Perspective
multiple perspective, non-linear projection
Multiple perspective
JANUSZ SZCZUCKI
Multiple perspective
Non-linear perspective
Getty Images
Outline
• Representation systems (projection and
perspective)
• Alternative perspectives in Computer Graphics
(multiple perspective, non-linear)
Animation Nonlinearly projected
n∗
Karan Singh†
ersity of Toronto
Plate 2. A multiprojection still life containing 10 camera groups took
phive
urper
hite
usdg
ns.
chbrown
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mevan
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erde
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a views,
ese
green.
he
cts
ed
—
ee-
on-
onwdel
of
about an hour to create with our system. The impressionist style
painting was created in a post!process using Hertzmann’s [7] image
processing algorithm.
(a) Multiple Oblique Projections
Figure 1: A nonlinear projection rendering from Ryan, designed
with our interactive(b)
system
True Perspective Projection
Plate 3.Multiple oblique projections create an artificial sense of
perspective, but still allow some area comparisons. In (b) the pink
building’s rooftop area (arrow) is exaggerated. In (a) it correctly
artists
suchtoas
of linear
perspective
appears
bePicasso
about thebroke
samefrom
size asthe
theconfines
gray rooftop
next to
it.
to integrate the temporal view of a scene as a nonlinear projection.
Linear perspective has the primary advantage of being a simple
and approximate model of the projections associated with both real
cameras and the human visual system. The model also provides
simple, consistent, and easily understood depth cues to the spatial relationships in a three-dimensional scene. From a mathematical standpoint, the pinhole camera model is a linear transformation
that provides an efficient foundation for current graphics pipelines
within which rendering issues such as clipping, shadowing, and illumination are well understood. While a linear perspective view is
• Glassner - multiple perspective rendering
• Agrawala - multiple perspective projection
• Singh - non-linear projection (based on multiple
perspectives and based on distortions of
geometry)
alternate views. This results in an appearance discrepancy between
the local regions of the nonlinearly projected image and the linear
perspective projections used by the animator to define the nonlinear
projection. We introduce two methods for incorporating the multiple views into illumination calculations, and compare these with
the single view illumination model. While both are appropriate, we
argue for the use of the model that is both more predictable with
respect to controlling multiple linear perspective cameras and has
Wyvill and Mcnaughton 1990; Glassner 2000] render scenes correctly, but can be difficult to control by artists and are not well
suited to interactive rendering. Multi-perspective panoramas capture three-dimensional camera paths into a single image [Wood
et al. 1997; Rademacher 1999; Peleg et al. 2000]. While these
approaches render correctly, they provide little control over varying
the importance and placement of different objects in a scene and are
also not well suited to interactive manipulation. Levene describes a
framework for incorporating multiple non-realistic projections defined as radial transformations in eye-space[Levene 1998]. Illumi-
Representation Systems
Consider the problem of
representing or depicting
a scene in a lower
dimension - projection.
Singh, RYAN
Object-based
• Parallel Systems - orthographic, oblique, axonometric,
View-based
• Perspective - linear perspective (single-point, many-
Mixed
• Mixed systems and multiple perspectives in a single
Figure 3: Preproduction artwork for Ryan incorporating an artistic
combination of projection techniques
Figure 4: A preproduction composite of multiple nonlinear deformations
and isometric projections. 130
point), non-linear perspective/projection.
representation.
Parallel Projection
In this case, orthogonals
(projection rays in third
dimension) are parallel.
Borromini
• Object-centric, viewer independent.
• Used in early art, Eastern art, mechanical and
architectural drawings, used in modern art
(cubism, surrealism, expressionism).
Perspective
Bruce MacEvoy, 2004
Linear single-point perspective
Non-linear perspective
• View point-centric.
• Linear vs. curvilinear: uses vanishing curves
instead of vanishing lines or points.
Mixed Systems
School of Athens, Raphael
Breakfast 1914, Juan Gris
• Multiple oblique projections and multiple
perspectives combined into one drawing system.
Done intentionally for different reasons.
3D Representation
• Nonlinear and
multiple
perspective in
3D... time
dimension or
spatial
location as
view point
can be
multiple or
non-linear.
Frank O. Gehry
MIT Strata Center
Alternative Projections/Perspectives to
CG staple (single-point perspective)
• Alternatives are: mixed systems (religious art),
multiple projections (cubism), multiple perspectives
(Hockney, camera path drawings), curvilinear
perspectives (imax, wide-angle, fish eye), non-linear
and 3D warps.
• Used especially for expressive or representational
reasons (i.e. divorce from viewer, make unreal, give
best view).
Applications
"Space Module"
Kasa Usterhjusa
•
Projection on large surfaces
(reduce distortion).
•
•
•
•
Emmersive environments.
•
Expressive CG imagery and
animation.
•
Simultaneous views of scene
and data.
IBR (warping, mapping).
View-independent rendering.
Better representation of data
and local regions.
Non-linear Perspective
Projections in CG
Cubism and Cameras, Glassner
RYAN, Coleman and Singh
• Lots of work being done: image
warping, 3D projections, multiperspective panoramas.
Multiperspective Panoramas for Cel
Animation, Wood et al
• Multi-projection rendering.
Cubism and Cameras... : Glassner
• “Suppose you could take a camera - lens, film, and all
- and stretch it like a blob of Silly Putty. You could wrap
it around people, simultaneously capturing them from
all directions.”
• Cubist Camera: presents many interpretations and
points of view simultaneously.
Cubism and cameras... : Glassner
• Glassner combines multiple non-linear perspectives to
make the imagery seamless and continuous. The nonlinearity of the perspectives allows them to be merged
more easily.
• Implemented as a material plug-in that alters the ray by an
‘eye’ surface and a ‘lens’ surface. Example of nonlinear ray
tracing.
• Nonlinear raytracing handles lighting, but can cause
artifacts.
occlusion relationship
Artistic multiprojection rendering:
Visibility Ord
Agrawala, Zorin,4.1Munzer
•
With
single linear pr
A tool for creating multi-projection
(of a
multiple
perspectives) images and animations.
points in 3D space tha
•
that locallie
on
Given scene geometry, UI to position
and master
cameras. Algorithm for multi-projection that solves
occlusions.
Algorithm must: resolve visibility,
constrain cameras (choose best
projections or perspectives), and
perform interactive rendering.
A
the
C
B
Artistic multiprojection rendering:
Agrawala, Zorin, Munzer
(a) Single projection master
camera view
(b) Multiprojection with
depth compositing only
(c) Multiprojection with occlusion
constraints and depth compositing
• Each scene object is assigned to a local camera.
• Visibility is difficult because of inconsistent depth ordering.
Fig. 2. To reduce the distortion of the column in the single projection image (a) we alter its projection as
shown in figure 5(c). In the multiprojection image, the point on the column (triangle) and the point on the
floor (circle) coincide. With depth-based compositing alone (b) the floor point “incorrectly” occludes the
column point since it is closer to the master camera. However, the column occludes the floor in the master
view. Applying this object-level occlusion constraint during compositing yields the desired image (c).
Use a ‘master camera’ and object-based occlusion
It may be tempting to resolve visibility for the multiprojection image by directly comparing
constraints.
local depth values stored with each image layer. The rendering algorithm would then be quite
simple: we could add the local camera projection to the modeling transformation matrix for each
object and render the scene using a standard z-buffer pipeline without resorting to layer based
compositing. However, this approach would lead to objectionable occlusion artifacts. Suppose
our scene consists of a vase sitting on a table. If we simply add the vase’s local camera projection
into its modeling transform, in most cases the vase will intersect the table. Our algorithm handles
these situations more gracefully by using the master camera to impose visibility ordering while
employing local cameras to provide shape distortion. In our example, the master camera would
be the original projection, in which the vase and table do not intersect, and the local projection
would affect only the shape of the vase without affecting visibility.
• Camera Constraints: for use in animations, based on best
camera placement or movement for local scenes (object
size, fixed-view, fixed-position, direction and orientation).
Artistic multiprojection rendering:
Agrawala, Zorin, Munzer
Plate 2. A mu
about an hour
painting was
processing alg
white
bldg
brown
bldg
van
•
•
•
Plate 1. Our reconstruction of Giorgio de
Fixes
distortions,
creates ofsurrealist
Chirico’s
Mystery and Melancholy
a Street.
The thumbnails show the 5 local camera views,
toony
styles.geometry highlighted in green.
with attached
and
Plate
2. A multiprojection
still lifedisjoint.
containing 10 camera groups too
Good
when
objects
about an hour to create with our system. The impressionist style
painting was created in a post!process using Hertzmann’s [7] imag
Doesn’t
solve lighting and shadow
processing
algorithm.
problems.
Plate 3.Mult
perspective,
A fresh perspecive: Karan Singh
• Creates images from a
nonlinear perspective by
combining the perspectives of
multiple cameras.
• Different from Agrawala
because resulting image of
each object is potentially
influenced by all cameras.
Figure 6: UI Framework (directional influence)
• Can create more continuous
multi-perspectives to actually
attain a ‘non-linear’
perspective.
A fresh perspecive: Karan Singh
(e)
(f)
(g)
Figure 9: Case Study I
Figure 10: Case Study II: Camera configuration
[4] S. Carpendale. A framework for elastic presentation
space. PhD. Dissert., Simon Fraser University, 1999.
[5] J. Dorsey, F. Sillion and D. Greenberg. Design and
Simulation of Opera Lighting and Projection Effects.
Computer Graphics, 25(4):41–50, 1991.
[6] J. Foley, A. van Dam, S. Feiner, J. Hughes. Computer
Graphics, Principles and practice. Addison-Wesley,
Chapter 6:229–284 ,1990.
[7] C. Fu, T. Wong, P. Heng. Warping panorama correctly with triangles. Eurographics Rendering Workshop,1999.
[8] E. H. Gombrich. Art and Illusion. Phaidon, 1960.
[9] J. Gomez, L. Darca, B. Costa and L. Velho. Warping
and Morphing of Graphical Objects . Morgan Kauf(a)
(b)
mann, San Francisco,
California, 1998.
(c)
• Interactive and familiar approach.
• Can weight cameras based on distance from object
cent squares are the 8 viewports. Some of them are horizontally scaled by -1 to unravel the object better but are
also responsible for the twisted behavior of the orange
and royal blue edges. The relative depth of the purplemagenta-pink corner’s viewport is changed to make the
cube projection appear turned inside out in Figure 11b.
Note that the silhouette of the edges remains unchanged.
Changing the depth on the green-limegreen-blue corner’s
viewport in Figure 11c results in a 2D projection of a 3D
projection of a tesseract (a common Escher motif, held
by the jester in his work titled Belvedere).
4
[10] E. Gröller. Nonlinear ray tracing: visualizing
11: Case Study II
strange worlds. The Visual Computer,Figure
11(5):263–
276, 1995.
[11] M. Inakage. Non-Linear Perspective Projections.
Proceedings of the IFIP WG 5.10), 203–215, 1991.
[12] J. Levene. A Framework for Non-Realistic Projections. Master of Engineering Thesis, MIT, 1991.
[13] D. Martin, S. Garcia and J. C. Torres.
Observer dependent deformations in illustration. NonPhotorealistic Animation and Rendering 2000 , Annecy, France, June 5-7, 2000.
[14] N. Max. Computer Graphics Distortion for IMAX
and OMNIMAX Projection. Nicograph ’83 Proceedings 137–159.
[15] S. Peleg, B. Rousso, A. Rav-Acha and A. Zomet.
Mosaicing on Adaptive Manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence,
22(10):1144–1154, 2000.
[16] P. Rademacher and G. Bishop. Multiple-Center-ofProjection Images. SIGGRAPH, 199–206, 1998.
[17] P. Rademacher. View-Dependent Geometry. Computer Graphics, 439–446, 1999.
[18] T. Sederberg and S. Parry. Free-form deformation of solid geometric models. Computer Graphics,
20:151–160, 1986.
[19] S. Seitz and C. Dyer. View Morphing: Synthesizing
3D Metamorphoses Using Image Transforms. Computer Graphics, 21–30, 1996.
[20] Edward Tufte. Envisioning Information. Graphics
Press, 1990.
[21] D. Wood, A. Finkelstein, J. Hughes, C. Thayer and
D. Salesin. Multiperspective Panoramas for Cel Animation. Computer Graphics, 243–250, 1997.
[22] George Wolberg. Digital Image Warping. IEEE
Computer Society Press, 1992.
or viewing direction of camera (localizes effect of
camera).
Conclusion
We have presented a new interactive approach for exploring and rendering 3D objects. Our chief contribution is an
intuitive way for artists to experiment with a 3D subject
and subsequently convey it expressively in a 2D rendering. Aside from its applicability to non-photorealistic and
painterly rendering, the model has wider applications in
scientific visualization, where the limitations of the traditional linear perspective are well recognized [20]. Interaction of illumination models with such non-linear projections is fertile area for future research. In conclusion,
the approach presented in this paper marks a step towards
overcoming the limited expressive potential of existing
projection models. We hope that this work will motivate
further discussion and open the door to an interesting new
type of computer generated imagery.
• Does not handle illumination issues and does not
control global scene coherence.
References
[1] M. Agrawala, D. Zorin and T. Munzner. Artistic
Multiprojection Rendering. Eurographics Rendering
RYAN: Rendering your animation
nonlinearly projected
• Nonlinear projection system that integrates
(a)
(b)
(c)
(d)
into the conventional animation workflow.
• Interactive techniques to control and
render scenes using nonlinear projections.
(e)
(f)
(g)
(h)
• A linear combination of linear perspectives.
Figure 15: Ryan Bathroom Set
RYAN: Rendering your animation
nonlinearly projected
1. Distorts scene geometry so under linear
perspective appears nonlinearly perspective.
2. Provides interactive authoring of nonlinear
projections with scene constraints and linear
perspective cameras.
3. Addresses nonlinear projection’s effect on
rendering and illumination.
In a mixed perspective scene, the goal is to keep qualities of global coherence and local
distortions of geometry and shading result from the changes in perspective.
RYAN: Rendering your animation
nonlinearly projected
• Boss camera is the traditional linear perspective.
Lackey cameras represent local linear views.
• Lackey camera deforms objects (in scene space) so
that through the boss camera, they have view
properties of the lackey, depending on weight of
lackey for the objects.
• Incorporate the multiple views of the lackey
cameras into the illumination calculations.
(a) Pillar, Rt (lackey view)
(b) Constraint deformed
pillar, Rt , Rf (boss view)
RYAN: Rendering your animation
nonlinearly projected
Figure 7: Constraint setup
The resulting spatial constraint matrix1 is:
(a) With constraints
Con = (Cartesianize(Rf Ci Mi Vi )) −1Cartesianize(Rt CbMbVb) (3)
The resulting deformation transform for the lackey camera with a
constraint is similar to Equation 1, but with the constraint matrix
appropriately inserted:
Ai = Ci Mi Vi (Con)(CbMbVb) −1.
(4)
• Constraints maintain global
Con is most often a per object constraint defined for all lackey
cameras, but it can also be global for all objects or even defined on
a selective basis per object per lackey camera.
Figure 7 demonstrates the use of a position constraint on a pillar
seen from an alternate point of view. Figure 7a shows the original
pillar geometry from the lackey camera’s point of view, as well as
a reference frame Rf that indicates a positional constraint on the
geometry. Figure 7b shows the column deformed to have the projective appearance of the lackey camera’s point of view, but seen
from the boss camera, which is located to the right of the lackey
camera. Without application of the constraint, the column would
appear at the same location in screen space in each view. The additional reference frame Rt indicates the deformed position of the
constrained point, and the constraint effects the image space transformation necessary to hold the column in place relative to Rf .
For complex objects it might be necessary to define multiple
constraints. Points on the object are constrained to proximal reference frames. Formally stated, a set of constraints Con1, ..,Conm
are defined using frames Rf 1, .., Rf m and Rt1, .., Rtm. The constraint
matrix Con(P) for a point P is defined using frames Rf (P) and
Rt (P). Rf (P) and Rt (P) are computed as weighted interpolations
of frames Rf 1, .., Rf m and Rt1, .., Rtm, respectively. The weight for
constraint j is inversely proportional 2 to the Euclidean distance
Rt (lackey
view)R . We
(b)precompute
ConstraintApre
deformed
from P(a)
toPillar,
the origin
of frame
i = Ci Mi Vi
fj
pillar,
R
,
R
(boss
view)
t
f
−1
and Aposti = (CbMbVb) to represent the deformation of a point
P, combining Equations 2 and 4 as:
coherence (and stop walls from
collapsing).
(b) Without constraints
Figure 6: Removal of scene constraints: wall and ceiling collapse
into scene
2.1
Constraints
Agrawala et. al.[2002] demonstrate that for multiple linear projections, it can be desirable to constrain objects in space to preserve
their relative position and size in a composited scene. They handle
these constraints with a translation and scale in screen space after
the object has been projected. Singh[2002] allows a user to control
the relative position and size of camera projections through viewport transformations within the canvas.
Figure 6 shows the importance of constraints in our system. The
removal of sceneconstraintscausesthetableon theleft to undergo a
large vertical translation due to the differing positions of the lackey
(a) With constraints
• Camera weights restrict influence.
• Chained lackeys (in-betweens) for
better interpolation between boss
and lackey and for better illumination
blending.
P" = P+
Figure 7: Constraint setup
P(w
∑ iP((Aprei )(Con(P))(Aposti ) − I )).
n
i= 1
The resulting spatial constraint matrix1 is:
2.2
Camera Weight Computation
−1
(5)
Figure
5b, theisgeom
isolated from the projection model,In
and
projection
disa
camera5viewpoint.
Note
As a comparative example, Figure
shows three
var
thewith
viewer
illumination for an object viewed
tworeflecting
cameras.spo
T
rectly
illuminated
by bos
spo
has been deformed such that when
viewed
from the
tocombination
the presenceofofprojecti
the lac
it
appears
as
an
equally
weighted
(a) Camera setup
(b) Boss camera view
by whichview.
the illu
boss camera’s view and to the methods
lackey camera’s
incorporated.
In Figure
5
shows the layout of the cameras,
the undeformed
geom
lationThe
between
boss ar
two spotlights used for illumination.
virtualthe
camera
in the illumination calcul
an interpolated viewpoint.
two modified
In Figure 5b, the geometry is illuminated
withhighlights
respect t
respectone
to directly
both thei
camera viewpoint. Note the two with
highlights:
resulting
the viewer reflecting spotlight 1 are
andblended,
one halfway
to thein
close
to appe
rectly illuminated by spotlight 2,arewith
noenough
illumination
e
this illumination
techniq
to the presence of the lackey camera.
Figures 5c and
5d
(b) Boss camera view
ation
ofthe
images
such
as s
methods by which the illumination
from
lackey
camera
incorporated. In Figure 5c a virtual camera representing a
lation between the boss and lackey cameras is used as the
4 theImplementat
(c) Nonlinear projection
(d) Nonlinear
projection (corin the illumination
calculations for
entire object, result
(wrong shadows and shadrect shadows
and shading)
two modified
highlights. In Figure 5d, the object is ill
This
section
describes
ing)
with respect to both the boss and
lackey
cameras,
and tht
ten as a plug-in
to the
a
are blended, resulting in four attenuated
highlights,
alth
the interface
the system
are close enough to appear as single
stretchedtohighlight.
Figure 10: Shadows
Singh[2002].
defor
this illumination technique to stylized
shaders The
allows
fo
rated into
ation of images such as seen in Figure
9. Maya is then p
e next section).
RYAN: Rendering your animation
nonlinearly projected
• Use original geometry so
shading is not based on the
deformed geometry.
• Illuminate by blending
illumination of boss and lackey
cameras, or set a single view
point for lighting.
3.3 (d)Illumination
Nonlinear projection (cor-
4
Implementation
4.1
User Interface
RYAN: Rendering your animation
!"#$%&!'()'*+(,&"-.
nonlinearly projected
!
References
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Figure 13: Bringing occluded regions into view with nonlinear perspective
progresses. We recognize, however, that manipulating many cameras can be a complicated task. The development of higher level
techniques for manipulating multiple cameras is a subject of future
work.
In summary, this paper presents a new formulation for interactive
nonlinear projections that addresses spatial scene coherence, shadows, and illumination, as well as their integration into current production pipelines. Practical methods of constructing various nonlinear projection effects are shown. Our results showcase the use of
our technique in the commercial animation production Ryan.
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References
•
Maneesh Agrawala, Denis Zorin, Tamara Munzner - Artistic Multiprojection Rendering, Appears in
Eurographics Rendering Workshop 2000,
•
Fred Dubery and John Willats, Perspective and Other Drawing Systems by in Back Flap
•
Patrick Coleman and Karan Singh, RYAN: Rendering Your Animation Nonlinearly projected. NPAR 2004.
•
Karan Singh, A Fresh Perspective.
•
Daniel N. Wood, Adam Finkelstein, John F. Hughes, Craig E. Thayer, David H. Salesin - Multiperspective
Panoramas for Cel Animation, Proceedings of SIGGRAPH 97
•
Glassner, Andrew S., "Cubism and Cameras: Free-form Optics for Computer Graphics", Microsoft
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•
http://www.gettyimags.com
•
EECS MIT http://www.eecs.mit.edu
•
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•
Janusz Szczucki - Multiple perspective. Mala Gallery, Warsaw, Poland, January 2000