Non-edge computer image generation system

• US 4,677,576 A
• Filed: 06/27/1983
• Issued: 06/30/1987
• Est. Priority Date: 06/27/1983
• Status: Expired due to Fees

• Current Assignee: Grumman Aerospace Corporation (Northrop Grumman Corporation)
• Original Assignee: Grumman Aerospace Corporation (Northrop Grumman Corporation)
• Inventors: Berlin, Edwin P. Jr., Gardner, Geoffrey Y., Gelman, Robert M., Gershowitz, Michael N.
• Primary Examiner(s): Gruber, Felix D.
• Application Number:
  US06/508,260
• Time in Patent Office: 1,464 Days
• Field of Search: 364/521, 364/522, 364/520, 340/720-728, 340/701, 340/703, 340/744, 340/798, 354/292, 353/30, 353/35, 355/40, 355/79, 358/104, 358/105
• US Class Current: 345/420
• CPC Class Codes: G09B 9/301 by computer-processed or -generated image

Abstract

A computer image generation system is described which models objects without the necessity of linear edges. The system is adaptable for dynamic (real time) image generation from a compact model base for use in, for example, flight training systems. Scene content is enhanced by a novel texture generator. The system is designed for use with standard video display equipment.

Claims

1. A method for generating an image, said method comprising:

storing a multiplicity of parameters defining quadric and planar surfaces, said parameters being in terms of a three-axis reference coordinate system;

computing coefficients defining image plane curves from the parameters, the image plane curves being a perspective projection into an image plane of silhouette curves of the quadric surfaces, and intersections between the planar surfaces and between the planar surfaces and quadric surfaces;

transforming the stored parameters into terms of an eyepoint coordinate system having an origin at a focal point of the perspective projection and first and third axes paralleling the image plane, the coefficients being computed from the transformed parameters;

grouping the stored parameters into object lists, each object list including all of the parameters defining no more than one of the quadric surfaces and no more than a predetermined number of the planar surfaces, the parameters defining each planar surface including an inequality identification defining a half space bounded by the planar surface, such that the parameters grouped in each object list define an object lying in a conjunction of all the associated half spaces and having a boundary surface defined by the associated quadric and planar surfaces; and

displaying desired portions of the image plane curves on a display means in accordance with said coefficients.

2. The method of claim 1, wherein the desired portion of the image plane curves for each object are limited to those portions related to a part of the object'"'"'s boundary surface which faces a focal point of the perspective projection.

3. The method of claim 1, further comprising

generating a set of eyepoint coordinates defining a focal point of the perspective projection and a line-of-sight normal to the image plane;

and

transforming the stored parameters into terms of an eyepoint coordinate system having an origin at the focal point of the perspective projection, first and third axes paralleling the image plane, and a second axis coincidental with the line-of-sight normal to the image plane, the coefficients being computed from the transformed parameters.

4. The method of claim 3, wherein the transformed parameters for each quadric surface include ten quadric surface coefficients from which a quadric surface equation may be written, utilizing Cartesian coordinates defining a point in terms of the eyepoint coordinate system, and the quadric surface is a locus of all the points whose Cartesian coordinates satisfy the associated quadric surface equation, and wherein the transformed parameters for each planar surface include four bounding plane coefficients and an inequality identification from which a bounding plane relation

5. The method of claim 4, wherein the step of computing coefficients defining image plane curves comprises computing limb curve coefficients for each quadric surface in accordance with six limb curve coefficient formulas:

space="preserve" listing-type="equation">a.sub.1 =q.sub.7.sup.2 -4q.sub.0 q.sub.1

space="preserve" listing-type="equation">a.sub.2 =q.sub.9.sup.2 -4q.sub.0 q.sub.3

space="preserve" listing-type="equation">a.sub.3 =2q.sub.7 q.sub.9 -4q.sub.0 q.sub.6

space="preserve" listing-type="equation">a.sub.4 =(2q.sub.7 q.sub.8 -4q.sub.0 q.sub.4)f

space="preserve" listing-type="equation">a.sub.5 =(2q.sub.8 q.sub.9 -4q.sub.0 q.sub.5)f

space="preserve" listing-type="equation">a.sub.6 =(q.sub.8.sup.2 -4q.sub.0 q.sub.2)f.sup.2

where a₁ through a₆ are the limb curve coefficients, f is a distance from the focal point of the perspective projection to the image plane, q₀ through q₉ are the associated quadric surface coefficients, and the perspective projection of the silhouette curve associated with the quadric surface includes a locus of all points in the image plane satisfying an equation;

space="preserve" listing-type="equation">a.sub.1 h.sup.2 +a.sub.2 v.sup.2 +a.sub.3 hv+a.sub.4 h+a.sub.5 v+a.sub.6 =0

where h and v are Cartesian coordinates defining a point in terms of an image plane coordinate system having an origin at an intersection of the second eyepoint coordinate axis with the image plane, and horizontal and vertical axes paralleling the first and third eyepoint coordinate axes respectively.

6. The method of claim 4, further comprising grouping the stored parameters into object lists, each object list including all of the parameters defining no more than one of the quadric surfaces and no more than a predetermined number of the planar surfaces, such that the parameters grouped in each object list define an object lying in a conjunction of all the associated half spaces and having a boundary surface defined by the associated quadric and planar surfaces.

7. The method of claim 6, wherein the step of computing coefficients defining image plane curves further comprises computing intersection curve coefficients for each planar surface associated by one of the object lists with one of the quadric surfaces, the intersection curve coefficients being computed in accordance with six intersection curve coefficient formulas:

space="preserve" listing-type="equation">e.sub.1 =B.sub.4.sup.2 q.sub.1 -B.sub.1 B.sub.4 q.sub.7 +q.sub.0 B.sub.1.sup.2

space="preserve" listing-type="equation">e.sub.2 =B.sub.4.sup.2 q.sub.3 -B.sub.3 B.sub.4 q.sub.9 +q.sub.0 B.sub.3

space="preserve" listing-type="equation">e.sub.3 =B.sub.4.sup.2 q.sub.6 -B.sub.1 B.sub.4 q.sub.9 -B.sub.3 B.sub.4 q.sub.7 +2B.sub.1 B.sub.3 q.sub.0

space="preserve" listing-type="equation">e.sub.4 =(B.sub.4.sup.2 q.sub.4 -B.sub.2 B.sub.4 q.sub.7 -B.sub.1 B.sub.4 q.sub.8 +2B.sub.1 B.sub.2 q.sub.0)f

space="preserve" listing-type="equation">e.sub.5 =(B.sub.4.sup.2 q.sub.5 -B.sub.2 B.sub.4 q.sub.9 -B.sub.3 B.sub.4 q.sub.8 +2B.sub.2 B.sub.3 q.sub.0)f

space="preserve" listing-type="equation">e.sub.6 =(B.sub.4.sup.2 q.sub.2 -B.sub.2 B.sub.4 q.sub.8 +B.sub.2.sup.2 q.sub.0)f.sup.2

where e₁ through e₆ are the intersection curve coefficients, q₀ through q₉ and B₁ through B₄ are the associated quadric and planar surface coefficients, and the perspective projection of the intersection between the planar surface and the associated quadric surface includes a locus of all points in the image plane satisfying an equation

space="preserve" listing-type="equation">e.sub.1 h.sup.2 +e.sub.2 v.sup.2 +e.sub.3 hv+e.sub.4 h+e.sub.5 v+e.sub.6 =0.

8. The method of claim 7, wherein the step of computing coefficients defining image plane curves further comprises computing intersection line coefficients for pairs of the planar surfaces, the intersection line coefficients being computed for each pair of planar surfaces associated with the same object list in accordance with three intersection line coefficient formulas:

space="preserve" listing-type="equation">l.sub.1 =B.sub.1-1 /B.sub.4-1 -B.sub.1-2 /B.sub.4-2

space="preserve" listing-type="equation">l.sub.2 =B.sub.3-1 /B.sub.4-1 -B.sub.3-2 /B.sub.4-2

space="preserve" listing-type="equation">l.sub.3 =B.sub.2-1 /B.sub.4-1 -B.sub.2-2 /B.sub.4-2

where l₁, l₂ and l₃ are the intersection line coefficients, B_1-1 through B_4-1 are the bounding plane coefficients for one of the planar surfaces from the pair, B_1-2 through B_4-2 are the bounding plane coefficients for the other planar surface from the pair, and the perspective projection of the intersection between the planar surfaces in the pair includes a locus of all points in the image plane satisfying an equation;

space="preserve" listing-type="equation">l.sub.1 h+l.sub.2 v+l.sub.3 =0.

9. The method of claim 8, wherein the desired portions of the image plane curves for each object are limited to those portions related to a part of the object'"'"'s boundary surface which faces the focal point of the perspective projection.

10. The method of claim 9, further comprising

calculating visibility test line coefficients for each intersection curve, the visibility test line coefficients being computed in accordance with three visibility test line coefficient formulas:

space="preserve" listing-type="equation">t.sub.1 =B.sub.1 /B.sub.4 -q.sub.7 /2q.sub.0

space="preserve" listing-type="equation">t.sub.2 =B.sub.3 /B.sub.4 -q.sub.9 /2q.sub.0

space="preserve" listing-type="equation">t.sub.3 =(B.sub.3 /B.sub.4 -q.sub.8 /2q.sub.0)f

where t₁, t₂ and t₃ are the visibility test line coefficients, B₁ through B₄ are the associated bounding plane coefficients and q₇, q₈, q₉ and q₀ are four of the associated quadric surface coefficients; and

determining the desired portions of each object'"'"'s associated image plane curves by referring to the associated limb curve, intersection curve, intersection line and visibility test line coefficients and the inequality identification for the associated planar surfaces.

11. The method of claim 10, wherein the image plane curves include limb curves, each defined by the limb curve coefficients for an associated one of the quadric surfaces;

intersection curves, each defined by the intersection curve coefficients for an associated one of the planar surfaces; and

intersection lines, each defined by the intersection line coefficients for an associated one of the pairs of planar surfaces.

12. The method of claim 11, wherein the step of determining desired portions of each object'"'"'s image plane curves comprises:

testing minimum and maximum points on the associated limb curve against the object'"'"'s associated visibility test line coefficients and intersection lines;

testing contact points between the associated limb curve and each associated intersection curve against the associated intersection lines; and

testing points of intersection between intersection curves and intersection lines against the object'"'"'s associated intersection line.

13. A method for simulating visual images encountered during nap-of-the-earth flight, said method comprising:

(a) gathering elevation data from a geographic region of which the visual images are to be simulated, the elevation data being measured with respect to a ground plane;

(b) digitizing the elevation data;

(c) determining major terrain features;

(d) isolating the major terrain features;

(e) fitting a single quadric surface to the elevation data corresponding to each isolated major terrain feature;

(f) determining bounding planes for each isolated major terrain feature to maximize continuity between adjoining quadric surfaces;

(g) determining the texture function parameters sufficient to define a desired texture pattern for each quadric surface and bounding plane by Fourier analysis of the elevation data;

(h) generating further quadric surfaces, bounding planes and texture function parameters corresponding to solid surface objects typically found in the geographic region for which the visual images are to be simulated;

(i) generating yet further quadric surfaces, bounding planes and texture function parameters corresponding to dynamic objects typically found in the geographic region for which the visual images are to be simulated;

(j) generating yet further texture function parameters corresponding to a desired cloud texture pattern;

(k) constructing a data base containing quadric surface parameters indicating the size and shape of each quadric surface and its location and orientation with respect to the ground plane, bounding plane parameters indicating the orientation of each bounding plane with respect to the ground plane, and the texture function parameters;

(l) defining a line of sight corresponding to an observation platform location and orientation with respect to the ground plane;

(m) assigning a subframe processor to each of a plurality of uniform subframe areas covering a field of view corresponding to the line of sight;

(n) determining which of the major terrain features, solid surface objects, and dynamic objects are within the field of view;

(o) assigning an object processor to each quadric surface corresponding to the major terrain features, solid surface objects, and dynamic objects within the field of view;

(p) causing each assigned object processor to compute intraobject visibility information for its assigned quadric surface and associated bounding planes from the quadric surface and bounding plane parameters in the data base, and the line of sight;

(q) causing the subframe processors to generate image information from the intraobject visibility information and the texture function parameters;

(r) displaying the image information;

(s) updating the line of sight to simulate a desired incremental in the location and orientation of the observation platform with respect to the ground plane;

(t) varying the texture function parameters corresponding to the dynamic objects and the cloud texture pattern to simulate a desired motion and agitation of the associated texture patterns; and

(u) repeating the steps (n) to (t) at a sufficient rate to simulate the desired visual images.

14. A flight training system for simulating an image plane viewable from a flight platform over a region modeled by a multiplicity of objects, said system comprising:

bulk memory means for storing an object list for each object, each such object list including a plurality of object parameters indicating the size, shape, orientation, location, and texture of the associated object;

viewpoint tracking means for providing an updated set of viewpoint parameters indicating the location and orientation of the flight platform with respect to the objects, said viewpoint parameters being updated once during each passage of a desired frame period;

a plurality of object processors for providing an updated set of coefficients for image plane curves representing a perspective projection of the objects into the image plane, said coefficients being computed once per frame period from the viewpoint parameters and the shape, size, orientation and location object parameters;

pixel generating means for providing an updated set of intensity levels for a multiplicity of pixels corresponding to sample points on the image plane, said intensity levels being computed once per frame period from the image plane curve coefficients, the viewpoint parameters, and the texture and location object parameters, said pixel generating means comprising a plurality of subframe processors each associated with a subframe defined by one of a plurality of uniform contiguous areas forming the image plane, and means for transferring the updated image plane curve coefficient from each object processor to at least one subframe processor whose associated area in the image plane is intersected by the projection of an object associated with that object processor, the updated image plane curve coefficients being transferred once per frame period;

fast memory means for storing those object lists corresponding to objects currently viewable in the image plane, the object lists being organized into blocks in the bulk memory means, each block including all the object lists for one of a plurality of predetermined contiguous areas making up the modeled region, the number of objects in each contiguous area being nearly equal; and

display means for arranging the pixels on a video raster.

15. A system as claimed in claim 14, wherein the bulk memory means comprises a magnetic disc memory.

16. A system as claimed in claim 14, further comprising means for transferring the blocks of object lists from the bulk memory means to the fast memory means.

17. A system as claimed in claim 14, wherein the object location parameters for each object comprise three Cartesian coordinates identifying a location point within a reference coordinate system having perpendicular width and depth axes defining a ground plane and an elevation axis perpendicular to the ground plane.

18. A system as claimed in claim 17, wherein the location point for at least one of the objects defines the geometric center of a quadric surface associated with that object.

19. A system as claimed in claim 18, wherein the quadric surface is an ellipsoid.

20. A system as claimed in claim 19, wherein the object shape and size parameters for the object having the associated quadric surface comprise three radial lengths taken from the location point to the surface of the ellipsoid along orthogonal object axes.

21. A system as claimed in claim 18, wherein the quadric surface is a two-sheeted hyperboloid.

22. A system as claimed in claim 21, wherein the object shape and size parameters for the object having the associated quadric surface comprise three linear measurements taken from a vertex of one part of the hyperboloid to a bounding cone associated with the hyperboloid along orthogonal object axes, one of the object axes passing through the location point.

23. A system as claimed in claim 17, wherein the object orientation parameters for each object comprise three angular measurements specifying an angular relationship of three orthogonal object axes to the width, depth and elevation axes, the object axes intersecting at a given point with respect to the object'"'"'s location point.

24. A system as claimed in claim 17, wherein the location point for at least one of the objects provides a reference for at least one bounding plane associated with that object.

25. A system as claimed in claim 24, wherein the object shape parameters for the object having the associated bounding plane comprise three vector coordinates defining a line segment normal to the bounding plane and extending from the location point to a point on the bounding plane.

26. The system of claim 14, wherein the pixel generating means comprises

means for identifying which of the pixels are superimposed by at least one area in the image plane circumscribed by the image plane curves of one of the objects;

means for determining, for each superimposed pixel, first, second and third Cartesian coordinates identifying a location in an object model space of a point on the associated object for which a projection of the point into the image plane falls within the superimposed pixel; and

computing means for modulating the intensity level of each superimposed pixel in accordance with a value of a texture function computed using the object texture parameters of the associated object and the Cartesian coordinates determined for the superimposed pixel, the object texture parameters for each object being statistically related to the object, and the object texture parameters for the object associated with the superimposed pixels including spatial frequencies related by the texture function to the first and second Cartesian coordinates determined for each superimposed pixel.

27. The system of claim 26, wherein the object texture parameters for the object associated with the superimposed pixels further include phase shift factors related by the texture function to the spatial frequencies.

28. The system of claim 27, wherein the value of the texture function, T, at each superimposed pixel is computed according to an expression:

##EQU6## wherein a number, m, of the spatial frequencies, indicated as ω

_xi, i being an integer varying from 1 to m, are related to the first Cartesian coordinate determined for the pixel, indicated as x, and a number, n, of the spatial frequencies, indicated as ω

_yj, j being an integer varying from 1 to n, are related to the second Cartesian coordinate determined for the pixel, indicated as y, each of the m spatial frequencies ω

_xi having an associated weighting factor, C_xi, and an associated phase shift component, P_xi, the phase shift component P_xi being derived from the associated phase shift factors, each of the n spatial frequencies ω

_yj having an associated phase shift component, P_yj, the phase shift component P_yj being derived from the associated phase shift factors.

29. The system of claim 28, wherein the phase shift components P_xi and P_yj are derived in accordance with two expressions:

space="preserve" listing-type="equation">P.sub.xi =J.sub.xi sin (φ

.sub.xi y)+K.sub.xi sin (ψ

.sub.xi z)

and

space="preserve" listing-type="equation">P.sub.yj =J.sub.yj sin (φ

.sub.yj x)+K.sub.yj sin (ψ

.sub.yj z)

wherein the phase shift factors related to each spatial frequency ψ

_xi are indicated as J_xi, φ

_xi, K_xi and ψ

_xi, the phase shift factors related to each spatial frequency ω

_yj are indicated as J_yj, φ

_yj, K_yj and ψ

_yj, and the third Cartesian coordinate determined for the pixel for which the texture function is being evaluated is indicated as z.

30. The system of claim 19, wherein the pixel generating means comprises:

means for computing the intensity levels of each of the pixels in accordance with a desired image plane intensity function evaluated at the sample points, the pixels being square and of a uniform pixel width, each sample point substantially corresponding with a center of its associated pixel;

alias testing means for identifying as aliased pixels those pixels intersected by a predetermined area about at least one of the image plane curves, said one of the image plane curves defining a boundary curve between first and second regions in the image plane, the image plane intensity function varying at relatively low frequencies within each region and at relatively high frequencies from one region to another;

distance measuring means for determining a weighting distance for each aliased pixel, each weighting distance being defined as a distance measured in pixel widths from the center of the associated pixel to the boundary curve;

weight calculating means for deriving first and second intensity weights for each aliased pixel, the first intensity weight for each aliased pixel whose center lies in the first region and the second intensity weight for each aliased pixel whose center lies in the second region being derived from the associated weighting distance, the remaining intensity weight for each aliased pixel being derived from a negative of the associated weighting distance; and

intensity adjusting means for assigning an antialiased intensity level to each aliased pixel, the antialiased intensity level being substantially equal to a value of the image plane intensity function in the first region multiplied by the first intensity weight plus a value of the image plane intensity function in the second region multiplied by the second intensity weight.

31. The system of claim 30, wherein the predetermined area about the boundary curve includes all points in the image plane within a cutoff distance of the boundary curve, the cutoff distance being substantially equal to the pixel width multiplied by a square root of two.

32. The system of claim 31, wherein the weight calculating means comprises an arithmetic unit which derives each first intensity weight, W₁, and each second intensity weight, W₂, in substantial accordance with two expressions:

space="preserve" listing-type="equation">W.sub.1 =1/2+(3√

2/8)d-(√

2/16)d.sup.3

and

space="preserve" listing-type="equation">W.sub.2 =1-W.sub.1

wherein the weighting distance of the associated pixel is indicated as d.

33. The system of claim 31, wherein the weight calculating means comprises a look up table.

34. The system of claim 14, wherein each subframe processor comprises:

a program unit including means for storing and executing a program in accordance with a data structure of the object lists and the image plane curve coefficients;

a program bus responsive to the program unit for conducting data words and address words;

a plurality of data registers responsive to the program bus;

an accumulator bus responsive to at least one of the data registers;

a multiplier responsive to the program bus and the accumulator bus, said multiplier receiving clocking and control signals from the program unit;

an adder responsive to the program bus and the accumulator bus, said adder receiving clocking and control signals from the program unit;

a buffer bus responsive to the multiplier, the adder, and at least one of the data registers;

a data buffer responsive to the buffer bus and receiving clocking and control signals from the program unit, the buffer bus being further responsive to the data buffer;

a mask register responsive to the buffer bus and receiving clocking and control signals from the program unit, the program bus being further responsive to the mask register;

an address register responsive to the program bus and receiving timing and control signals from the program unit;

an input memory having an address input responsive to the address register, the data buffer being further responsive to the input memory;

an output memory having an address input responsive to the address register, the output buffer being responsive to the data buffer;

an input buffer for receiving object lists and updated image plane curve coefficients for those objects whose projections intersect the associated subframe area in the image plane, the input buffer further receiving the updated viewpoint parameters and transferring the received object lists, updated image plane curve coefficients, and updated viewpoint parameters to the input memory;

an output buffer responsive to the output memory for storing the updated pixel intensity levels;

a scratchpad memory;

at least one scratchpad buffer responsive to the scratchpad memory and the buffer bus, the accumulator bus being further responsive to the scratchpad buffer, the scratchpad memory being responsive to the scratchpad buffer; and

at least one scratchpad address register responsive to the program bus and receiving clocking and control signals from the program unit, the scratchpad memory having an address input responsive to the program bus.

35. The system of claim 14, wherein each object processor comprises:

a program unit including means for storing and executing an algorithm in accordance with a data structure of the object lists;

a program bus responsive to the program unit for conducting data words and address words;

a plurality of data registers responsive to the program bus;

an accumulator bus responsive to at least one of the data registers;

a multiplier responsive to the program bus and the accumulator bus, said multiplier receiving clocking and control signals from the program unit;

an adder responsive to the program bus and the accumulator bus, said adder receiving clocking and control signals from the program unit;

a sine lookup table responsive to the program bus, the accumulator bus being further responsive to the sine lookup table, the sine lookup table receiving clocking and control signals from the program unit;

a buffer bus responsive to the multiplier, the adder, the sine lookup table and at least one of the data registers;

a data buffer responsive to the buffer bus and receiving clocking and control signals from the program unit, the buffer bus being further responsive to the data buffer;

a mask register responsive to the buffer bus and receiving clocking and control signals from the program unit, the program bus being further responsive to the mask register;

an address register responsive to the program bus and receiving timing and control signals from the program unit;

an input memory having an address input responsive to the address register, the data buffer being further responsive to the input memory;

an output memory having an address input responsive to the address register, the output memory being responsive to the data buffer;

an input buffer for receiving assigned object lists from the bulk memory means and updated viewpoint parameters from the viewpoint tracking means, the input buffer transferring the assigned object lists and updated viewpoint parameters to the input memory; and

an output buffer responsive to the output memory for storing the updated image plane curve coefficients.

36. In a system for displaying an image plane on a video raster made up of pixels arranged on scan lines, each pixel having an associated intensity level, a portion of the pixels being superimposed by a projection into the image plane of at least one simulated object, a texture generator comprising

memory means for storing parameters for a texture function, the parameters being statistically related to the simulated object;

means for determining, for each superimposed pixel, first, second and third Cartesian coordinates identifying a location in an object model space of a point on the simulated object for which a projection of the point into the image plane falls within the superimposed pixel;

computing means for modulating the intensity level of each superimposed pixel in accordance with a value of the texture function computed using the Cartesian coordinates determined for that pixel;

comparing means for comparing the value of the texture function for each superimposed pixel with a translucence limit for the object and substituting a background intensity level for the intensity level of each superimposed pixel whose texture function value is less than the translucence limit.

37. The system of claim 36, wherein the translucence limit varies from a maximum value at a center of the object'"'"'s projection to a minimum value at an edge of the object'"'"'s projection, the maximum and minimum values of the translucence limit corresponding to maximum and minimum values of the texture function.

38. The system of claim 36, wherein said parameters include spatial frequencies related by the texture function to the first and second Cartesian coordinates determined for each superimposed pixel, and further include phase shift factors related by the texture function to the spatial frequencies.

39. The system of claim 38, wherein the value of the texture function, T, at each superimposed pixel is computed according to an expression:

##EQU7## wherein a number, m, of the spatial frequencies, indicated as ω

_xi, i being an integer varying from 1 to m, are related to the first Cartesian coordinate determined for the pixel, indicated as x, and a number, n, of the spatial frequencies, indicated as ω

_yj, j being an integer varying from 1 to n, are related to the second Cartesian coordinate determined for the pixel, indicated as y, each of the m spatial frequencies ω

_xi having an associated weighting factor, C_xi, and an associated phase shift component, P_xi, the phase shift component P_xi being derived from the associated phase shift factors, each of the n spatial frequencies ω

_yj having an associated weighting factor, C_yj, and an associated phase shift component, P_yj, the phase shift component P_yj being derived from the associated phase shift factors.

40. The system of claim 39, wherein the phase shift components P_xi and P_yj are derived in accordance with two expressions:

space="preserve" listing-type="equation">P.sub.xi =J.sub.xi sin (φ

.sub.xi y)+K.sub.xi sin (ψ

.sub.xi z)

and

space="preserve" listing-type="equation">P.sub.yj =J.sub.yj sin (φ

.sub.yj x)+K.sub.yj sin (ψ

.sub.yj z)

wherein the phase shift factors related to each spatial frequency ω

_xi are indicated as J_xi, φ

_xi, K_xi and ψ

_xi, the phase shift factors related to each spatial frequency ω

_yj are indicated as J_yj, φ

_yj, K_yj and ψ

_yj, and the third Cartesian coordinate determined for the pixel is indicated as z.

41. In a system for generating a display of an image plane characterized by at least one boundary curve separating first and second regions of different intensities, the display being formed by a plurality of square pixels of uniform width arranged on a video raster, each pixel having an associated intensity level, an antialiasing filter comprising:

alias testing means for identifying as aliased pixels those pixels intersected by a predetermined area about the boundary curve, said predetermined area about said boundary curve including all points in the image plane within a cutoff distance of said boundary curve, the cutoff distance being substantially equal to the pixel width multiplied by a square root of two

distance measuring means for determining a weighting distance for each aliased pixel, each weighting distance being defined as a distance measured in pixel widths from a center of the associated pixel to the boundary curve;

weight calculating means for deriving first and second intensity weights for each aliased pixel, the first intensity weight for each aliased pixel whose center lies in the first region and the second intensity weight for each aliased pixel whose center lies in the second region being derived from the associated weighting distance, the other intensity weight for each aliased pixel being derived from a negative of the associated weighting distance; and

intensity adjusting means for assigning an antialiased intensity level to each aliased pixel, the antialiased intensity level being substantially equal to the associated first intensity weight multiplied by the intensity of the first region plus the associated second intensity weight multiplied by the intensity of the second region.

42. The system of claim 41, wherein the weight calculating means comprises an arithmetic unit which derives each first intensity weight, W₁, and each second intensity weight, W₂, in substantial accordance with two expressions:

space="preserve" listing-type="equation">W.sub.1 =1/2+(3√

2/8)d-(√

2/16)d.sup.3

and

space="preserve" listing-type="equation">W.sub.2 =1-W.sub.1

wherein the weighting distance of the associated pixel is indicated as d.

43. The system of claim 41, wherein the weight calculating means comprises a look-up table.