1. FIO/LS 2006 ece Task-Specific Information Amit Ashok 1, Pawan K Baheti 1 and Mark A. Neifeld 1,2 Optical Computing and Processing Laboratory 1 Dept. of Electrical and Computer Engineering, 2 College of Optical Sciences, University of Arizona, Tucson.

2. FIO/LS 2006 ece Presentation Outline Images and Information Task-specific information (TSI) Detection and Localization tasks Comparison for conventional and compressive imagers Results and Conclusions

3. FIO/LS 2006 ece Information content of an image 512 512 × 512 × 3 × 8 = 6.2 Mb 64 × 64 × 1 × 8 = 32 Kb 64 Compression 2.1 Mb Compression 24 Kb More precise measure requires source probability density ρ PROBLEM: ρ is very complex/unknown

4. FIO/LS 2006 ece 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)

5. FIO/LS 2006 ece Task specific source encoding Y = C(X) Virtual source Encoding X 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

6. FIO/LS 2006 Imager is characterized by channel H and noise n Imager does not add entropy to the relevant task Definition for Task-specific information: ece Task specific information (TSI) Imaging chain block diagram R = n(H(C(X))) Y = C(X) Virtual source Encoding X 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

7. FIO/LS 2006 ece TSI (continued) Measurement can be written as n and s denote additive Gaussian noise and snr respectively Computing TSI is difficult for non-Gaussian source Use Verdu’s relation between mutual information and minimum estimation error

8. FIO/LS 2006 Encoding matrix: selects target at one of the P positions in M×M scene Clutter weighted by β ~ N(m β, Σ β ) ece Target detection Virtual source X is binary indicating the presence/absence of tank Measurement: s is signal to noise ratio c is the clutter to noise ratio

9. FIO/LS 2006 ece Simulation details Detection task: probability of occurrence = ½ TSI will be bounded by 1 bit H = IH = sinc 2 (.) 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

10. FIO/LS 2006 ece 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 MMSE snr TSI for both H = I & sinc 2 () H = I Nyquist blur Twice the Nyquist blur Task-specific information snr

11. FIO/LS 2006 ece 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

12. FIO/LS 2006 Detection and localization H = I Nyquist blur Twice the Nyquist blur 14.47 dB 15.45 dB 16.53 dB 20 dB Task-specific information snr ece Results TSI saturates at 2 bits Degradation in performance due to blur as expected H = I: TSI = 1.8 bits for snr = 28 H = sinc 2 (0.5x): TSI = 1.8 bits for snr = 35 H = sinc 2 (0.25x): TSI = 1.8 bits for snr = 45

13. FIO/LS 2006 ece 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×M 2 ) Source Encoding Channel Noise R = N(P(H(C(X)))) Projection XR IMAGE (M×M ) P (F×1 )

14. FIO/LS 2006 ece 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 Task-specific information snr snr = 25 Rollover starts at F = 24 onwards (trade-off between TSI and measurement snr) Task-specific information F (# of projections) Too few measurements Too few photons per measurement

15. FIO/LS 2006 ece 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