Implicit Spline Research Articles

Binary segmentation of medical images using implicit spline representations and deep learning

We propose a novel approach to image segmentation based on combining implicit spline representations with deep convolutional neural networks. This is done by predicting the control points of a bivariate spline function whose zero-set represents the segmentation boundary. We adapt several existing neural network architectures and design novel loss functions that are tailored towards providing implicit spline curve approximations. The method is evaluated on a congenital heart disease computed tomography medical imaging dataset. Experiments are carried out by measuring performance in various standard metrics for different networks and loss functions. We determine that splines of bidegree (1,1) with 128×128 coefficient resolution performed optimally for 512×512 resolution CT images. For our best network, we achieve an average volumetric test Dice score of close to 92%, which reaches the state of the art for this congenital heart disease dataset.

A new high-resolution two-level implicit method based on non-polynomial spline in tension approximations for time-dependent quasi-linear biharmonic equations with engineering applications

In this present work, a new two-level implicit non-polynomial spline in tension method is proposed for the numerical solution of the 1D unsteady quasi-linear biharmonic problem. Using the continuity of the first-order derivative of the spline in tension function, a fourth-order accurate implicit finite-difference method is developed in this manuscript. By considering the linear model biharmonic problem, the given implicit spline method is unconditionally stable. Since the proposed method is based on half-step grid points, so it can be directly applied to 1D singular biharmonic problems without alteration in the scheme. Finally, the numerical experiments of the various biharmonic equation such as the generalized Kuramoto–Sivashinsky equation, extended Fisher–Kolmogorov equation, and 1D linear singular biharmonic problems are carried out to show the efficacy, accuracy, and reliability of the method. From the computational experiments, improved numerical results obtained as compared to the results obtained in earlier research work.

Isogeometric analysis on implicit domains using weighted extended PHT-splines

The urging quest for diminishing the gap between finite element methods and computer-aided design is growing every day. Isogeometric analysis is driving that search a step forward in contrast with traditional finite element methods. One of the areas that need more investigation is using isogeometric analysis on domains that contain singularities. In this case, the parameterization process is occasionally non-trivial, and some of the inherited basis functions are not well defined producing rough solutions. Moreover, NURBS basis functions are not the best choice to perform local refinement since adaptivity is indispensable. In this study, a new approach that serves as a generalization for some of the ideas in both the isogeometric analysis and the WEB method is introduced. Isogeometric analysis on implicit domains replaces mesh generation with implicit spline weight function generation and construct weighted extended PHT-splines for analysis. The geometry and the analysis both are built starting from the same function space avoiding parameterization. The PHT-splines are adaptive and allow local refinement naturally leading to a higher accuracy while dealing with singularities. An illustration with numerical examples demonstrates the efficiency and performance of the proposed method.

On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations

A boundary value problem for linear partial differential algebraic systems of equations with multiple characteristic curves is examined. It is assumed that the pencil of matrix functions associated with this system is smoothly equivalent to a special canonic form. The spline collocation is used to construct for this problem a difference scheme of an arbitrary approximation order with respect to each independent variable. Sufficient conditions are found for this scheme to be absolutely stable.

Implicit Reconstruction of Vasculatures Using Bivariate Piecewise Algebraic Splines

Vasculature geometry reconstruction from volumetric medical data is a crucial task in the development of computer guided minimally invasive vascular surgery systems. In this paper, a technique for reconstructing the geometry of vasculatures using bivariate implicit splines is developed. With the proposed technique, an implicit geometry representation of the vascular tree can be accurately constructed based on the voxels extracted directly from the surface of a certain vascular structure in a given volumetric medical dataset. Experimental results show that the geometric representation built using our method can faithfully represent the morphology and topology of vascular structures. In addition, both the qualitative and the quantitative validations have been performed to show that the reconstructed vessel geometry is of high accuracy and smoothness. An virtual angioscopy system has been implemented to indicate one of the strengths of our proposed method.

Numerical Methods Based on Non-Polynomial Sextic Spline for Solution of Variable Coefficient Fourth-Order Wave Equations

A new technique based on non-polynomial sextic spline functions connecting spline functions values at mid knots and their corresponding values of the fourth-order derivatives is developed. We derive various classes of three level implicit spline methods for solution of fourth-order non-homogeneous parabolic partial differential equation with variable coefficient. These new numerical methods are based on non-polynomial sextic spline in space and finite difference in time directions. The linear stability of the presented methods is investigated. We solve test problems numerically to validate the derived methods. Numerical comparison with other existing methods shows the superiority of our presented scheme.

Un résultat de consistance pour des SVM fonctionnels par interpolation spline

This Note proposes a new methodology for function classification with Support Vector Machine (SVM). Rather than relying on projection on a truncated Hilbert basis as in our previous work, we use an implicit spline interpolation that allows us to compute SVM on the derivatives of the studied functions. To that end, we propose a kernel defined directly on the discretizations of the observed functions. We show that this method is universally consistent. To cite this article: N. Villa, F. Rossi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Three-dimensional analysis of thermal resistance of exterior basement insulation systems (EIBS)

A consortium 1 1 The consortium included the Expanded Polystyrene Association of Canada, the Canadian Urethane Foam Contractors Association, Owens Corning Inc. and Roxul Inc. research project was initiated to determine the field performance of various thermal insulation products as applied in the exterior insulation basement system (EIBS). Initially a two-dimensional (2-D) analytical tool was used to derive the thermal transmission characteristics from an array of temperature measurements performed over a period of two years. Results immediately showed the influence of lateral heat flux between various products that differed in thermal transmission properties. Therefore, development of a three-dimensional (3-D) model became imperative. This paper gives the theoretical and numerical approaches adopted to develop a 3-D computer model of heat transfer. The implicit Spline Method was selected for the problem solver. The applicability of the model was verified using measured data on temperature distributions at several material interfaces. Then the model was used to estimate the effect of lateral heat flows. The paper also briefly reports the experimental details and presents results on model verification.

Quadric and cubic bitetrahedral patches

Algebraic surface patches bounded by tetrahedra are promising building blocks for low-degree implicit spline surfaces. General algebraic surface patches may have topological anomalies such as singular points and multiple sheets. The A-patch technique avoids many of these anomalies but does not ensure that the surface patches are connected and have no holes. Requiring these qualities often places overly restrictive conditions on the patch. We present a technique for building a bitetrahedral patch that is single sheeted, smooth, and singly connected in a pair of face-adjacent tetrahedra. The bitetrahedral patch technique is applicable to most low-degree algebraic surface patches, and it establishes single sheetedness, smoothness, and single connectness when existing techniques fail.

A note on two-sided holes in implicit quadric spline surfaces

Some implicit spline surface schemes require filling two-sided holes [Dahmen 1989; Guo 1993] and a blending technique of Warren (1987) can be used to fill two-sided holes in quadric spline surfaces. This note shows that there are important cases where the blending technique does not apply. These cases are identified and an applicatble technique is proposed.

Nonsplitting Macro Patches for Implicit Cubic Spline Surfaces

AbstractMacro patches are important for generating quadric or cubic implicit spline surfaces from the input of a polyhedron. All existing macro patches split the triangular facets of the polyhedron; this paper presents cubic nonsplitting macro patches (NMP) that do not split these facets. The NMP's are based on a necessary and sufficient condition for nonsplitting constructions of implicit cubic spline surfaces. This condition can be satisfied for most practical applications, so the NMP's lead to an efficient and powerful spline surface scheme using implicit cubics. The free parameters in an NMP are set using a new technique for excluding topological anomalies such as extraneous sheets, splits, unwanted holes, self‐intersections, and unwanted handles. Each cubic patch obtained by this technique best approximates, in a least‐squares sense, a quadric patch from a single algebraic component of a monotone polynomial derived from the input data.