fio/ls 2006 ece task-specific information amit ashok 1, pawan k baheti 1 and mark a. neifeld 1,2...
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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
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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 )
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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
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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