Abstract

Humans as designers have quite versatile problem-solving strategies. Computer agents on the other hand can access large-scale computational resources to solve certain design problems. Hence, if agents can learn from human behavior, a synergetic human-agent problem-solving team can be created. This paper presents an approach to extract human design strategies and implicit rules, purely from historical human data, and use that for design generation. A two-step framework that learns to imitate human design strategies from observation is proposed and implemented. This framework makes use of deep learning constructs to learn to generate designs without any explicit information about objective and performance metrics. The framework is designed to interact with the problem through a visual interface as humans did when solving the problem. It is trained to imitate a set of human designers by observing their design state sequences without inducing problem-specific modeling bias or extra information about the problem. Furthermore, an end-to-end agent is developed that uses this deep learning framework as its core in conjunction with image processing to map pixel-to-design moves as a mechanism to generate designs. Finally, the designs generated by a computational team of these agents are then compared with actual human data for teams solving a truss design problem. Results demonstrate that these agents are able to create feasible and efficient truss designs without guidance, showing that this methodology allows agents to learn effective design strategies.

References

References
1.
Campbell
,
M.
,
Hoane
,
A. J.
, and
Hsu
,
F.
,
2002
, “
Deep Blue
,”
Artif. Intell.
,
134
(
1
), pp.
57
83
. 10.1016/S0004-3702(01)00129-1
2.
Mnih
,
V.
,
Kavukcuoglu
,
K.
,
Silver
,
D.
,
Rusu
,
A. A.
,
Veness
,
J.
,
Bellemare
,
M. G.
,
Graves
,
A.
,
Riedmiller
,
M.
,
Fidjeland
,
A. K.
,
Ostrovski
,
G.
,
Petersen
,
S.
,
Beattie
,
C.
,
Sadik
,
A.
,
Antonoglou
,
I.
,
King
,
H.
,
Kumaran
,
D.
,
Wierstra
,
D.
,
Legg
,
S.
, and
Hassabis
,
D.
,
2015
, “
Human-Level Control Through Deep Reinforcement Learning
,”
Nature
,
518
(
7540
), p.
529
. 10.1038/nature14236
3.
Silver
,
D.
,
Huang
,
A.
,
Maddison
,
C. J.
,
Guez
,
A.
,
Sifre
,
L.
,
van den Driessche
,
G.
,
Schrittwieser
,
J.
,
Antonoglou
,
I.
,
Panneershelvam
,
V.
,
Lanctot
,
M.
,
Dieleman
,
S.
,
Grewe
,
D.
,
Nham
,
J.
,
Kalchbrenner
,
N.
,
Sutskever
,
I.
,
Lillicrap
,
T.
,
Leach
,
M.
,
Kavukcuoglu
,
K.
,
Graepel
,
T.
, and
Hassabis
,
D.
,
2016
, “
Mastering the Game of Go With Deep Neural Networks and Tree Search
,”
Nature
,
529
(
7587
), p.
484
. 10.1038/nature16961
4.
Brown
,
N.
, and
Sandholm
,
T.
,
2018
, “
Superhuman AI for Heads-Up No-Limit Poker: Libratus Beats Top Professionals
,”
Science
,
359
(
6374
), pp.
418
424
. 10.1126/science.aao1733
5.
Vinyals
,
O.
,
Babuschkin
,
I.
,
Chung
,
J.
,
Mathieu
,
M.
,
Jaderberg
,
M.
,
Czarnecki
,
W. M.
,
Dudzik
,
A.
,
Huang
,
A.
,
Georgiev
,
P.
,
Powell
,
R.
,
Ewalds
,
T.
,
Horgan
,
D.
,
Kroiss
,
M.
,
Danihelka
,
I.
,
Agapiou
,
J.
,
Oh
,
J.
,
Dalibard
,
V.
,
Choi
,
D.
,
Sifre
,
L.
,
Sulsky
,
Y.
,
Vezhnevets
,
S.
,
Molloy
,
J.
,
Cai
,
T.
,
Budden
,
D.
,
Paine
,
T.
,
Gulcehre
,
C.
,
Wang
,
Z.
,
Pfaff
,
T.
,
Pohlen
,
T.
,
Yogatama
,
D.
,
Cohen
,
J.
,
McKinney
,
K.
,
Smith
,
O.
,
Schaul
,
T.
,
Lillicrap
,
T.
,
Apps
,
C.
,
Kavukcuoglu
,
K.
,
Hassabis
,
D.
, and
Silver
,
D.
,
2019
, AlphaStar: Mastering the Real-Time Strategy Game StarCraft II.
6.
Cross
,
N.
,
2004
, “
Expertise in Design: An Overview
,”
Des. Stud.
,
25
(
5
), pp.
427
441
. 10.1016/j.destud.2004.06.002
7.
Newell
,
A.
, and
Simon
,
H. A.
,
1972
,
Human Problem Solving
,
Prentice-Hall Inc.
,
Upper Saddle River, NJ
.
8.
Daly
,
S.
,
McKilligan
,
S.
,
Christian
,
J.
,
Seifert
,
C.
, and
Gonzalez
,
R.
,
2012
, “
Design Heuristics in Engineering Concept Generation
,”
J. Eng. Edu.
,
101
(
4
), pp.
601
629
.
9.
Ross
,
S.
,
2013
, “
Interactive Learning for Sequential Decisions and Predictions
,” Ph.D. thesis,
Carnegie Mellon University
.
10.
Yannakakis
,
G. N.
, and
Togelius
,
J.
,
2018
,
Artificial Intelligence and Games
,
Springer Publishing Company, Inc.
, New York
.
11.
Payne
,
J. W.
,
Bettman
,
J. R.
, and
Johnson
,
E. J.
,
1993
,
The Adaptive Decision Maker
,
Cambridge University Press
,
New York, NY
.
12.
Busemeyer
,
J. R.
, and
Townsend
,
J. T.
,
1993
, “
Decision Field Theory: A Dynamic-Cognitive Approach to Decision Making in an Uncertain Environment
,”
Psychol. Rev.
,
100
(
3
), pp.
432
459
. 10.1037/0033-295X.100.3.432
13.
Singer
,
D. J.
,
Doerry
,
N.
, and
Buckley
,
M. E.
,
2009
, “
What Is Set-Based Design?
,”
Naval Eng. J.
,
121
(
4
), pp.
31
43
. 10.1111/j.1559-3584.2009.00226.x
14.
McComb
,
C.
,
Cagan
,
J.
, and
Kotovsky
,
K.
,
2017
, “
Capturing Human Sequence-Learning Abilities in Configuration Design Tasks Through Markov Chains
,”
ASME J. Mech. Des.
,
139
(
9
), p.
091101
. 10.1115/1.4037185
15.
McComb
,
C.
,
Cagan
,
J.
, and
Kotovsky
,
K.
,
2017
, “
Mining Process Heuristics From Designer Action Data via Hidden Markov Models
,”
ASME J. Mech. Des.
,
139
(
11
), p.
111412
. 10.1115/1.4037308
16.
Finger
,
S.
, and
Dixon
,
J. R.
,
1989
, “
A Review of Research in Mechanical Engineering Design. Part II: Representations, Analysis, and Design for the Life Cycle
,”
Res. Eng. Des.
,
1
, pp.
121
137
. 10.1007/BF01580205
17.
McComb
,
C.
,
Cagan
,
J.
, and
Kotovsky
,
K.
,
2017
, “Utilizing Markov Chains to Understand Operation Sequencing in Design Tasks,”
Design Computing and Cognition
, Vol.
16
,
J. S.
Gero
, ed.,
Springer International Publishing
,
Cham
, pp.
401
418
.
18.
Raina
,
A.
,
McComb
,
C.
, and
Cagan
,
J.
,
2018
, “
Design Strategy Transfer in Cognitively-Inspired Agents
,”
44th Design Automation Conference
,
Quebec City Convention Center
,
Quebec City, Canada
.
19.
Brooks
,
R. A.
,
1991
, “
New Approaches to Robotics
,”
Science
,
253
(
5025
), pp.
1227
1232
. 10.1126/science.253.5025.1227
20.
Athavankar
,
U. A.
,
1997
, “
Mental Imagery As a Design Tool
,”
Cybern. Syst.
,
28
(
1
), pp.
25
42
. 10.1080/019697297126236
21.
Goldschmidt
,
G.
,
1992
, “
Serial Sketching: Visual Problem Solving in Designing
,”
Cyber. Syst.
,
23
(
2
), pp.
191
219
. 10.1080/01969729208927457
22.
Yin
,
Y. H.
,
Xie
,
J. Y.
,
Xu
,
L. D.
, and
Chen
,
H.
,
2012
, “
Imaginal Thinking-Based Human-Machine Design Methodology for the Configuration of Reconfigurable Machine Tools
,”
IEEE Trans. Ind. Inf.
,
8
(
3
), pp.
659
668
. 10.1109/TII.2012.2188900
23.
Yin
,
Y. H.
,
Zhou
,
C.
, and
Zhu
,
J. Y.
,
2010
, “
A Pipe Route Design Methodology by Imitating Human Imaginal Thinking
,”
CIRP Ann.
,
59
(
1
), pp.
167
170
. 10.1016/j.cirp.2010.03.096
24.
Diez
,
M.
,
Campana
,
E. F.
, and
Stern
,
F.
,
2015
, “
Design-Space Dimensionality Reduction in Shape Optimization by Karhunen–Loève Expansion
,”
Comput. Methods Appl. Mech. Eng.
,
283
(
2015
), pp.
1525
1544
. 10.1016/j.cma.2014.10.042
25.
D’Agostino
,
D.
,
Serani
,
A.
,
Campana
,
E. F.
, and
Diez
,
M.
,
2018
, “Nonlinear Methods for Design-Space Dimensionality Reduction in Shape Optimization BT—Machine Learning, Optimization, and Big Data,”
Machine Learning, Optimization, and Big Data: Second International Workshop
,
G.
Nicosia
,
P.
Pardalos
,
G.
Giuffrida
,
R.
Umeton
, eds.,
Springer International Publishing
,
Cham
, pp.
121
132
.
26.
Chen
,
W.
,
Fuge
,
M.
, and
Chazan
,
J.
,
2017
, “
Design Manifolds Capture the Intrinsic Complexity and Dimension of Design Spaces
,”
ASME J. Mech. Des.
,
139
(
5
), p.
051102
. 10.1115/1.4036134
27.
Yumer
,
M. E.
,
Asente
,
P.
,
Mech
,
R.
, and
Kara
,
L. B.
,
2015
, “
Procedural Modeling Using Autoencoder Networks
,”
Proceedings of the 28th Annual ACM Symposium on User Interface Software & Technology
,
New York, NY
,
Nov. 11–15
, pp.
109
118
.
28.
Guo
,
T.
,
Lohan
,
D.
,
Cang
,
R.
,
Ren
,
Y.
, and
Allison
,
J.
,
2018
,
An Indirect Design Representation for Topology Optimization Using Variational Autoencoder and Style Transfer
,
American Institute of Aeronautics and Astronautics Inc, AIAA
,
Kissimmee, FL
.
29.
D’Agostino
,
D.
,
Serani
,
A.
,
Campana
,
E.
, and
Diez
,
M.
,
2018
, “
Deep Autoencoder for Off-Line Design-Space Dimensionality Reduction in Shape Optimization
,”
AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
,
Kissimmee, FL
,
Jan. 8–12
.
30.
Hinton
,
G. E.
, and
Salakhutdinov
,
R. R.
,
2006
, “
Reducing the Dimensionality of Data With Neural Networks
,”
Science
,
313
(
5786
), p.
504
. 10.1126/science.1127647
31.
Bengio
,
Y.
,
Courville
,
A. C.
, and
Vincent
,
P.
,
2013
, “
Representation Learning: A Review and New Perspectives
,”
IEEE Trans. Patttern. Anal. Mac. Intell
,
35
(
8
), pp.
1798
1828
.
32.
McComb
,
C.
,
2018
, “Towards the Rapid Design of Engineered Systems Through Deep Neural Networks,”
Design Computing and Cognition '18. DCC 2018
,
J. S.
Gero
, ed.,
Springer
,
Cham
.
33.
LeCun
,
Y.
,
Bottou
,
L.
, and
Haffner
,
P.
,
1998
, “
Gradient-Based Learning Applied to Document Recognition
,”
Proceedings of the IEEE
,
86
(
11
), pp.
2278
2324
.
34.
Kingma
,
D. P.
, and
Welling
,
M.
,
2013
, “
Auto-Encoding Variational Bayes
,” arXiv preprint. https://arxiv.org/abs/1412.6980
35.
Zhang
,
Y.
,
Chen
,
A.
,
Peng
,
B.
,
Zhou
,
X.
, and
Wang
,
D.
,
2019
, “
A Deep Convolutional Neural Network for Topology Optimization With Strong Generalization Ability
,”
arXiv preprint
. https://arxiv.org/abs/1901.07761
36.
Banga
,
S.
,
Gehani
,
H.
,
Bhilare
,
S.
,
Patel
,
S.
, and
Kara
,
L.
,
2018
, “
3D Topology Optimization Using Convolutional Neural Networks
,”
arXiv preprint
. https://arxiv.org/abs/1808.07440
37.
Burnap
,
A.
,
Liu
,
Y.
,
Pan
,
Y.
,
Lee
,
H.
,
Gonzalez
,
R.
, and
Papalambros
,
P. Y.
,
2016
, “
Estimating and Exploring the Product Form Design Space Using Deep Generative Models
,”
ASME. International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 2A: 42nd Design Automation Conference
,
Charlotte, NC
,
Aug. 21–24
, p.
V02AT03A013
.
38.
Carroll
,
J. D.
,
1963
, “
Functional Learning: The Learning of Continuous Functional Mappings Relating Stimulus and Response Continua
,”
ETS Res. Bull. Ser.
,
1963
(
2
), p.
i-144
. 10.1002/j.2333-8504.1963.tb00958.x
39.
Koh
,
K.
, and
Meyer
,
D. E.
,
1991
, “
Function Learning: Induction of Continuous Stimulus-Response Relations
,”
J. Exp. Psychol. Learn Mem. Cogn.
,
17
(
5
), pp.
811
836
. 10.1037/0278-7393.17.5.811
40.
DeLosh
,
E. L.
,
Busemeyer
,
J. R.
, and
McDaniel
,
M. A.
,
1997
, “
Extrapolation: The Sine qua Non for Abstraction in Function Learning
,”
J. Exp. Psychol. Learn Mem. Cogn.
,
23
(
4
), pp.
968
986
. 10.1037/0278-7393.23.4.968
41.
Busemeyer
,
J. R.
,
Byun
,
E.
,
Delosh
,
E. L.
, and
McDaniel
,
M. A.
,
1997
, “Learning Functional Relations Based on Experience With Input-Output Pairs by Humans and Artificial Neural Networks,”
Knowledge, Concepts and Categories
,
K.
Lamberts
,
D. R.
Shanks
, eds.,
The MIT Press
,
Cambridge, MA
, pp.
408
437
.
42.
Spelke
,
E. S.
,
Gutheil
,
G.
, and
Van de Walle
,
G.
,
2009
, “
The Development of Object Perception in Humans
,”
F1000 Biology Reports
,
1
(
56
).
43.
Baillargeon
,
R.
,
Li
,
J.
,
Ng
,
W.
, and
Yuan
,
S.
,
2008
, “An Account of Infants’ Physical Reasoning,”
Learning and the Infant Mind
,
A.
Woodward
,
A.
Needham
, eds.,
Oxford University Press
,
New York, NY, US
, pp.
66
116
.
44.
Bates
,
C.
,
Yildirim
,
I.
,
Tenenbaum
,
J. B.
, and
Battaglia
,
P. W.
,
2015
, “
Humans Predict Liquid Dynamics Using Probabilistic Simulation
,”
Cog. Sci.
, pp.
171
177
.
45.
Gershman
,
S. J.
,
Horvitz
,
E. J.
, and
Tenenbaum
,
J. B.
,
2015
, “
Computational Rationality: A Converging Paradigm for Intelligence in Brains, Minds, and Machines
,”
Science
,
349
(
6245
), pp.
273
278
. 10.1126/science.aac6076
46.
Kulkarni
,
T. D.
,
Narasimhan
,
K.
,
Saeedi
,
A.
, and
Tenenbaum
,
J. B.
,
2016
, “Hierarchical Deep Reinforcement Learning: Integrating Temporal Abstraction and Intrinsic Motivation,”
Proceedings of the 30th International Conference on Neural Information Processing Systems
, Vol.
NIPS'16
,
Curran Associates Inc.
,
New York
, pp.
3682
3690
.
47.
Biederman
,
I.
,
1987
, “
Recognition-by-Components: A Theory of Human Image Understanding
,”
Psychol. Rev.
,
94
(
2
), pp.
115
147
. 10.1037/0033-295X.94.2.115
48.
Thrun
,
S.
, and
Pratt
,
L.
,
1998
,
Learning to Learn: Introduction and Overview
,
Kluwer Academic Publishers
,
Norwell, MA
.
49.
Lake
,
B. M.
,
Ullman
,
T. D.
,
Tenenbaum
,
J. B.
, and
Gershman
,
S. J.
,
2017
, “
Building Machines That Learn and Think Like People
,”
Behav. Brain Sci.
,
40
(
e253
).
50.
Pomerleau
,
D. A.
,
1989
, “ALVINN: An Autonomous Land Vehicle in a Neural Network,”
Advances in Neural Information Processing Systems 1
,
D. S.
Touretzky
, ed.,
Morgan-Kaufmann
,
San Francisco, CA
, pp.
305
313
.
51.
Billard
,
A.
, and
Matarić
,
M. J.
,
2001
, “
Learning Human Arm Movements by Imitation:: Evaluation of a Biologically Inspired Connectionist Architecture
,”
Rob. Auton. Syst.
,
37
(
2
), pp.
145
160
. 10.1016/S0921-8890(01)00155-5
52.
Finn
,
C.
,
Yu
,
T.
,
Zhang
,
T.
,
Abbeel
,
P.
, and
Levine
,
S.
,
2017
, “One-Shot Visual Imitation Learning via Meta-Learning,”
Proceedings of the 1st Annual Conference on Robot Learning
,
S.
Levine
,
V.
Vanhoucke
,
K.
Goldberg
, eds.,
PMLR
,
Long Beach, CA
, pp.
357
368
.
53.
Hester
,
T.
,
Vecerik
,
M.
,
Pietquin
,
O.
,
Lanctot
,
M.
,
Schaul
,
T.
,
Piot
,
B.
,
Sendonaris
,
A.
,
Dulac-Arnold
,
G.
,
Osband
,
I.
,
Agapiou
,
J.
,
Leibo
,
J. Z.
, and
Gruslys
,
A.
,
2017
,
Learning from Demonstrations for Real World Reinforcement Learning
, arXiv preprint.
54.
Abbeel
,
P.
,
Coates
,
A.
, and
Ng
,
A. Y.
,
2010
, “
Autonomous Helicopter Aerobatics Through Apprenticeship Learning
,”
Int. J. Rob. Res.
,
29
(
13
), pp.
1608
1639
. 10.1177/0278364910371999
55.
Liu
,
Y.
,
Gupta
,
A.
,
Abbeel
,
P.
, and
Levine
,
S.
,
2018
, “
Imitation From Observation: Learning to Imitate Behaviors From Raw Video via Context Translation
,”
IEEE International Conference on Robotics and Automation (ICRA)
,
Brisbane, Australia
,
May 21–25
.
56.
Ha
,
D.
, and
Schmidhuber
,
J.
,
2018
, World Models. https://worldmodels.github.io
57.
Pretz
,
J. E.
,
2008
, “
Intuition Versus Analysis: Strategy and Experience in Complex Everyday Problem Solving
,”
Mem. Cognit.
,
36
(
3
), pp.
554
566
. 10.3758/MC.36.3.554
58.
Cagan
,
J.
,
Dinar
,
M.
,
Shah
,
J. J.
,
Leifer
,
L.
,
Linsey
,
J.
,
Smith
,
S.
, and
Vargas Hernandez
,
N.
,
2013
, “
Empirical Studies of Design Thinking: Past, Present, Future
,”
Proceedings of the ASME Design Engineering Technical Conference
, Vol.
5
,
American Society of Mechanical Engineers
,
Portland, OR
.
59.
Björklund
,
T. A.
,
2013
, “
Initial Mental Representations of Design Problems: Differences Between Experts and Novices
,”
Des. Stud.
,
34
(
2
), pp.
135
160
. 10.1016/j.destud.2012.08.005
60.
Egan
,
P.
, and
Cagan
,
J.
,
2016
, “Human and Computational Approaches for Design Problem-Solving,”
Experimental Design Research: Approaches, Perspectives, Applications
,
P.
Cash
,
T.
Stanković
,
M.
Štorga
, eds.,
Springer International Publishing
,
Cham
, pp.
187
205
.
61.
Cagan
,
J.
, and
Kotovsky
,
K.
,
1997
, “
Simulated Annealing and the Generation of the Objective Function: A Model of Learning During Problem Solving
,”
Comput. Intell.
,
13
(
4
), pp.
534
581
. 10.1111/0824-7935.00051
62.
McComb
,
C.
,
Cagan
,
J.
, and
Kotovsky
,
K.
,
2016
, “
Drawing Inspiration From Human Design Teams For Better Search and Optimization: The Heterogeneous Simulated Annealing Teams Algorithm
,”
ASME J. Mech. Des.
,
138
(
4
), p.
044501
. http;//dx.doi.org/10.1115/1.4032810
63.
Matthews
,
P.
,
Blessing
,
L.
, and
Wallace
,
K. M.
,
2002
, “
The Introduction of a Design Heuristics Extraction Method
,”
Adv. Eng. Inform.
,
16
(
1
), pp.
3
19
.
64.
Fuge
,
M.
,
Peters
,
B.
, and
Agogino
,
A.
,
2014
, “
Machine Learning Algorithms for Recommending Design Methods
,”
ASME J. Mech. Des.
,
136
(
10
), p.
101103
. http://dx.doi.org/ 10.1115/1.4028102
65.
Sexton
,
T.
, and
Ren
,
M. Y.
,
2017
, “
Learning an Optimization Algorithm Through Human Design Iterations
,”
ASME J. Mech. Des.
,
139
(
10
), p.
101404
. 10.1115/1.4037344
66.
McComb
,
C.
,
Cagan
,
J.
, and
Kotovsky
,
K.
,
2018
, “
Data on the Design of Truss Structures by Teams of Engineering Students
,”
Data Brief
,
18
(
2018
), pp.
160
163
. 10.1016/j.dib.2018.02.078
67.
Springenberg
,
J. T.
,
Dosovitskiy
,
A.
,
Brox
,
T.
, and
Riedmiller
,
M. A.
,
2015
, “Striving for Simplicity: The All Convolutional Net,”
ICLR (workshop track)
,
ICLR
,
USA
.
68.
Fergus
,
R.
,
Zeiler
,
M. D.
,
Taylor
,
G. W.
, and
Krishnan
,
D.
,
2010
, “
Deconvolutional Networks
,”
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR)
, pp.
2528
2535
.
69.
Nair
,
V.
, and
Hinton
,
G. E.
,
2010
, “
Rectified Linear Units Improve Restricted Boltzmann Machines Vinod Nair
,”
Proceedings of the 27th International Conference on International Conference on Machine Learning
, Vol.
ICML'10
,
Omnipress
,
Madison, WI
, pp.
807
814
.
70.
Kingma
,
D. P.
, and
Ba
,
J.
,
2015
, “
Adam: A Method for Stochastic Optimization
,”
3rd International Conference for Learning Representations
,
San Diego, CA
.
71.
Maas
,
A. L.
,
Hannun
,
A. Y.
, and
Ng
,
A. Y.
,
2013
, “
Rectifier Nonlinearities Improve Neural Network Acoustic Models
,”
ICML Workshop on Deep Learning for Audio, Speech and Language Processing.
,
Atlanta, GA
,
June 16
.
72.
Bengio
,
Y.
,
Lamblin
,
P.
,
Popovici
,
D.
, and
Larochelle
,
H.
,
2006
, “
Greedy Layer-Wise Training of Deep Networks
,”
Proceedings of the 19th International Conference on Neural Information Processing Systems
,
MIT Press
,
Cambridge, MA
, pp.
153
160
.
73.
Franklin
,
S.
, and
Graesser
,
A.
,
1997
, “Is It an Agent, or Just a Program?: A Taxonomy for Autonomous Agents,”
Intelligent Agents III Agent Theories, Architectures, and Languages
,
J. P.
Müller
,
M. J.
Wooldridge
, and
N. R.
Jennings
, eds.,
Springer
,
Berlin, Heidelberg
, pp.
21
35
.
74.
Tarjan
,
R.
,
1971
, “
Depth-First Search and Linear Graph Algorithms
,”
12th Annual Symposium on Switching and Automata Theory (Swat 1971)
,
East Lansing, MI
,
Oct. 13–15
, pp.
114
121
.
75.
Rodriguez
,
J.
, and
Ayala
,
D.
,
2001
, “
Erosion and Dilation on 2-D and 3-D Digital Images: A New Size-Independent Approach
,”
Vision Modeling and Visualization Conference 2001 (VMV-01)
,
Stuttgart, Germany
,
Nov. 21–23
.
76.
Wang
,
Z.
,
Bovik
,
A. C.
,
Sheikh
,
H. R.
, and
Simoncelli
,
E. P.
,
2004
, “
Image Quality Assessment: From Error Visibility to Structural Similarity
,”
IEEE Trans. Image Process.
,
13
(
4
), pp.
600
612
. 10.1109/TIP.2003.819861
You do not currently have access to this content.