A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions

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London Math. From Euler to Eisenstein. A second course in elementary number theory. Vinogradov and Ju. Linnik , Hypoelliptic curves and the least prime quadratic residue , Dokl. MR [14] I. They do not necessarily correspond to the angles of the slides from which the three-dimensional model was originally constructed. Thus, the medical diagnostition may review images taken along any desired plane within the three-dimensional model. This provides for a great deal of flexibility in the use of the three-dimensional model as a diagnostic tool. As those skilled in the art will appreciate, such three-dimensional images provide a valuable tool to the medical diagnostition in a manner which is non-invasive, and which is therefore considered to be of very low risk to the patient.

However, although such three-dimensional imaging techniques have proven extremely useful for their intended purposes, they still possess inherent deficiencies which detract from their overall effectiveness. The undesirable presence of such superfluous imagery only complicates the image, making it much more difficult to view and interpret the desired imagery. For example, viewing delicate portions of the vascular system is typically difficult since veins, arteries, and capillaries are intermixed with surrounding tissue. This makes it very difficult to distinguish the desired portions of the vascular system from surrounding tissue.

Often, only slight changes in the intensity of the image distinguish a desired anatomical structure from surrounding tissue.

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Buy A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions (Cambridge Tracts in Mathematics) on compcommanalporth.gq ✓ FREE. A higher-dimensional sieve method with procedures for computing sieve functions by Harold G. Diamond; H. Halberstam; William F. Galway. Article in.

Thus, it is desirable to provide a method for isolating anatomical structures of interest such that surrounding tissue is not displayed along therewith. In this manner, the medical diagnostition may view only the unobstructed anatomical structures of interest. This vastly reduces the complexity of the image and thus minimizes confusion as to precisely what portions of the image relate to the anatomical structure of interest.

Summary of the Invention. In the analysis and review of three-dimensional medical imaging, it is of critical importance to be able to measure and analyze image features having various fractal dimensionalities from zero dimensions to three dimensions.

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For example, veins and arteries are characterized as one-dimensional curvilinear forms, while capillaries exhibit one plus fractal dimensions, typically exhibiting fractional fractal dimensionality. Tumors have three-dimensional fractal forms and exhibit smaller fractal dimensions if metastases are considered.

The present invention specifically addresses and alleviates the above-mentioned deficiencies with the prior art. More particularly, the present invention comprises a method for isolating anatomical structures contained within a three-dimensional data set, e.

The method comprises the steps of forming a morphological skeleton of the three-dimensional data set, selecting a seed data point within the morphological skeleton so as to identify a desired anatomical structure to be displayed or analyzed, and utilizing fuzzy connectivity to define additional data points of the desired anatomical structure so as to reconstruct substantially only the desired anatomical structure.

Reconstruction of substantially only the desired anatomical structure facilitates the review and analysis of the anatomical structure. For example, if it is desirable to obtain a three-dimensional data set containing only data points which are representative of the brain, then the patient's head may be imaged via MRI, CAT, PET scanning techniques or the like to provide a three-dimensional model of substantially the entire head.

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The three-dimensional data set which defines this model is then processed so as to form a morphological skeleton thereof. An operator then selects a seed data point within the morphological skeleton corresponding to the patient's brain.

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To further benefit readers, the Appendix describes methods for computing sieve functions. Harold G. Responsibility Alina Carmen Cojocaru, M. His mathematical interests are number theory and classical analysis. List this Seller's Books. Skip to main content. This book provides a motivated introduction to sieve theory.

This is typically accomplished by viewing the morphological skeleton on a display such as a CRT. The morphological skeleton maintains all of the data available in the original three-dimensional data set. However, in the morphological skeleton, anatomical structures are separated from one another, based upon the fractal dimensionality thereof.

A higher-dimensional sieve method with procedures for computing sieve functions

Thus, anatomical structures having a fractal dimensionality of less than one dimension are separated from those having a fractal dimensionality of less than two dimensions and the anatomical structures are separated from those having a fractal dimensionality of less than three dimensions. After selecting a seed data point within the brain, fuzzy connectivity is utilized to define the additional data points which are required to provide a substantially complete image of the brain. Reconstruction of the brain is simply the reverse of the process utilized to form the morphological skeleton.

With the use of fuzzy connectivity to define the set of points defining the brain, it appears that all of the features thereof are substantially utilized in the reconstruction process. Reconstruction of the brain without the use of fuzzy connectivity would result in the loss of substantial surface details thereof. For example, the surface texture and even, to a lesser degree, the convolutions of the brain, would tend to be degraded or smoothed. The morphological skeleton is formed by recursive opening and erosion of the three-dimensional data set so as to form a plurality of residuals which define the morphological skeleton.

Reconstructing a desired anatomic structure from the morphological skeleton comprises performing the opposite procedure from that utilized to form the morphological skeleton. Thus, reconstruction comprises recursive dilation and closing of the morphological skeleton. As those skilled in the art are aware, each step of opening comprises an erosion followed by a dilation and each step of the closing comprises a dilation followed by an erosion.

The use of fuzzy connectivity during the reconstruction process assures that substantially all of the data points associated with the desired anatomical structure are utilized in the reconstruction process. According to the preferred embodiment of the present invention, a generally spherical structuring element is utilized in both the formation of the morphological skeleton and the reconstruction process. However, those skilled in the art will appreciate that various other shapes of structuring elements are likewise suitable.

Indeed, it has been found that various different shapes of structuring elements are particularly suited for use with various different dimensionalities or shapes of anatomical structures. Generally, the seed data point is selected by positioning a cursor at a desired point on an image being displayed upon a monitor. Thus, the operator may simply visually identify and manually select a seed within the organ or anatomical structure of interest.

However, as those skilled in the art will appreciate, various different computer algorithms may be utilized in the selection of such a seed.

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For example, the operator may simply initiate an algorithm which selects the largest organ within a given volume. Thus, if the operator desires to select the brain for reconstruction, the operator could merely select the largest organ within the head. The use of fuzzy connectivity to define additional data points of the desired anatomical structure comprises defining connectivity based upon the size and shape of a structuring element utilizing a fuzzy generalization of mathematically defined distances between sets of data points as a criterion.

This is accomplished based upon a modified Hausdorff metric. Thus, separation of such anatomical features from one another according to the present invention is accomplished via dimensional sieving. Dimensional sieving results in the formation of a morphological skeleton utilizing the recursive opening and erosion processing according to well known principles.

According to the present invention, a cascade of data dimensional sieving filters are used directly with a three-dimensional image from an MRI device or the like to isolate structures such as arteries and veins from surrounding tissue for unobstructed visualization. This cascade of data dimensional sieving filters comprises the use of a generally spherical structuring element, followed by the use of a two-dimensional surface structuring element, followed by the use of a curvilinear structuring element, followed by the use of a point structuring element.

Thus, to provide for the identification of desired dimensional features within the multi-dimensional data set provided by a tomographic imaging device, a data dimensional sieving algorithm separates the data based upon the dimensional characteristics of the anatomical structures contained therein. The algorithm utilizes filters which resemble geometric constructions such as lines, disks, and spheres, to sieve multi-dimensional features of curves, surfaces, and regions, as well as features of fractal dimensions in between.

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A hierarchy of dimensional filters is thus utilized to first remove features of less than one fractal dimension, then to remove features of less than two fractal dimensions, and finally to remove features of less than three fractal dimensions from the original three-dimensional data set as the morphological skeleton is being formed. Thus, the cascade of filters is used directly with a tomographic image to isolate anatomical structures from surrounding tissues to facilitate analysis and review thereof.

By utilizing the residuals of morphological erosion and opening, the morphological skeleton is formed. This process is ideal for processing data with fractal dimensional components. For example, the recursive formation of the morphological skeleton utilizing alternating opening and erosion transforms a 3. Once the morphological skeleton has been formed via recursive development utilizing alternating opening and erosion processes, then fuzzy connectivity is utilized in the reconstruction of those anatomical structures of interest. Reconstruction of anatomical structures without utilizing fuzzy connectivity results in the loss of significant features such as surface textures and roughness.

These features must be reconstructed from the residuals defining the morphological skeleton utilizing fuzzy connectivity. The reconstruction of such anatomical features requires the satisfaction of a fuzzy connectivity criteria such that only those tissue features connected to the dimensional features isolated by the sieving process are utilized. The final result of both the sieving and fuzzy connectivity processes is a classification and clear visualization of the anatomical structures of interest, e.

Additionally, quantification of the volume of organs and tumors as well as other measurements of interest, such as the diameter of arteries and veins, are easily facilitated as a direct result of the use of dimensional sieving and fuzzy connectivity. Connectivity is a mathematical concept which states that a set of points is connected if and only if every pair of points in the set can be connected by a line which is contained within the set. The algorithm described in this invention generalizes this concept of connectivity to the discrete topological grids utilized by a computer to store the digital image data by utilizing fuzzy set operators.

A fuzzy set is itself a generalization of a discrete set by defining a function over a set representing degrees of membership such that membership varies from zero which indicates no membership to one which indicates complete membership. To define connectivity, this algorithm utilizes a fuzzy generalization of mathematically defined distances between sets as a connectivity criterion.

Sieving Method

This criterion establishes that if two points or two sets of points are within a specified distance of one another, then they have membership to the same set of points. The prior art attempted to isolate anatomical features from one another based solely upon the intensity of pixels within the three-dimensional data set. The present invention facilitates the distinguishing or isolation of anatomical features based upon such criteria such as size.

Thus, more flexibility in designating those features to isolate is provided and improved accuracy of such isolation is attained.

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These, as well as other advantages of the present invention will be more apparent from the following description and drawings. It is understood that changes in the specific structure shown and described may be made within the scope of the claims without departing from the spirit of the invention.

Figure 1 is an illustration of the recursive alternating opening and erosion processes for two dimensions utilized to define the residuals of which the morphological skeleton is constructed;.