Joint Sparsity for Target Detection Nasser M. Nasrabadi Nasser M. - - PowerPoint PPT Presentation

UNCLASSIFIED

Nasser M. Nasrabadi

Joint 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 classify classes (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 classes images) 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-w ise Sparsity Model

• Background pixels approximately lie in a low-

di i l b dimensional subspace

,1 1 ,2 2 ,

b b

b b b b b b b b i i i i N N i

        a a A a α x 

• Target pixels also lie in a low-dimensional

subspace

t t t t t t t t

A

• A test sample

can be sparsely represented b

,1 1 ,2 2 ,

t t

t t t t i i i i N t t i t N t

         a a a α x A

i

x

by

b b b t t b t i i i i i t

              A A A A A x    

i i i i t i

      

Ilustration: Pixel-Wise Sparse Model

0 . 1 2 0 . 1 4

0 . 0 6 0 . 0 8 0 . 1 b a c k g ro u n d d ic t io n a ry





5 0 1 0 0 1 5 0 0 . 0 2 0 . 0 4 t a rg e t d ic t io n a ry

Target Pixel

b

 

Test Spectrum

i

x

Nonzero entries Spectral dictionary A

b b b t t b t i i i i i t i

              A A A A A x       

Sparse Recovery

• Sparse coefficient is recovered by

ˆ arg min subject to   A x   

• For empirical data

arg min subject to

i i i i

  A x   

2

ˆ arg min subject to ˆ arg min subject to

i i i i

K        A x A x      

• NP-hard problem

Greedy algorithms: MP OMP SP C S MP LARS

2

arg min subject to

i 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

1

ˆ arg min subject to

i i i i

  A x   

Classification Based on Residuals

• Recover sparse coefficient

ˆ ˆ

b i i

      

Recover sparse coefficient

• Compute the residuals (approximation errors

ˆ

i t i 

    

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

   

ˆ ˆ

b b t t

   

2 2

and ˆ ˆ

b b i i i i t b t i i t

r r     x A x A x x  

• Class of test pixel is made by comparing the

residuals

i

x

Example: Reconstruction

0 . 6 0 . 7 0 . 8 0 . 9 R e c o v e r e d s p a r s e c o e ffi c i e n t s

ˆ b   

2 0 . 3 0 . 4 0 . 5

ˆ ˆ

b i i t i

         

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

0 . 1 2 0 . 0 8 0 . 1

ˆ ˆ

t t t

 A x 

0 . 0 2 0 . 0 4 0 . 0 6 O r ig in a l R e c o n s t r u c t e d fr o m b g d ic .

i i

A x  ˆ ˆi

b i b b

 A x 

5 0 1 0 0 1 5 0 0 0 e c o s t u c t e d

• b g d c

R 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 characteristics g 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

 A A x 

2 2

 A x  

   

1 2 1 2 T T

   

S

X x x x A AS         

–

’s: sparse vectors with same support, different magnitude

T T

 A x 

i



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

i

S

Illustration: T=3x3 Neighborhood

0.14 0.14

9

0.1 0.12 0.08 0.1 0.12 0 04 0.06 0.08 0.04 0.06





T=9

50 100 150 0.02 0.04 50 100 150 0.02

X

Spectral dictionary A Row-sparse t i

S

Data matrix matrix S

Joint Sparse Recovery

• is recovered by

S

• Solved by greedy algorithms: Simultaneous OMP

row, 0

ˆ arg min subject to   S S AS X

• Solved by greedy algorithms: Simultaneous OMP

(SOMP) , Simultaneous SP (SSP) or Convex

• ptimization to find the same active set

1,2

ˆ 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 single Input a single background pixel x

ˆ arg min subject to   A x    Input nine put e neighboring background pixels X

row, 0

ˆ arg min subject to   S S AS X

Results on HYDICE FR-I

Original image (averaged Proposed detector output g g ( g

• ver 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 resolution 220 bands, 20 meters pixel resolution

Extension to Multiple Classes

Multi-View Target Classification

• In ATR applications we can have multiple
• bservations of the same physical target from

p y g different platforms or from the same platform at different viewing angles (aspects). g g ( p )

ˆ arg min subject to

i i i i

  A y   

ˆ

(Single-Measurement)

row, 0

ˆ arg min subject to   S S AS Y

(Multi-Measurements)

Experimental Results on Multi- View Target Classification

• MSTAR SAR data-base

consists of 10 military consists 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.

ˆ arg min subject to

i i i i

  A x   

1 1

ˆ arg min subject to and                               A x x A x     

row, 0 1

ˆ arg min subject to Note [ ]

M

    S S AS X S  

M M

        x 

Experimental Results on Number

• f View s 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 y N=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 ti the regularization part of the optimization

• Investigated performance of multiple measurements on

classification performance on several data bases. p

Thank You

THANK YOU