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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Distributed Sensing and Perception via Sparse Representation

Allen Y. Yang yang@eecs.berkeley.edu CIS Seminar, Johns Hopkins, 2010

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Distributed Sensing and Perception: A Comparison

Centralized Perception Up: powerful processors Up: unlimited memory Up: unlimited bandwidth Down: single modality Distributed Perception Down: mobile processors Down: limited onboard memory Down: band-limited communications Up: distributed, multi-modality

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Distributed Sensing and Perception: A Comparison

Centralized Perception Up: powerful processors Up: unlimited memory Up: unlimited bandwidth Down: single modality Distributed Perception Down: mobile processors Down: limited onboard memory Down: band-limited communications Up: distributed, multi-modality Design an intelligent system over a network that performs better than the sum of its parts?

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Challenges

1

Making real-time decisions on portable mobile devices is difficult.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Challenges

1

Making real-time decisions on portable mobile devices is difficult.

2

Applications demand extremely high accuracy: 99% Precision, 99% Recall?

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Challenges

1

Making real-time decisions on portable mobile devices is difficult.

2

Applications demand extremely high accuracy: 99% Precision, 99% Recall?

3

Scenarios demand the ability to reconstruct 3-D environments.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Smart Camera Platform: CITRIC v1

CITRIC platform Available library functions

1

Full support Intel IPP Library and OpenCV.

2

JPEG compression: 10 fps.

3

Edge detector: 3 fps.

4

Background Subtraction: 5 fps.

5

SIFT detector: 10 sec per frame.

Academic users:

Reference: AY, et al. “CITRIC: A low-bandwidth wireless camera network platform.” (submitted) ACM Trans. Sensor Networks, 2010.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Body Sensor Platform: DexterNet

1

Body Sensor Layer (BSL)

2

Personal Network Layer (PNL)

3

Global Network Layer (GNL)

Reference: AY, et al. “DexterNet: An open platform for heterogeneous body sensor networks and its applications.” Body Sensor Networks, 2009.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Outline

1

Robust face recognition with low-resolution, distorted, and disguised images

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Outline

1

Robust face recognition with low-resolution, distorted, and disguised images

2

Fast ℓ1-Minimization Algorithms x∗ = arg min

x

x1

• subj. to b = Ax.

Augmented Lagrange Multiplier

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Outline

1

Robust face recognition with low-resolution, distorted, and disguised images

2

Fast ℓ1-Minimization Algorithms x∗ = arg min

x

x1

• subj. to b = Ax.

Augmented Lagrange Multiplier

3

Distributed object recognition using a camera network

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Robust Face Recognition

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Classification of Mixture Subspace Model

1

Face-subspace model [Belhumeur et al. ’97, Basri & Jacobs ’03] Assume b belongs to Class i in K classes. b = αi,1vi,1 + αi,2vi,2 + · · · + αi,n1vi,ni , = Aiαi.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Classification of Mixture Subspace Model

1

Face-subspace model [Belhumeur et al. ’97, Basri & Jacobs ’03] Assume b belongs to Class i in K classes. b = αi,1vi,1 + αi,2vi,2 + · · · + αi,n1vi,ni , = Aiαi.

2

Nevertheless, Class i is the unknown label we need to solve: Sparse representation b = [A1, A2, · · · , AK ] 2 4

α1 α2

. . .

αK

3 5 = Ax.

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Classification of Mixture Subspace Model

1

Face-subspace model [Belhumeur et al. ’97, Basri & Jacobs ’03] Assume b belongs to Class i in K classes. b = αi,1vi,1 + αi,2vi,2 + · · · + αi,n1vi,ni , = Aiαi.

2

Nevertheless, Class i is the unknown label we need to solve: Sparse representation b = [A1, A2, · · · , AK ] 2 4

α1 α2

. . .

αK

3 5 = Ax.

3

x∗ = [ 0 ··· 0 αT

i

0 ··· 0 ]T ∈ Rn.

Sparse representation x∗ encodes membership!

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Image Corruption

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Image Corruption

1

Sparse representation + sparse error b = Ax + e

2

Occlusion compensation: b = `A | I´ „x e « = Bw

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Performance on the AR database

Reference: AY, et al. Robust face recognition via sparse representation. IEEE PAMI, 2009.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Face Alignment

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Face Alignment

Seek a 2-D transformation b ◦ τi = Aix + e. (1) Although x1 is no longer penalized, the problem becomes nonlinear.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Face Alignment

Seek a 2-D transformation b ◦ τi = Aix + e. (1) Although x1 is no longer penalized, the problem becomes nonlinear. Linear approximation: b ◦ τi + ∇τ(b ◦ τi) · ∆τi ≈ Aix + e. (2)

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Face Alignment

Seek a 2-D transformation b ◦ τi = Aix + e. (1) Although x1 is no longer penalized, the problem becomes nonlinear. Linear approximation: b ◦ τi + ∇τ(b ◦ τi) · ∆τi ≈ Aix + e. (2) Convert to a linear equation: b(k)

i

= [Ai, −J(k)

i

]w + e, (3) where w . = [xT , ∆τ T

i ]T .

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Demo I: Misalignment & Corruption Compensation

Alignment Demo

Reference: Wagner, et al. Towards a Practical Face Recognition System: Robust Registration and Illumination via Sparse Representation. CVPR, 2009.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Question: How to effectively estimate HD sparse signals?

“Black gold” age [Claerbout & Muir 1973, Taylor, Banks & McCoy 1979]

Figure: Deconvolution of spike train.

Basis pursuit [Chen-Donoho 1999]: x∗ = arg min x1, subject to b = Ax The Lasso (least absolute shrinkage and selection operator) [Tibshirani 1996] x∗ = arg min b − Ax2, subject to x1 ≤ k

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

ℓ0/ℓ1 Equivalence Relationship

ℓ0-Minimization over an underdetermined system (NP-Hard) x∗ = arg min

x

x0

• subj. to b = Ax.

· 0 simply counts the number of nonzero terms.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

ℓ0/ℓ1 Equivalence Relationship

ℓ0-Minimization over an underdetermined system (NP-Hard) x∗ = arg min

x

x0

• subj. to b = Ax.

· 0 simply counts the number of nonzero terms. ℓ1-Minimization (Linear Program) [Candes & Tao 2006, Donoho 2006] x∗ = arg min

x

x1

• subj. to b = Ax.

x1 = |x1| + |x2| + · · · + |xn|.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

ℓ1-Minimization via Linear Programming

Using interior-point methods [Karmarkar ’84] Log-Barrier: min

x

1T x − µ

n

X

i=1

log xi,

• subj. to Ax = b, x ≥ 0.

(4) Using the Karush-Kuhn-Tucker (KKT) conditions 1 − µX −11 − AT y = 0. (5) where x ≥ 0 are the primal variables, and y are the dual variables. Update by solving a linear system with O(n3) [Monteiro & Adler ’89] Z (k)∆x + X (k)∆z = ˆ µ1 − X (k)z(k), A∆x = 0, AT ∆y + ∆z = 0, (6)

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Fast ℓ1-minimization is still a difficult problem!!

Interior-point methods are very expensive in HD space.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

References

1

Primal-Dual Interior-Point Methods

Log-Barrier [Frisch 1955, Karmarkar 1984, Megiddo 1989, Monteiro-Adler 1989, Kojima-Megiddo-Mizuno 1993]

2

Homotopy Methods:

Homotopy [Osborne-Presnell-Turlach 2000, Malioutov-Cetin-Willsky 2005, Donoho-Tsaig 2006] Polytope Faces Pursuit (PFP) [Plumbley 2006] Least Angle Regression (LARS) [Efron-Hastie-Johnstone-Tibshirani 2004]

3

Gradient Projection Methods

Gradient Projection Sparse Representation (GPSR) [Figueiredo-Nowak-Wright 2007] Truncated Newton Interior-Point Method (TNIPM) [Kim-Koh-Lustig-Boyd-Gorinevsky 2007]

4

Iterative Thresholding Methods

Soft Thresholding [Donoho 1995] Sparse Reconstruction by Separable Approximation (SpaRSA) [Wright-Nowak-Figueiredo 2008]

5

Proximal Gradient Methods [Nesterov 1983, Nesterov 2007]

FISTA [Beck-Teboulle 2009] Nesterov’s Method (NESTA) [Becker-Bobin-Cand´ es 2009]

6

Augmented Lagrange Multiplier Methods [Yang-Zhang 2009, AY et al 2010]

YALL1 [Yang-Zhang 2009] Primal ALM, Dual ALM [AY et al 2010]

References: AY, et al., A review of fast ℓ1-minimization algorithms for robust face recognition. Submitted to SIAM Imaging Sciences, 2010.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Iterative Soft-Thresholding (IST) Methods

Objective: x∗ = arg min x1

• subj. to e = b − Ax < ǫ

F(x) . = 1 2 b − Ax2

2 + λx1 = f (x) + λg(x)

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Iterative Soft-Thresholding (IST) Methods

Objective: x∗ = arg min x1

• subj. to e = b − Ax < ǫ

F(x) . = 1 2 b − Ax2

2 + λx1 = f (x) + λg(x)

IST iteratively approximate the composite objective function x(k+1) ≈ arg minx{f (x(k)) + (x − x(k))T ∇f (x(k)) + ∇2f (x(k))

2

x − x(k)2

2 + λg(x)}

= arg minx{(x − x(k))T ∇f (x(k)) + α(k)I

2

x − x(k)2

2 + λg(x)}

where the hessian ∇2f (x) is approximated by a diagonal matrix αI.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Iterative Soft-Thresholding (IST) Methods

Objective: x∗ = arg min x1

• subj. to e = b − Ax < ǫ

F(x) . = 1 2 b − Ax2

2 + λx1 = f (x) + λg(x)

IST iteratively approximate the composite objective function x(k+1) ≈ arg minx{f (x(k)) + (x − x(k))T ∇f (x(k)) + ∇2f (x(k))

2

x − x(k)2

2 + λg(x)}

= arg minx{(x − x(k))T ∇f (x(k)) + α(k)I

2

x − x(k)2

2 + λg(x)}

where the hessian ∇2f (x) is approximated by a diagonal matrix αI. A closed-form solution exists element-wise [Donoho ’95, Wright et al. ’08] x(k+1)

i

= arg min

xi { (xi − u(k) i

)2 2 + λ|xi| α(k) } = soft(u(k)

i

, λ α(k) )

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Augmented Lagrange Multiplier

ALM considers an augmented Lagrange function Lµ(x, y) = x1 + y, b − Ax + µ 2 b − Ax2

2,

where y are the Lagrange multipliers for the constraint b = Ax.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Augmented Lagrange Multiplier

ALM considers an augmented Lagrange function Lµ(x, y) = x1 + y, b − Ax + µ 2 b − Ax2

2,

where y are the Lagrange multipliers for the constraint b = Ax. It can be shown if y∗(µ) is optimal [Hestenes ’69, Powell ’69, Bertsekas ’03] x∗(µ) = arg min

x

Lµ(x, y∗); x∗∗ = lim

µ→∞ x∗(µ)

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Augmented Lagrange Multiplier

ALM considers an augmented Lagrange function Lµ(x, y) = x1 + y, b − Ax + µ 2 b − Ax2

2,

where y are the Lagrange multipliers for the constraint b = Ax. It can be shown if y∗(µ) is optimal [Hestenes ’69, Powell ’69, Bertsekas ’03] x∗(µ) = arg min

x

Lµ(x, y∗); x∗∗ = lim

µ→∞ x∗(µ)

Iteratively update x, y, and µ with O(dn): 8 < : xk+1 = arg minx Lµk (x, yk), yk+1 = yk + µk(b − Axk+1), µk+1 → ∞. .

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Demo II: Speed of ALM vs Interior-Point

Table: Source signal in 1000-D: sparsity = 200; random projection = 600-D.

Algorithm Estimate Runtime PDIPA 63 s ALM 0.16 s

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Distributed Object Recognition: Problem Statement

1

L camera sensors observe a single object in 3-D.

2

The relative positions between cameras are unknown, cross-sensor communication is prohibited.

3

On each camera, seek an encoding function for a high-dim, sparse xi (SIFT histogram) f : xi ∈ RD → bi ∈ Rd

4

At the base station, upon receiving b1, b2, · · · , bL, simultaneously recover x1, x2, · · · , xL, and classify the object class in space.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Key Observations: Scale Invariant Feature Transform

(a) Histogram 1 (b) Histogram 2

All SIFT histograms are nonnegative and sparse.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Key Observations: Scale Invariant Feature Transform

(a) Histogram 1 (b) Histogram 2

All SIFT histograms are nonnegative and sparse. Multiple-view histograms share joint sparse patterns.

Reference: AY, et al. Multiple-view object recognition in smart camera networks. Springer, 2010.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

The System

bi = Axi, where x is assumed sparse.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Joint Sparsity Model

Definition: Joint Sparsity Model [Baron et al. 2005] x1 = xc + z1, . . . xL = xc + zL. xc is called the common component, and zi is called an innovation.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Joint Sparsity Model

Definition: Joint Sparsity Model [Baron et al. 2005] x1 = xc + z1, . . . xL = xc + zL. xc is called the common component, and zi is called an innovation. Recovery of the JS model 2 4

b1

. . .

bL

3 5 = 2 4

A1 A1 ···

. . . ... ...

AL ··· AL

3 5 2 6 4

xc z1

. . .

zL

3 7 5 ⇔ b′ = A′x′ ∈ RdL.

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Joint Sparsity Model

Definition: Joint Sparsity Model [Baron et al. 2005] x1 = xc + z1, . . . xL = xc + zL. xc is called the common component, and zi is called an innovation. Recovery of the JS model 2 4

b1

. . .

bL

3 5 = 2 4

A1 A1 ···

. . . ... ...

AL ··· AL

3 5 2 6 4

xc z1

. . .

zL

3 7 5 ⇔ b′ = A′x′ ∈ RdL.

1

New histogram vector remains nonnegative and sparse.

2

Joint sparsity xc is automatically determined by ℓ1-min: No prior training, no assumption about fixing cameras and calibration.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Berkeley Multiview Wireless (BMW) Database

20 landmarks at UC Berkeley. 16 different vantage points (large baseline); five images at one location (short baseline). Low-quality images: low resolution, inaccurate focal length, dusty lenses.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Experiment: Accuracy on BMW Database

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Experiment: Accuracy on BMW Database

Better recognition rate is achieved using multiple views but less bandwidth!

Reference: AY, et al. Towards an efficient distributed object recognition system in wireless smart camera networks. in Information Fusion, 2010.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

A distributed network shall be greater than the sum of its parts!

Centralized Perception Up: powerful processors Up: unlimited memory Up: unlimited bandwidth Down: single modality Distributed Perception Down: mobile processors Down: limited onboard memory Down: band-limited communications Up: distributed, multi-modality

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

A distributed network shall be greater than the sum of its parts!

Centralized Perception Up: powerful processors Up: unlimited memory Up: unlimited bandwidth Down: single modality Distributed Perception Down: mobile processors Down: limited onboard memory Down: band-limited communications Up: distributed, multi-modality Our approach to the unique challenges:

1

How to design real-time recognition systems in sensor networks? A: Efficient numerical solvers plus new computational models: parallel & cloud computing.

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

A distributed network shall be greater than the sum of its parts!

Centralized Perception Up: powerful processors Up: unlimited memory Up: unlimited bandwidth Down: single modality Distributed Perception Down: mobile processors Down: limited onboard memory Down: band-limited communications Up: distributed, multi-modality Our approach to the unique challenges:

1

How to design real-time recognition systems in sensor networks? A: Efficient numerical solvers plus new computational models: parallel & cloud computing.

2

How to achieve extremely high accuracy? A: Pay attention to special structures in HD data imposed by applications; Simple solutions are often the best (e.g., sparse representation).

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation

Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

A distributed network shall be greater than the sum of its parts!

Centralized Perception Up: powerful processors Up: unlimited memory Up: unlimited bandwidth Down: single modality Distributed Perception Down: mobile processors Down: limited onboard memory Down: band-limited communications Up: distributed, multi-modality Our approach to the unique challenges:

1

How to design real-time recognition systems in sensor networks? A: Efficient numerical solvers plus new computational models: parallel & cloud computing.

2

How to achieve extremely high accuracy? A: Pay attention to special structures in HD data imposed by applications; Simple solutions are often the best (e.g., sparse representation).

3

How to provide the ability to reconstruct 3-D using mobile smart cameras? A: Don’t just label the images; Take advantage of available information in 3-D geometry (e.g., joint sparsity).

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Sparse Representation is “the Next Wave”?

Single-Pixel Camera for Deep-Space Imaging [Baraniuk 2008] Background Subtraction [Chellappa 2008] MRI Imaging [Lustig 2007] Robust PCA [Candes 2009]

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

References

Acknowledgments UC Berkeley

Dr S. Sastry, Dr R. Bajcsy, Dr E. Seto, Dr T. Darrell, Dr J. Malik, N. Naikal, V. Shia,

• P. Yan.
• Univ. Illinois
• A. Ganesh, Z. Zhou, A. Wagner

MSR Asia

Dr Y. Ma, Dr J. Wright

Funding Support

ARO MURI: Heterogeneous Sensor Networks in Urban Terrains ARL: Micro Autonomous Systems and Technology

Patents

Yang, et al. “Recognition via High-Dimensional Data Classification.” US & China Patent, 2009. Yang, et al. “System for Detection of Body Motion.” US Patent, 2010.

Publications

Wright, Yang, Ganesh, Sastry, Ma. “Robust face recognition via sparse representation,” IEEE PAMI, 2009. Yang, Gastpar, Bajcsy, Sastry. “Distributed Sensor Perception via Sparse Representation.” Proceedings of IEEE, 2010. Naikal, Yang, Sastry. “Towards an efficient distributed object recognition system in wireless smart camera networks.” Information Fusion, 2010. Yang, Ganesh, Zhou, Sastry, Ma.“A review of fast ℓ1-minimization algorithms in robust face recognition”, arXiv, 2010. Ganesh, Ma, Wagner, Wright, Yang, “Robust face recognition by sparse representation,” (submitted) Cambridge Press, 2010.

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Feasibility and Uniqueness: ℓ0-Minimization

Spark Condition Spark(A): smallest number of columns that are linearly dependent

1

Example I: Identity matrix I ∈ Rd×d, Spark(A) = d+1;

2

Example II: » 1 1 1 1 – , Spark(A) = 2;

3

Example III: Random matrix [v1, v2, · · · , vn] ∈ Rd×n, Spark(A) = d+1 (with high probability);

Sparse signal x can be uniquely recovered by ℓ0-min if x0 < Spark(A) 2 Proof.

1

Suppose x1 = x2 both satisfy the spark condition, and b = Ax1, b = Ax2.

2

A(x1 − x2) . = Ay = b − b = 0.

3

But y0 < Spark(A)

2

+ Spark(A)

2

= Spark(A). Contradiction.

Estimating Spark(A) is as expensive as ℓ0-min itself!

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Introduction Face Recognition Fast ℓ1-Minimization Algorithms Distributed Object Recognition Conclusion

Feasibility and Uniqueness: ℓ1-Minimization

k-Neighborliness Condition Define cross polytope C and quotient polytope P such that P = AC. x is k-sparse ⇔ x lie in a unique (k − 1)-face of C. Necessary and Sufficient:

1

If the (k − 1)-face where x lies maps to a face of P, then ℓ1/ℓ0 holds for this specific x.

2

If all (k − 1)-faces of C map to the faces of P on the boundary, ℓ1/ℓ0 holds for all k-sparse signals.

http://www.eecs.berkeley.edu/~yang Distributed Sensing and Perception via Sparse Representation