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BACKGROUND OF THE INVENTION
A. Field of the Invention
This invention relates generally to digital image processing and computer
graphics. More particularly, it is concerned with simulating free movement
within a multidimensional environment, which can be either
computer-generated or real.
B. Description of the Related Art
Computer-generated (CG) environments are typically created by representing
objects with polygons and associated computer-generated or photographic
surfaces, or texture maps. Rendering, the construction of an image from a
CG model, can be done from any point of view. See Foley ›J. D. Foley, A.
van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: principles and
practice, 2nd ed., Addison-Wesley, 1987!. As such, it provides
unrestricted simulated movement within the environment. However, the
temporal resolution of unrestricted movement within a realistic CG
environment that one can achieve on today's personal computers is severely
limited by the computational requirements and by the labor of constructing
realistic imagery.
U.S. Pat. No. 4,807,158 to Blanton, et. al discloses a method for reducing
the computational requirements of rendering CG images, which could also be
applied to natural images. First they build a database of images at
selected positions, or "keypoints", within the environment by rendering
them in an off-line process. They store these panoramic images as conic
projections. Then in real time, the application approximates the image at
any position from that at the nearest keypoint. This approach works well
when all objects are about the same distance from the viewer. This is a
good assumption in their application, a flight simulator, but the loss of
parallax would be a severe limitation in many environments. Objects at
different distances move as a unit within the domain of a keypoint, and
parallax is only evident when the keypoint changes.
U.S. Pat. No. 5,396,583 to Chen, et. al captures panoramas and project them
onto cylindrical surfaces for storage. They are able to rapidly project
images from a cylinder to a plane using "scanline coherence".
Unfortunately, like Blanton, their method does support parallax.
McMillan, et. al. ›L. McMillan and G. Bishop, Plenoptic Modeling: An
Image-Based Rendering System, Siggraph '95 Proceedings, 1995!, report a
method that supports parallax, and apply it to natural images. They also
produce a series of reference images off-line, which are captured with a
video camera and re-projected to cylinders for storage. To support
parallax, they calculate the image flow field between adjacent reference
images. Now, when an image is approximated from a nearby reference image,
different parts will move differently. Unfortunately, artifacts are quite
apparent unless the image flow field is extremely accurate. Occluded
regions cause additional artifacts.
The cylindrical surface (Chen and McMillan) is very inefficient for storing
panoramic imagery near the vertical.
Accordingly, the need remains for an improved method for simulating free
movement within a multidimensional environment.
SUMMARY OF THE INVENTION
It is the object of this invention to simulate movement in a
multidimensional space by approximating views at any viewpoint and
orientation, with correct perspective and parallax. An additional object
of this invention is to support stereography. It is a further object of
this invention to provide an efficient method for storing panoramic
imagery, especially for orientations near the vertical.
This invention captures panoramic views at many keypoints in the
environment, preferably using fisheye photography. It stores these views
as projections, from which one can produce views at any position and
orientation.
In the preferred embodiment, the projections are planar, and consist of
polygons that are projections of areas in the environment that are
approximately planar. The locations of these areas are stored, giving the
playback system the three-dimensional information necessary to infer how
the individual polygons move with the viewpoint, and thus simulate
parallax.
Because it simulates parallax, the invention can produce stereographic
images.
The preferred embodiment solves the occlusion problem in a novel way.
Imagery that is occluded at a keypoint but visible at a nearby viewpoint
is added to that keypoint, either by extending existing polygons or by
creating new ones.
BRIEF DESCRIPTION OF THE FIGURES
FIG. 1 is a flow chart illustrating an overview of the invention.
FIG. 2 is a flow chart illustrating an overview of the off-line processing
of the invention for a CG model that takes advantage of access to that
model.
FIG. 3 is a flow chart illustrating how the invention can be used to
eliminate off-line processing for a CG model.
FIG. 4 shows a two dimensional arrangement of keypoints (open circles) with
a potential viewpoint (closed circle).
FIG. 5 represents both a fisheye projection and a spherical projection of
that fisheye projection.
FIG. 6 illustrates the projection from a sphere to a plane.
FIG. 7 shows the relevant parameters in converting origins of a sphere.
FIG. 8 illustrates the projection of an object point to a plane.
DETAILED DESCRIPTION OF THE INVENTION
FIG. 1 shows the overall operation of the invention. Images are captured
either by a fisheye 101 or planar camera 108, and are used to form planar
projections 103 at keypoints; fisheye images are first projected to a
sphere 102. An analysis 104 is performed to segment these images into
polygons that are the projections of approximately planar areas in the
environment. These polygons are extended 105 with occluded imagery and
then compressed 106. The compressed polygons, together with
three-dimensional and other information, are written 107 to
computer-readable storage 120.
The playback system, on receiving a view request 130 specifying a position
and orientation in the environment, determines the nearest keypoint 131
and reads 132 the relevant data from storage 120. It decompresses 133 the
polygons at that keypoint, translates 134 them to the desired position,
rotates 135 them to the desired orientation, and displays 136 the
resulting image.
CG environments can be handled exactly as in FIG. 1, except that the views
are rendered from the model rather than being captured. However, if one
has access to the internals of the model, then one can simplify the
analysis, and even eliminate off-line processing. FIG. 2 shows the
off-line processing that takes advantage of the model. After the model is
created 21, this processing tailors the model 22 for the keypoint by
eliminating detail that is too fine to be realized at that keypoint. It
then extends and renders 23 and compresses 24 the polygons, and writes the
k-point data 25 to computer-readable storage 26.
FIG. 3 shows a self-contained playback system, one that does not require
off-line processing. It is similar to the playback system in FIG. 1,
except that it generates keypoint data as needed. It does this using the
methods illustrated in FIG. 2. In particular, on receiving a view request
321, it determines the nearest k-point 322. If this k-point is close
enough 323 to the viewpoint, it reads the k-point data 324 from
computer-readable storage 310, decompresses the polygons 325, translates
the polygons to the desired position 326, rotates them to the desired
orientation 327, and displays the resulting image 328. If it is determined
323 that no k-point is sufficiently close to the viewpoint, then the CG
model is tailored to the viewpoint 301, to make it a new k-point. It then
extends and renders the polygons 302, compresses them 303, and writes the
k-point data 304 to computer-readable storage 310.
FIG. 4 shows a possible two-dimensional environment with open circles 41,
42, 43 representing keypoints, and a closed circle 45 representing a
viewpoint. The view at the viewpoint 45 is preferably based on that of the
nearest keypoint 41.
The invention consists of the panoramic database, its creation, and its use
to map images to arbitrary positions and orientations.
A. Notation
CG, or Computer Graphics, refers to artificial environments.
A projection of the environment is specified by an orientation, or axis of
projection, a projection surface normal to that axis, and by the center of
projection. The center of projection, or point of view, is the viewpoint;
it corresponds to the nodal point of a camera. Unless otherwise specified,
the projection surface is planar. Other possibilities include cylindrical,
conic, spherical, and fisheye projections. The distance from the center of
projection to the plane of projection is the focal length, and is measured
in pixels.
Polygons, unless otherwise qualified, are projections of approximately
planar areas of the environment. These planar approximations are also
polygonal, and are called e-polygons (e is for environment). The
e-polygons form a "wire-frame model", and the polygons form the "texture
maps". In contrast to conventional CG modeling, however, the environment
description is redundant, with multiple e-polygons and polygons instead of
one.
Images are projections that are large enough to fill a display. Images may
be segmented into polygons.
Projections may be produced directly from the environment or from a CG
model, or they may be approximated from previously calculated
k-projections (key projections) by a mapping process. The viewpoints of
the k-projections are called k-points (keypoints). A k-polygon is a
k-projection that is a polygon, and a k-image is a k-projection that is an
image. We will sometimes omit the "k-" prefix when the context makes it
obvious.
The domain of a k-point is the range of viewpoints and orientations to
which its projections can be mapped. The domain of a k-projection is that
of its k-point. K-polygons with occluded areas may be extended to support
their domains. The domain of a k-point is supported when its k-polygons
contain all of the imagery needed to map to any viewpoint in that domain.
Domains can overlap. Non-overlapping domains of k-points can be constructed
as Voronoi regions, also known as Dirichlet and nearest neighbor regions,
or in other ways.
The mapper produces an image that can be displayed, and is part of the
playback system.
B. Description of the panoramic database
The database, which is in a computer-readable storage medium, consists of a
set of k-projections, together with their k-points, predetermined key
orientations, domains, and other information useful for the creation of
views of the environment. All of the data associated with a k-point is
collectively referred to as k-point data. The k-projections may be
uncompressed, or may be compressed with any of a variety of techniques,
such as MPEG or JPEG. For CG environments, the k-projections that are
rendered at a k-point might be compressed using a graphic representation.
(This is less compression than one could achieve by simply storing the
original CG model, but it leads to faster playback.)
The k-points may be arranged on a two or three dimensional lattice, may be
concentrated in areas of high detail, or may follow some other
arrangement.
In the preferred embodiment, the projections are all planar. However,
alternate embodiments use cylindrical, conic, spherical, or fisheye
projections. One embodiment uses a cylindrical projection for the
"equatorial" region and conic projections for the "poles"; the mathematics
for these projections is described by Blanton and McMillan.
There are many different ways to store the k-projections as planar
projections at a k-point. In the preferred embodiment, one stores a large
number of k-polygons at various orientations. In an alternate embodiment,
one stores one or more k-images at various orientations. The way this is
done will depend on the needs of the application, such as the importance
of the "polar regions". As few as four k-images, arranged tetrahedrally,
provide a full panorama. However, the more k-images, the fewer the total
number of pixels. An obvious arrangement consists of six k-images around
the equator (horizon), three to six k-images about 60.degree. above the
equator, and three to six below. Another possibility is four k-images
about 45.degree. above the equator, and four below.
There is a useful advantage in storing the panorama as planar images, as
opposed to, say, cylindrical ones: the images at the k-points can be
directly displayed without mapping. When one moves rapidly through the
environment, it may be sufficient to display an existing image that has
approximately the correct view point and orientation. This benefit is
somewhat reduced when the k-projections are similarly oriented polygons
rather than images, and reduced more if their orientations are
independent.
When the k-projections are stored as k-images, then these images are
completely segmented into non-overlapping polygons. Whether the polygons
are stored as units or as subunits of an image, each polygon represents
the projection of an area of the environment that is approximately planar.
The planar approximation of this area is also polygonal, and is known as
an e-polygon. The database records the planar equation of each e-polygon.
(Alternately, one can specify the coordinates of its vertices, or
calculate them from the planar equation, the coordinates of the vertices
of the k-polygon, and Eq. 5 and Eq. 10 below.) This permits the mapper to
translate the polygon to another viewpoint in its domain.
It is desirable for storage and computational efficiency that the
k-polygons and their corresponding e-polygons be large and few in number.
However, the larger they are, the worse the planar approximation becomes,
and the larger will be the parallax distortion when a polygon is mapped to
a viewpoint in its domain. This distortion limits their sizes. This
implies that an area of the environment represented by a single e-polygon
(and a single k-polygon) at a distant k-point will likely be represented
by multiple e-polygons at one nearby.
The e-polygons form a three-dimensional model for a k-point and its domain.
In the preferred embodiment only one k-polygon is stored for each
e-polygon. The k-polygon can be formed as a projection of the e-polygon at
any orientation; a convenient orientation is the one pointing to the
center of gravity of the e-polygon. In an alternate embodiment, an
e-polygon is represented by k-polygons at several orientations to minimize
the magnitude of the scale changes and shearing of a mapping, and thus to
minimize aliasing.
In the preferred embodiment, the polygons of a k-point are extended to
include parts of the environment that are occluded at that k-point but are
visible at some point in their domain. (The k-projections cannot be stored
as k-images in this embodiment because the extended polygons would overlap
if placed in a single plane.) Without this embodiment, the mapper must use
k-projections at more than one k-point to map the occluded areas.
In an alternate embodiment, the k-polygons are replaced by curved areas
that more naturally represent the environment.
To minimize the complexity of mapping, the polygons (or curved areas) at
each k-point are described in "list-priority" order. They are split as
necessary to preserve this order over their domain, as described by Foley
in chapter 15.
C. Preparation of the panoramic database for CG environments
One can calculate the k-projections for CG environments using the standard
graphics rendering techniques described by Foley. However, clipping will
be disabled, or at least relaxed, in order to extend the polygons to
support their domain. Similarly, the e-polygons follow directly from the
CG model. This assumes that the embodiment of the invention has access to
the CG model, and that it can control the rendering of that model.
Otherwise, some of the techniques described below for natural environments
will be needed here.
CG models tend to use a large number of e-polygons. These e-polygons are
important for mapping images at nearby viewpoints, but their numbers are
excessive for distant viewpoints. For those viewpoints, it is necessary to
consolidate e-polygons; a good criterion is to consolidate when the
resulting parallax error is below a predetermined limit, preferably 1
pixel. However, merging continues at least until the number of e-polygons
is reduced to a predetermined limit.
In the preferred embodiment the database is built off-line, as shown if
FIG. 2. In an alternate embodiment, projections are rendered from the CG
model as needed, as shown if FIG. 3. The viewer-selected viewpoints, or a
subset of them, become k-points. This alternate embodiment reduces
bandwidth and/or storage requirements, and is important when it is not
practical to pre-calculate the database, or to transmit it dynamically.
Suppose, for example, a game player enters a new environment. Because some
delay can be expected in this case, it may be acceptable to render the
first several projections dynamically. These projections and associated
information are saved as k-point data. As the number of k-points
increases, it will become increasingly likely that one can map
k-projections from existing ones, and response will improve.
D. Preparation of the panoramic database for natural environments
1. Construction of k-projections
There are various ways to construct panoramic views. One embodiment
combines the projections from a video camera (McMillan). The preferred
embodiment uses a camera with a fisheye lens. A single fisheye camera,
pointed forward, will capture about half of the panorama at a k-point,
which will be adequate for some applications, and two may provide a full
panorama. The preferred embodiment uses three horizontal fisheye
projections with 120.degree. between their axes, which provide a full
panorama with ample overlap to minimize boundary artifacts. An alternate
embodiment uses a single fisheye projection pointed upward; a 220.degree.
lens can capture everything except for imagery about 20.degree. or more
below the horizon.
Conceptually, one produces planar projections from the fisheye image by
using a spherical projection as an intermediary. The mapping to a sphere
can be understood from FIG. 5, which represents both the fisheye image and
the sphere, with slightly different interpretations. Subscripts will
distinguish these cases: f for the fisheye and s for the sphere. The point
to be mapped, p, is represented by h, its distance to the origin, O, and
.theta., the angle from the vertical. H is the maximum value of h.
H.sub.f is the radius of the fisheye image. The units of h.sub.f and
H.sub.f are pixels.
For the sphere, the view in FIG. 5 is along the axis of the camera, with
the viewpoint, or center of projection, at the center of the sphere. The
origin, O, is at the intersection of the axis and the surface of the
sphere; it is NOT the center of the sphere. h. is the angle of the arc of
the great circle between O.sub.s and p.sub.s, and H.sub.s is the maximum
angle of view, measured from O.sub.s. For example, for a 180.degree. lens,
H.sub.s =90.degree.=.pi./2. .theta..sub.s is the dihedral angle between 2
axial planes, one through the vertical and one through p.sub.s.
The mapping to the sphere is simply (see, e.g., R. Kingslake, Optical
System Design, p. 87, Academic Press, 1983, for a discussion of the
fisheye):
##EQU1##
One can readily project from a sphere to a plane at any orientation. FIG. 6
shows a plane in cross section tangent to the sphere at point O', which
specified the orientation. For any point in the plane, one finds the
corresponding point in the sphere by extending the radial line from the
sphere to the plane. Then, with f being the focal length and (h.sub.p,
.theta..sub.p) being the polar coordinates of a point in the plane (see
FIG. 5), the point on the sphere is:
##EQU2##
It is desirable to express this in terms of (h.sub.s, .theta..sub.s) so
that Eq. 1 can be used to get the fisheye coordinates. FIG. 7 shows the
quantities used to express the conversion, where all of the curves are
arcs of great circles, and the lengths are measured as angles. The "North
Pole" defines the vertical, and l and l' are the compliments of the
"latitudes" of O and O', respectively. .phi. is the azimuth angle between
O and O'.
From l, l', and .phi., which are independent of p and are assumed known, we
find the intermediate quantities: .PHI., .PHI.', and d from spherical
trigonometry:
##EQU3##
Of course, the entire panorama requires more than one fisheye, and each
fisheye will have its own origin, O. Typically one will select the O
closest to the tangent point, O'.
2. Analyzing images to find the k-polygons and e-polygons
The analysis is more difficult for natural environments. The analysis,
which is individually optimized for each k-point, determines the
three-dimensional environment in sufficient detail to accurately map
k-projections from the k-point to viewpoints in its domain. The
three-dimensional environment is specified by specifying the e-polygons
and their corresponding k-polygons. Since the three-dimensional
environment determines how the k-polygons move with the viewpoint, one can
determine the e-polygons, which approximate this environment, by observing
this motion.
In the preferred embodiment the analysis uses k-images, and one selects a
target image for each reference image being analyzed. The target and
reference images will be assumed to have the same orientation, and will be
associated with nearby k-points. Normally, the nearby k-point will be an
adjacent one; however, occlusion or other reasons may reduce the value of
the adjacent k-points, forcing the use of other ones. In FIG. 4, 42 is a
k-point adjacent to k-point 41, and 43 is a nearby k-point that is not
adjacent to 41.
One then segments the reference image into (non-degenerate) polygons that
are the projections of approximately planar areas in the environment. This
segmentation can be done manually using human understanding of the
environment. Alternatively, one can base it on a three-dimensional model
of the environment, which could be built using a commercially-available
three-dimensional modeling package. One would then have to consolidate
polygons, as with CG environments above.
Each polygon in the reference image is registered with the target image,
and the registration is fitted to a planar model. The following describes
this analysis.
First we need to know how a point in the environment projects to the two
k-points. In particular, we need to know how the two image points are
related. The coordinate system has x and y in the image plane and z
perpendicular to it. Then:
O=origin of the reference image; i.e., its k-point.
O'=origin of the target image
M=O'-O=the displacement between the k-points
m=(M.sub.x, M.sub.y, 0)=components of M in the plane of projection
Q=object point in the O coordinate system
q=(Q.sub.x, Q.sub.y, 0)=components of Q in the plane of projection
p=image point of Q in the reference image
where boldface indicates a vector or point in two to three dimensions, and
quantities with a prime refer to the target image. Then (see FIG. 8):
p/f=q/Q.sub.z Eq. 5
and similarly
p'/f=q'/Q'.sub.z Eq. 6
Also, from the definition of M, it follows:
Q'=Q-M Eq. 7
From these equations, it follows:
p'=p+(M.sub.z p-fm)/(Q.sub.z -M.sub.z) Eq. 8
It is useful to rewrite this as:
1/Q.sub.z =(p'-p)/(M.sub.z p'-p'-fm) Eq. 9
The vector division in Eq. 9 implies that the numerator and denominator
must be parallel, which is guaranteed by Eq. 8.
For Q in an e:polygon, which is assumed planar, we can write:
aQ.sub.x +bQ.sub.y +cQ.sub.z =1 Eq. 10
This equation explicitly constrains the constant term to be non-zero. When
zero, the plane of the e-polygon passes through O, and the e-polygon
projects as a line, or degenerate polygon, contradicting the segmentation.
It follows that 1/Q, is linear in p:
1/Q.sub.z =aQ.sub.x /Q.sub.z +bQ.sub.y /Q.sub.z +c=ap.sub.x /f+bp.sub.y
/f+cEq. 11
using Eq. 5. Combining Eq. 9 and Eq. 11:
ap.sub.x /f+bp.sub.y /f+c=(p'-p)/(M.sub.z p'-fm) Eq. 12
The next step is to determine p' as a function of p over the polygon. There
are various ways of doing this. In the preferred embodiment, the polygon
is broken into 8.times.8 blocks, and each block is compared with the
target to find the best match; e.g., the minimum mean-square-difference.
This is a well-known registration procedure in image coding and analysis.
The possible values of p' are subject to epipolar constraints; in
particular, the direction of p'-p is fixed by Eq. 8.
Eq. 12 is a system of linear equations in a, b, and c. If p' is known for
at least four points in the polygon, this system is over-determined, and
one can readily find the least-squares solution using standard techniques
in linear algebra.
In the preferred embodiment, weighting is first applied to these points to
reflect our confidence in them. One possibility is
weight=<quality of fit>*<reliability of fit>
<quality of fit>=activity/(RMSE+noise) Eq. 13
<reliability of fit>=<useful edge strength>
A quality fit has low RMSE (root-mean-square-error) relative to what is
possible; for various reasons, one cannot expect a very low RMSE for a
very "active" block, even with a very good vector. Activity measures the
roughness of the block.
Furthermore, one cannot find a reliable fit for a block without any
structure. If there is a strong edge, then the component of the vector
p'-p normal to direction of the edge will be reliable, but the parallel
component may not be. Since the direction of this vector is determined by
the epipolar constraint (Eq. 8), the useful edge strength is the component
of the edge direction normal to the constrained direction.
3. Extending the polygons
The polygon is then extended to support the viewpoints in its domain. This
is done by:
finding imagery that is not visible at the k-point because of occlusion,
but that is visible at some viewpoint in the domain;
finding a nearby k-point at which that imagery is visible; and
mapping that imagery from said nearby k-point to said k-point.
In the preferred embodiment, one analyzes the relative motion of adjacent
k-polygons by using Eq. 8 to examine pixels along their common border. If
the polygons separate for any vector M, occlusion results, and the
k-polygon corresponding to the more distant e-polygon must be extended.
This can be done by mapping imagery to it from a k-point in the M
direction. If that k-point does not have the occluded imagery, then it
will be necessary to find a nearby k-point that does. The amount the
k-point must be extended depends on the extent of the domain in the M
direction. In some cases, an additional k-polygon must be created to
support the occluded area.
In the embodiment that stores the polygons without extension, the extension
is done dynamically as needed. This is the case when the k-projections are
stored as k-images.
a) Mapping images to viewpoints
There are several steps in mapping an image at a viewpoint and orientation:
Select the appropriate k-point.
Translate the projections at that k-point to the viewpoint.
Rotate the orientations of the projections to the desired orientation.
Rotation and translation can be done be done in either order. If it is
likely that the viewer will stay at one location for a while and look
around, it is most efficient to translate first, to avoid repeating the
translation. Otherwise, if the application supports stereographic images,
rotation should probably be done first. The two stereographic images will
have the same orientation but slightly different translations.
Of course, rotation and translation can be done at the same time.
The polygons are mapped in list-priority order, with the most distant one
mapped first. Then the mapper needs not determine which pixels will
actually, be displayed. Alternatively, the polygons could be mapped in
reverse order, with the mapper taking care not to overwrite previously
mapped pixels. The extra logic prevents any pixel from being mapped more
than once.
This order is obviously only important when the polygons are combined into
an image. If all of the polygons have the same orientation, it is
desirable to combine then in the translation phase. Otherwise, they should
be combined in the second phase.
i) K-projection selection
The domain of the k-point must contain the viewpoint. If domains overall;,
the mapper will normally select the nearest one. However, if the mapper
has adequate resources, it may map the image more than once and calculate
a weighed average. This will reduce the discontinuity when the viewpoint
moves between domains.
For the embodiment in which k-projections are k-images, it is likely that
the domain of a k-image at a k-point will not cover the entire image at
the viewpoint, and that k-images with different orientations will be
required.
For the embodiment in which an e-polygon at a k-point is represented by
multiple polygons at various orientations, the closest orientation will be
preferable. However, the mapper can map more than once and average.
ii) Translation
Translation requires the use of polygons to achieve correct parallax. The
mapping of any point, p, in a k-polygon to p' at the viewpoint is given by
Eq. 8. M and m now refer to the displacement to the viewpoint, and Q.sub.z
is given by Eq. 11. When necessary to reduce computation, the mapper will
first consolidate polygons into larger ones.
In the preferred embodiment, Eq. 8 and Eq. 11 are only used to translate
the k-polygon's vertices to the viewpoint. Then for each pixel at that
viewpoint, p', the corresponding source pixel, p, is found from the
inverses of these equations:
p=p'-(M.sub.z p'-fm)/(Q.sub.z '+M.sub.z) Eq. 14
1/Q.sub.z '=a'p.sub.x '/f+b'p.sub.y '/f+c' Eq. 15
where:
a'=a/k Eq. 16
b'=b/k
c'=c/k
k=1-aM.sub.x -bM.sub.y -cM.sub.z
iii) Rotation of orientation
If the orientation at a viewpoint changes, then the projection will change.
This orientation change can be represented as a rotation of the coordinate
system. As is well known, the corresponding change in the planar
projection can be readily expressed in homogeneous coordinates (Foley,
McMillan). If R is the rotation matrix that changes the coordinates of an
object point Q to Q': Q'=R Q, then using Eq. 5:
(Q.sub.x ', Q.sub.y ', Q.sub.z ').sup.T =R(Q.sub.x, Q.sub.y, Q.sub.z).sup.T
=Q.sub.z R(Q.sub.x /Q.sub.z, Q.sub.y /Q.sub.z, 1).sup.T =Q.sub.z R(p.sub.x
/f, p.sub.y /f, 1).sup.T
so that:
R(p.sub.x /f, p.sub.y /f, 1).sup.T =(Q.sub.x ', Q.sub.y ', Q.sub.z ').sup.T
/Q.sub.z =(Q.sub.x '/Q.sub.z ', Q.sub.y '/Q.sub.z ', 1).sup.T Q.sub.z
'/Q.sub.z =(p.sub.x '/f, p.sub.y '/f, 1).sup.T w Eq. 17
where w=Q.sub.z '/Q.sub.z is the third component of R(p.sub.x /f, p.sub.y
/f, p/f, 1).sup.T.
Note this transformation is independent of the three-dimensional structure
of the environment, since there is no parallax change. If all of the
polygons have the same orientation and have been combined into an image,
then the image can be transformed as a unit. Otherwise, the polygons must
be rotated individually.
Rotations about an axis in the plane of projection is simpler. For example,
for a rotation of co about the y axis is:
p.sub.x /f=((p.sub.x '/f) cos .omega.+sin .omega.)/D Eq. 18
p.sub.y /f=(p.sub.y '/f)/D
where
D=-(p.sub.x '/f) sin .omega.+cos .omega.
This can be done fairly fast if done column-wise: to find all of the pixels
in the rotated projection for a column defined by fixed p.sub.x ', first
calculate D and p.sub.x, using the above formulas. Then, for each pixel in
that column, calculate p.sub.y, which is a (non-integer) multiple of
p.sub.y ', and interpolate the value from the original projection at
(p.sub.x ', p.sub.y). This is analogous to Chen's scanline coherence.
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