KR950001352B1 - Method and apparatus for rendering of geometric volumes - Google Patents

Abstract

None.

Description

Method and apparatus for rendering geometric volumes

1 is a block diagram of a computer system for rendering volume data on a computer disk.

2 is an exemplary diagram for mapping volume components in volume space to components in geometric space.

3a, 3b and 3c are application examples of the preferred embodiment of the present invention.

4A and 4B illustrate exemplary volumetric mapping and display limited by geometric cubes.

5 is an exemplary diagram of a linear mapping function of three lines.

[Field of Invention]

The method and apparatus of the present invention relate to the field of computer graphics and more particularly to rendering volume data on a computer display.

[Background of invention]

Volume-rendering techniques, three-dimensional data, are mainly used in computer graphics for applications such as X-ray crystallography, computed tomography, positron emission imaging, and display of images generated through computational hydrodynamics. . Volume data is represented by a three-dimensional array of volume components called VOXEL. There are three integer coordinates associated with each box that indicate the location of the box in volume space, and at least one integer value called the concentration of the box contains a certain property of that location, eg temperature or composite, and is not directly related to the rendering properties. Do not.

Geometric data, on the other hand, is represented by x, y, z coordinate positions and is associated with rendering attributes such as color. Geometry data, unlike volume data, can be easily rendered for display because geometry data provides the rendering information needed to display a geometry image and volume data includes the extent to which the volume itself is related.

Therefore, in order to display volume data, the volume data must be converted from volume space into a format compatible with the display system space. There are basically two techniques for rendering volumes on a computer graphics display; Geometric and Volumetric Techniques.

In the geometric approach, volumetric data is equipped with appropriate geometric primitives that yield a model consisting of geometric primitives. In the volumetric approach, the volume is rendered directly using light projection techniques or multilayer reprojection techniques.

Optimal Surface Recons truction for Planer Contours (CACM (1977)) reconstructs a three-dimensional surface from a set of contours drawn on a series of slices through volume. Describes geometric algorithms for rendering faces included in volumetric data. The contour is commonly obtained using a semi-automatic edge following the routine, which is then followed using geometric primitives such as triangles, polyhedrons, or bi-cubic patches. These primitives form a three-dimensional polymorphic model that can be rendered using the prior art. Similar techniques for face reconstruction are described in Mr. Klein's US Pat. No. 4,729,098, but extraction of face contours often requires operator intervention.

Many polymorphs need to be maintained in detail. In addition, since the volume data is now represented only as volume face data, the raw volume data is lost and no partial face or object can be displayed.

In an effort to keep the finer details at the box cell level, the Cuberille method (Gordon, Image Space Shading of 3-Dimensional Objects, computer Vision Graphics Image Processing 29, 377-393 (1985)) was developed. This technique creates a condensed binary volume for rendering effects that represent the boundary of the face. The face of the thin cell is rendered in contrast polymorph.

This technique is not allowed for translucent or partial aspects. The geometrical algorithm of FIG. 1, known as Marching Cubes, computes triangles that compute the intersections of faces into thin cell cubes and provide very detailed face approximations. Lawrence's Marching Cube; See “High Resolution Three-Dimensional Surface Composition Algorithms” (Computer Graphics, Siggraph 1987 Proceedings, page 163-169 (July, 1987)), for more details provided by the segmented cube algorithm, which is bounded by a series of points. See Kline's US patent (No. 4 719 585) and two algorithms for reconstructing faces from tomographic X-rays (Medical physics, June 1988).

However, these methods provide volume extraction of the volume and raw volume data is not maintained. Therefore, no further interaction with raw volumetric data is possible because only volumetric information is maintained. Thus, functions such as partial volume, transparency, and translucency and interaction with volumes such as picking (i.e. reading data from or writing data values from volume) and imposing geometric objects on the volume can be performed. none. In constructing the technique, volume data is preprocessed by classifying each thin cell according to various rendering attributes such as color, opacity and composition, thereby creating multiple volume representations. The different volume representations are then combined to form one volume representation to be rendered to the display device. “Volume Rendering” (Drebin, et al., Volume Rendering, Computer Graphics Siggraph Proceedings, pages 65-74, August 1988); See Lavoy, Display of Surfaces From Volume Data, IEEE Computer Graphics and Applications (May 1988), and US Patent (No. 4737921) by Mr. Gold Waser.

However, this technique does not provide a continuous mapping function.

Therefore, if the output image to be rendered is much larger than the original volume data, large aliasing occurs.

Raycasting is a simple method of direct box-selendering. "Roth, Raycasting For Modeling Solids, Computer Graphics and Image Processing 18, pages 109-144"; “Rendering Algorithm for 3-D Scaler Field Visualization” (Sabella, A Rendering Algorithm For Visualizing 3-D Scaler Fields, Computer Graphics Siggraph 1988 Proceedings, pages 51-55, August 1988); See Upson et al., V-buffer visible Volume Rendering, computer Graphics Siggraph 1988 Proceedings, pages 59-64, August 1988. In the light projection technique, light rays are emitted at each pixel through a volume along a view vector.

The values determined along the ray are combined into only one value and displayed at the display output. Many algorithms have been developed to determine the output rendered by a ray. For example, additional reprojection algorithms produce an image-like X-ray by averaging the intensity of the volume point along the ray.

Another light projection technique assigns color and opacity to a range of contrasts, making the volume appear to be a synthesis of translucent gels. See Schlusselberg, et al., 3-Dimensional Display of Medical Image Volumes, Proceedings of the NCGA, March 1986. Raytracing Volume Densities (Kjiya, et al., Computer Graphics Siggraph 1984 Proceedings, pages 165-173) describes the true ray tracing where additional light is generated at the plane boundaries to calculate the interfacial reflectivity. . In practice, however, ray tracing has not been used computationally extensive and expensive. In multi-plane reprojection techniques, one or more faces move through a volume that displays concentration values mapped to a function of adoption. However, like ray projection, this technique is cumbersome, extensive, and requires additional rendering for other perspectives.

[Purpose of invention]

It is therefore an object of the present invention to provide a method and apparatus for direct volume rendering using volume primitives.

It is an object of the present invention to provide an apparatus and method for rendering a volume containing a volume mapping geometric primitive.

It is another object of the present invention to provide an apparatus and method for rendering a volume in which a volume interacts with a geometric object.

In the apparatus and method of the present invention, a means of directly rendering a volume from volume data is provided such that the resolution of the volume is not lost and the volume data completely interacts with the geometric data.

The volume or portion of the volume to be rendered is aligned with the geometric primitive and a mapping function is generated that associates the geometric primitive with the volume or volume portion. The mapping function associates each box in volume space with a point or component of a primitive. Thus, the volume is displayed as a function of the geometric primitive to which the volume or volume portion is mapped.

In a preferred embodiment a mapping is generated in which the volume or portion of volume to be rendered is associated with one or more geometric primitives. Any geometry operation performed on the volume is then accomplished by using the mapping function to perform the operation on the geometric primitives and convert the volume into the geometric space so that volume data is displayed. Preferably, the polymorphism bounded by the volume is a rectilinear cube, such as a normalized cube with a vertex range of -1 to +1.

By defining a volume as a function of a geometric primitive such as polymorphism, geometric operations such as clipping or rotation are easily performed on the primitive and the data is used to check through mapping functions that determine the corresponding volume data to be displayed.

Moreover, geometric data easily interacts with the original volume data because raw volume data is directly rendered in the geometric space.

[Comment and notation]

The detailed description set forth below is chiefly described in terms of symbolic representations and algorithms of operations on data bits in computer memory.

The description and representation of this algorithm is the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. Algorithms are generally accepted here with ordered steps that result in desirable results. These steps are those requiring physical manipulation of physical quantities. Usually, though not necessary, these physical quantities take the form of electrical or magnetic signals that can be stored, moved, combined and compared or otherwise manipulated. It proves convenient at times to be referred to as such signals, such as bits, values, elements, signals, letters, numbers, or the like, mainly for common use.

However, it should be borne in mind that terms similar to all of these are simply required levels associated with and appropriate to these physical quantities. Moreover, the manipulations performed are often referred to in terms, such as additions or comparisons, which are associated with the mental activity performed by the operator. Some operations described herein do not require the ability of a person to operate in most cases, but form part of the present invention; Operation is mechanical operation.

Useful machines for performing the operations of the present invention include general purpose digital computers and other similar equipment. In all cases there must be a mental discernment between the method of operation and the calculation method itself in computer operation. The present invention relates to the steps of a method of operating a computer in processing electrical or other (eg mechanical, chemical) physical signals that produce other desirable physical signals. The invention also relates to a device for performing this operation. The apparatus includes a general purpose computer that is specifically configured for the necessary purpose and which is selectively moved or reconfigured by a computer program stored in the computer.

The algorithm here is not inherently related to any particular computer or other device. In particular, machines for various purposes are used as programs written according to the teachings here, but it is further demonstrated that it is necessary to construct more special devices in order to carry out the steps of the required method. The required structure for a number of these substrates will appear from the description below.

[General system configuration]

Figure 1 shows a computer based typical system for the rendering of volumetric data on a computer display according to the present invention. A computer 1 is shown consisting of three main components. The first of these is the input / output (I / O) circuit 2, which is used to exchange information in an appropriately configured form with or from other parts of the computer 1. Also shown is a central processing unit (CPU) 3 and a memory 4 as part of the computer 1. These two elements are typically found on most general purpose computers and almost all special purpose computers. In fact, the various components included in the computer 1 are intended to represent a broad category of data processors. A particular example of a suitable data processor to fulfill the role of the computer 1 is a machine manufactured by Sun Micro, Inc. of Mountain View, California. Of course, other computers having the same function can be employed in the same way to perform the following functions.

As shown in FIG. 1, the input device 5 is shown in the exemplary embodiment of the keyboard. However, it is understood that the input device may in fact be a card reader, a magnetic or paper tape reader, or other well known input device (including computerized data collection equipment such as a computer x-ray tomography device or other computer). shall. The mass memory device 6 is connected to the I / O circuit 2 and provides additional storage for the computer 1. Mass memory may include other programs and the like and may take the form of magnetic or papertape readers or other well-known devices. The data held in the mass memory 6 may be included in the standard format in the computer 1 as part of the memory 4 as appropriate.

In addition, an example display monitor 7 is used to display messages or other transmissions to the user. Such display monitors can take the form of several well-known and many CRT displays. Preferably, the display monitor 7 can display the rendered volume image in accordance with the process of the present invention with a high resolution computer raster display device. Cursor control 8 provides a more convenient means for selecting a command mode, editing input data such as typography, and entering information into the system.

[Overview of Process

In the apparatus and method of the present invention, a portion or volume of a volume object is aligned with one or more geometric primitives and a mapping function is generated between the geometric primitives in the geometric space and the volume objects in the volume space. Through the use of mapping functions, deformations, observations, and other geometries can simply be performed on the volume by performing geometry on the primitive to which the volume is mapped and transforming the corresponding volume data.

The volume data, which is a thin cell representing a volume object containing a volume attribute at a specific position in the volume space, is stored in a memory according to each Valsell body in a three-dimensional array of thin cells representing the volume object. In order to display the volume object, the volume object is described by at least one geometric primitive, for example one cube. The geometric primitives or primitives used must be aligned and sized in such a way that the volume or portion of the volume to be rendered can be mapped to the primitive.

The primitives used are preferably normalized to simplify the calculation of the continuation so performed, such as magnification, transformation, or rotation of the image.

For example, for FIG. 2, the volume representation portion of volume vector 80 may be mapped to a geometric vector and displayed in screen space. Once this vector is mapped, the volume vector is easily rotated and transformed in the geometric space simply by performing a rotation / transformation operation on the mapped geometric vector and searching the corresponding volume element using the mapping function. This is especially useful for techniques for extracting surfaces using volume vector primitives.

For example, for FIGS. 3A, 3B, and 3C, the volumetric object may represent human organs and / or tissues where the volume includes a capillary 100 located through the volume. It is sometimes desirable to observe and extract the capillary from various perspective views measuring the exact path and shape of the capillary through the organ / tissue representing the volume.

This is obtained by mapping the volume vector 110 from a point 120 to a geometric vector along a curved surface, for example, a capillary. When the mapping is performed, the geometric vector is rotated and the corresponding volume data is mapped to render the display showing the capillary from another perspective view, for example the perspective view shown in FIG. 3C.

Preferably, the volume is mapped to one or more geometric primitives that form a three-dimensional object, such as a linear cube. The cubes are arranged to form a volume boundary.

As shown in FIG. 4A, the volume object 150 is surrounded by the cube 160. A mapping function is generated between the box containing the volume and the {x, y, z} coordinates of the cube. When the mapping function is generated, several geometries are performed on the volume object through operations performed on the cube. For example, the portion 175 of the volume 170 is removed by performing a clipping operation on the cube 160 and the resulting image of the exposed volume can be easily displayed as shown in FIG. 4B.

The mapping function is the grid point of the geometric primitive in the volume space (also referred to as the (u, v, w) space) of each volume cell in the volume object and in the {x, y, z} space (geometric space). Provide a mathematical correspondence between vertices. It can be said that the volume boundary is mapped to the geometric primitive by fitting the volume object to the geometric primitive.

이다.Therefore, the location of the box cell stored in the memory is linked to a mapping function by a specific x, y, z coordinate position of the geometric primitive to which the volume data is mapped. The mapping function used may be any sequential function. The function may be a zero sequential function where each boxel is mapped to the nearest vertex in geometry. For example, if the volume is mapped to a normalized linear cube with dimensions (10 × 20 × 30) and a vertex located between -1 and 1, the boxels at the array (1,1,1) position are coordinates. Mapping to the vertex of the cube at position {0,0,0} and the boxel at the position of array 10,20,30 are mapped to the vertex at coordinate position {1,1,1}. Box cells at positions (1,1,1) and (10,20,30) in the volume space are mapped to vertices at coordinate positions {1,1,1}. The thin cells between positions (1,1,1) and (10,20,30) in the volume space are mapped to the nearest vertices of the calculated values using size proportional calculations between the volume space and the geometric space. In this example, the u: x proportionality is equal to 10: 2 because the number of thin cells in u is equal to 10 and the number of vertices of x is 2 (-1 to 1). So the mapping function

to be.

This function is also the first sequential function referred to as a 3-line function in which the weighted average calculation is performed at the eight nearest vertices in geometry to determine the position of the volume vertices in geometry. The first order function is used to create a softening effect, which reduces the amount of aliasing that occurs when volume data is mapped. With respect to FIG. 5, the thin cell 200 is mapped to the bounding body by determining the eight nearest vertices in the geometric space shown by vertices a, b, c, d, e, f, g and h. The coordinate position of the boxel is then determined according to the equation:

Where X is the geometric position of the thin cell 200 at position (u, v, w) in the volume space, and a, b, c, d, e, f, g and h are 8 of the cube corner surrounding the volume coordinate to be determined. {X, y, z} coordinate positions and FRACT is a function that determines the fractional fraction of the coordinates.

The distance of the boxel from eight vertices is weighted in the equation, i.e. FRACT (u), 1-FRACT (u), FRACT (v), 1-FRACT (v), FRACT (w) and 1-FRACT (w) Factorized by.

Once the mapping function is determined, modifications or manipulations of any geometries performed on the geometric object are performed on the volume object via geometric primitives to which the volume is mapped. In order to obtain a perspective or aspect ratio of the dimension, volume data from another perspective, the geometric primitive is multiplied by a transformation matrix that modifies the volume data mapped to the geometric primitive.

In addition, volume primitives are defined. Volume primitives are primitives, such as points, vectors, polymorphs, or rays, that correspond in geometry space and have definitions in volume space, depending on the mapping performed. Thus, for example, a point has positions in {x, y, z} space and (u, v, w) space, and a vector has {x, y, z} and (u, v, w) It has two points in space. Similarly, polymorphs have multiple points in the {x, y, z} and (u, v, w) spaces that define the boundaries of the polymorphs. The ray has only one point in the {x, y, z} space that maps to a number of points in the (u, v, w) space. Geometric primitives are used to add, subtract or otherwise interact with volumetric data. For example, primitives are used to cut a portion of a volume to expose the inner surface. This is illustrated in Figures 4a and 4b that polymorphs are used to cut a portion of the volume to expose the inner surface of the volume. Primitives are also used to generate geometric images to interact with the volume within the volume space in which the results are displayed. This is particularly useful for medical radiation therapy where medical practitioners can predict the interaction of laser beams or partial beams created using primitive human tissues created in volume space. Therefore, the more realistic images are provided because the geometric images interact with the original volume data rather than the volume extraction data of the volume.

While the invention has been described in connection with preferred embodiments, numerous alternatives, modifications, variations and handling will be apparent to those skilled in the art in the foregoing description.

Claims (48)

A method for rendering a volume object on a computer graphics display, the method comprising: storing volume data representing at least a portion of the volume object in a memory, the geometric primitives corresponding to the volume data comprising a plurality of geometries comprising a plurality of boundary coordinates; And a plurality of boundary coordinates define corresponding geometric primitives and the corresponding geometric primitives are represented in a {x, y, z} coordinate space with a reference frame, the geometric coordinates of the reference frame. Representing a plurality of corresponding displaying pixels on a computer graphics display, generating at least one corresponding geometric primitive, aligning a volume object with the corresponding geometric primitive, and using the bounding coordinates of the volume data. To map the geometric coordinates of the corresponding geometric primitive Generating a mapping function, resulting in a modified geometric coordinate by performing a geometric operation on the corresponding geometric primitive to which the volume object is mapped, and volume data according to the geometric coordinates of the modified geometric primitive using the determined mapping function. And displaying the displayed volume object in accordance with the geometric operation performed on the corresponding geometric primitive. The method of claim 1, wherein the corresponding geometric primitive is a linear cube and the mapping function is a zero (zero) sequential function, and the boundary coordinates defining the linear cube are geometric coordinates of the vertices of the linear cube. The function maps the volume data into a number of ideal geometric coordinates closest to the geometric coordinates, the ideal geometric coordinates between the plurality of volume components and the vertices of the linear cube in each dimension of the {x, y, z} reference frame. A method for rendering a volumetric object, characterized by a proportional function with distance. The method of claim 1, wherein the corresponding vaporization coordinates are linear cubes, the mapping function is a first sequential function, the boundary coordinates defining the linear cubes are geometric coordinates of the linear cubes, and the first sequential function is 3. A method for rendering a volume object characterized by interpolating volume data in three lines to determine the geometric coordinates to which the volume data is mapped. 4. The method of claim 3, wherein the method of interpolating the geometric data into three lines comprises determining the geometric coordinates of the eight nearest vertices of the linear cube surrounding the volume data to be mapped, the geometric coordinates and the volume data of the eight nearest vertices. Calculating a geometric coordinate to which volumetric data is mapped as a function of the distance of these vertices from

Where x is the filled geometric coordinates of the volumetric data (u, v, w), and a, b, c, d, e, f, g and h are the eight linear cubes that surround the volume object to be determined. {X, y, z} geometric coordinates of the nearest vertex and FRACT is a function for determining the fractional part of volume data (u, v, w). The method of claim 1, wherein the volume object is one of a plurality of volume primitives, the volume primitives consisting of points, vectors, polymorphs, and light rays. 10. The method of claim 1, wherein the geometry performed is a multiplication of a corresponding geometric primitive and a transformation matrix that rotates the displayed geometric primitive and thus rotates the displayed volume primitive. The rendering of the volume object of claim 1, wherein the geometry performed is a multiplication of the corresponding geometric primitive and the transformation matrix changing the dimension of the corresponding geometric primitive, thereby changing the size of the displayed volume object. Way. The method of claim 1, wherein the geometry performed is a multiplication of a corresponding geometric primitive and a transformation matrix that transforms the corresponding geometric primitive, thereby modifying the displayed volume object. The method of claim 1, wherein the geometric action performed is an interaction between a corresponding geometric primitive mapped to a volume object and the acting geometric primitive such that a volume object displaying a volume primitive matched to the interacting geometric primitive is displayed. A method for rendering a volume object, characterized in that it acts. 10. The method of claim 9, wherein the interaction of the interacting geometric primitives overlaps the interacting geometric primitives on the corresponding geometric primitive to which the volume is mapped, thereby displaying the volume primitives mapped to the interacting geometric primitives on the displayed volumetric object. A method for rendering a volume object, characterized in that overlapping. 10. The method of claim 9, wherein the interaction is a geometric primitive's interaction is the movement of a portion of the corresponding geometric primitive to which the volume is mapped, wherein the portion is determined according to the size, shape, and position of the interacting geometric primitive and thus displayed. A corresponding portion of the volumetric object is moved, characterized in that for moving the volumetric object. 10. The method of claim 9 wherein the interaction of interacting geometric primitives is the extraction of a portion of the corresponding geometric primitive to which the volume is mapped, wherein the portion is determined according to the size, shape and location of the interacting geometric primitives and thus extracted. A corresponding portion of the volumetric object is displayed. CLAIMS 1. A method for rendering a volume object on a computer graphics display in which the volume object is displayed in screen space, the method comprising: storing volume data in memory at least a portion of the volume object represented in the (u, v, w) volume space; A geometric primitive corresponding to a volume object forming a boundary with respect to a volume object in three dimensions is represented by a plurality of geometric coordinates including a plurality of boundary coordinates, and the plurality of boundary coordinates define corresponding geometric primitives. Primitives are represented in {x, y, z} geometry, where the geometry has a frame of reference, the geometry is defined as a function of the frame of reference, and the geometry is defined within a screen space on a computer graphics display. Represent corresponding display pixels, and generate at least one of the corresponding geometric primitives. Generating a mapping function that maps the volumetric data to one of the geometric coordinates of the corresponding geometric primitive forming a boundary for the volume object using boundary coordinates; and a corresponding geometric primitive to which the volume is mapped. Resulting in a modified geometric primitive by performing on the image and displaying the volume data according to the geometric coordinates of the modified geometric primitive using the determined mapping function to modify the displayed volume object in accordance with the geometric operation performed on the corresponding geometric primitive. Method for rendering a volume object, characterized in that consisting of. 15. The method of claim 13, wherein the geometric primitive to which the volume object is mapped is a linear cube, and the boundary coordinates defining the linear cube are geometric coordinates of the vertices of the linear cube. 15. The method of claim 14, wherein the mapping function is a zero sequential function, and the zero sequential function maps volume data into a geometric coordinate closest to a plurality of ideal geometric coordinates, so that the ideal geometric coordinate is a plurality of volume components and {x, y, z} method for rendering a volume object, characterized in that it is determined by a proportional function of the distance between the vertices of the linear cube in each dimension of the reference frame. 15. The method of claim 14, wherein the mapping function is a first sequential function, and the first sequential function interpolates volume data in three lines to determine a geometric coordinate to which volume data is mapped. Way. 17. The method of claim 16, further comprising the steps of: determining geometric coordinates of the eight nearest vertices of a linear cube surrounding the volume object being mapped, geometric coordinates of the eight nearest vertices, and Computing a geometric coordinate to which the volume data is mapped as a function of the distance of these vertices from the volume data, wherein the function comprises;

Where X is the mapped geometric coordinate of the volume data (u, v, w), and a, b, c, d, e, f, g and h are eight pieces of linear cubes surrounding the volume object to be determined. {X, y, z} geometric coordinates of the nearest vertex and FRACT is a function for determining the fractional part of volume data (u, v, w). 14. The method of claim 13, wherein the geometric coordinates performed are a multiplication of the corresponding geometric coordinates and a change matrix that rotates the corresponding geometric primitives, thus rotating the displayed volume object. 14. The rendering of a volume object according to claim 13, wherein the geometric operation performed is a multiplication of a corresponding geometric primitive and a transformation matrix that changes the dimension of the corresponding geometric primitive, thereby changing the size of the displayed volume object. Way. 14. The method of claim 13, wherein the geometry performed is a multiplication of a corresponding geometric primitive and a transform matrix transforming the corresponding geometric primitive, thereby modifying the displayed volume object. The method of claim 13, wherein the geometry performed is an interaction of interacting geometric primitives and corresponding geometric primitives to which volumetric objects are mapped, such that volume primitives mapped to the interacting geometric primitives interact with the displayed volumetric object. A method for rendering a volumetric object, characterized in that it is operated. The method of claim 21, wherein the interaction of the interacting geometric primitives is to superimpose the interacting geometric primitives on the corresponding geometric primitive to which the volume is mapped and thus the volume mapped to the interacting geometric primitives on the displayed volume object. A method for rendering a volume object characterized by overlapping primitives. The method of claim 21, wherein the interaction of the interacting geometric primitives is the movement of a portion of the corresponding geometric primitive to which the volume is mapped, wherein the portion is determined in accordance with the size, shape, and position of the interacting geometric primitives and thus displays. A method for rendering a volumetric object, characterized in that the corresponding part of the volumetric object is moved. The method of claim 21, wherein the interaction of the interacting geometric primitives is the extraction of the portion of the corresponding geometric primitive to which the volume is mapped, the portion determined according to the size, shape, and location of the interacting geometric primitives. A corresponding portion of the volumetric object is displayed. An apparatus for rendering a volume object on a computer graphics display, the apparatus comprising: means for storing volume data in memory, the volume data representing at least a portion of the volume object being represented in the (u, v, w) volume space, a geometry corresponding to the volume object A primitive is represented by a plurality of geometric coordinates representing a plurality of boundary coordinates, the plurality of boundary coordinates defining corresponding geometric primitives, and the corresponding geometric primitives within a {x, y, z} geometric coordinate space having a reference frame. Wherein the geometric coordinates are defined as a function of a frame of reference, the geometric coordinates representing a plurality of pixels on a computer graphics display, means for generating at least one of the geometric coordinates, aligning a volume object with a corresponding geometric primitive. Means, the geometric coordinates of the corresponding geometric primitive using boundary coordinates for the volumetric data; Means for generating a mapping function that maps, means for performing a volumetric motion on a mapped geometric primitive to which the volume object is mapped, and thus a modified geometric primitive result, and means for determining the volume object according to the modified geometric primitive using the determined mapping function. Display and wherein the displayed volume data consists of means that are modified in accordance with a geometric operation performed on the corresponding geometric primitive. 26. The method of claim 25, wherein the corresponding geometric primitive is a linear cube and the means for generating a mapping function generates a zero sequential function and the boundary coordinates define the linear cube that is a geometric coordinate of a vertex of the linear cube. The means for generating a means consists of a zero sequential function mapping means for mapping the optimal data to the geometric coordinates closest to the plurality of ideal geometric coordinates, the ideal geometric coordinates are based on a number of volume components and {x, y, z} An apparatus for rendering a volumetric object, characterized in that it is determined by a proportional function with the distance between the vertices of the straight cube in each dimension of the frame. 26. The method of claim 25, wherein the corresponding geometric primitive is a linear cube and the means for generating a mapping function generates a first sequential function and the boundary coordinates define the linear cube that is a geometric coordinate of a vertex of the linear cube. And said means for generating a function comprises first-order function mapping means for interpolating the volume data in three lines to determine the geometric coordinates. 28. The method of claim 27, wherein the first sequential function means for interpolating the volume object in three lines is a means for determining the geometric coordinates of the eight nearest vertices of the linear cube surrounding the volume object being mapped, the geometry of the eight nearest vertices. Means of calculating the geometric coordinates to which each volume data is mapped as a function of the distance of the eight nearest vertices from the coordinates and volume data, said function being given by

Where x is the mapped geometric coordinate of the volume data (u, v, w) and a, b, c, d, f, g and h are the {x of the eight nearest vertices of the linear cube surrounding the volume object to be determined. , y, z} An apparatus for rendering a volume object, wherein the geometric coordinate is a function of determining the fractional part of the volume data (u, v, w). 27. The apparatus of claim 25, wherein the volume object is one of a number of volume primitives consisting of points, vectors, polymorphs, and light rays. 27. The volumetric object of claim 25, wherein said means for performing a geometrical operation comprises means for multiplying a corresponding geometric primitive with a deformation matrix for rotating the corresponding geometric primitive and thus rotating the displayed volumetric object. Device for rendering. 27. The method of claim 25, wherein said means for performing a geometry consists of means for multiplying a corresponding geometric primitive with a deformation matrix that changes the dimension of the corresponding geometric primitive, thereby changing the size of the displayed volume object. An apparatus for rendering a volume object. 27. The volumetric object of claim 25, wherein the means for performing the geometric operation comprises means for multiplying a corresponding geometric primitive with a deformation matrix for modifying the corresponding geometric primitive, thereby transforming the displayed volumetric object. Device for rendering. 27. The method of claim 25, wherein said means for performing a geometrical operation comprises means for interacting with a corresponding geometric primitive mapped with a corresponding geometric primitive and with a volume primitive mapped to the interacting geometric primitive. Apparatus for rendering a volume object characterized in that it interacts with the volume object. 34. The method of claim 33, wherein the means for interacting with the interacting geometric primitives superimposes the interacting geometric primitives on the corresponding geometric primitive to which the volume object is mapped and thereby interacts with the interacting geometric primitives on the displayed volume object. Apparatus for rendering a volume object characterized by overlapping mapped volume primitives. 34. The method of claim 33, wherein said means for interacting with interacting geometric primitives comprises means for removing a portion of a corresponding geometric primitive to which a volume is mapped, said portion being the size, shape, and location of interacting geometric primitives. And a corresponding portion of the displayed volume object is removed in accordance with the arrangement. 34. The method of claim 33, wherein the means for interacting with the interacting geometric primitives consists of means for extracting a portion of the corresponding geometric primitive to which the volume is mapped, the portion being the size, shape and And a corresponding portion of the extracted volume object is displayed according to the position. An apparatus for rendering a volume object on a computer graphics display, comprising: means for storing volume data in memory indicative of at least a portion of the volume object represented in (u, v, w) volume space, the volume object in three dimensions Corresponding geometric coordinates forming a boundary are represented by geometric coordinates including a plurality of boundary coordinates, the plurality of boundary coordinates defining corresponding geometric primitives, and the corresponding geometric primitives being represented in {x, y, z} geometric space. The geometric space has a reference frame, the corresponding geometric primitive is defined as a function of the reference frame, the geometric coordinates representing a plurality of corresponding display pixels in a computer graphics display screen space, and at least one corresponding geometric primitive. Means for generating a boundary of the volume object using boundary coordinates of the volume data Means for generating a mapping function for mapping the corresponding geometric primitives to the geometric coordinates, means for performing a geometric operation on the corresponding geometric primitive to which the optimal object is mapped, which becomes a modified geometric primitive, and a modified mapping function. And a means for displaying the volume object in accordance with the geometric coordinates of the geometric primitive, thereby allowing the displayed volume object to be modified in accordance with the geometric motion performed on the corresponding geometric primitive. 38. The apparatus of claim 37, wherein the geometric primitive to which the volume object is mapped is a straight cube, and the boundary coordinates defining the straight cube are geometric coordinates of the vertices of the straight cube. 39. The method of claim 38, wherein the means for generating a mapping function comprises a zero sequential function mapping means for generating a zero sequential function and mapping volume data to an ideal geometric coordinate closest to the ideal geometric coordinate. An apparatus for rendering a volume object, characterized in that it is determined by a proportional function of the distance between the plurality of volume components and the vertices of the linear cube in each dimension of the {x, y, z} reference frame. 39. The apparatus of claim 38, wherein the means for generating a mapping function comprises a first sequential function means for generating a first sequential function and interpolating in three lines the volume data for determining the geometric coordinates to which the volume data is mapped. Apparatus for rendering a volume object, characterized in that. 41. The apparatus of claim 40, wherein the first sequential function means for interpolating the volume data in three lines comprises: means for determining the geometric coordinates of the eight nearest vertices of the linear cube surrounding the volume object to be mapped; Computing the geometric coordinates to which each volume data is mapped as a function of the distance of the eight nearest vertices from the geometric coordinates and the volume data, wherein the function is defined as follows.

Where x is the mapped geometric coordinate of the volume data (u, v, w) and a, b, c, d, e, f, g and h are the eight nearest neighbor vertices of the linear cube surrounding the volume object to be determined. {x, y, z} An apparatus for rendering a volume object, wherein the geometric coordinate is a function of determining the fractional part of the volume data (u, v, w). 38. The rendering of claim 37 wherein said means for performing a geometry consists of means for multiplying a corresponding geometric primitive with a transformation matrix for rotating the corresponding geometric primitive and thus rotating the displayed volume object. Device for 38. The method of claim 37, wherein said means for performing a geometrical operation comprises means for multiplying a corresponding geometric primitive with a deformation matrix for changing the dimensions of the corresponding geometric primitive, thereby changing the size of the displayed volume object. Device for rendering a volume object. 38. The volume object of claim 37, wherein the means for performing the geometric operation comprises means for multiplying a corresponding geometric primitive with a transformation matrix that transforms the corresponding geometric primitive, thereby transforming the displayed volume object. Device for rendering. 38. The method of claim 37, wherein the means for performing the geometry consists of interacting geometric primitives and means of interacting with corresponding geometric primitives mapped to the volume object, thereby interacting with the displayed volume object. Apparatus for rendering a volume object, characterized in that the interaction. 46. The method of claim 45, wherein the means for interacting with the interacting geometric object consists of means for superimposing the interacting geometric primitives on the corresponding geometric primitive to which the volume is mapped, thereby displaying the mapped volume primitive. An apparatus for rendering a volume object characterized by superposition of interacting geometric primitives on a phase. 46. The method of claim 45, wherein said means for interacting with interacting geometric primitives comprises means for removing a portion of a corresponding geometric primitive to which a volume is mapped, wherein said removed portion is the size, shape of the interacting geometric primitive. And a corresponding portion of the displayed volume object is determined according to the position and thus removed. 46. The method of claim 45, wherein said means for interacting with interacting geometric primitives consists of means for extracting a portion of a corresponding geometric primitive to which volumetric data is mapped, said extracting portion being the size, shape of the interacting geometric primitive. And a corresponding portion of the extracted volumetric object is displayed according to the position, and the volumetric object is rendered.

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