UNCLASSIFIED

Nasser M. NasrabadiJoint Sparsity for Target Detection

Nasser M. Nasrabadi

UNCLASSIFIED

U.S. Army Research Laboratory

Introduction

• Objective: Segmentation of HSI into multiple classes (target and background) or classifyclasses (target and background) or classify individual objects (military targets) from multiple views of the same physical target.

• Assumptions– Training data: known spectral characteristics (or

images) of different classesimages) of different classes– Test data: a sparse linear combination of all

training data– In HSI Neighboring pixels: similar materials– Mutiple views of targets are similar

• Results compared to classical SVM classifiers

Hyperspectral Imagery

Pixel-wise Sparsity Model

• Background pixels approximately lie in a low-di i l bdimensional subspace

,1 1 ,2 2 , b b

b b b b b b b bi i i i N N i a a Aa αx

• Target pixels also lie in a low-dimensional subspace

t t t t t tt tA• A test sample can be sparsely represented

b

,1 1 ,2 2 , t t

t t t ti i i i N

t ti

tN

t a a a αx A

ixby

bb b t t b t i

i i i it

A A A A Ax

i i i it

i

Ilustration: Pixel-Wise Sparse Model

0 . 1 2

0 . 1 4

0 . 0 6

0 . 0 7

0 . 0 8

0 . 0 9

0 . 1

0 . 1 1t e s t s a m p l e

0 . 0 6

0 . 0 8

0 . 1

b a c k g ro u n d d ic t io n a ry

0 5 0 1 0 0 1 5 0

0 . 0 4

0 . 0 5

0 5 0 1 0 0 1 5 00 . 0 2

0 . 0 4

t a rg e t d ic t io n a ry

Target Pixel

b

Test Spectrum ix Nonzero entries

Spectral dictionary A

bb b t t b t i

i i i iti

A A A A Ax

Sparse Recovery

• Sparse coefficient is recovered by

ˆ arg min subject to A x

• For empirical data0

arg min subject toi i i i A x

0 2ˆ arg min subject to

ˆ arg min subject toi i i i

K

A x

A x

• NP-hard problemGreedy algorithms: MP OMP SP C S MP LARS

02 0arg min subject toi i i i K A x

– Greedy algorithms: MP, OMP, SP, CoSaMP, LARS

– Convex relaxation: Iterative Thresholding, Primal-Dual Interior-Point, Gradient Projection, Proximal Gradient, Augmented Lagrange Multiplier

1ˆ arg min subject toi i i i A x

Classification Based on Residuals

• Recover sparse coefficientˆ

ˆbi

i

Recover sparse coefficient

• Compute the residuals (approximation errors

ˆi ti

Compute the residuals (approximation errors w.r.t. the two sub-dictionaries)

ˆ ˆb b t t 2 2

and ˆ ˆ b bi i i i

tb t i i

tr r x A x Ax x

• Class of test pixel is made by comparing the residuals

ix

Example: Reconstruction

0 . 6

0 . 7

0 . 8

0 . 9R e c o v e r e d s p a r s e c o e f f i c i e n t s

ˆ b

0 2

0 . 3

0 . 4

0 . 5ˆˆ

bi

i ti

0 5 0 1 0 0 1 5 0 2 0 0 2 5 00

0 . 1

0 . 2

0 . 1 2

0 . 0 8

0 . 1

ˆ ˆt t t Ax

0 . 0 2

0 . 0 4

0 . 0 6

O r i g i n a lR e c o n s t r u c t e d fr o m b g d i c .

i iAx

ˆ ˆib

ib b Ax

0 5 0 1 0 0 1 5 00

0 0

e c o s t u c t e d o b g d cR e c o n s t r u c t e d fr o m t a r g e t

Joint Sparsity Model(Joint Structural Sparsity Prior)

• Use of contextual information– Neighboring pixels: similar spectral characteristicsg g p p– Approximated by the same few training samples, weighed

differently• Consider T pixels in a small neighborhood• Consider T pixels in a small neighborhood

1 1 AA

x

2 2 Ax

1 2 1 2T T S

X x x x A AS

– ’s: sparse vectors with same support, different magnitude

T T Ax

i p pp , g– : sparse matrix with only a few nonzero rows

i

S

Illustration: T=3x3 Neighborhood

0.140.14

9

0.1

0.12

0.08

0.1

0.12

0 04

0.06

0.08

0.04

0.06 T=9

0 50 100 1500.02

0.04

0 50 100 1500

0.02

X Spectral dictionary A Row-sparse t i S

Data matrixmatrix S

Joint Sparse Recovery

• is recovered byS

• Solved by greedy algorithms: Simultaneous OMProw, 0

ˆ arg min subject to S S AS X

• Solved by greedy algorithms: Simultaneous OMP (SOMP) , Simultaneous SP (SSP) or Convex optimization to find the same active set

1,2ˆ arg min subject to S S AS X

• Decision obtained by comparing total residuals

Comparison of single pixel sparsity model VS Joint Sparsity Recovery

Model (k=5 atoms active)

Input a singleInput a single background pixel x

0ˆ arg min subject to A x

Input nine put e

neighboringbackground pixels X

row, 0ˆ arg min subject to S S AS X

Results on HYDICE FR-I

Original image (averaged Proposed detector outputg g ( gover 150 bands)

p p

Results on FR-I: ROC Curves

Extension to Multiple Classes

• AVIRIS HSI data set with 16 classes, 220 bands, 20 meters pixel resolution220 bands, 20 meters pixel resolution

Extension to Multiple Classes

Multi-View Target Classification

• In ATR applications we can have multiple observations of the same physical target from p y gdifferent platforms or from the same platform at different viewing angles (aspects).g g ( p )

0ˆ arg min subject toi i i i A y

ˆ

(Single-Measurement)

row, 0ˆ arg min subject to S S AS Y (Multi-Measurements)

Experimental Results on Multi-View Target Classification

• MSTAR SAR data-base consists of 10 militaryconsists of 10 military targets at roughly 1-3 interval azimuth angles (0-360 ) t t diff t

360 ) at two different depression angles 15 and 17 . Data from 17 is used for

training (dictionary design) 15 is used for testing

Experimental Results on Multi-View Target Classification

• Three class (BMP2, BTR70, T72) target classification C=3 with multiple views M=3 . Features are incoherent random projections dimension range from d=128 to1024.

0ˆ arg min subject toi i i i A x

1 1

0ˆ arg min subject to

and

A x

xA x

row, 0

1

ˆ arg min subject to

Note [ ]M

S S AS X

S

M M x

Experimental Results on Number of Views and Angle Size

• Effect of different number of views M

• Effect of the angle size between the views

Experimental Results on Multi-View Target Classification

• 10 class classification results using M=3 views with dictionary of size yN=2747 tested on 15 degree depression g p

Multi-Pose Face Recognition

• Scenarios where we have multiple poses of the same face as input to the classifier.

• UMIST database consists of 564 images of 20 individuals with a range of poses.

• Randomly select 10 poses for each individual to construct the dictionary.

Conclusions

• Formulated target and object recognition as joint sparsity underdetermined regression problem.

• Investigated the effect single vs multiple measurements • Included the idea of joint structured sparsity prior into

th l i ti t f th ti i tithe regularization part of the optimization• Investigated performance of multiple measurements on

classification performance on several data bases. p

Thank You

THANK YOU