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FIO/LS 2006ece

Task-Specific Information

Amit Ashok1, Pawan K Baheti1 and Mark A. Neifeld1,2

Optical Computing and Processing Laboratory

1Dept. of Electrical and Computer Engineering,

2College of Optical Sciences,

University of Arizona, Tucson.

FIO/LS 2006ece

Presentation Outline

• Images and Information

• Task-specific information (TSI)

• Detection and Localization tasks

• Comparison for conventional and compressive imagers

• Results and Conclusions

FIO/LS 2006ece

Information content of an image

512

512

512 × 512 × 3 × 8

= 6.2 Mb

64 × 64 × 1 × 8

= 32 Kb

64

64

Compression

2.1 Mb

Compression

24 Kb

• More precise measure requires source probability density ρ

• dZ logentropy source Compute

• PROBLEM: ρ is very complex/unknown

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Motivation

• Information content is task specific

Detection task: For equal probability of presence/absence the information content < 1 bit

Detection & Localization task: Probability of tank being absent = ½ ; Probability of occurrence in each region: ⅛

Information content < 2 bits

Classification task: For equal probability of each target the information content < 1 bit

• How to quantify task specific information (TSI)

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Task specific source encoding

Y = C(X)Virtual source

EncodingX C(X) stochastically

encodes X and produces scene Y

• Detection task: presence/absence of target is of interest

• Virtual source variable must be binary

• X = 1/0 implies tank present/absent

X = 1 (Tank present) X = 0 (Tank absent)

SCENE

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• Imager is characterized by channel H and noise n

• Imager does not add entropy to the relevant task

• Definition for Task-specific information:

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Task specific information (TSI)

• Imaging chain block diagram

R = n(H(C(X)))Y = C(X)Virtual source

EncodingX H(Y)

Channel Noise

IMAGERSCENE

Entropy Z(X) – maximum task-specific information content

Mutual information between X and R

Always bounded by the entropy of X

)X();X( ZRITSI

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TSI (continued)

• Measurement can be written asn and s denote additive Gaussian noise and snr respectively

)and over dconditione (MMSE

) over dconditione (MMSE

X .X,/YYX,/YY

, /YY/YY

(X)}{Y ,)( where

,)(),;X(

XY,

Y

X,YY1

0

RREREE

RREREE

CTracemmse

sdsmmsesRITSI

T

T

nT

s

E

E

EEHH

nCsR )X(H

• Computing TSI is difficult for non-Gaussian source

• Use Verdu’s relation between mutual information and

minimum estimation error

FIO/LS 2006

Encoding matrix: selects target at one of the P positions in M×M scene

Clutter weighted by β ~ N(mβ ,Σβ)

cVT

H

csC

nCR

X)X( where

,X)(

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Target detection

• Virtual source X is binary indicating the presence/absence of tank

• Measurement: s is signal to noise ratioc is the clutter to noise ratio

FIO/LS 2006ece

Simulation details

• Detection task: probability of occurrence = ½

• TSI will be bounded by 1 bit

H = I H = sinc2(.)

Example scenes

• Ideal and diffraction limited

• TSI and MMSE estimation – Monte Carlo• Scene dimension: 80 × 80• Number of clutter components: K = 6• Possible positions of tank: P = 64• Comparison will be versus s (called as snr)

SCENE MODEL

IMAGER MODEL

bit121log2

12

1log21

22

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Detection Task results

• MMSE is small in low and high snr region

• MMSE component conditioned on X improves faster through medium snr

• TSI saturates at 1 bit with increasing snr

• Degradation in performance due to blur as expected

MMSE

MMSE conditioned over R and X

MMSE plots for H = I

MMSE conditioned over R

MM

SE

snr

TSI for both H = I & sinc2()

H = I

Nyquist blur

Twice the Nyquist blur

Tas

k-sp

ecifi

c in

form

atio

n

snr

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Detection and Localization Task• Example scene when considering localization task

• Scene divided into 4 regions with P/4 possible positions

in each region for the tank

• Task is to localize the tank in one of the regions if present

• Probability of occurrence in each region: 1/8

• Probability of target not present: 1/2

• Modifications to the encoding matrix T

TSI will be bounded by 2 bits in this case

Region 1Region 2

Region 3Region 4 bits28

1log8142

1log21

22

FIO/LS 2006

Detection and localization

H = I

Nyquist blur

Twice the Nyquist blur

14.47 dB 15.45 dB16.53 dB

20 dB

Tas

k-sp

ecifi

c in

form

atio

n

snr

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Results

• TSI saturates at 2 bits

• Degradation in performance due to blur as expected

H = I: TSI = 1.8 bits for snr = 28

H = sinc2(0.5x): TSI = 1.8 bits for snr = 35

H = sinc2(0.25x): TSI = 1.8 bits for snr = 45

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Projective imager

• Modification to the imaging model

• P transforms high dimension image to low dimension measurement

• Principal component projections

• Training set of the scenes is created using the encoder

• Correlation matrix from the training set

• Eigenvector decomposition of the correlation matrix

• Choose dominant F eigenvectors to form P (dimension: F×M2)

Source Encoding Channel Noise

R = N(P(H(C(X))))

ProjectionX R

IMAGE (M×M )

P

.

.

.

.

.

(F×1 )

FIO/LS 2006ece

Detection and localization: PC Projections

• Projective imager performs better than conventional at low snr

• TSI improves with F increasing from 8 to 24 due to increasing signal fidelity

Conventional Imager (P = I)

snr = 19

snr = 35

Tas

k-sp

ecifi

c in

form

atio

n

snr

snr = 25

Rollover starts at F = 24 onwards

(trade-off between TSI and measurement snr)

Tas

k-sp

ecifi

c in

form

atio

nF (# of projections)

Too few measurements

Too few photons per measurement

FIO/LS 2006ece

Conclusions

• Information content of an image is associated with a task

• Introduced the framework for task-specific information

• TSI confirms our intuition about ideal, diffraction-limited and

projective imagers

• Can be used as a metric to optimize the systems based

on task specificity