WO1995030209A1

WO1995030209A1 - Graphics system using parametric grid paradigm

Info

Publication number: WO1995030209A1
Application number: PCT/US1995/004944
Authority: WO
Inventors: Grigory Kushnir
Original assignee: Geographix, Inc.
Priority date: 1994-04-29
Filing date: 1995-04-21
Publication date: 1995-11-09
Also published as: AU2393595A

Abstract

The graphics system uses a parametric grid paradigm to accurately model complex free-form surfaces, even when the data that defines the surface is incomplete. This system utilizes input data, such as data points and cross section views of an object, to identify the locus and extent of objects and surfaces that are extant in a predefined volume. These objects and surfaces, termed elements herein, may be multiple-z in nature and may even fold back on themselves. The graphics system transforms the multiple-z elements into a single-z elements representation that accurately maps the features of the free-form elements with following interpolation into a two-dimensional parametric grid that is created in a predefined coordinate system. This reduced representation of the multiple-z elements enables the processor to manipulate the data more efficiently using conventional two-dimensional interpolation methods, yet also provides sufficient data to produce accurate representations of the multiple-z elements. The processor can manipulate the reduced data set to model structures that correspond to the free-form elements and thereby produce an idealized view of these elements, even if the input data incompletely define such elements.

Description

GRAPHICS SYSTEM USING PARAMETRIC GRID PARADIGM

FIELD OF THE INVENTION

This invention relates to graphics systems and in particular to a data storage and display system that uses a parametric grid paradigm to accurately model an object that is extant in a predefined volume and display its features, wherein the features can include complex freeform surfaces.

PROBLEM It is a problem in existing display systems to accurately produce representations of complex three-dimensional multiple-z geometric surfaces. There are a number of applications for this technology, including but not limited to: mapping geologic strata, controlling milling machines, producing visual representations of engineered objects, underground utilities recording and mapping systems. These existing display systems are spline-based and must process significant quantities of data to produce accurate output images. Such processing required to accomplish these results is burdensome and many system modifications have been implemented to either improve accuracy or reduce the processing burden. The patents listed below are indicative of such improvements. U. S. Patent No. 5,261,029 discloses a method and apparatus for the dynamic tessellation of curved surfaces. This system determines a uniform step size with respect to pre-transforation derivative bounds for tessellation of a graphic primitive, wherein the step size results in triangles which meet post-transformation thresholds. To maximize the efficiency of rendering curved surfaces while ensuring that the tessellation criteria are met, a maximum scale value for the non-linear transformation between device coordinate and lighting coordinate space is determined and utilized to translate the tessellation threshold in the device coordinate space to a tessellation threshold value in the lighting space.

U. S. Patent No. 5,255,352 discloses a system that maps two-dimensional surface detail on mathematically defined three-dimensional surfaces, while preserving the specific dimensional integrity of the surface detail image being mapped to provide dimensionally correct surface detail. This is accomplished by performing the intermediate step of mapping the surface detail image to a two-dimensional flattened pattern piece representation of the surface and thereafter mapping this representation to the three-dimensional surface.

U. S. Patent No. 5,175,806 discloses a method and apparatus that applies surface detail to a two-dimensional image to thereby provide a three-dimensional appearance to the surface. A perspective mesh is applied to the two-dimensional image and then scanned pixel by pixel to build several tables of data that are derived from the characteristics of each pixel. A processing step is then activated to apply detail from a stored pattern to the pixels of the image to provide surface detail to the surface without compromising the visual appearance of the resulting image. U.S. Patent No.5, 128,870 discloses a system that automatically fabricates three- dimensional objects of complex and unique geometry. This system acquires data that defines the object and its surroundings, which data is used to construct a computer- based three-dimensional model of the object. The system superimposes an ideal geometry on the computer generated model to adapt the form and fit of the object to a predefined application. The computer guides a milling machine to reproduce the object without human intervention.

U. S. Patent No. 5,121,333 discloses a method and apparatus for manipulating a computer based representation of objects of complex and unique geometry. This system acquires data that defines the object and its surroundings, which data is used to construct a computer-based three-dimensional model of the object. The system superimposes an ideal geometry on the computer generated model to adapt the form and fit of the object to a predefined application.

U. S. Patent No. 5,107,444 discloses a method and apparatus for flattening previously built three-dimensional surfaces into two-dimensional pattern pieces. This is accompUshed by creating a mesh using two-dimensional curvelinear coordinates on this surface, defining surface elements and mapping these elements in groups to a flat plane where they are reassembled. The position of data points on the flat plane are recursively adjusted to cancel errors.

U. S. Patent No. 4,991,095 discloses a process for three-dimensional mathematical modelling of underground geologic volumes that has inclined, stacked layers of sedimentary deposits. This process uses a model volume with layers of cells which are inclined and stacked analogous to the layers of sedimentary deposits as well as being arranged in vertical columns. The existing patented systems either record the features of well defined objects, or add detail to existing definitions of object surfaces, or mathematically model theoretical flawless objects. These systems are generally spline-based and require significant engineering knowledge and experience to model freeform surfaces. None of the existing or proposed graphics systems can effectively manage the task of modeling complex multiple-z surfaces, which surfaces may even fold back on themselves, which surfaces may also be defined by incomplete data. This problem is commonly encountered in the field of geologic strata mapping and there is presently no system available that can effectively assist the geologist in identifying the locus and extent of underground strata, especially where the strata includes complex freeform surfaces that are defined by seismic data that only partially represent the strata. There is a need for a system that is both simple to use and can be used to model freeform surfaces in a variety of applications.

SOLUTION

The above described problems are solved and a technical advance achieved in the field by the graphics system of the present invention that uses a parametric grid paradigm to accurately model complex multiple-z freeform surfaces, even when the data that defines the surface is incomplete. This system may utilize, for example, input data, such as seismic data and/or well data, to identify the locus and extent of objects and surfaces that are extant in a predefined volume. These objects and surfaces, termed elements herein, may be multiple-z in nature.

The graphics system uses a parametric gird paradigm which converts the input data, which only incompletely defines the multiple-z elements, into a plurality of three-dimensional curves, each of which accurately represents a cross-section view of an element. This plurality of curves and data points is then used to map the features of the multiple-z element as defined by the data into a two-dimensional parametric grid that is created in a predefined Cartesian coordinate system using conventional single-z inteφolation algorithms. This reduced representation of the multiple-z element enables the processor to manipulate the data more efficiently, yet also provides sufficient data to produce an accurate representation of the multiple-z element. Each datum point represents an accurate measurement of a point on the surface of an element, yet the intervening areas of the element's surface may be undefined. If the input data is too sparse, significant features of the surface can be absent from the representation of the element. The datum point can represent multiple dimensions, in that the element may not be a static object and can therefore include characteristics such as velocity, acceleration, etc., that reflect the movement of the object in space, or even temporal changes. The processor manipulates the reduced data set to model structures that correspond to the multiple-z elements and thereby produce an accurate view of these elements, even if the input data incompletely define such elements. This is accomphshed by using rules, which can be provided by the user on a dynamic basis, to generate cross section views of an idealized view of the multi-dimensional elements. Each such cross-section view represents the use of a subset of the input data to represent known points on the surface of the element to be represented. The rules serve to enable the processor to interconnect these isolated datum points on the surface via a continuous line, which serves to interpolate the region between adjacent datum points. By generating a plurality of these cross section views, a three-dimensional view of the object can be created. This created three-dimensional view can also be modulated by the use of datum points that do not lie on a cross section view and by data indicative of an idealized shape of the element. The processor incorporates all this data to produce an accurate representation of the element, with the accuracy being a function of the number of cross section views generated as well as the pertinency of the idealized shape data and interpretation model.

BRIEF DESCRIPTION OF THE DRAWING

Figure 1 illustrates in block diagram form the overall architecture of the graphics system of the present invention;

Figures 2-5 illustrates in flow diagram form the operational steps taken by the graphics system of the present invention to generate a three-dimensional view of an object that is extant in a volume;

Figure 6 illustrates in perspective view an object that is to be represented by the graphics system and the projection plane overlaid in the volume of interest;

Figure 7 illustrates in cross-section view, the object of Figure 6; Figure 8 illustrates a cross-section view of an idealized representation of the salt dome element of Figure 6; and

Figures 9-18 illustrate perspective and corresponding plan views of a plurality of representations of the salt dome element as a function of the number of cross section views (Figure 8) generated by the graphics system.

DETAILED DESCRIPTION

The graphics system of the present invention is illustrated in block diagram form in Figure 1. This system uses a parametric grid paradigm to accurately model complex multiple-z freeform surfaces, even when the data that defines the object surface is incomplete. This graphics system may utilize input data, such as cross- sectional views of the object, seismic data, user generated data, data output by other processors, etc. to identify the locus and extent of objects that are extant in a predefined volume. These objects, also termed elements herein, may be multiple-z in nature.

The graphics system converts the multiple-z elements into a single-z surface representation that accurately maps the features of the original elements with following interpolation on a two-dimensional parametric grid that is created in a predefined coordinate system. This reduced representation of the complex elements enable the processor to manipulate the data more efficiently using conventional single- z surface interpolation/gridding algorithms, yet also provides sufficient data to produce accurate representations of the complex elements. The processor can manipulate the reduced data set, using the parametric grid paradigm, u model the objects that correspond to the multiple-z elements and thereby produce an idealized view of these elements, even if the input data incompletely define the complex elements.

Graphics System Hardware Architecture

The graphics system includes a processor 20 that is used to perform the data manipulations that are described herein. The processor typically includes a hard drive

28 that is used to store data and programs for processor 20. Resident on processor

20 or outboard on hard disk 28 is a set of graphics libraries 30 and Tenderer control software 31 that implements the parametric grid paradigm. The processor 20 interfaces with the user, sources of data, and output devices via a number of peripheral devices, some of which are shown in Figure 1. A keyboard 10 and mouse 12 are provided to enable the user to interact with processor 20. Also provided are two-dimensional digitizer 14 which converts two-dimensional objects to a digital representation. The three-dimensional digitizer 16 functions in a manner analogous to the two-dimensional digitizer 14, but on three-dimensional objects. A graphics monitor 18 is provided to enable the user to obtain a visual representation of the output of processor 20, whether the output be queries from processor 20 or images produced by processor 20. A plurality of output devices are also shown, and include graphics monitor 22, printer 24 and plotter 26. These various peripheral devices are well-known elements that are presently commercially available and are illustrated therein to indicate a typical architecture of the graphics system of the present invention. It is expected that alternate configurations of this system can be implemented to perform the functions of the preferred embodiment of the invention disclosed herein. There are numerous applications of this system, and two possible applications are noted below to illustrate the broad spectrum of environments in which this system can function.

Meteorological Phenomena Example Adverse weather conditions, especially those affecting airport operation, are a significant safety concern for airline operators. Low level wind shear is of significant interest because it has caused a number of major air carrier accidents. Wind shear is a change in wind speed and/or direction between and two points in the atmosphere. It is generally not a serious hazard for aircraft en route between airports at normal cruising altitudes but strong, sudden low-level wind shear in the terminal area can be deadly for an aircraft on approach or departure from an airport. The most hazardous form of wind shear is the microburst, an outflow of air from a small scale but powerful downward gush of cold, heavy air that can occur beneath or from the storm or rain shower or even in rain free air under a harmless looking cumulus cloud. As this downdraft reaches the earth's surface, its spreads out horizontally hke a stream of water sprayed straight down on a concrete driveway from a garden hose. An aircraft that flies through a microburst at low altitude first encounters a strong headwind, then a downdraft, and finally a tailwind that produces a sharp reduction in air speed and a sudden loss of lift. This loss of lift can cause an airplane to stall and crash when flying at a low speed, such as when approaching an airport runway for landing or departing on takeoff. It is therefore desirable to provide pilots with a runway specific alert when a fifteen knot or greater headwind loss or gain situation is detected in the region where the aircraft are below one thousand feet above ground level and within three nautical miles of the runway ends.

The form of the windshear is a circular or elliptical column of air that travels in a downward direction from a thunderstorm to the ground. A thunderstorm produces a powerful downward gush of cold heavy air which spreads out horizontally as it reaches the earth's surface. One segment of this downflow spreads out away from the airport's TDWR radar while an opposing segment spreads out towards the TDWR radar. It is generally assumed that these outflows are symmetrical for the purpose of detecting microburst wind shears. Because most microbursts do not have purely symmetrical horizontal outflows, the TDWR system can have problems detecting or estimating the true intensity and extent of asymmetrical microburst outflows. The graphics system of the present invention can utilize the data that can be obtained from the TDWR radar and generate a representation of the windshear event using the techniques illustrated below. The wind shear phenomena represents a dynamically varying element that must be modelled in real time with a moderate degree of spatial and temporal accuracy.

Geologic Strata Example

Another application of the graphics system of the present invention is to static objects, such as in mapping and modelling the topology of geologic strata. For example, various mineral bearing strata have a characteristic geometry that can be used to identify the presence and locus of potential mining sites to retrieve the minerals from the subterranean strata. One commonly used method of mineral exploration makes use of seismic data to ascertain the cross section of the subterranean regions at a specific site. This seismic data is obtained by activating a source of mechanical disturbance (discharge of explosives, mechanical hammering, etc.) to create shock waves that are propagated through the ground. As each wavefront encounters a discontinuity in the subterranean volume, a reflection is created and a segment of the wavefront is reflected toward the surface. Detectors spaced about the site detect the reflected components of the wavefront and can be used to identify the depth and characteristics of the discontinuity and the surrounding strata. The time interval between the initiation of the disturbance and its receipt back at the surface is used to determine the depth of the discontinuity, while the magnitude of the reflected signal can be used to identify various characteristics of the discontinuity. The speed at which the wavefront travels through the various formations is a function of the materials that comprise the formation, as is the magnitude of the reflection. The discontinuity can be a fault line, mineral vein, oil bearing strata, salt dome etc. that is of interest to the user. These various discontinuities typically have predefined properties, including geometric shape.

The complexity of geological properties combined with irregular and scattered distribution of data points may produce geologically unrealistic features in computer generated maps. One manifestation of this problem is the production of extended local anomalies that have improper dips between them. Another manifestation is the production of mathematically induced anomalies in areas of sparse data distribution. Yet another manifestation is the representation of unrealistic contours at the edges of the mapped area. The resultant map may not reflect the geological concepts and hypotheses for the volume of interest. The use of the user's expert knowledge to inject additional information into the mapping process produces improved results. The use of geological knowledge relating to sedimentation, structural development, geomorphology, trend surfaces, etc. of the area provides additional data that enables the system to render a significantly more accurate representation of the object that is extant in the volume of interest.

One example of a typical oil or gas bearing strata is illustrated in simplified perspective view in Figure 6 and cross-section view in Figure 7, taken along line H-H', wherein a mushroom shaped strata 61 is located in a predefined volume. This mushroom-shaped object 61 represents a salt dome that has intruded past the existing sedimentary strata 62-63. Under the overhang of the salt dome is a hydrocarbon accumulation 60, such as oil. It is common to find that the general orientation of the sedimentary layers 62-63 is either parallel or peφendicular to the contours of the salt dome structure 61. However, the salt dome 61 can overhang the sedimentary strata 62-63 is shown in Figures 6 & 7 and as a result, seismic information may not be enough to produce quality inteφretation under the overhang of the salt dome 61. Therefore, while the locus and extent of the top surface of the salt dome 61 can be determined with a certain degree of accuracy, the underlying surfaces of the salt dome 61 are difficult to determine. These surfaces are shown in dotted line form on Figure 7 to illustrate this uncertainty. When well information is not sufficient, the surface determination may result from the use of seismic data. A seismic wavefront transmitted from a source creates a number of reflections as the wavefront encounters each surface of the strata. These reflected wavefronts are detected at the surface by a plurality of receivers. By cross- correlating and processing the data indicative of the signals produced at each of these receivers, the presence and locus of the surfaces of this strata can be ascertained. However, the precise geometry, extent and features of this strata can not always be determined from the seismic data obtained to represent the object of Figure 6. Therefore, a geologist must utilize whatever data is available to approximate the extent of the strata. A geologist typically compares the strata definition that is available with various typical characteristic shapes of this class of strata. The data is then adapted to fit a selected characteristic shape and the defined shape inteφolated and/or extrapolated to extend the available data (wells, seismic). If this approximation is flawed, it can have a deleterious effect in that a well 6A that is drilled may completely miss the strata of interest. There is no existing system that can assist a user in performing this data analysis task. Therefore, it is difficult to accurately drill a well 6B, 6C into hydrocarbon bearing strata that underlie a salt overhang, since the extent and geometry of the overlie is unknown or does not reflect geologic concepts.

The system of the present invention can be used to map data, such as the seismic and well data, to model objects of unknown extent and contours, such as the geologic strata illustrated in Figure 6. This system makes use of the available data to generate a representation of the object, typically on an iterative processing basis. In addition, a library of characteristic shapes and/or rules can be used to adapt the available data to further refine the definition of the object of interest and provide a user with information relating to the correlation between the selected characteristic shape and the received data. Sof ware Architecture

The overall architecture of the Tenderer control software 31 (herein Tenderer) of the graphics system of the present invention is disclosed in flow diagram form in Figures 2-5. This flow diagram outlines the fundamental steps taken by the graphics system to process input data to create the rendering of a selected element, whose precise geometry, shape and extent may be incompletely defined by the input data. To illustrate the operational capabilities of this software, it is described in the context of a geologic strata mapping system that operates on seismic data and/or well log data to define the presence, locus and extent of an n-dimensional (three-dimensional) strata such as the salt dome illustrated in Figure 6.

At step 201, the rendering process reads input data that is obtained from a source of data, such as the three-dimensional digitizer 16 illustrated in Figure 1. This digitized data can be processed seismic data or any interactive inteφretation that define the presence, locus and extent of the element illustrated in Figure 6, which is located in the volume V of interest. These data are typically individual datum points obtained from wells 6B-6D that define a site on the surface of the element or lines (curves) that he on a three-dimensional surface of the element. From this input data, the user builds or digitizes a plurahty of lines that follow the shapes of the surface of the element in three-dimensional foπn. This surface being modeled can be represented at step 201A by a plurahty of three-dimensional lines that represent individual cross-section views of the element's surface or by three-dimensional lines produced at step 201B in the graphics environment to display the element's overall shape.

In the digitizing step, it important to note that a curve is not just a set of three or more individual datum points, but represents a continuum that tracks the shape of the surface of the element. Individual datum points are suspect, although a collection of datum points can be used to create a curve to represent a three-dimensional cross- section view of the surface. Once such a curve is created, the values of individual datum points can be adjusted to conform to a surface defined by the remaining datum points or curves. A line is a substance that carries information about the shape of the surface. A curve is created by taking a plurahty (at least three) of datum points in a predefined order and creating a continuous curve through these plurahty of datum points. The curves can be created by selecting a predefined shape or algorithmic representation of the surface that is expected, or this process can be done dynamically without constraining the data to predefined rules. The plurality of curves that are created outlines the shape of the element. These curves produce a framework which can be modified by additional data input to represent the surface of the element in the volume of interest. Each curve provides a different cross-section view of the element and the greater the number of curves (cross-sections), the more accurate the rendering of the surface of the element. The framework need not be rectilinear in content, but can be so if desired by the user. The various curves that are created need not intersect, they can represent substantially parallel cross-section views of the element.

At step 202, Tenderer 31 creates a flat plane which represents the projection plane on which the surface being created is projected for further processing. This is accomplished by defining a three-dimensional volume V, such as a cube, at step 202A which volume V contains the element that is of interest, all of the element's relevant surfaces, and typically some of the surrounding environment. At step 202B, a two- dimensional projection plane P is defined within the selected volume V. Typically, the projection plane P is comprised of a horizonta: plane, with the Z coordinate of the projection plane being selected to be equal to the Z coordinate of the bottom of the cube V. Thus, the projection plane P comprises the "floor" of the cube V. A projection grid G, consisting of grid lines in a selected coordinate system is then overlaid on the projection plane P.

For example, Figure 6 illustrates in perspective view the projection grid G overlaid in the volume of interest on the projection plane P. The projection plane P can be placed at any position in the volume V, and is illustrated as parallel to ground level (Z-axis constant value) at the bottom of the volume V for simphcity of description. The coordinate system illustrated is a rectangular Cartesian (X,Y,Z axis) coordinate system. The data points that represent sites on the surface of the element therefore have a definition in terms of X, Y, Z coordinates as well as other characteristics that relate to the nature of the element surface being modeled. These additional characteristics can include the position of this datum point within the curve of which it is a part, materials characteristics of the strata that comprises this surface, time variation, etc. At step 203, Tenderer 31 creates for each three-dimensional curve or point on the surface a coπesponding line or point on the projection plane. This is accomplished by selecting at step 203A one of the plurality of lines that were created at step 201. At step 203B, for each line selected on step 203A a coπesponding projection line is drawn on the projection plane P. This step will later coπespond to a precise transformation of the plurahty of datum points from the created curve to coπesponding points of the projection line on the projection plane P. At step 203C, the original curve and the projection line are associated. This is accomplished by noting the coordinates in the three-dimensional volume V of each datum point located on the original three-dimensional curve, as well as the precise order of the points that are used to create the original three-dimensional cross-section curve and noting this data at the coπesponding points of the projection line on the projection plane P. Thus, the line on the projection plane P includes sufficient data to enable the system to recreate the original three-dimensional curve. At step 203D, a determination is made of whether all the curves have been mapped to the projection plane P. If not, processing returns to step 203A and steps 203A-D are repeated until all such curves and data points have been mapped. If additional original datum points are available to Tenderer 31 to assist in the production of the surface of the element, processing proceeds to step 213, wherein these original datum points are also processed. The order of processing the input data is a matter of design choice, with curves and original datum points being processed in a selected order, or being processed in an interspersed manner. In any case, at step 213, Tenderer 31 creates for each original datum point on the surface a coπesponding point on the projection plane P. This is accomplished by selecting at step 213 A one of the plurahty of input original datum points. At step 213B, for each original datum point selected as step 213A a coπesponding point (projection point) on the projection plane P is created. This step will later coπespond to a precise transformation of the selected datum point to a coπesponding projection point on the projection plane P. At step 213C, the original datum point and the projection point are associated. This is accomphshed by noting the coordinates in the three-dimensional volume of each original datum point, as well as the precise order of the datum points if they are used to create a three-dimensional cross-section curve and noting this data at the coπesponding points on the projection plane P. Thus, each point on the projection plane P includes sufficient data to enable the system to use this datum point along with the other input data to recreate the original three-dimensional surface. At step 213D, a determination is made of whether all the original datum points have been mapped to the projection plane P. If not, processing returns to step 213A and steps 213A-D are repeated until all such original datum points have been mapped. Once the association is complete, processing advances to step 204.

In performing the above steps, the projection line or point must be associated with the appropriate original curve or point in a manner that preserves the continuity of the surface being modeled. For example, if two curves on the three-dimensional surface intersect, their projection lines on the projection plane P must also intersect or at least produce an indication that the three-dimensional curves on the element surface intersect. At steps 204 A,B, the Tenderer 31 transforms the three-dimensional curves of the surface into extended projection lines wherein the extended projection lines have extra properties of the original three-dimensional curve. The set of coordinates associated with each point mapped on to the projection plane P include, but are not limited to: X**^, Y_origi_nai. Z^,,, X_projected, Y_prθjected. Z_projected, Sequence Number, Cross-Section Number.

Renderer 31 at step 205 defines a grid mesh G, shown in Figure 6 as being overlaid on the projection plane P. This process is initiated at step 205A by defining an area within the volume that is to be gridded on to the projection plane P. At step 205B, a grid mesh G is created, whose extent should be such that the area of interest in the volume V is covered by the grid mesh G. In addition, the properties of the grid mesh G should be selected to obtain the data required from the projection plane P. The grid mesh G represents a coordinate system matrix, such as X-Y grid, that is used to relate the various projection points to the rendering of the element into a three- dimensional representation. At steps 206-208, renderer 31 seriatim extracts X _{π u}__ai_> ^_ori_gi_nal ^an^ 2-^^ properties of the associated three-dimensional curves from the extended projection lines. The extracted data is then used to build a coπesponding X ri_gi_nai-g^ήd. Y_ori_gi_nargrid, an^d

on projection plane P which have the parameters of the grid mesh G that was created at step 205. This is accomphshed at each of steps 206-208 by extracting the coπesponding X _πgmai/Y_oπgi_nai/Z_bri i_nai property at the coπesponding step 206A-208A from the extended projection lines on the projection plane P. At step 206B-208B, a determination is made of whether all projection lines located on the projection plane P have been processed by the coπesponding step 206A-208A. If not, processing returns to the coπesponding step 206A-208A for the next selected projection line on the projection plane P. When all such lines have been processed, the process advances to the coπesponding one of steps 206C-208C where each (X^^, or Y_original or Z^^) property is inteφolated and gridded separately using conventional single-z gridding methods over the entire area of interest to provide data values for the regions between adjacent datum points. These inteφolated values are then stored in the grid mesh G that is created for the coπesponding property.

Parametric Grid Paradigm

The processes defined by steps 206-208 represent the creation of parametric grids for each of the variables X, Y, Z. The parametric grids represent mapping of the three-dimensional surface of the object on to the respective grids. In order to understand the concept of a parametric grid paradigm, it should be noted that a continuous curve in three-dimensional Euclidean space is defined as a set of data points which have coordinates (X, Y, Z) that satisfy the following system of parametric equations:

X = X(T); Y = Y(T); Z = Z(T) where X(T), Y(T), Z(T) are all functions of a real parameter T over a given interval (this is a parameterized representation of the continuous curve). The use of a parameterized representation of the continuous curve is beneficial when the continuous curve is not a single-Z curve, in that there is more than one Z axis value for the curve for a given set of coordinates X, Y. When the task is to create the continuous curve from a set of data points, the selection of the proper sequence of coordinate values to use for this task can become a difficult task. For example, the sequence of data points can be provided in a tabular representation and the continuous curve must be recreated between these given data points, such as in the smoothing of a contour line that is used to define a surface of the object. Traditionally, the three-dimensional continuous curve may be recreated separately in each of the following three domains: (X, T), (Y, T), (Z, T). These curves that are created in each of these three domains are termed domain curves and are individually refeπed to as: X-curve, Y-curve, Z-curve. Conventional one- dimensional inteφolation techniques such as polynomial, or spline techniques can be used to recreate the inteφolated one-dimensional curve, which can then be represented analytically or in tabular form, termed domain tables: X-table, Y-table, Z-table. If the analytical representation of the domain curves is selected, then the X, Y, and Z coordinates of any point P on the three-dimensional curve can be defined as follows: for a given value T(k) of the parameter T, the coπesponding values of X(T(k)), Y(T(k)), Z(T(k)), calculated from analytical expressions, are the Euclidean coordinates of the data point P on the three-dimensional continuous curve. If a tabular representation of the domain curves is selected, then the X, Y, and Z coordinates of any point P on the three-dimensional curve can be defined as follows: for a given value T(k), k= l,...n, of the parameter T, the coπesponding values of X(T(k)), Y(T(k)), Z(T(k)) obtained from the X-table, Y-table, Z-table, respectively, are the Euclidean coordinates of the data point P on the three-dimensional continuous curve. For convenience of computation, the three domain tables, X-table, Y-table, Z- table can be combined into a single table, XYZ-table, which contains the X, Y, Z values for each given value T(k), k= l,...n, of the parameter T.

In the system of the present invention, this above process is expanded into a process that designates continuous anc_ discontinuous surfaces rather than continuous curves in a three-dimensional space. The surface is defined in Euclidean space as a set of data points P, each of which have coordinates X, Y, Z that satisfy the following system of parametric equations:

X = X(Ti, Tj); Y = Y(Ti, Tj); Z = Z(Ti, Tj) where X = X(Ti, Tj), Y = Y(Ti, Tj), Z = Z(Ti, Tj) are all functions of the real parameters Ti, Tj over given intervals. Thus, the resultant ametric grids are a parameterized representation of the surface, which is a useful method of representing surfaces that are not single-Z surfaces. Such a surface may be represented, as described earlier, as a set of continuous curves and scattered data points. To accurately recreate the surface in the three-dimensional Euclidean space, the following process is applied to the collection of continuous curves and data points: The three- dimensional surface is first flattened and then recreated separately in each of the following domain surfaces: (X, Ti, Tj), (Y, Ti, Tj), (Z, Ti, Tj), which are respectively termed: X-surface, Y-surface, Z-surface. These respective domain surfaces are recreated using conventional two-dimensional inteφolation techniques, such as minimum curvature, moving weighted average, spline, etc. The recreated surface in each domain can be represented analytically or in grid form, termed domain grids: X- grid, Y-grid, Z-grid. If the analytical representation of the domain surfaces is selected, then the X, Y, and Z coordinates of any point P on the three-dimensional surface can be defined as follows: for a given value Ti(k), Tj(l) of the parameters Ti, Tj, the coπesponding values of X(Ti(k), Tj^'(l)), Y(Ti(k),Tj(l)), Z(Ti(k), Tj(l)), calculated from analytical expressions, are the Euclidean coordinates of the data point P on the three- dimensional continuous surface. If a tabular representation of the domain curves is selected, then the X, Y, and Z coordinates of any point P on the three-dimensional surface can be defined as follows: for a given value Ti(k), k= l,...n, of the parameter Ti and Tj(l), 1 = l,...m, of the parameter Tj, the coπesponding values of X(Ti(k), Tj(l)), Y(Ti(k), Tj(l)), Z(Ti(k), Tj(l)) obtained from the X-grid, Y-grid, Z-grid, respectively, are the Euclidean coordinates of the data point P on the three-dimensional continuous surface. For convenience of computation, the three domain grids, X-grid, Y-grid, Z- grid can be combined into a single grid, XYZ-grid, which contains the X, Y, Z values for each given value Ti(k), k= l,...n, of the parameter Ti and Tj(l), l = l,...m, of the parameter Tj.

The resultant _origi_nai-grid, Y_0ri_gi_nargrid, and

(each is a parametric grid of the same size) are merged by renderer 31 at step 209 into an XYZ-grid mesh that contains triples in each grid node that represent the data values obtained from the coπesponding grid nodes of X^^,- grid, Y_originaI- grid, and Z^^,- grid. The XYZ-grid mesh is then stored on hard drive 28 and can be retrieved by renderer 31 to produce at step 210 an image that illustrates the three-dimensional surface as well as the remainder of the volume. This image can be displayed on graphics monitor 22, output in tangible foπn on print media via printer 24 or plotter 26.

Graphics System Shape Generation

The theoretical, or user input, idealized representation of a typical cross-section of the salt dome element illustrated in Figure 6 is shown in Figure 8. This straight line approximation of the mushroom shaped salt dome 61 provides renderer 31 with a set of rules 30, input either as a mathematical representation by the user, or a graphical input by the user, or preprogrammed in memory. The characteristic shape of Figure 8 can be used as additional data by renderer 31 to assist in creating the three-dimensional representation of the salt dome element. The result of the processing of the cross-section curves, taken in three dimensions is illustrated in Figures 9-18. These Figures represent a plan view of the salt dome element, showing via bold lines drawn on the plan view, the various cross-section views taken through the salt dome element to map its surface, and a corresponding three-dimensional perspective view of the resultant representation of the salt dome element in frame outline form. As can be seen from this progression of Figures, the detail presented by the renderer 31 is increased as the number of cross-section views increases. Thus, Figure 9 illustrates a plan view of the salt dome element, as generated using two cross-section lines 9A, 9B, which are spaced at 90° increments around the axis of the salt dome element. This plan view includes an extensive region of ambiguity, illustrated by the oval-shaped "white spaces" centered in the region between the cross- section lines 9 A, 9B. The perspective view of this representation using the two cross- section lines is illustrated in Figure 10. As can be seen from this Figure, the general shape of the salt dome element is recognizable and corresponds to the characteristic shape of Figure 8. This perspective view lacks detail, especially in the region where the dome of salt dome element overlaps the stem of the mushroom shape.

Figure 11 illustrates a plan view of the salt dome element, as generated using three cross-section lines 11A, 11B, 11C which are spaced at 60° increments around the axis of the salt dome element. This plan view includes less of a region of ambiguity, illustrated by the oval-shaped "white spaces" centered in the region between the cross- section lines 11 A, 11B, 11C, than the previous plan view of Figure 9. The perspective view of this representation using the three cross-section lines is illustrated in Figure 12. As can be seen from this Figure, the general shape of the salt dome element is recognizable and corresponds to the characteristic shape of Figure 8. This perspective view includes additional detail, especially in the region where the dome overlaps the stem of the mushroom shape, not shown in the perspective view of Figure 10. Figure 13 illustrates a plan view of the element, as generated using four cross-section lines 13A, 13B, 13C, 13D which are spaced at 4.5° increments around the axis of the salt dome element. This plan view has significantly reduced the region of ambiguity, illustrated by the oval-shaped "white spaces" centered in the region between the four cross-section lines 13A, 13B, 13C, 13D. The perspective view of this representation using the four cross-section lines is illustrated in Figure 14. As can be seen from this Figure, the general shape of the salt dome element is recognizable and coπesponds closely to the characteristic shape of Figure 8. This perspective view includes additional detail, especially in the region where the dome overlaps the stem of the mushroom shape, which detail is provided by the additional cross-section views. Figure 15 illustrates a plan view of the salt dome element, as generated using six cross-section lines 15A, 15B, 15C, 15D, 15E, 15F which are spaced at 3CP increments around the axis of the salt dome element. This plan view includes a small region of ambiguity, illustrated by the oval-shaped "white spaces" centered in the region between the cross-section lines 15A, 15B, 15C, 15D, 15E, 15F. The perspective view of this representation using the six cross-section lines is illustrated in Figure 16. As can be seen from this Figure, the general shape of the salt dome element is recognizable and closely coπesponds to the characteristic shape of Figure 8. This perspective view provides clear detail, especially in the region where the dome overlaps the stem of the mushroom shape, with the contours of the surface being represented in fine detail.

Finally, Figure 17 illustrates a plan view of the salt dome element, as generated using twelve cross-section lines 17A-17I, which are spaced at 15° increments around the axis of the salt dome element. This plan view includes little ambiguity, illustrated by the oval-shaped "white spaces" centered in the region between the cross-section lines 17A-17I. The perspective view of this representation using the twelve cross- section lines is illustrated in Figure 18. As can be seen from this Figure, the general shape of the salt dome element is recognizable and coπesponds to the characteristic shape of Figure 8. This perspective view provides a significant amount of detail, especially in the region where the dome overlaps the stem of the mushroom shape. The salt dome element is now represented by a smooth, continuous surface, as contrasted with the incomplete rendering of Figure 10.

As can be seen from this succession of Figures, the level of detail of the presentation of the salt dome element is a function of the number of cross-sections taken through the salt dome element. Additional cross-sections provide additional data, which can be used by renderer 31 to resolve the ambiguities in the regions of sparse data. Each curve provides a different cross-section view of the element and the greater the number of curves (cross-sections), the more accurate the rendering of the surface of the salt dome element. The framework need not be rectilinear in content, but can be so if desired by the user. Also, the various curves that are created need not intersect, they can represent substantially parallel cross-section views of the salt dome element, or a combination of intersecting and non-intersecting curves.

Additional Data Input

There are two fundamental requirements that are operative in this system. A first requirement is that the surface of the element 61 substantially pass through the datum points that are computed from the input data. A second requirement is that the mapping surface should be controlled, if necessary, between data clusters, in areas that are defined by sparse data, and on the periphery of the area of interest.

The surface substitution process is implemented by the creation, generally manually by an expert, of a regional map that defines the gross features that exist in the area of interest. On this first step, this regional map is projected on to a parametric grid to provide additional data for the rendering process. The second step in this process is to use this surface during the gridding process in the areas where insufficient data is present. Thus, the default is to use the input data when sufficient data density is available. The substitute data is incoφorated into the input data to supplement it where the data density is insufficient to accurately portray the surface of the element of interest. While the parametric grid paradigm process is described as applied to a static object, it is evident that this concept can be extended to a time-varying object, wherein the representation produced can illustrate the movement of the object in the predefined volume. Thus, while a specific embodiment of the invention has been disclosed herein, it is expected that one skilled in the art can produce alternate embodiments that fall within the expected scope of the appended claims.

Claims

I CLAIM;

1. A system for representing a surface of an n-dimensional object that is extant in a predefined volume, comprising: means for storing a set of data, each member of said set of data being indicative of a locus in said volume of a datum point located on said surface; means for interconnecting members of said set of data in order to define a plurality of curves representative of a cross-section of said surface; means for creating a projection plane having a predefined coordinate system in said volume; means for mapping each of a plurahty of said curves created by said interconnecting means on to said projection plane; and means for generating an n-dimensional rendering of said element represented by said curves mapped on to said projection plane.

2. The system of claim 1 wherein said generating means comprises: means for creating a plurahty of grids, one for each of said n dimensions, each of said grids comprising a mapping of data values from said projection plane on to said grid for a selected one of said n dimensions.

3. The system of claim 2 wherein said generating means further comprises: means for incoφorating said plurality of grids into a multi-dimensional representation of said element.

4. The system of claim 2 wherein each of said grids comprises a parameterized representation of said surface for a selected one of said n dimensions.

5. The system of claim 4 wherein said creating means comprises: means for flattening said surface on said projection plane; and means for recreating said surface in each of said n dimensions via inteφolation.

6. The system of claim 5 wherein said creating means further comprises: means for representing said recreated surface in each of said n dimensions via a technique selected from the class of surface representations including: analytical representation, grid representation.

7. The system of claim 6 wherein said representing means comprises: parametric grid generating means, responsive to said data values, for representing members of said set of data as a product of a set of parametric functions.

8. The system of claim 7 wherein said grid creating means further comprises: means for mapping each of said set of parametric functions on to a coπesponding domain surface.

9. The system of claim 6 wherein said grid creating means further comprises: means for representing each said domain surface as a coπesponding domain grid comprising a table of data values.

10. The system claim 9 wherein said grid creating means further comprises: means for combining said domain grids into a single composite grid that contains the data from each of said domain grids.

11. The system of claim 6 wherein said generating means further comprises: means for incoφorating said plurahty of recreated surfaces into a multi¬ dimensional representation of said element.

12. The system of claim 3 further comprising: means for generating at least one supplemental grid comprising a mapping of additional element defining data; and wherein said incoφbrating means additionally incoφorates said at least one supplemental grid with said plurahty of grids.

13. The system of claim 1 wherein said interconnecting means comprises: means for producing data indicative of a continuous curve that runs through at least three of said members of said set of data in a predetermined order.

14. The system of claim 13 wherein said interconnecting means further comprises: means for storing data indicative of a set of rules that define the operational steps used by said producing means to construct said curve.

15. The system of claim 1 wherein said creating means comprises: means for defining a two-dimensional plane located at a predetermined position in said volume.

16. The system of claim 15 wherein said creating means further comprises: means for creating a grid system that is overlaid on said two-dimensional plane to define all points located in said two-dimensional plane.

17. The system of claim 16 wherein said mapping means comprises: means for translating each of said plurahty of said members of said set of data that define a curve representative of a cross-section of said surface on to a coπesponding point on said grid system overlaid on said two-dimensional plane.

18. The system of claim 17 wherein said translating means produces a set of coordinates for each of said plurahty of said members of said set of data that define said locus in said volume of said datum point, said point on said grid system overlaid on said two-dimensional plane, a relation of said datum point to other datum points in said plurality of members of said set of data.

19. The system of claim 18 wherein said translating means further produces information in said set of coordinates that define characteristics of said surface.

20. The system of claim 1 wherein said multi-dimensional rendering comprises a three-dimensional representation of said surface, further comprising: wherein said generating means comprises: means for creating three grids, one for each of said three dimensions, each of said grids comprising a mapping of data values from said projection plane on to said grid for a selected one of said three dimensions; and means for incoφorating said three grids into a three dimensional representation of said element.

21. The system of claim 20 wherein said creating means comprises: means for defining a two-dimensional plane located at a predetermined position in said volume, means for creating a grid system that is overlaid on said two-dimensional plane to define all points located in said two-dimensional plane; and wherein said mapping means comprises: means for translating each of said plurahty of said members of said set of data that define a curve representative of a cross-section of said surface on to a coπesponding point on said grid system overlaid on said two-dimensional plane.

22. The system of claim 21 wherein said translating means produces a set of coordinates for each of said plurahty of said members of said set of data that define said locus in said volume of said datum point, said point on said grid system overlaid on said two-dimensional plane, a relation of said datum point to other datum points in said plurahty of members of said set of data.

23. The system of claim 22 wherein said translating means further produces information in said set of coordinates that define characteristics of said surface.

24. A method for representing a surface of an n-dimensional object that is extant in a predefined volume, comprising the steps of: storing a set of data, each member of said set of data being indicative of a locus in said volume of a datum point located on said surface; interconnecting members of said set of data in order to define a plurality of curves representative of a cross-section of said surface; creating a projection plane having a predefined coordinate system in said volume; mapping each of a plurality of said curves created by said step of interconnecting on to said projection plane; and generating an n-dimensional rendering of said element represented by said curves mapped on to said projection plane.

25. The method of claim 24 wherein said step of generating comprises: creating a plurahty of grids, one for each of said n dimensions, each of said grids comprising a mapping of data values from said projection plane on to said grid for a selected one of said n dimensions.

26. The method of claim 25 wherein said step of generating further comprises: incoφorating said plurahty of grids into a multi-dimensional representation of said element.

27. The method of claim 25 wherein each of said grids comprises a parameterized representation of said surface for a selected one of said n dimensions.

28. The method of claim 27 wherein said step of creating comprises: flattening said surface on said projection plane; and recreating said surface in each of said n dimensions via inteφolation.

29. The method of claim 28 wherein said step of creating further comprises: representing said recreated surface in each of said n dimensions via a technique selected from the class of surface representations including: analytical representation, grid representation.

30. The method of claim 29 wherein said step of representing comprises: generating, in response to said data values, a parametric grid representation of members of said set of data as a product of a set of parametric functions.

31. The method of claim 30 wherein said step of grid creating further comprises: mapping each of said set of parametric functions on to a coπesponding domain surface.

32. The method of claim 29 wherein said step of grid creating further comprises: representing each said domain surface as a coπesponding domain grid comprising a table of data values.

33. The method claim 32 wherein said step of grid creating further comprises: for combining said domain grids into a single composite grid that contains the data from each of said domain grids.

34. The method of claim 29 wherein said step of generating further comprises: incoφorating said plurahty of recreated surfaces into a multi¬ dimensional representation of said element.

35. The method of claim 26 further comprising the step of: generating at least one supplemental grid comprising a mapping of additional element defining data; and wherein said step of incoφorating additionally incoφorates said at least one supplemental grid with said plurality of grids.

36. The method of claim 24 wherein said step of interconnecting comprises: producing data indicative of a continuous curve that runs through at least three of said members of said set of data in a predetermined order.

37. The method of claim 36 wherein said step of interconnecting further comprises: storing data indicative of a set of rules that define the operational steps used by said step of producing to construct said curve.

38. The method of claim 24 wherein said step of creating comprises: defining a two-dimensional plane located at a predetermined position in said volume.

39. The method of claim 38 wherein said step of creating further comprises: creating a grid system that is overlaid on said two-dimensional plane to define all points located in said two-dimensional plane.

40. The method of claim 39 wherein said step of mapping comprises: translating each of said plurahty of said members of said set of data that define a curve representative of a cross-section of said surface on to a coπesponding point on said grid system overlaid on said two-dimensional plane.

41. The method of claim 40 wherein said step of translating produces a set of coordinates for each of said plurahty of said members of said set of data that define said locus in said volume of said datum point, said point on said grid system overlaid on said two-dimensional plane, a relation of said datum point to other datum points in said plurality of members of said set of data.

42. The method of claim 41 wherein said step of translating further produces information in said set of coordinates that define characteristics of said surface.

43. The method of claim 24 wherein said multi-dimensional rendering comprises a three-dimensional representation of said surface, further comprising the steps of: wherein said step of generating comprises: creating three grids, one for each of said three dimensions, each of said grids comprising a mapping of data values from said projection plane on to said grid for a selected one of said three dimensions; and incoφorating said three grids into a three dimensional representation of said element.

44. The method of claim 43 wherein said step of creating comprises: defining a two-dimensional plane located at a predetermined position in said volume, creating a grid system that is overlaid on said two-dimensional plane to define all points located in said two-dimensional plane; and wherein said step of mapping comprises: translating each of said plurahty of said members of said set of data that define a curve representative of a cross-section of said surface on to a coπesponding point on said grid system overlaid on said two-dimensional plane.

45. The method of claim 44 wherein said step of translating produces a set of coordinates for each of said plurahty of said members of said set of data that define said locus in said volume of said datum point, said point on said grid system overlaid on said two-dimensional plane, a relation of said datum point to other datum points in said plurality of members of said set of data.

46. The method of claim 45 wherein said step of translating further produces information in said set of coordinates that define characteristics of said surface.