WO2000052629A1 - Mesure de formes tridimensionnelles au moyen d'une analyse de courbure statistique - Google Patents

Mesure de formes tridimensionnelles au moyen d'une analyse de courbure statistique Download PDF

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WO2000052629A1
WO2000052629A1 PCT/US2000/005596 US0005596W WO0052629A1 WO 2000052629 A1 WO2000052629 A1 WO 2000052629A1 US 0005596 W US0005596 W US 0005596W WO 0052629 A1 WO0052629 A1 WO 0052629A1
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curvature
vertex
triangle
normal
subset
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PCT/US2000/005596
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John E. Stewart
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Virginia Commonwealth University
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Priority to AU37207/00A priority patent/AU3720700A/en
Priority to CA002364176A priority patent/CA2364176A1/fr
Priority to JP2000602979A priority patent/JP2003502723A/ja
Publication of WO2000052629A1 publication Critical patent/WO2000052629A1/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
    • G06T17/20—Finite element generation, e.g. wire-frame surface description, tesselation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00—Image analysis
    • G06T7/60—Analysis of geometric attributes
    • G06T7/64—Analysis of geometric attributes of convexity or concavity
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/30—Subject of image; Context of image processing
    • G06T2207/30004—Biomedical image processing
    • G06T2207/30016—Brain
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/30—Subject of image; Context of image processing
    • G06T2207/30004—Biomedical image processing
    • G06T2207/30101—Blood vessel; Artery; Vein; Vascular

Definitions

  • the present invention generally relates to a computer implemented method for applying differential computational geometry and statistics to detect shapes on three dimensional (3-D) computer models representative of a sequential series of two dimensional images of a structure. Twelve new curvature measures are also described. These measurements are applied to the surface of a 3-D computer model made up of triangles. Collections of 3-D images of known structures having aberrant surface curvature characteristics associated with dysfunction or defects are then used to train the software to recognize the particular curvature properties of an aberrant structure. Multiple linear regression is applied to the curvature measures of the 3-D test models and the results are optimized to remove multicolinearity and maintain a high coefficient of regression.
  • An integral step in many medical diagnostic protocols is the detection of structural abnormalities associated with anatomical regions of interest.
  • cerebral aneurysms also known as "berry aneurysms,” are typically found as sacular outpoutchings of major intracranial arteries. They are known to enlarge progressively in many individuals and are thought to arise from arterial blood flow striking a weak area of the internal elastic lamina in the wall of an artery. Based on autopsy studies, these aneurysms occur in the general population at a rate anywhere between 1-8 %. For a given individual with an unruptured aneurysm, the annual risk of rupture is 1-2% while the lifetime risk of such a rupture is approximately 50%. At present time there do exist methods for elective treatment of such aneurysms in order to prevent rupture. Accordingly, effective diagnosis can potentially greatly reduce the morbidity and mortality associated with the occurrence of aneurysms.
  • Cerebral angiography is an invasive procedure in which a catheter is advanced through the intracranial vessels and a contrast material is injected to enable visualization of the vessel walls. This procedure is costly, time consuming, and involves serious potential complications that can lead to aneurysm rupture, stroke, and death, although the risk of such complications is small (less than 1%). MRA, on the other hand, is less invasive, less time consuming, and presents less risk than cerebral angiography.
  • Differential geometry has been used extensively as a means of analyzing a variety of geometric shapes and computational geometry is a means of rapidly determining surface characteristics using the mathematics of differential geometry.
  • the techniques of computational geometry have been applied to the prediction of protein-protein interactions (Duncan et al, “Shape Analysis of Molecular Surfaces,” Biopolymers, 33: 231-238, 1993), analysis of dose-effect surfaces of combined agents (Lam, “The Combined Actions of Agents Using Differential Geometry of Dose-effect Surfaces,” Bulletin of Mathematical Biology, 54:813-826, 1992), and the study of biological surface growth.
  • (2-D) cross sectional images Arteries in these images are highlighted using a 3-D time-of-flight algorithm or its equivalent that is included as a matter of course with typical magnetic resonance scanner software.
  • the images are displayed on sheets of 2-D radiological film for review on light boxes.
  • Each image is read by a radiologist or surgeon who "creates" a three-dimensional interpretation of the vasculature through his or her knowledge of the anatomy of the blood vessels. This image may not be reduced to a three dimensional interpretation that can be reviewed by third parties. Due to the branching nature of the vasculature, this interpretation can be very difficult and time consuming. Branching vessels often appear as an aneurysm on a single image but analysis of surrounding images reveals normal vasculature. Similarly, an aneurysm might appear as a branching vessel unless it is carefully followed through a series of images to its termination.
  • skeletal representations of the vasculature have been used to determine vessel branch points and topological features in vessels (Puig et al., "An Interactive Cerebral Blood Vessel Exploration System” Visualization '97, Proceedings, pp.443- 446, 1996). While these techniques are useful in some fields, such as virtual colonoscopy, the small vessel radii and large directional changes encountered in the topography of blood vessels are problematic to the application of skeletal representations for the identification of aneurysms. Smoothing and filtering of MRA images have been employed in an attempt to overcome this problem but the result is often a distorted view of the vasculature.
  • Morphometric analysis has been proposed as a means of automatically detecting aneurysms but this procedure can only analyze small regions of the vasculature at a time (Matsutani, et al, Quantitative Vascular Shape Analysis for 3-D MR-Angiography Using Mathematical Morphology," Computer Vision, Virtual Reality and Robotics in Medicine, CVRMed '95, Proceedings, pp.449-454, 1995).
  • the present invention provides a method for objectively determining a 3-D analysis of a series of 2-D images.
  • One particular embodiment of the invention is based on the simple observation that cerebral vessels are roughly cylindrical while aneurysms contained within such vessels are roughly spherical in shape. This particular embodiment exploits the spherical nature of aneurysms and computational differential geometry to identify and highlight aneurysms in unread 2-D MRA images.
  • first and second fundamental forms There are two classical mathematical entities used to analyze smooth surfaces which are known as the first and second fundamental forms. These measures are useful because they are intrinsic to the surface and therefore invariant to transform (rotation, translation, scaling). The mean and Gaussian curvatures are based upon the fundamental forms.
  • the principal curvatures k, and k 2 are averaged to produce the mean curvature (H) and multiplied to produce the Gaussian curvature (G).
  • cylindrical surfaces are parabolic and spherical surfaces are elliptic.
  • Most algorithms that classify real surfaces according to curvature utilize an approach known as thresholding, wherein the principal curvatures are represented as a continuum and a particular threshold is used to separate the surface into different curvature types.
  • the technique described above has been modified to allow multiple rows of surface vertices to be added to the platelet and used in the determination of the bivariate quadratic polynomial.
  • the result is a much more accurate approximation of local surface shape and subsequently a more accurate approximation of surface curvature around each vertex.
  • the determination of the number of additional rows of surface vertices to include is optimized by finding the number of rows of surface vertices that produces the largest coefficient of regression (described later).
  • the classical curvature approximations provide a great deal of information about the surface shape and rate of change in surface shape.
  • the classic measures are not, however, optimized to recognize and distinguish between cylinders and spheres.
  • It is a further object of the invention to provide a method of evaluating determining three dimensional structures comprising the steps of: a) obtaining a computerized three dimensional representation of a structure or structures; b) identifying a first set of regions on the three dimensional structure or structures and assigning a numeric value to said structure or structures; c) identifying a second set of regions and assigning a numerical value to said regions; d) determining values for a plurality of curvature measures for each vertex on a surface of the structure or structures; e) performing multiple linear regression analysis on said values determined in said determining step to obtain a coefficient of regression for all curvatures for all vertices; f) determining the variance inflation factor for each of said curvature measures; g) if all variance inflation factors are less than 10, go to step 1; h) if any variance inflation factor is greater than 10, sequentially reduce the subset of curvature measures used in multiple linear regression by 1 ; i) performing multiple linear regression on all combinations of curvature measures
  • Figure 1 is a flow chart representing a method in accordance with the present invention of applying novel and classical curvature measures and multiple linear regression to optimize structural differences in a series of sequential vertices.
  • Figure 2 is a perspective view of the first principal curvature vector.
  • Figure 3 is a perspective view indicating the normal vectors that comprise a normal triangle on the surface of a unit sphere.
  • Figure 4 is a perspective view indicating surface normal vectors lying in a plane perpendicular to a cylinder.
  • Figure 5 is a perspective view indicating the normal triangle characteristics for a sphere.
  • Figure 6 is a perspective view showing the radius of an inscribed and a circumscribed circle in relation to the normal triangle.
  • Figure 7 is a graphical representation of the unit normal vectors of a surface triangle and normal triangle.
  • Figure 8 is a graphical representation of the correlation between the correlation coefficient and platelet radius.
  • Figure 9 is a perspective view of a platelet constructed from vertices and triangles.
  • the present invention generally relates to a method and system as schematically represented in Figure 1 , of applying differential computational geometry to the analysis of surface curvature.
  • the method accepts as input a 3-D computer model of a structure that can be generated either automatically or semi-automatically from a collection of cross sectional images.
  • Medical applications of the invention disclosed herein can be adapted to the study of much of the human body, as well as the bodies of other animals.
  • Much of the human body is composed of organs that are roughly spherical or roughly cylindrical in shape. In many instances, derivation from one of these shapes is strongly correlated with disease.
  • the present techniques can be applied to the measurement of arterial stenosis, the detection of arteriovenous malformations, colonic polyps, and the detection of lung and liver cancers.
  • the present invention can also be applied to the analysis of spherical and cylindrical machine parts that cannot otherwise be readily taken apart for inspection.
  • non-destructive testing of machinery via detecting anomalies in structure of internal parts of a machine via detecting anomalies in structure of internal parts of a machine.
  • the invention is also applicable to the analysis of 3-D imagery generated from weather patterns to detect funnel clouds or thunderstorms, for example.
  • the method can be applied to the technique of molecular modeling to detect abnormal protein-ligand interactions and in confocal microscopy detecting cancerous cells in a collection of images taken from a microscopic field or fields.
  • Sequential cross sectional images of a selected anatomical structure are acquired with the use, for example, of a scanner such as a magnetic resonance imaging scanner (MRI).
  • MRI magnetic resonance imaging scanner
  • Any type of digital image scanner can be used in place of MRI images such as a helical computer tomography scanner (CT), ultrasound, or PET images.
  • CT helical computer tomography scanner
  • PET images PET images.
  • the 2-D images are arranged in a computer memory to create a 3-D data volume set.
  • the image data to be analyzed can be generated and or stored in any of a variety of image formats.
  • the present invention is ideally suited for use with Picture Archiving and Communication system (PACS) format.
  • the image data can be stored in the digital imaging and communications in medicine standard (DICOM), or as raw binary slices, or in a variety of volume formats.
  • DICOM medicine standard
  • the image data can be stored in the computer memory in an internal data format which allows the image files to be saved as a single data volume instead of individual image files
  • a Web-based PACS software package acts as an automated filing system accepts and stores digital images created by traditional 2-D means.
  • a second software component IsoView
  • IsoView reads DICOM images stored in the PACS format and creates manifold (closely and singly connected )3-D triangulated surfaces from these images using a variant of the marching cubes algorithm.
  • the surface images can be interactively displayed on a local workstation, stored as Virtual Reality Modeling Language (VRML) for later review, or stored as video recording or photographs for future viewing.
  • VRML Virtual Reality Modeling Language
  • the 3-D images created in the IsoView format serve as the input for the curvature measurement method disclosed herein.
  • the calculation of the bivariate quadratic polynomial at each surface vertex is computed exactly as described by Hamman et al.
  • the principal curvature measures, kl and k2 are then computed on these bivariate quadratic polynomials as described in the same.
  • Hamman's technique determines the principal curvature magnitudes, kl and k2, but does not determine the principal curvature vectors, vl and v2. Therefore, equations and a mathematical method are described below to determine the principal curvature vectors vl and v2.
  • the first step is to compute the eigenvectors (which were not described or computed by Hamman, et al.) of the same matrix for which the eigenvalues were determined,
  • the eigenvectors are calculated by solving,
  • the 3-D principal curvature vectors vl and v2 are computed by normalizing the parametric principal curvature vectors and multiplying them times the 3D basis vectors determined in the method as described by Hamman,
  • the classical curvature approximations provide a great deal of information about the surface shape and rate of change of surface shape, they are not optimized to recognize and distinguish cylinders from spheres.
  • curvature magnitude also masks surface curvature analysis with the classical approximations.
  • the present invention describes twelve novel scalar measures that are applicable to measuring curvatures. Two of these scalars combine the Gaussian and mean curvatures into a single value, three measure the change in principal curvature vector direction and seven make use of the surface normal vectors to predict surface shape.
  • ) A ratio of the two principal curvatures is calculated. Since both kl and k2 can assume positive (concave) or negative (convex) values, the unsigned magnitude of kl and k2 is determined. The smaller of the two numbers becomes the numerator and the larger becomes the denominator. This ration should be 0 for a cylinder, 1 for a sphere, and undefined for a plane.
  • AVGdlV j ⁇ Vi j :!), j l,n: The average of the dot product between the first principal curvature vector 200 at the center platelet vertex 201 and the principal curvature vectors 202 of those vertices immediately connected to this vertex is calculated.
  • 200 will be parallel to the axia of a cylinder and will be randomly oriented on the surface of a sphere. Therefore this curvature measure should be small for cylinders and large for spheres. It also will be large in regions where the surface shape is changing dramatically such as at the branch point of vessels.
  • ), j l,n: The average of the dot product between the second principal curvature vector at the center platelet vertex 201 and those vertices immediately connected to this vertex is calculated.
  • v 2 will be perpendicular to the axia of a cylinder and will be randomly oriented on the surface of a sphere. Therefore this curvature measure should be large for spheres, small for large cylinders, and large for small cylinders. It should also be large at vessel branch points.
  • a cylinder is defined by a circle extruded along a vector running perpendicular to the plane of the circle.
  • surface normals 401, 402, and 403 for a cylinder lie in planes perpendicular to the axis of the cylinder. If the unit normal vectors for three points on the surface of a cylinder are translated towards one another parallel to the axis of this cylinder, the three planes will eventually coincide and all three unit normal vectors will lie in this coincident plane.
  • the "tips" of these unit normal vectors will form a circle 404 with a radius 1 unit larger than the radius of the cylinder 405.
  • each unit normal vector is then translated in 2-D with the coincident plane, they can be positioned such that they originated from the same point but still existed entirely within the coincident plane.
  • the tips of these unit normal vectors will therefore form triangle 406 that lies in the coincident plane.
  • This normal triangle will typically have a large aspect ratio that will increase to infinity as the dot product of the normal vectors approaches 1.0. In other words, the more triangles on the surface of the cylinder, the larger the aspect ratio of the cylinder's normal triangles. Because the radius of the cylinder does not influence the aspect ratio of the normal triangle, this measure will be useful in detecting cylinders of any radius.
  • the size, aspect ratio, and tilt of the normal triangle are used to predict the shape of the underlying triangles.
  • the normal triangle aspect ratio will be the same as the surface triangle aspect ratio and will therefore also be an equilateral triangle, as shown in Figure 6.
  • the normals will fan out in a straight line forming a triangle orthogonal to the axis of the cylinder. If the surface under examination is planar, all three normals will coincide and their tips will form a point.
  • the size of the normal triangle also reflects the magnitude of curvature on the surface. Large normal triangles are seen in highly curved regions while small normal triangles are seen in relatively flat regions. Curvature measures that use these normal triangles are referred to as "normal triangle curvatures.”
  • NAREA The area of the normal triangle is calculated by taking the cross product of two sides of this triangle and dividing by 2.0. It should be large for a sphere, small for a cylinder and 0 for a plane.
  • Perimeter The perimeter of a normal triangle should be large for a sphere, of intermediate length for a cylinder and 0 for a plane.
  • TILT is measured by first finding the unit normal vector of the surface triangle. This can be done by calculating the cross product of two of the three sides of the surface triangle and then normalizing (converting the vector to unit length) the normal vector of the surface triangle. The unit normal vector of the surface triangle is illustrated in
  • n s the unit normal vector, n nt of the normal triangle is also determined.
  • TILT is calculated by finding the dot product of these two vectors, n s and n nt This scalar value will be 1 for convex spherical surfaces, 0 for cylindrical surfaces and -1 for concave spherical surfaces.
  • the above equations produce a group of statistical measures to determine curvature.
  • a useful technique for combining multiple variables into a single equation is multiple linear regression. Multiple linear regression minimizes the sum of the squares of the residual between the regression equation and data to produce the linear equation that best fits the data.
  • the x m ⁇ are the 16 curvature approximations consisting of the four classical curvature measures and the twelve novel measures reported herein, while y, are values assigned to the vertices of the 3-D computer models that "teach"multiple linear regression what aneurysms look like.
  • EXAMPLE 1 In order to test the usefulness of the disclosed techniques in predicting aneurysms, MRA studies of 11 patients with normal arterial vasculature and 11 patients with diagnosed aneurysms were reconstructed. The 3-D models of all 22 patients were created at a grayscale threshold of 325. The total number of vertices for all 3-D computer models is 290,802. The aneurysm patient group was contained nine females and two males while the normal group consisted of eight females and two males. The average age of the aneurysms group was
  • the surface vertices belonging to the aneurysm group were partitioned into three groups- aneurysm, transition region, or normal. This was done interactively with IsoView. The entire dome of each aneurysm was partitioned and given a value of 1.0. The transition regions for each aneurysm were marked separately and given a value of 0.5. All other points on the model were assigned a value of 0.0. All vertices that make up the 3-D computer models of the normal patients were assigned a value of 0.0. These values constitute the y, dependent variables of the two previous equations.
  • transition regions lying between the aneurysm and normal vasculature is included because it has different surface curvature characteristics than the aneurysm dome and is often seen accompanying an aneurysm.
  • the transition region is typically concave and is commonly referred to as the neck of the aneurysm.
  • the first step in this process is to compute all 16 scalar curvatures at each vertex of all 22 3-D computer models.
  • the curvature approximations of equations 1-12 are measured for each model and stored in separate files for each patient and curvature type.
  • a platelet radius determines the number of points to use in the least squares fit. The larger this radius, the more points are used.
  • This modification solves the problem of directional bias that can occur when the image pixel size is much smaller than the distance between consecutive MRA images.
  • Directional bias would cause the preferential selection of points in the direction normal to the image planes since triangles would tend to be "stretched" in this direction. No matter what radius is chosen, the first concentric row of points around the central vertex are always used.
  • a useful measure of the goodness fit of the regression line to the data is the correlation coefficient as in,
  • r2 is the coefficient of determination (equal to the correlation coefficient to the power of 2)
  • m is the number of curvature measures
  • n is the number of vertices analyzed.
  • the next step in confirming the model is a test of the individual partial regression coefficients.
  • the test statistic for this hypothesis is also the F statistic but the equation to test a single partial coefficient 3. is,
  • Equation 17 Equation 17
  • VIF variance inflation factor
  • VIF — 1 2 r j Equation 19
  • VIF of 1 indicates that independent variable x. is not multicolinear with other independent variables. Anything larger than 1 indicates some degree of multicolinearity. Although there are no firm rules as to the magnitude that VIF can attain before multicolinearity is demonstrated, many statisticians have adopted a value of 10 as the VIF cutoff. Thus, independent variables greater than 10 are considered multicolinear while those less than 10 are considered independent.
  • Table II lists the VIFs for the five classical curvature measures and the twelve novel measures of curvature described herein. Columns labeled subset size 16-5 represent the test of the correlation coefficient for a combination of
  • the determination of the coefficient of regression for 2m subsets where m is the number of independent variables is called an exhaustive search. Multiple linear regression was performed on all 22 3-D computer models using all combinations of curvatures for a platelet radius of 3.0 mm. The results were sorted first by subset size (number of curvature measures used in the regression analysis) and then by the coefficient of regression. The curvatures that constitute the subset are those that produce the maximum coefficient of regression for a given subset size. Once the optimal subset is determined, the
  • VIF for each curvature in the subset is computed.
  • the coefficient of regression, F statistic and VIF for each subset is displayed in Table II. Empty entries indicate that the curvature measure was not used in the subset.
  • optimization of subsets can be performed by the process of backward elimination, in which each independent variable is tested to determine the variable that can be eliminated while still maintaining the largest coefficient of regression.
  • Table II indicates that curvatures H, DK1K2V, RINCIR, NPER, and TILT are ideal for identifying aneurysms when only five curvature measures are to be used. All five of these curvature measures were computed at each vertex of all
  • the optimal curvature measures for the detection of aneurysms have been found through the optimization steps described above.
  • the five curvature measures determined are not only optimal for the detection of aneurysms, but are also non-collinear. Therefore, each measure provides some useful information that can be used to recognize an aneurysm based on its shape.
  • a 3-D, triangulated computer model of the cerebral vasculature is created by any technique (e.g., marching cubes).
  • a bivariate quadratic polynomial is fit to the collection of vertices, i.e., the platelet, immediately surrounding a central vertex as shown in Figure 2.
  • the subset of 5 curvature measures are calculated for each 3D surface vertex by determining their values on the bivariate quadratic polynomial.
  • Equation 20 is then used to determine y for each vertex using the 5 individual curvature measures calculated for that vertex.
  • a table of color values (a color table) is then created to represent the expected range of y ; and the surface vertices or surface triangles are colored according to the color table.
  • the color table can be interactively adjustable by the user or it can be fixed. Regardless of the technique used, the intention of the color table is to provide a simple means to color the 3-D computer model such that an observer's attention is directed to an area that has curvature properties consistent with an aneurysm.
  • the 3-D computer models can be displayed in any number of different formats including 2D black-and-white images, 2D color images, stereographic images, movie format or in 3-D computer model format.

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Abstract

L'invention concerne un algorithme de courbure tridimensionnelle utilisant la régression linéaire afin de modéliser une matière biologique. Pour obtenir une représentation tridimensionnelle d'une structure, on balaie cette matière. On attribue des valeurs numériques à des régions choisies de la structure balayée. Sur la base des sommets, on effectue un certain nombre de mesures de courbure. On utilise l'analyse par régression linéaire afin d'obtenir un coefficient de régression pour toutes les courbures. On calcule des facteurs d'extension de variance pour les mesures de courbure. On effectue plusieurs régressions pour obtenir un modèle optimal.
PCT/US2000/005596 1999-03-03 2000-03-03 Mesure de formes tridimensionnelles au moyen d'une analyse de courbure statistique WO2000052629A1 (fr)

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EP00916040A EP1236156A1 (fr) 1999-03-03 2000-03-03 Mesure de formes tridimensionnelles au moyen d'une analyse de courbure statistique
AU37207/00A AU3720700A (en) 1999-03-03 2000-03-03 3-d shape measurements using statistical curvature analysis
CA002364176A CA2364176A1 (fr) 1999-03-03 2000-03-03 Mesure de formes tridimensionnelles au moyen d'une analyse de courbure statistique
JP2000602979A JP2003502723A (ja) 1999-03-03 2000-03-03 統計的湾曲解析を用いた三次元形状計測

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