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A chronology of interpolation: From ancient astronomy to modern signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into histo ..."
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Cited by 88 (0 self)
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This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolutionbased interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.
Surface Interpolation With Radial Basis Functions for Medical Imaging
 IEEE Transactions on Medical Imaging
, 1997
"... Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from threedimensional (3D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial ..."
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Cited by 86 (2 self)
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Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from threedimensional (3D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial basis functions impose few restrictions on the geometry of the interpolation centers and are suited to problems where the interpolation centers do not form a regular grid. However, their high computational requirements have previously limited their use to problems where the number of interpolation centers is small (! 300). Recently developed fast evaluation techniques have overcome these limitations and made radial basis interpolation a practical approach for larger data sets. In this paper radial basis functions are fitted to depthmaps of the skull's surface, obtained from Xray CT data using raytracing techniques. They are used to smoothly interpolate the surface of the skull across defe...
A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... Abstract—A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a kdimensional binary image in time linear in the total number of voxelsN. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for co ..."
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Cited by 83 (3 self)
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Abstract—A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a kdimensional binary image in time linear in the total number of voxelsN. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the Lp and chamfer metrics. At each dimension level, the DT is computed by constructing the intersection of the Voronoi diagram whose sites are the feature voxels with each row of the image. This construction is performed efficiently by using the DT in the next lower dimension. The correctness and linear time complexity are demonstrated analytically and verified experimentally. The algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast (not only does it run in O N time but the time constant is small). A simple modification of the algorithm computes the weighted Euclidean DT, which is useful for images with anisotropic voxel dimensions. A parallel version of the algorithm runs in O
N=p time with p processors. Index Terms—Euclidean distance transform, closest feature transform, Voronoi diagram. æ
An Active Contour Model For Mapping The Cortex
 IEEE TRANS. ON MEDICAL IMAGING
, 1995
"... A new active contour model for finding and mapping the outer cortex in brain images is developed. A crosssection of the brain cortex is modeled as a ribbon, and a constant speed mapping of its spine is sought. A variational formulation, an associated force balance condition, and a numerical approac ..."
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Cited by 79 (15 self)
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A new active contour model for finding and mapping the outer cortex in brain images is developed. A crosssection of the brain cortex is modeled as a ribbon, and a constant speed mapping of its spine is sought. A variational formulation, an associated force balance condition, and a numerical approach are proposed to achieve this goal. The primary difference between this formulation and that of snakes is in the specification of the external force acting on the active contour. A study of the uniqueness and fidelity of solutions is made through convexity and frequency domain analyses, and a criterion for selection of the regularization coefficient is developed. Examples demonstrating the performance of this method on simulated and real data are provided.
Quantitative evaluation of convolutionbased methods for medical image interpolation
 Medical Image Analysis
, 2001
"... Abstract—Interpolation is required in a variety of medical image processing applications. Although many interpolation techniques are known from the literature, evaluations of these techniques for the specific task of applying geometrical transformations to medical images are still lacking. In this p ..."
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Cited by 49 (2 self)
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Abstract—Interpolation is required in a variety of medical image processing applications. Although many interpolation techniques are known from the literature, evaluations of these techniques for the specific task of applying geometrical transformations to medical images are still lacking. In this paper we present such an evaluation. We consider convolutionbased interpolation methods and rigid transformations (rotations and translations). A large number of sincapproximating kernels are evaluated, including piecewise polynomial kernels and a large number of windowed sinc kernels, with spatial supports ranging from two to ten grid intervals. In the evaluation we use images from a wide variety of medical image modalities. The results show that spline interpolation is to be preferred over all other methods, both for its accuracy and its relatively low computational cost. Keywords—Convolutionbased interpolation, spline interpolation, piecewise polynomial kernels, windowed sinc kernels, geometrical transformation, medical images, quantitative evaluation. 1
Efficient semiautomatic segmentation of 3d objects in medical images
 In Proc. of Medical Image Computing and Computerassisted Intervention (MICCAI
, 2000
"... Abstract. We present a fast and accurate tool for semiautomatic segmentation of volumetric medical images based on the live wire algorithm, shapebased interpolation and a new optimization method. While the usersteered live wire algorithm represents an efficient, precise and reproducible method for ..."
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Cited by 36 (4 self)
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Abstract. We present a fast and accurate tool for semiautomatic segmentation of volumetric medical images based on the live wire algorithm, shapebased interpolation and a new optimization method. While the usersteered live wire algorithm represents an efficient, precise and reproducible method for interactive segmentation of selected twodimensional images, the shapebased interpolation allows the automatic approximation of contours on slices between userdefined boundaries. The combination of both methods leads to accurate segmentations with significantly reduced user interaction time. Moreover, the subsequent automated optimization of the interpolated object contours results in a better segmentation quality or can be used to extend the distances between usersegmented images and for a further reduction of interaction time. Experiments were carried out on hepatic computer tomographies from three different clinics. The results of the segmentation of liver parenchyma have shown that the user interaction time can be reduced more than 60% by the combination of shapebased interpolation and our optimization method with volume deviations in the magnitude of interuser differences. 1
Geometric Methods for Vessel Visualization and Quantification  A Survey
 IN GEOMETRIC MODELLING FOR SCIENTIFIC VISUALIZATION
, 2002
"... ... This paper surveys several geometric methods to solve basic visualization and quantification problems like centerline computation, boundary detection, projection techniques, and geometric model generation. ..."
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Cited by 32 (1 self)
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... This paper surveys several geometric methods to solve basic visualization and quantification problems like centerline computation, boundary detection, projection techniques, and geometric model generation.
Surface Interpolation From Sparse CrossSections Using Region Correspondence
, 1999
"... The ability to estimate a surface from a set of crosssections allows calculation of the enclosed volume and the display of the surface in threedimensions (3D). This process has increasingly been used to derive useful information from medical data. However, extracting the crosssections (segmentin ..."
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Cited by 14 (2 self)
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The ability to estimate a surface from a set of crosssections allows calculation of the enclosed volume and the display of the surface in threedimensions (3D). This process has increasingly been used to derive useful information from medical data. However, extracting the crosssections (segmenting) can be very difficult, and automatic segmentation methods are not sufficiently robust to deal with all situations. Hence, it is an advantage if the surface reconstruction algorithm can work effectively on a small number of crosssections. In addition, crosssections of medical data are often quite complex. In this paper, an algorithm is presented which can interpolate a surface through sparse, complex crosssections. This is an extension of maximal disc guided interpolation [25], which is itself based on shape based interpolation [8, 21]. The performance of this algorithm is demonstrated on various types of medical data (Xray Computed Tomography, Magnetic Resonance Imaging and threedimen...
The Image Foresting Transformation
 IEEE Trans. on Pattern Analysis and Machine Intelligence
, 2000
"... In this paper, we introduce an image processing operator called Image Foresting Transformation (IFT ). The image foresting transformation maps an input image into a graph, computes a shortestpath forest in this graph, and outputs an annotated image, which is basically an image and its associated ..."
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Cited by 12 (2 self)
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In this paper, we introduce an image processing operator called Image Foresting Transformation (IFT ). The image foresting transformation maps an input image into a graph, computes a shortestpath forest in this graph, and outputs an annotated image, which is basically an image and its associated forest. We describe the application of IFT to region growing, edge detection, Euclidean distance transform, geodesic distance computation, and watershed transformation. All the operators are eciently computed using the same IFT algorithm based on the same set of parameters by changing only their meaning and values. We also present a new interactive image segmentation paradigm based on the region growing operator and discuss other applications of the IFT for boundarybased object denition and shapebased interpolation. 1 Introduction The use of graph in computer vision and image processing has been investigated for many years now. Its motivation stems from a solid theory with many e...
Binary morphological shapebased interpolation applied to 3D tooth reconstruction
, 2002
"... In this paper we propose an interpolation algorithm using a mathematical morphology morphing approach. The aim of this algorithm is to reconstruct the ndimensional object from a group of (n  1)dimensional sets representing sections of that object. The morphing transformation modifies pairs of con ..."
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Cited by 12 (1 self)
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In this paper we propose an interpolation algorithm using a mathematical morphology morphing approach. The aim of this algorithm is to reconstruct the ndimensional object from a group of (n  1)dimensional sets representing sections of that object. The morphing transformation modifies pairs of consecutive sets such that they approach in shape and size. The interpolated set is achieved when the two consecutive sets are made idempotent by the morphing transformation. We prove the convergence of the morphological morphing. The entire object is modeled by successively interpolating a certain number of intermediary sets between each two consecutive given sets. We apply the interpolation algorithm for 3D tooth reconstruction.