Abstract

Imagery is often used to accomplish some computational task. In such cases there are some aspects of the imagery that are relevant to the task and other aspects that are not. In order to quantify the task-specific quality of such imagery, we introduce the concept of task-specific information (TSI). A formal framework for the computation of TSI is described and is applied to three common tasks: target detection, classification, and localization. We demonstrate the utility of TSI as a metric for evaluating the performance of three imaging systems: ideal geometric, diffraction-limited, and projective. The TSI results obtained from the simulation study quantify the degradation in the task-specific performance with optical blur. We also demonstrate that projective imagers can provide higher TSI than conventional imagers at small signal-to-noise ratios.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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2007 (1)

D. P. Palomar and S. Verdu, "Representation of mutual information via input estimates," IEEE Trans. Inf. Theory 53, 453-470 (2007).
[CrossRef]

2006 (2)

S. P. Awate, T. Tasdizen, N. Foster, and R. T. Whitaker, "Adaptive, nonparametric Markov modeling for unsupervised, MRI brain-tissue classification," Med. Image Anal 10, 726-739 (2006).
[CrossRef] [PubMed]

D. P. Palomar and S. Verdu, "Gradient of mutual information in linear vector Gaussian channels," IEEE Trans. Inf. Theory 52, 141-154 (2006).
[CrossRef]

2005 (3)

D. Guo, S. Shamai, and S. Verdu, "Mutual information and minimum mean-square error in Gaussian channels," IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

L. Zhen and Karam, "Mutual information-based analysis of JPEG2000 contexts," IEEE Trans. Image Process. 14, 411-422 (2005).
[CrossRef]

J. Ahlberg and I. Renhorn, "An information-theoretic approach to band selection," Proc. SPIE 5811, 15-23 (2005).
[CrossRef]

2003 (2)

2001 (2)

J. Liu and P. Moulin, "Information-theoretic analysis of interscale and intrascale dependencies between image wavelet coefficients," IEEE Trans. Image Process. 10, 1647-1658 (2001).
[CrossRef]

N. Towghi and B. Javidi, "Optimum receivers for pattern recognition in the presence of Gaussian noise with unknown statistics," J. Opt. Soc. Am. A 18, 1844-1852 (2001).
[CrossRef]

1999 (1)

F. O. Huck and C. L. Fales, "Information-theoretic assessment of sampled imaging systems," Opt. Eng. (Bellingham) 38, 742-762 (1999).
[CrossRef]

1998 (3)

J. A. O'Sullivan, R. E. Blahut, and D. L. Snyder, "Information-theoretic image formation," IEEE Trans. Image Process. 44, 2094-2123 (1998).

A. Ortega and K. Ramchandran, "Rate-distortion methods for image and video compression," IEEE Signal Process. Mag. 15, 23-50 (1998).
[CrossRef]

H. H. Barrett, C. K. Abbey, and E. Clarkson, "Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions," J. Opt. Soc. Am. A 15, 1520-1535 (1998).
[CrossRef]

1996 (1)

F. O. Huck, C. L. Fales, and Z. Rahman, "An information theory of visual communication," Philos. Trans. R. Soc. London, Ser. A 354, 2193-2248 (1996).
[CrossRef]

1995 (1)

1990 (1)

1974 (1)

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284-287 (1974).
[CrossRef]

Appl. Opt. (1)

IEEE Signal Process. Mag. (1)

A. Ortega and K. Ramchandran, "Rate-distortion methods for image and video compression," IEEE Signal Process. Mag. 15, 23-50 (1998).
[CrossRef]

IEEE Trans. Image Process. (3)

J. Liu and P. Moulin, "Information-theoretic analysis of interscale and intrascale dependencies between image wavelet coefficients," IEEE Trans. Image Process. 10, 1647-1658 (2001).
[CrossRef]

L. Zhen and Karam, "Mutual information-based analysis of JPEG2000 contexts," IEEE Trans. Image Process. 14, 411-422 (2005).
[CrossRef]

J. A. O'Sullivan, R. E. Blahut, and D. L. Snyder, "Information-theoretic image formation," IEEE Trans. Image Process. 44, 2094-2123 (1998).

IEEE Trans. Inf. Theory (4)

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284-287 (1974).
[CrossRef]

D. Guo, S. Shamai, and S. Verdu, "Mutual information and minimum mean-square error in Gaussian channels," IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

D. P. Palomar and S. Verdu, "Gradient of mutual information in linear vector Gaussian channels," IEEE Trans. Inf. Theory 52, 141-154 (2006).
[CrossRef]

D. P. Palomar and S. Verdu, "Representation of mutual information via input estimates," IEEE Trans. Inf. Theory 53, 453-470 (2007).
[CrossRef]

J. Opt. Soc. Am. A (4)

Med. Image Anal (1)

S. P. Awate, T. Tasdizen, N. Foster, and R. T. Whitaker, "Adaptive, nonparametric Markov modeling for unsupervised, MRI brain-tissue classification," Med. Image Anal 10, 726-739 (2006).
[CrossRef] [PubMed]

Opt. Eng. (Bellingham) (1)

F. O. Huck and C. L. Fales, "Information-theoretic assessment of sampled imaging systems," Opt. Eng. (Bellingham) 38, 742-762 (1999).
[CrossRef]

Opt. Express (1)

Philos. Trans. R. Soc. London, Ser. A (1)

F. O. Huck, C. L. Fales, and Z. Rahman, "An information theory of visual communication," Philos. Trans. R. Soc. London, Ser. A 354, 2193-2248 (1996).
[CrossRef]

Proc. SPIE (1)

J. Ahlberg and I. Renhorn, "An information-theoretic approach to band selection," Proc. SPIE 5811, 15-23 (2005).
[CrossRef]

Other (10)

D. S. Taubman and M. W. Marcellin, JPEG2000: Image Compression Fundamentals, Standards and Practice (Springer, 2002).
[CrossRef]

T. Cover and J. Thomas, Elements of Information Theory (Wiley, New York, 1991).
[CrossRef]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

M. Tanner, Tools for Statistical Inference, 2nd ed. (Springer, 1993).

J. S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2001).

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 2004).

A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice (Springer, 2001).

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I: Detection, Estimation, and Linear Modulation Theory (Wiley, 1968).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.

S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice Hall, 1993).

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Figures (21)

Fig. 1
Fig. 1

(a) 256 × 256 image, (b) compressed version of the image in (a) using JPEG2000, and (c) 64 × 64 image obtained by rescaling the image in (a).

Fig. 2
Fig. 2

Block diagram of an imaging chain.

Fig. 3
Fig. 3

Example scenes from the deterministic encoder.

Fig. 4
Fig. 4

Example scenes from the stochastic encoder.

Fig. 5
Fig. 5

Plot of (a) mmse and (b) TSI versus the signal-to-noise ratio for the scalar detection task.

Fig. 6
Fig. 6

Illustration of stochastic encoding C det : (a) target profile matrix T and position vector ρ and (b) clutter profile matrix V c and mixing vector β .

Fig. 7
Fig. 7

Structure of T and ρ matrices for the two-class problem.

Fig. 8
Fig. 8

Structure of T and Λ matrices for the joint detection/localization problem.

Fig. 9
Fig. 9

Structure of T and Ω matrices for the joint classification/localization problem.

Fig. 10
Fig. 10

Example scenes: (a) tank in the middle of the scene, (b) tank at the top of the scene, (c) jeep at the bottom of the scene, and (d) jeep in the middle of the scene.

Fig. 11
Fig. 11

Detection task: (a) mmse versus signal-to-noise ratio for an ideal geometric imager and (b) TSI versus signal-to-noise ratio for geometric and diffraction-limited imagers.

Fig. 12
Fig. 12

Scene partitioned into four regions: (a) tank in the top-left region of the scene, (b) tank in the top-right region of the scene, (c) tank in the bottom-left region of the scene, and (d) tank in the bottom-right region of the scene.

Fig. 13
Fig. 13

Joint detection/localization task: (a) mmse versus signal-to-noise ratio for an ideal geometric imager and (b) TSI versus signal-to-noise ratio for geometric and diffraction-limited imagers.

Fig. 14
Fig. 14

Classification task: TSI versus signal-to-noise ratio for geometric and diffraction-limited imagers.

Fig. 15
Fig. 15

Joint classification/localization task: TSI versus signal-to-noise ratio for geometric and diffraction-limited imagers.

Fig. 16
Fig. 16

Example scenes with optical blur: (a) tank at the top of the scene, (b) tank in the middle of the scene, (c) jeep at the bottom of the scene, and (d) jeep in the middle of the scene.

Fig. 17
Fig. 17

Block diagram of a projective imager.

Fig. 18
Fig. 18

Detection task: TSI for PC projective imager versus signal-to-noise ratio.

Fig. 19
Fig. 19

Joint detection/localization task: TSI for PC projective imager versus signal-to-noise ratio.

Fig. 20
Fig. 20

Detection task: TSI for MF projective imager versus signal-to-noise ratio.

Fig. 21
Fig. 21

Joint detection/localization task: TSI for MF projective imager versus signal-to-noise ratio.

Equations (68)

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C S 1 ( X ) = V target X + V bg ,
C S 2 ( X ) = V target X + V bg + V tree β 1 + V shrub β 2 ,
TSI I ( X ; R ) = J ( X ) J ( X R ) ,
R = s X + N ,
d d s I ( X ; R ) = 1 2 mmse = 1 2 E [ X E ( X R ) 2 ] ,
R = s H X + N ,
d d s I ( X ; R ) = 1 2 E [ H X E [ H X R ] 2 ] .
d d s I ( X ; R ) = 1 2 mmse H = 1 2 Tr ( H Σ N 1 H E ) .
R = s H C ( X ) + N .
d d s I ( X ; R ) = 1 2 mmse H ,
mmse H = Tr ( H Σ N 1 H ( E Y E Y X ) ) ,
E Y = E [ ( Y E ( Y R ) ) ( Y E ( Y R ) ) T ] ,
E Y X = E [ ( Y E ( Y R , X ) ) ( Y E ( Y R , X ) ) T ] .
R = s t X + N ,
I ( X ; R ) J ( X ) 1   bit ,
pr ( R ) = 1 2 π σ 2 ( p exp [ ( R s t ) 2 2 σ 2 ] + ( 1 p ) exp [ R 2 2 σ 2 ] ) .
E ( X R ) = [ 1 + 1 p p exp ( s t ( s t 2 R ) 2 σ 2 ) ] 1 .
R = H C det ( X ) + N ,
C det ( X ) = s T ρ X + c V c β ,
R = s H T ρ X + N c .
TSI = I ( X ; R ) = 1 2 0 s mmse H ( s ) d s ,
mmse H ( s ) = Tr ( H Σ N c 1 H ( E Y E Y X ) ) ,
Y = T ρ X .
ρ = [ ρ H 0 0 ρ H ] ,
R = s H T Λ ( X ) ρ + N c ,
Λ ( X = i ) = [ [ 0 ] P 1 × P i [ 0 ] P i 1 × P i [ I ] P i × P i [ 0 ] P i + 1 × P i [ 0 ] P Q × P i ] .
R = s H T Λ ( X ) ρ α + N c ,
TSI = I ( X ; R ) = 1 2 0 s mmse H ( s ) d s ,
mmse H ( s ) = Tr ( H Σ N c 1 H ( E Y E Y X ) ) ,
X { X , 0 } , Y = T Λ ( X ) ρ α .
J ( X ) = ( 1 p ) log ( 1 p ) q = 1 Q Pr ( X = q ) log Pr ( X = q ) ,
R = s H T Ω ( X ) ρ α + N c .
Ω ( X = i ) = [ Λ ( X = i ) 0 0 Λ ( X = i ) ] ,
ρ = [ ρ H 0 0 ρ H ] ,
J ( X ) = i = 1 2 q = 1 P Pr ( X = q , α i ) log Pr ( X = q , α i ) ,
h i , j = Δ 2 Δ 2 Δ 2 Δ 2 sinc 2 ( ( x i Δ ) W ) sinc 2 ( ( y j Δ ) W ) d x d y ,
R = N ( P { H [ C ( X ) ] } ) .
R = s P H T ρ X + N c ,
N c = c P H V c β + N .
TSI I ( X ; R ) = 1 2 0 s mmse H ( s ) d s ,
mmse H ( s ) = Tr ( H P Σ N c 1 P H ( E Y E Y X ) ) .
R = s P H T Λ ( X ) ρ α + N c ,
TSI I ( X ; R ) = 1 2 0 s mmse H ( s ) d s ,
mmse H ( s ) = Tr ( H P Σ N c 1 P H ( E Y E Y X ) ) .
R O O = E ( o o T ) ,
P = 1 cs P * ,
P = T ¯ Σ N c 1 ,
E ( Y R ) = l Y l Pr ( Y = Y l R ) = l Y l pr ( R Y = Y l ) Pr ( Y = Y l ) m pr ( R Y = Y m ) Pr ( Y = Y m ) ,
E ( Y R ) = l = 1 P p Y l pr ( R Y = Y l ) m = 1 P p [ pr ( R Y = Y m ) ] + ( 1 p ) P [ pr ( R Y = 0 ) ] .
pr ( R Y = Y l ) = 1 ( 2 π ) K 2 det Σ R Y exp [ 1 2 ( Θ 1 + Θ 2 l + Θ 3 l + Θ 4 ) ] ,
Σ R Y = c P H V c Σ β ( P H V c ) T + σ N 2 I ,
Θ 1 = R T Σ R Y 1 R 2 c R T Σ R Y 1 P H V c μ β ,
Θ 2 l = 2 s R T Σ R Y 1 P H Y l ,
Θ 3 l = s Y l T H T P T Σ R Y 1 P H Y l + 2 s c Y l T H T P T Σ R Y 1 P H V c μ β ,
Θ 4 = c μ β T V c T H T P T Σ R Y 1 P H V c μ β .
E ( Y R ) = l = 1 P p Y l exp [ 1 2 ( Θ 2 l + Θ 3 l ) ] m = 1 P p exp [ 1 2 ( Θ 2 m + Θ 3 m ) ] + ( 1 p ) P .
E ( Y R ) = l = 1 ; X = [ 1 , 0 ] T P p Y l exp [ 1 2 ( Θ 2 l + Θ 3 l ) ] + l = 1 ; X = [ 0 , 1 ] T P ( 1 p ) Y l exp [ 1 2 ( Θ 2 l + Θ 3 l ) ] m = 1 ; X = [ 1 , 0 ] T P p exp [ 1 2 ( Θ 2 m + Θ 3 m ) ] + m = 1 ; X = [ 0 , 1 ] T P ( 1 p ) exp [ 1 2 ( Θ 2 m + Θ 3 m ) ] .
E ( Y R ) = i = 1 Q l = 1 P i Pr ( X = i ) P i Y i , l exp [ 1 2 ( Θ i , 2 l + Θ i , 3 l ) ] j = 1 Q m = 1 P j Pr ( X = j ) P j exp [ 1 2 ( Θ j , 2 m + Θ j , 3 m ) ] + ( 1 p ) ,
E ( Y R ) = i = 1 Q l = 1 P i α = [ 1 , 0 ] T , [ 0 , 1 ] T Pr ( X = i , α ) P i Y i , l , α exp [ 1 2 ( Θ i , 2 l + Θ i , 3 l ) ] j = 1 Q m = 1 P j α = [ 1 , 0 ] T , [ 0 , 1 ] T Pr ( X = j , α ) P j exp [ 1 2 ( Θ j , 2 m + Θ j , 3 m ) ] ,
E ( Y R , X ) = l Y l Pr ( Y = Y l R , X ) .
Pr ( Y = Y l R , X ) = pr ( R , X Y l ) Pr ( Y = Y l ) pr ( R , X )
= pr ( R Y = Y l , X ) Pr ( X Y = Y l ) Pr ( Y = Y l ) m pr ( R Y = Y m , X ) Pr ( X Y = Y m ) Pr ( Y = Y m ) .
E ( Y R , X = 1 ) = l = 1 P Y l exp [ 1 2 ( Θ 2 l + Θ 3 l ) ] m = 1 P exp [ 1 2 ( Θ 2 m + Θ 3 m ) ] ,
E ( Y R , X = 0 ) = 0 ,
E ( Y R , X ) = l = 1 P Y l exp [ 1 2 ( Θ 2 l + Θ 3 l ) ] m = 1 P exp [ 1 2 ( Θ 2 m + Θ 3 m ) ] ,
E ( Y R , X = i , α = 1 ) = l = 1 P i Y i , l exp [ 1 2 ( Θ i , 2 l + Θ i , 3 l ) ] m = 1 P i exp [ 1 2 ( Θ i , 2 m + Θ i , 3 m ) ] ,
E ( Y R , α = 0 ) = 0 ,
E ( Y R , X = i , α ) = l = 1 P i Y i , l , α exp [ 1 2 ( Θ i , 2 l + Θ i , 3 l ) ] m = 1 P i exp [ 1 2 ( Θ i , 2 m + Θ i , 3 m ) ] ,

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