generalized matryoshka computational design of nesting … · 2017. 10. 13. · generalized...
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Alec JacobsonUniversity of Toronto
Generalized MatryoshkaComputational Design of Nesting Objects
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Previous work enables computational design of reconfigurables
7[Garg et al. 2016]
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Previous work enables computational design of reconfigurables
8[Garg et al. 2016]
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Previous work enables computational design of reconfigurables
9[Garg et al. 2016]
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[Zvyozdochkin & Malyutin 1890]
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We present a method to generalize Matryoshkato arbitrary shapes
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We present a method to generalize Matryoshkato arbitrary shapes
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Nesting requires strict enclosure…
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loose enclosure
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Nesting requires strict enclosure…
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enclosure
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Nesting also requires removal
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enclosed, but not removable loose enclosure
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Nesting also requires removal
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enclosed, but not removable
cut
loose enclosure
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Nesting also requires removal
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enclosed, but not removable loose enclosure
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Nesting also requires removal
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enclosed, but not removable enclosed and removable loose enclosure
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We present highly parallelizable methods to…
• determine feasibility of nesting,
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We present highly parallelizable methods to…
• determine feasibility of nesting,• find maximum scale,
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We present highly parallelizable methods to…
• determine feasibility of nesting,• find maximum scale, and• optimize nesting scale
over some or all parameters
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Our optimization utilizes rigid motionfor tighter nesting
39% 53% 63%
fixed position+rotation fixed rotation free22
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Our optimization utilizes rigid motionfor tighter nesting
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39% 53% 63%
fixed position+rotation fixed rotation free
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Our optimization utilizes rigid motionfor tighter nesting
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39% 53% 63%
fixed position+rotation fixed rotation free
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We define valid self-nesting
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A
AGiven:1. shape ,
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We define valid self-nesting
Given:1. shape ,2. similarity transform ,
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A
A
T T (A)
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We define valid self-nesting
Given:1. shape , 2. similarity transform ,3. cut plane , and
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AT (A)T
A
PP
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We define valid self-nesting
Given:1. shape , 2. similarity transform ,3. cut plane , and4. removal trajectories
28
AT (A)T
A
PP
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We define valid self-nesting
Given:1. shape , 2. similarity transform ,3. cut plane , and4. removal trajectories directions
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AT (A)T
A
PP
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We define valid self-nesting
Must have:1. , and2. no collisions along
either direction after cutting by
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AT (A)
A
P
T (A)⇢
A P
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We define valid self-nesting
Must have:1. , and2. no collisions along
either direction after cutting by
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AT (A)
A
P
T (A)⇢
A PDefinition depends on choice of cut plane and removal directions.
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Some configurations admit perfect self-nesting
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convex shapes?
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Some configurations admit perfect self-nesting
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convex shapes?enclosure is easy ….
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Some configurations admit perfect self-nesting
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convex shapes?enclosure is easy ….but removal depends on cut plane!
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[Zvyozdochkin & Malyutin 1890]
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invalid
valid36
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invalid
valid37
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Perfect self-nesting requires visibility of cut plane at all points along removal directions
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Our tool exploresnesting of arbitrary solid 3D shapes
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Our tool exploresnesting of arbitrary solid 3D shapes
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Our tool exploresnesting of arbitrary solid 3D shapes
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We cast this as a computational design problem
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Manual design with traditional tools would be tortuous
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We cast this as a computational design problem
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Manual design with traditional tools would be tortuous
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We cast this as a computational design problem
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Manual design with traditional tools would be tortuous
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We cast this as a computational design problem
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Manual design with traditional tools would be tortuous
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Take a clue from order-independent transparency by “depth peeling”
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Take a clue from order-independent transparency by “depth peeling”
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Take a clue from order-independent transparency by “depth peeling”
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Take a clue from order-independent transparency by “depth peeling”
[Everitt 2001, Bavoil et al. 2007][Baldacci et al. 2016]
[Shade et al. 1998][Inui & Ohta 2007][Faure et al. 2008]
[Kim et al. 2002][Myszkowski et al.1995, Knott & Pai 2003, Heidelberger et al. 2004]
[Goldfeather et al. 1986, Kelley et al. 1994, Hable & Rossignac 2005]
a.k.a. K-Buffer, Layered Depth Imagestransparencyshape diameterimage-based renderingCNC millingintersection volumeswept volumescollision detectionCSG operations
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Take a clue from order-independent transparency by “depth peeling”
[Everitt 2001, Bavoil et al. 2007][Baldacci et al. 2016]
[Shade et al. 1998][Inui & Ohta 2007][Faure et al. 2008]
[Kim et al. 2002][Myszkowski et al.1995, Knott & Pai 2003, Heidelberger et al. 2004]
[Goldfeather et al. 1986, Kelley et al. 1994, Hable & Rossignac 2005]
a.k.a. K-Buffer, Layered Depth Imagestransparencyshape diameterimage-based renderingCNC millingintersection volumeswept volumescollision detectionCSG operations
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Bad “codes”:• blue before orange• orange before green• orange before front-facing blue
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
Feasible!• all green
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Step 1: we determine feasibility in real-timeby exploiting orthographic rendering
Feasible!• all green
“ping-pong” with 2 buffersGL_SAMPLES_PASSED
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Step 2: binary search to maximize scale
Assume momentarily that shape is convex
Fix cut plane, center of mass, rotation
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Step 2: binary search to maximize scale
Assume momentarily that shape is convex
Fix cut plane, center of mass, rotation
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Step 2: binary search to maximize scale
Assume momentarily that shape is convex
Fix cut plane, center of mass, rotation
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Step 2: binary search to maximize scale
Assume momentarily that shape is convex
Fix cut plane, center of mass, rotation
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Step 2: binary search to maximize scale
Assume momentarily that shape is convex
Fix cut plane, center of mass, rotation
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Step 2: binary search to maximize scale
Assume momentarily that shape is convex
Fix cut plane, center of mass, rotation
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Step 2: binary search to maximize scale
For non-convex shapes binary search is conservative,
but in practice optimal
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77
Step 2: binary search to maximize scale
For non-convex shapes binary search is conservative,
but in practice optimal
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78
Step 2: binary search to maximize scale
For non-convex shapes binary search is conservative,
but in practice optimal
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79
Step 2: binary search to maximize scale
For non-convex shapes binary search is conservative,
but in practice optimal
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Step 2: binary search to maximize scale
For non-convex shapes binary search is conservative,
but in practice optimal
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Step 3: optimize over all parameters
81
70%
0%
70%
maximize scale subject to nesting constraint
non-convex energy landscape
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Step 3: optimize over all parametersvia particle swarm optimization
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k parameter vector as point in nD
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Step 3: optimize over all parametersvia particle swarm optimization
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k parameter vector as point in nD
update each iteration according to “velocity”
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Step 3: optimize over all parametersvia particle swarm optimization
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pull velocity toward personal best and global best of swarm
k parameter vector as point in nD
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Step 3: optimize over all parametersvia particle swarm optimization
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random perturbations
k parameter vector as point in nD
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Naive P-Swarm would treat scale as just another parameter (coordinate)…
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… instead optimize over all others,
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all other parameters
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… instead optimize over all others,
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… instead optimize over all others, and search for max scale
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… instead optimize over all others, and search for max scale
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abort search early if upper bound < best
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Our optimization enables fully automatic Matryoshka generation…
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Automatic Upright Custom Cut Plane
63%60% 56%
fully optimized
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… or partially constrained interactive design
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Automatic Upright Custom Cut Plane
63%60% 56%fixed upright orientation
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… or partially constrained interactive design
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Automatic Upright Custom Cut Plane
63%60% 56%
+ fixed cut plane
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Tool performs fast enough for interaction
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Tool performs fast enough for interaction
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We validate our results via 3D printing
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We accommodate printer tolerances by nesting within an offset surface
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Our tools trivially generalize to nesting disparate shapes
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Limitations & Future Work• no global optimum guarantee
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Limitations & Future Work• no global optimum guarantee• search assumption too conservative
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Limitations & Future Work• no global optimum guarantee• search assumption too conservative• thin shapes don’t rigidly nest well
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Limitations & Future Work• no global optimum guarantee• search assumption too conservative• thin shapes don’t rigidly nest well• deformable nesting?
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Limitations & Future Work• no global optimum guarantee• search assumption too conservative• thin shapes don’t rigidly nest well• deformable nesting?
1. deform during design
112Prevost et al.
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Limitations & Future Work• no global optimum guarantee• search assumption too conservative• thin shapes don’t rigidly nest well• deformable nesting?
1. deform during design2. nest soft physical objects
113Bickel et al.
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Acknowledgements…David LevinNSERC Discovery Grants (RGPIN-2017-05235 & RGPAS-2017-507938)Connaught Fund (NR-2016-17)Adobe Systems Inc.Kevin Gibson, Masha Shugrina, Michael Tao, and Alex Tessier
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