accomplished through a uniform, parallel, and modular recognition architecture. Concurrent and stacked parameter transforms reconstruct a variety of features from the input scene. At each stage, constraint satisfaction networks collect and fuse the evidence obtained through the parameter transforms, ensuring a globally consistent interpretation of the input scene and allowing for the integration of diverse types of information. The final interpretation of the scene is a small consistent subset of the many initial hypotheses about partial features, primitive features, feature assemblies, and 3D objects computed by the various parameter transforms. A complete, integrated, and implemented system that extracts planar surfaces, patches of quadrics of revolution, and planar intersection curves of these surfaces from a depth map viewing 3D objects is described. Experimental results on the recognition behavior of the system are presented     

Tracking in object action space

In this paper we focus on the joint problem of tracking humans and recognizing human action in scenarios such as a kitchen scenario or a scenario where a robot cooperates with a human, e.g., for a manufacturing task. In these scenarios, the human directly interacts with objects physically by using/manipulating them or by, e.g., pointing at them such as in “Give me that…”. To recognize these types of human actions is difficult because (a) they ought to be recognized independent of scene parameters such as viewing direction and (b) the actions are parametric, where the parameters are either object-dependent or as, e.g., in the case of a pointing direction convey important information. One common way to achieve recognition is by using 3D human body tracking followed by action recognition based on the captured tracking data. For the kind of scenarios considered here we would like to argue that 3D body tracking and action recognition should be seen as an intertwined problem that is primed by the objects on which the actions are applied. In this paper, we are looking at human body tracking and action recognition from a object-driven perspective. Instead of the space of human body poses we consider the space of the object affordances, i.e., the space of possible actions that are applied on a given object. This way, 3D body tracking reduces to action tracking in the object (and context) primed parameter space of the object affordances. This reduces the high-dimensional joint-space to a low-dimensional action space. In our approach, we use parametric hidden Markov models to represent parametric movements; particle filtering is used to track in the space of action parameters. We demonstrate its effectiveness on synthetic and on real image sequences using human-upper body single arm actions that involve objects.     

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

The standard way of solving numerically a polynomial eigenvalue problem (PEP) is to use a linearization and solve the corresponding generalized eigenvalue problem (GEP). In addition, if the PEP possesses one of the structures arising very often in applications, then the use of a linearization that preserves such structure combined with a structured algorithm for the GEP presents considerable numerical advantages. Block-symmetric linearizations have proven to be very useful for constructing structured linearizations of structured matrix polynomials. In this scenario, we analyze the eigenvalue condition numbers and backward errors of approximated eigenpairs of a block symmetric linearization that was introduced by Fiedler (Linear Algebra Appl 372:325–331, 2003) for scalar polynomials and generalized to matrix polynomials by Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004). This analysis reveals that such linearization has much better numerical properties than any other block-symmetric linearization analyzed so far in the literature, including those in the well known vector space \(\mathbb {DL}(P)\) of block-symmetric linearizations. The main drawback of the analyzed linearization is that it can be constructed only for matrix polynomials of odd degree, but we believe that it will be possible to extend its use to even degree polynomials via some strategies in the near future.