Computer graphics processing and selective visual display system – Computer graphics processing – Three-dimension
Reexamination Certificate
2000-04-11
2003-04-22
Nguyen, Phu K. (Department: 2671)
Computer graphics processing and selective visual display system
Computer graphics processing
Three-dimension
Reexamination Certificate
active
06552725
ABSTRACT:
FIELD OF THE INVENTION
The invention relates generally to the area of computer graphics. It more particularly relates to the use of Coons patches to represent surfaces and the preparation of these patches prior to the rendering of a raster image.
BACKGROUND OF THE INVENTION
Computer graphics have become an integral part of many people's lives. The use of graphics in computers has risen dramatically since the use of graphical interfaces for both operating systems and the programs that run on them became commonplace. The incorporation of more detailed graphics has also been a major factor in the rapid increase in memory capacity and processing power in the computer industry. The increased use of graphics and more powerful computing resources have also increased the expectations of the end user. Computer generated graphics must be of high quality with little if any noticeable degradation of image compared to the ‘real thing’. These demands for high quality graphics have penetrated all levels of computing. This places great demands on the computer systems that are running these graphics intensive programs.
An integral step in the production and manipulation of computer graphics is rendering. In this step an image is often converted into polygons with linear line segments for edges. When the image is in this form the edges of the polygon can be manipulated numerically and the image is altered on the display. However, rendering of an image is not always straight forward. The image should be prepared for rendering such that it appears that same after it is rendered.
Surfaces in a graphics image are often represented by a mesh of surface ‘patches’. A Coons patch can be used for this representation. A Coons patch is a specialization of a bicubic (Bezier) patch that is bounded by four arbitrary curves.
An n
th
-order Bezier curve is defined parametrically by the following equation: where p
1
are the control points.
σ
(
t
)
=
∑
1
n
ρ
i
(
n
!
i
!
(
n
-
i
)
!
)
(
t
′
(
1
-
t
)
)
′
A Coons patch is a mapping of points in a unit square in (u,v) co-ordinate space, FIG.
1
(
a
), to a patch in (x,y) co-ordinate space, FIG.
1
(
b
). The patch in (x,y) co-ordinate space is bounded by four cubic Bezier curves; c
1
, c
2
, d
1
and d
2
. Line
100
(FIG.
1
(
a
)) is a line of constant-u while line
110
is a line of constant-v. The (v=0) and (v=1) edges of the unit square map to curves c
1
and c
2
, respectively. Similarly the (u=0) and (u=1) edges map to curves d
1
and d
2
, respectively. Following the notation used in documentation of Adobe Postscript Coons patch fills (Smooth Shading, Adobe Technical Note #5600, Oct. 10, 1997, Adobe Systems Incorporated), the mapping from a point (u,v) to a point (x,y) equals
S(u,v)=(S
x
(u,v), S
y
(u,v))
and is given by the equations
S
c
(u,v)=(1−v)*c
1
(u)+(v)*c
2
(u)
S
d
(u,v)=(1−u)*d
1
(v)+(u)*d
2
(v)
S
b
(u,v)=(1−v)*[(1−v)c
1
(0)+(u)*c
1
(1)]+(1−v)*[(1−u)c
1
(0)+(u)*c
2
(1)]
S=S
c
+S
d
−S
b
Surfaces patches are used to represent the shape, shading and colours of surfaces. The surface whose shape these patches are representing will quite likely have some contouring. This may take the form of slight waves in the surface or it may take the form of folds.
A Coons patch may contain regions that contain folds which will cause one part of the patch to lie on top of the other. A fold is defined as a region in (x,y) co-ordinate space in which multiple points in (u,v) co-ordinate space map to the same point in (x,y) space. Two Coons patches that contain folds are shown in FIG.
2
. The fold in FIG.
2
(
a
) is created by a loop (
200
) in the patch edge curve while the fold in FIG.
2
(
b
) is created by the edge bending back on itself.
Because a fold maps several points from one co-ordinate space to a single point in another co-ordinate space there may be errors in point priority. Point priority is defined as follows. Assume points (u
1
,v
1
) and (u
2
,v
2
) in (u,v) co-ordinate space map to the same point in (x,y) co-ordinate space. The point (u
1
,v
1
) has a higher priority than point (u
2
,v
2
) if v
3
>v
1
or if v
2
=v
1
and u
2
>u
1
. While the presence of folds gives Coons patches a 3-D appearance it is not a 3-D surface as the point priority is determined by a 2-D quantity. The mapping of points associated with a fold can cause point priority errors during rendering that will result in a noticeable reduction in image quality. Regions that do not contain folds are referred to as a flat.
A Coons patch can also used to represent the colours of a surface. A colour is assigned to each of the corners of the unit square in (u,v) co-ordinate space and colours inside the square are calculated by bilinear interpolation of the corner colours. The colour of a point in (x,y) space is then determined by the colour at the (u,v) point mapped to it. When multiple points in (u,v) space map to a single point in (x,y) space the colour of that point in (x,y) space is taken to be the colour of the (u,v) source point having the highest priority.
These are several methods that have been used to prepare a Coons patch for rendering. The patch can be subdivided along curves of constant-u and constant-v. This method is illustrated in FIG.
3
. These lines in (u,v) co-ordinate space map to Bezier curves in (x,y) co-ordinate space. The sub-patches are also Coons patches. Colours at the new corner points of the sub-patches are calculated from the original patch colours by bilinear interpolation in (u,v) co-ordinate space. Patch subdivision is terminated when the patch shape and colour meet some linearity criteria across the patch and/or when the sub-patch size, as measured by its bounding rectangle, falls below a given threshold. This threshold is typically a small multiple of the dimensions of a raster image pixel. The sub-patches can then be approximated as quadrilaterals and rendered with Gouraud shading (i.e. bilinear colour interpolation). This method is computationally intensive as the resultant sub-patches, which are converted to polygons for rendering are very small.
A method of rendering bicubic patches using Coons patch approximations was disclosed in U.S. Pat. No. 5,063,375 of Lien et. al., issued on Nov. 5, 1991. Multiple single-pixel-wide Bezier curves are drawn to render the surface. The curves are drawn close enough to one another that there are no gaps between them. Each curve is drawn with the v-co-ordinate fixed. The v-co-ordinate is stepped from zero to one with the step size determined dynamically to ensure that are no gaps between the curves. For each curve the endpoint colours are determined by interpolation between corner colours. Pixel colours along the curve are then interpolated between the endpoint colours according to the u-value at each point along the curve.
Goel et.al. disclose a method for rendering surfaces using Bezier patches in U.S. Pat. No. 5,995,109 issued on Nov. 30, 1999. The proposed method is able to render patches with contours but it does not discuss the presence of folds in the patches. The patch is first tested for flatness. If the patch is flat it is divided into triangles for rendering. If the patch is not found to be flat i.e. the patch normal deviates from vertical by more than a prescribed amount, it is divided into sub-patches. The testing and division is continued until the sub-patches are considered to be flat. Once the patches are found to have the desired surface quality they are converted into sets of triangles for which “established rendering hardware exists”.
The rendering of a Coons patch that contains fold involves considerable computational effort. This computation is associated with the subdivision of the Coons patch or part of a Coons patch containing folds. It is also associated with maintaining point priorities throughout. This is not the case for flat patches. Therefo
Houtman Reynold
Sheikh Haroon
Whytock Chris
Corel Corporation
Hayes & Soloway P.C.
Nguyen Phu K.
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