Of late we have seen an increase in the use of subdivision techniques forboth modeling and animat... more Of late we have seen an increase in the use of subdivision techniques forboth modeling and animation. They have given rise to a class of surfaces calledsubdivision surfaces. These have many advantages over traditional Non UniformRational B-spline (NURB) surfaces. Subdivision surfaces easily addressthe issues related to multiresolution, refinement, scalability and representationof meshes. Many schemes have been introduced that take a
The concentration edge detection and Gegenbauer image reconstruction methods were previously show... more The concentration edge detection and Gegenbauer image reconstruction methods were previously shown to improve the quality of segmentation in magnetic resonance imaging. In this study these methods are utilized as a pre-processing step to the Weibull E-SD field segmentation. It is demonstrated that the combination of the concentration edge detection and Gegenbauer reconstruction method improves the accuracy of segmentation for
A new approach for segmenting 3D voxel data sets is presented. It is semi-automatic and consists ... more A new approach for segmenting 3D voxel data sets is presented. It is semi-automatic and consists of two phases. First, the initial seg- mentation is accomplished by voxel labeling using statistical max- imum likelihood estimation techniques. The novelty of the present approach is that the type probability distribution function is not re- quired a priori. A multi-parameter distribution which includes
Communications in Numerical Methods in Engineering, 2009
In this paper we extend the method of inter-modality image registration using the maximization of... more In this paper we extend the method of inter-modality image registration using the maximization of normalized mutual information (NMI) for the registration of [ 18 F]-2-fluoro-deoxy-D-glucose (FDG)positron emission tomography (PET) with T1-weighted magnetic resonance (MR) volumes. We investigate the impact on the NMI maximization with respect to using coarse-to-fine grained B-spline bases and to the number of bins required for the voxel intensity histograms of each volume. Our results demonstrate that the efficiency and accuracy of elastic, as well as rigid body, registration is improved both through the use of a reduced number of bins in the PET and MR histograms, and of a limited coarse-to-fine grain interpolation of the volume data. To determine the appropriate number of bins prior to registration, we consider the NMI between the two volumes, the mutual information content of the two volumes, as a function of the binning of each volume. Simulated data sets are used for validation and the registration improves that obtained with a standard approach based on the Statistical Parametric Mapping software. this paper, is the mapping of functional data from [ 18 F]-2-fluoro-deoxy-D-glucose (FDG) positron emission tomography (PET) volumes with volumetric T1-weighted magnetic resonance (MR) volumes. Thus, throughout, when we refer to PET and MR volumes, it is understood that these are FDG-PET and T1-weighted MR.
A common reverse engineering problem is to convert several hundred thousand points collected from... more A common reverse engineering problem is to convert several hundred thousand points collected from the surface of an object via a digitizing process, into a coherent geometric model that is easily transferred to a CAD software such as a solid modeler for either design improvement or manufacturing and analysis. These data are very dense and make data-set manipulation difficult and
Given a three-dimensional (3D) array of function values F i,j,k on a rectilinear grid, the marchi... more Given a three-dimensional (3D) array of function values F i,j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation % approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C 0 -continuous surface consisting of rational-quadratic surface patches interpolating the triangles in %. We determine the Bézier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.
A classic problem in geometric modelling is curve interpolation to data points. Some of the exist... more A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.
Most numerical simulations require techniques for the representation and manipulation of complex,... more Most numerical simulations require techniques for the representation and manipulation of complex, three-dimensional geometries. This paper provides a short historical survey and an overview of state-of-the-art geometric modeling techniques and research issues, and discusses a few selected applications of geometric modeling in computational areas.
We present a novel and computationally fast method for automatic human face authentication. Takin... more We present a novel and computationally fast method for automatic human face authentication. Taking a 3D triangular facial mesh as input, the approach first automatically extracts the bilateral symmetry plane of the facial surface. The intersection between the symmetry plane and the facial surface, namely the symmetry profile, is then computed. Using both the mean curvature plot of the facial surface and the curvature plot of the symmetry profile curve, three essential points of the nose on the symmetry profile are automatically extracted. The three essential points uniquely determine a Face Intrinsic Coordinate System (FICS). Different faces are aligned based on the FICS. The symmetry profile, together with two transverse profiles, composes a compact representation, called the SFC representation, of a 3D face surface. The face authentication and recognition steps are finally performed by comparing the SFC representations of the faces. The proposed method was tested on 382 face surfaces, which come from 166 individuals and cover a wide ethnic and age variety. The equal error rate (EER) of face authentication on scans with variable facial expressions is 10.8%. For scans with normal expression, the ERR is 0.8%.
IEEE Transactions on Visualization and Computer Graphics, 2003
This paper presents a coarse-grain approach for segmentation of objects with gray levels appearin... more This paper presents a coarse-grain approach for segmentation of objects with gray levels appearing in volume data. The input data is on a 3D structured grid of vertices vði; j; kÞ, each associated with a scalar value. In this paper, we consider a voxel as a   cube and each voxel is assigned two values: expectancy and standard deviation (E-SD). We use the Weibull noise index to estimate the noise in a voxel and to obtain more precise E-SD values for each voxel. We plot the frequency of voxels which have the same E-SD, then 3D segmentation based on the Weibull E-SD field is presented. Our test bed includes synthetic data as well as real volume data from a confocal laser scanning microscope (CLSM). Analysis of these data all show distinct and defining regions in their E-SD fields. Under the guide of the E-SD field, we can efficiently segment the objects embedded in real and simulated 3D data.
IEEE Transactions on Visualization and Computer Graphics, 2000
This paper presents a shape-based approach in extracting thin structures, such as lines and sheet... more This paper presents a shape-based approach in extracting thin structures, such as lines and sheets, from threedimensional (3D) biomedical images. Of particular interest is the capability to recover cellular structures, such as microtubule spindle fibers and plasma membranes, from laser scanning confocal microscopic (LSCM) data. Hessian-based shape methods are reviewed. A synthesized linear structure is used to evaluate the sensitivity of the multiscale filtering approach in extracting closely positioned fibers. We find that the multiscale approach tends to fuse lines together, which makes it unsuitable for visualizing mouse egg spindle fibers. Single-scale Gaussian filters, balanced between sensitivity and noise resistance, are adopted instead. In addition, through an ellipsoidal Gaussian model, the eigenvalues of the Hessian matrix are quantitatively associated with the standard deviations of the Gaussian model. Existing shape filters are simplified and applied to LSCM data. A significant improvement in extracting closely positioned thin lines is demonstrated by the resultant images. Further, the direct association of shape models and eigenvalues makes the processed images more understandable qualitatively and quantitatively.
interactive techniques are often praised as the optimal tool to achieve aesthetic shapes. Yet the... more interactive techniques are often praised as the optimal tool to achieve aesthetic shapes. Yet the interactive "tweaking of control polygons" cannot produce shapes of production quality. We discuss the use of curvature plots for the design of curves that have to meet aesthetic requirements. We also discuss a local method to fair cubic B-spline curves and surfaces.
Sibson's interpolant uses Voronoi diagrams in the plane to interpolate a set of scattered data po... more Sibson's interpolant uses Voronoi diagrams in the plane to interpolate a set of scattered data points. This paper presents an extension of this method to handle the interpolation of a set of functional curves (transfinite surface interpolation). We derive a simple formula for this new surface type which can interpolate to any number of boundary curves. In addition, a unique surface may be created from a set of discontinuous curves. Finally, we present a form of the interpolant which uses convex or concave polygonal domains.
The objective of this work is to give biologists the ability to visualize Multicolor Laser Confoc... more The objective of this work is to give biologists the ability to visualize Multicolor Laser Confocal Microscope (MLCM) data sets. We present the application of volume ray casting techniques to create fast and accurate visualization of the spatial structures in MLCM data sets. We provide various volume pre-processing routines for noise removal, smoothing and edge enhancement.
Algorithms are presented for locally fairing a B-spline curve in order to produce a pleasant curv... more Algorithms are presented for locally fairing a B-spline curve in order to produce a pleasant curvature plot. The methods are based on inversions of the knot insertion algorithm for B-spline curves.
Finding the intersection of two surfaces is important for many Computer Aided Design tasks concer... more Finding the intersection of two surfaces is important for many Computer Aided Design tasks concerned with surface modeling. An adaptive algorithm is developed for finding the intersection curve(s) of pairs of rectangular parametric patches which are continuously differentiable. The balance between robustness and efficiency of the algorithm is controlled by a set of tolerances. A suite of examples concludes the paper.
Of late we have seen an increase in the use of subdivision techniques forboth modeling and animat... more Of late we have seen an increase in the use of subdivision techniques forboth modeling and animation. They have given rise to a class of surfaces calledsubdivision surfaces. These have many advantages over traditional Non UniformRational B-spline (NURB) surfaces. Subdivision surfaces easily addressthe issues related to multiresolution, refinement, scalability and representationof meshes. Many schemes have been introduced that take a
The concentration edge detection and Gegenbauer image reconstruction methods were previously show... more The concentration edge detection and Gegenbauer image reconstruction methods were previously shown to improve the quality of segmentation in magnetic resonance imaging. In this study these methods are utilized as a pre-processing step to the Weibull E-SD field segmentation. It is demonstrated that the combination of the concentration edge detection and Gegenbauer reconstruction method improves the accuracy of segmentation for
A new approach for segmenting 3D voxel data sets is presented. It is semi-automatic and consists ... more A new approach for segmenting 3D voxel data sets is presented. It is semi-automatic and consists of two phases. First, the initial seg- mentation is accomplished by voxel labeling using statistical max- imum likelihood estimation techniques. The novelty of the present approach is that the type probability distribution function is not re- quired a priori. A multi-parameter distribution which includes
Communications in Numerical Methods in Engineering, 2009
In this paper we extend the method of inter-modality image registration using the maximization of... more In this paper we extend the method of inter-modality image registration using the maximization of normalized mutual information (NMI) for the registration of [ 18 F]-2-fluoro-deoxy-D-glucose (FDG)positron emission tomography (PET) with T1-weighted magnetic resonance (MR) volumes. We investigate the impact on the NMI maximization with respect to using coarse-to-fine grained B-spline bases and to the number of bins required for the voxel intensity histograms of each volume. Our results demonstrate that the efficiency and accuracy of elastic, as well as rigid body, registration is improved both through the use of a reduced number of bins in the PET and MR histograms, and of a limited coarse-to-fine grain interpolation of the volume data. To determine the appropriate number of bins prior to registration, we consider the NMI between the two volumes, the mutual information content of the two volumes, as a function of the binning of each volume. Simulated data sets are used for validation and the registration improves that obtained with a standard approach based on the Statistical Parametric Mapping software. this paper, is the mapping of functional data from [ 18 F]-2-fluoro-deoxy-D-glucose (FDG) positron emission tomography (PET) volumes with volumetric T1-weighted magnetic resonance (MR) volumes. Thus, throughout, when we refer to PET and MR volumes, it is understood that these are FDG-PET and T1-weighted MR.
A common reverse engineering problem is to convert several hundred thousand points collected from... more A common reverse engineering problem is to convert several hundred thousand points collected from the surface of an object via a digitizing process, into a coherent geometric model that is easily transferred to a CAD software such as a solid modeler for either design improvement or manufacturing and analysis. These data are very dense and make data-set manipulation difficult and
Given a three-dimensional (3D) array of function values F i,j,k on a rectilinear grid, the marchi... more Given a three-dimensional (3D) array of function values F i,j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation % approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C 0 -continuous surface consisting of rational-quadratic surface patches interpolating the triangles in %. We determine the Bézier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.
A classic problem in geometric modelling is curve interpolation to data points. Some of the exist... more A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.
Most numerical simulations require techniques for the representation and manipulation of complex,... more Most numerical simulations require techniques for the representation and manipulation of complex, three-dimensional geometries. This paper provides a short historical survey and an overview of state-of-the-art geometric modeling techniques and research issues, and discusses a few selected applications of geometric modeling in computational areas.
We present a novel and computationally fast method for automatic human face authentication. Takin... more We present a novel and computationally fast method for automatic human face authentication. Taking a 3D triangular facial mesh as input, the approach first automatically extracts the bilateral symmetry plane of the facial surface. The intersection between the symmetry plane and the facial surface, namely the symmetry profile, is then computed. Using both the mean curvature plot of the facial surface and the curvature plot of the symmetry profile curve, three essential points of the nose on the symmetry profile are automatically extracted. The three essential points uniquely determine a Face Intrinsic Coordinate System (FICS). Different faces are aligned based on the FICS. The symmetry profile, together with two transverse profiles, composes a compact representation, called the SFC representation, of a 3D face surface. The face authentication and recognition steps are finally performed by comparing the SFC representations of the faces. The proposed method was tested on 382 face surfaces, which come from 166 individuals and cover a wide ethnic and age variety. The equal error rate (EER) of face authentication on scans with variable facial expressions is 10.8%. For scans with normal expression, the ERR is 0.8%.
IEEE Transactions on Visualization and Computer Graphics, 2003
This paper presents a coarse-grain approach for segmentation of objects with gray levels appearin... more This paper presents a coarse-grain approach for segmentation of objects with gray levels appearing in volume data. The input data is on a 3D structured grid of vertices vði; j; kÞ, each associated with a scalar value. In this paper, we consider a voxel as a   cube and each voxel is assigned two values: expectancy and standard deviation (E-SD). We use the Weibull noise index to estimate the noise in a voxel and to obtain more precise E-SD values for each voxel. We plot the frequency of voxels which have the same E-SD, then 3D segmentation based on the Weibull E-SD field is presented. Our test bed includes synthetic data as well as real volume data from a confocal laser scanning microscope (CLSM). Analysis of these data all show distinct and defining regions in their E-SD fields. Under the guide of the E-SD field, we can efficiently segment the objects embedded in real and simulated 3D data.
IEEE Transactions on Visualization and Computer Graphics, 2000
This paper presents a shape-based approach in extracting thin structures, such as lines and sheet... more This paper presents a shape-based approach in extracting thin structures, such as lines and sheets, from threedimensional (3D) biomedical images. Of particular interest is the capability to recover cellular structures, such as microtubule spindle fibers and plasma membranes, from laser scanning confocal microscopic (LSCM) data. Hessian-based shape methods are reviewed. A synthesized linear structure is used to evaluate the sensitivity of the multiscale filtering approach in extracting closely positioned fibers. We find that the multiscale approach tends to fuse lines together, which makes it unsuitable for visualizing mouse egg spindle fibers. Single-scale Gaussian filters, balanced between sensitivity and noise resistance, are adopted instead. In addition, through an ellipsoidal Gaussian model, the eigenvalues of the Hessian matrix are quantitatively associated with the standard deviations of the Gaussian model. Existing shape filters are simplified and applied to LSCM data. A significant improvement in extracting closely positioned thin lines is demonstrated by the resultant images. Further, the direct association of shape models and eigenvalues makes the processed images more understandable qualitatively and quantitatively.
interactive techniques are often praised as the optimal tool to achieve aesthetic shapes. Yet the... more interactive techniques are often praised as the optimal tool to achieve aesthetic shapes. Yet the interactive "tweaking of control polygons" cannot produce shapes of production quality. We discuss the use of curvature plots for the design of curves that have to meet aesthetic requirements. We also discuss a local method to fair cubic B-spline curves and surfaces.
Sibson's interpolant uses Voronoi diagrams in the plane to interpolate a set of scattered data po... more Sibson's interpolant uses Voronoi diagrams in the plane to interpolate a set of scattered data points. This paper presents an extension of this method to handle the interpolation of a set of functional curves (transfinite surface interpolation). We derive a simple formula for this new surface type which can interpolate to any number of boundary curves. In addition, a unique surface may be created from a set of discontinuous curves. Finally, we present a form of the interpolant which uses convex or concave polygonal domains.
The objective of this work is to give biologists the ability to visualize Multicolor Laser Confoc... more The objective of this work is to give biologists the ability to visualize Multicolor Laser Confocal Microscope (MLCM) data sets. We present the application of volume ray casting techniques to create fast and accurate visualization of the spatial structures in MLCM data sets. We provide various volume pre-processing routines for noise removal, smoothing and edge enhancement.
Algorithms are presented for locally fairing a B-spline curve in order to produce a pleasant curv... more Algorithms are presented for locally fairing a B-spline curve in order to produce a pleasant curvature plot. The methods are based on inversions of the knot insertion algorithm for B-spline curves.
Finding the intersection of two surfaces is important for many Computer Aided Design tasks concer... more Finding the intersection of two surfaces is important for many Computer Aided Design tasks concerned with surface modeling. An adaptive algorithm is developed for finding the intersection curve(s) of pairs of rectangular parametric patches which are continuously differentiable. The balance between robustness and efficiency of the algorithm is controlled by a set of tolerances. A suite of examples concludes the paper.
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Papers by Gerald Farin