Part I: Modeling image curves using invariant 3-D object curvemodels-a path to 3-D recognition and shape estimation from imagecontours

This paper and its companion are concerned with the problems of 3-D object recognition and shape estimation from image curves using a 3-D object curve model that is invariant to affine transformation onto the image space, and a binocular stereo imaging system. The objects of interest here are the ones that have markings (e.g., characters, letters, special drawings and symbols, etc.) on their surfaces. The 3-D curves on the object are modeled as B-splines, which are characterized by a set of parameters (the control points) from which the 3-D curve can be totally generated. The B-splines are invariant under affine transformations. That means that the affine projected object curve onto the image space is a B-spline whose control points are related to the object control points through the affine transformation. Part I deals with issues relating to the curve modeling process. In particular, the authors address the problems of estimating the control points from the data curve, and of deciding on the “best” order B-spline and the “best” number of control points to be used to model the image or object curve(s). A minimum mean-square error (mmse) estimation technique which is invariant to affine transformations is presented as a noniterative, simple, and fast approach for control point estimation. The “best” B-spline is decided upon using a Bayesian selection rule. Finally, we present a matching algorithm that allocates a sample curve to one of p prototype curves when the sample curve is an a priori unknown affine transformation of one of the prototype curves stored in the data base. The approach is tried on a variety of images of real objects     

基于B 样条Snake模型的人体运动跟踪方法研究*

提出了一种基于B-样条曲线Snake模型的新的人体运动跟踪方法.Snake算法是通过最小能量来逼近物体的轮廓.采用改进的B-样条曲线Snake模型,每一帧图像中的目标轮廓用三次样条曲线准确地表示,使Snake模型更加稳定和具有较快的收敛速度.计算相邻帧之间的差分图像,通过利用一种基于统计关系双阈值分割方法,有效地检测出图像中运动人体,初步确定目标在每帧图像中的粗略位置.把从上一帧图像中得到的目标轮廓置于该位置,作为B-样条曲线Snake算法中轮廓提取的初始值,经运算后可得到对人体目标的准确分割与跟踪.     

曲线的整数型生成算法

本文提出了一个用光栅显示器或数字化绘图仪等显示设备中选择曲线上最佳点的通过算法，该算法由几部分组成，分别对应曲线的不同走向段，其最大的特点是可以根据实际曲线的走向，在算法的各部分实现自动跳动，由此算法可生成所有常用曲线，本文给出Ｂｅｚｉｅｒ曲线和Ｂ样条曲线的生成算法，这些算法选择距离实际曲线最近的网格点，并且只有整数运算。     

A B-spline curve extension algorithmB 样条曲线延伸算法

B-spline curve extension is an important operation in computer aided design systems. In this paper, we present a new extension algorithm for B-spline curves. The algorithm uses curve unclamping to generate a uniform B-spline curve segment from the original curve and gradually extends the segment to pass through every target point. Algorithms of uniform B-spline curves are used such that our algorithm has a low time cost and can easily handle arbitrary-order derivative constraints at the target points. Generalization for non-uniform rational B-spline curve extension is also discussed, and examples show the efficiency of our method.     

基于Freeman链码的B样条曲线轮廓拟合

提出了一种Freeman链码与B样条曲线误差控制相结合实现轮廓拟合的算法,首先利用Freeman链码法进行边界跟踪,根据相邻像素点间的不同的链码变化关系,排除伪特征点,提取出轮廓中绝大多数特征点,然后结合基于误差控制的B样条曲线法,取得能够精确表示轮廓信息的特征点。本文算法即避免了使用曲率来进行求取特征点的复杂计算,提高了特征点检测速度,又提取出能够精确拟合轮廓的局部支撑点,实现了基于误差控制的轮廓曲线拟合。实验结果证明了本文算法的正确性。     

B样条驱动纹理生成的构造式信息隐藏

目的 传统构造式图像信息隐藏算法通常直接将图像空域特征与秘密信息关联,对算法的安全性造成威胁。因此,本文将曲线绘制函数与信息隐藏相结合,提出一种以B样条控制点为特征,在图像空域间接隐藏信息的算法。方法 算法主要分为信息隐藏及信息提取两阶段。在信息隐藏阶段,发送方首先通过选取初始控制点、仿射变换及B样条曲线绘制生成多条参考曲线,然后利用曲线控制点的位置隐藏信息,最后为图像填充颜色,即完成含密纹理图像的构造。在信息提取阶段,提取方根据纹理曲线和图像颜色获得含密曲线及参考曲线,经对照计算即可提取出秘密信息。结果 本算法具有较高的隐藏容量、鲁棒性和安全性。实验结果表明,由本文算法生成的800×800像素图像,其最高隐藏容量可达2870bits,分别是另两种典型构造式信息隐藏算法的6.7和3.4倍,且在质量因子为10的JPEG（joint photographic experts group）压缩攻击下的提取误码率可低至0,优于鲁棒较强的选择式信息隐藏算法LDA-DCT（robust coverless image steganography based on DCT and LDA topic classification）以及与之类似的构造式信息隐藏算法。同时,抗隐写分析检测实验表明,在隐藏容量小于250bits时检测误差趋近于0.5。结论 本文以B样条曲线控制点为特征,在纹理图像的绘制过程中隐藏信息,有效提高了传统构造式图像信息隐藏算法的安全性、隐藏容量和鲁棒性。     

An extension algorithm for B-splines by curve unclamping

This paper presents an algorithm for extending B-spline curves and surfaces. Based on the unclamping algorithm for B-spline curves, we propose a new algorithm for extending B-spline curves that extrapolates using the recurrence property of the de Boor algorithm. This algorithm provides a nice extension, with maximum continuity, to the original curve segment. Moreover, it can be applied to the extension of B-spline surfaces. Extension to both single and multiple target points/curves are considered in this paper.     

The conversion of a dynamic B-spline curve into piecewise polynomials in power form

The evaluation of points and the computations of inflection points or cusps on a curve are often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed into a set of piecewise polynomial curves in power form. The usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in power form is done either by a knot refinement followed by basis conversions, or by applying a Taylor expansion on each knot span of a B-spline curve.Presented in this paper is a new algorithm to convert a B-spline curve into a set of piecewise polynomial curves in power form. Experiment shows that the proposed algorithm significantly outperforms the conventional approach when one or more control points of a B-spline curve are continuously moving.     

带法向约束的隐式T样条曲线重构

目的 隐式曲线能够描述复杂的几何形状和拓扑结构,而传统的隐式B样条曲线的控制网格需要大量多余的控制点满足拓扑约束。有些情况下,获取的数据点不仅包含坐标信息,还包含相应的法向约束条件。针对这个问题,提出了一种带法向约束的隐式T样条曲线重建算法。方法 结合曲率自适应地调整采样点的疏密,利用二叉树及其细分过程从散乱数据点集构造2维T网格;基于隐式T样条函数提出了一种有效的曲线拟合模型。通过加入偏移数据点和光滑项消除额外零水平集,同时加入法向项减小曲线的法向误差,并依据最优化原理将问题转化为线性方程组求解得到控制系数,从而实现隐式曲线的重构。在误差较大的区域进行T网格局部细分,提高重建隐式曲线的精度。结果 实验在3个数据集上与两种方法进行比较,实验结果表明,本文算法的法向误差显著减小,法向平均误差由10^-3数量级缩小为10^-4数量级,法向最大误差由10^-2数量级缩小为10^-3数量级。在重构曲线质量上,消除了额外零水平集。与隐式B样条控制网格相比,3个数据集的T网格的控制点数量只有B样条网格的55.88%、39.80%和47.06%。结论 本文算法能在保证数据点精度的前提下,有效降低法向误差,消除了额外的零水平集。与隐式B样条曲线相比,本文方法减少了控制系数的数量,提高了运算速度。     

Morphing active contours

A method for deforming curves in a given image to a desired position in a second image is introduced. The algorithm is based on deforming the first image toward the second one via a partial differential equation (PDE), while tracking the deformation of the curves of interest in the first image with an additional, coupled PDE; both the images and the curves on the frame/slices of interest are used for tracking. The technique can be applied to object tracking and sequential segmentation. The topology of the deforming curve can change without any special topology handling procedures added to the scheme. This permits, for example, the automatic tracking of scenes where, due to occlusions, the topology of the objects of interest changes from frame to frame. In addition, this work introduces the concept of projecting velocities to obtain systems of coupled PDEs for image analysis applications. We show examples for object tracking and segmentation of electronic microscopy     

Approximate merging of B-spline curves via knot adjustment and constrained optimization

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the end k knots of a kth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.     

Contour tracking with automatic motion model switching

In this paper we present an efficient contour-tracking algorithm which can track 2D silhouette of objects in extended image sequences. We demonstrate the ability of the tracker by tracking highly deformable contours (such as walking people) captured by a static camera. We represent contours (silhouette) of moving objects by using a cubic B-spline. The tracking algorithm is based on tracking a lower dimensional shape space (as opposed to tracking in spline space). Tracking the lower dimensional space has proved to be fast and efficient. The tracker is also coupled with an automatic motion-model switching algorithm, which makes the tracker robust and reliable when the object of interest is moving with multiple motion. The model-based tracking technique provided is capable of tracking rigid and non-rigid object contours with good tracking accuracy.     

Smooth surface approximation to serial cross-sections

The reconstruction of the surface model of an object from 2D cross-sections plays an important role in many applications. In this paper, we present a method for surface approximation to a given set of 2D contours. The resulting surface is represented by a bicubic closed B-spline surface with C² continuity. The method performs the skinning of intermediate contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The proposed method provides a smooth and accurate surface model, yet realizes efficient data reduction. Some experimental results are given using synthetic and MRI data.     

实平面奇异代数曲线的全局B样条逼近

提出了一种用k次B样条曲线全局逼近实平面k次代数曲线的算法,每个连通部分用一条B样条曲线逼近.它适合于任意亏格的不可约的实平面代数曲线(包括含奇异点的曲线).这种逼近建立在所提出的代数曲线胀开采样的基础上,这种胀开采样算法从本质上解决了奇异点周围采样难的问题.实验结果表明,该方法的逼近精度高于已有算法.     

基于遗传算法的B样条曲线拟合改进算法

B样条曲线拟合应用于绘制离散数据点的变化趋势,一般采用数据逼近或者迭代的方法得到,是图像处理和逆向工程中的重要内容。针对待拟合曲线存在多峰值、尖点、间断等问题,提出一种基于遗传算法的B样条曲线拟合算法。首先利用惩罚函数将带约束的曲线优化问题转换为无约束问题,然后利用改进的遗传算法来选择合适的适应度函数,再结合模拟退火算法自适应调整节点的数量和位置,在寻优的过程中找到最优的节点向量,持续迭代直到产生最终的优良重建曲线为止。实验结果表明,该算法有效地提高了精度并加快了收敛速度。     

Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method

In this paper, we consider the problem of fitting the B-spline curves to a set of ordered points, by finding the control points and the location parameters. The presented method takes two main steps: specifying initial B-spline curve and optimization. The method determines the number and the position of control points such that the initial B-spline curve is very close to the target curve. The proposed method introduces a length parameter in which this allows us to adjust the number of the control points and increases the precision of the initial B-spline curve. Afterwards, the scaled BFGS algorithm is used to optimize the control points and the foot points simultaneously and generates the final curve. Furthermore, we present a new procedure to insert a new control point and repeat the optimization method, if it is necessary to modify the fitting accuracy of the generated B-spline fitting curve. Associated examples are also offered to show that the proposed approach performs accurately for complex shapes with a large number of data points and is able to generate a precise fitting curve with a high degree of approximation.     

基于Mumford-Shah模型和开样条曲线的边界检测

受Cremers方法启发, 本文提出了一种新的开边界自动检测算法, 如图像中海岸线和天际线的检测. 这一算法的设计主要是基于样条函数、曲线演化理论和Mumford-Shah图像分割泛函模型. 由于所要检测的目标为图像区域中开曲线, 在一般Mumford-Shah模型中引入了两个约束条件. 这就将开边界的检测问题转化为一般的曲线最小分割问题. 通过样条曲线控制点所满足的微分方程和约束条件, 曲线将演化至所要求的边界. 如果图像中有一条开曲线将图像分为两个明显不同质区域, 这一算法将能有效地自动检测出该边界曲线, 且不需要边界的梯度信息. 即使在图像中有大量噪声情况下, 该算法同样有效. 此外, 通过两条曲线演化方程, 该算法可推广到图像中带状区域的(如河流、道路等)自动检测.     

任意NUBS曲线的小波分析和造型技术

为了对任意NUBS曲线进行精确的分解和重构，提出了半正交B样条小波分解和重构的新算法，同时给出了处理非均匀B样条曲线的非整数阶分辨率的小波分解和重构算法，并实现了任意非均匀B样条曲线的多分辨率表示，对于任意非均匀B样条或NUBS曲线，无论它有多少个控制点，均可以对它进行半正交分解和重构，而不受控制点数必须等于2＋3的限制，从这个意义上讲，该方法不仅可以实现连续分辨率水平（continuous-resolutionlevel)的非均匀B样条曲线造型，还可以对非均匀B样条和NURBS曲线进行精确的分解和重构，这对于B样条曲线曲面的多分辨率造型与显示具有重大应用价值。     

两种带形状参数的曲线

本文构造了两种带参数的三角样条基,基于这两组基定义了两种三角样条曲线。与二次B样条曲线类似,这两种曲线的每一段都由相继的三个控制顶点生成。这两种曲线具有许多与二次B样条曲线类似的性质,但它们的连续性都比二次B样条曲线更好。对于等距节点,在一般情况下,这两种曲线都整体C3连续,在特殊条件下,它们都可达C5连续。两种曲线中的形状参数均有明确的几何意义,参数越大,曲线越靠近控制多边形。另外,当形状参数满足一定条件时,这两种曲线都具有比二次B样条曲线更好的对控制多边形的逼近性。运用张量积方法,将这两种曲线推广后所得到的曲面也具有较好的连续性。