ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-Bsplines that reduces to the de BoorYMansionYCox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as NevilleYAitken weights of certain generalized interpolation problems. For multiple knots they are limits of NevilleYAitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval ½a; b connected at inner knots all of multiplicities zero by full connection matrices A ½i that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECTspline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves.

Separation of the femur head and acetabulum is one of main difficulties in the diseased hip joint due to deformed shapes and extreme narrowness of the joint space. To improve the segmentation accuracy is the key point of existing automatic or semi-automatic segmentation methods. In this paper, we propose a new method to improve the accuracy of the segmented acetabulum using surface fitting techniques, which essentially consists of three parts: (1) design a surface iterative process to obtain an optimization surface; (2) change the ellipsoid fitting to two-phase quadric surface fitting;(3) bring in a normal matching method and an optimization region method to capture edge points for the fitting quadric surface. Furthermore, this paper cited vivo CT data sets of 40 actual patients (with 79 hip joints). Test results for these clinical cases show that: (1) the average error of the quadric surface fitting method is 2.3 (mm); (2) the accuracy ratio of automatically recognized contours is larger than 89.4%;(3) the error ratio of section contours is less than 10% for acetabulums without severe malformation and less than 30% for acetabulums with severe malformation. Compared with similar methods, the accuracy of our method, which is applied in a software system, is significantly enhanced.

Cardinal ECT-spline curves are generated from one ECT-system of order n which is shifted by integer translations via one connection matrix. If this matrix is nonsingular, lower triangular and totally positive, there exists an ECT-B-spline function N n 0 (x) having minimal compact support [0, n] whose integer translates span the cardinal ECT-spline space. This B-spline is computed explicitly piece by piece. Involved is the characteristic polynomial of a certain matrix which is the product of a matrix related to the connection matrix and of the generalized Taylor matrix of the basic ECT-system. This approach extends results for polynomial cardinal splines via connection matrices [6] to the more general setting of cardinal ECT-splines. The method is illustrated by two examples based on ECT-systems of rational functions with prescribed poles. Also, a Green's function involved is expressed explicitly as an ECT-B-splines series.

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Abstract. Contour extraction of image stacks is a basic task in medical modeling. The existing level-set methods usually suffer from some problems (e.g. serious errors around sharp features, incorrect split of topology and contour occlusions). This paper proposes a novel method of multi-constraint level-set evolution to avoid above-mentioned problems. Interpolation constraint and deviation constraint are added to evolution processin addition to existing constraints (such as edge points and areas). In order to prevent occlusions, it proposes a method of three-phase level-set evolution. The first phase obtains a rough contour according to selected edge points. The second phase applies an expanding LSE (level-set evolution). Missing edge points in the first phase are added when occlusions probably appear. In the third phase, occlusions are deleted and a refining evolution is implemented. As proved by final experiments, our method can steadily extract contours slice by slice when the shapes of previous contours (contours in the previous slice) are similar to current contours (contours in the current slice). Furthermore, there is no error propagation during the process of contour extraction.