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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. ece. Presentation Outline.

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  1. ece 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 2006

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

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

  4. 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) FIO/LS 2006

  5. ece Task specific source encoding Y = C(X) C(X) stochastically encodes X and produces scene Y X Virtual source Encoding SCENE • Detection task: presence/absence of target is of interest • Virtual source variable must be binary • X = 1/0 implies tank present/absent X = 0 (Tank absent) X = 1 (Tank present) FIO/LS 2006

  6. Virtual source R = n(H(C(X))) X H(Y) Y = C(X) Noise Encoding Channel SCENE IMAGER Mutual information between X and R Always bounded by the entropy of X ece Task specific information (TSI) • Imaging chain block diagram • Imager is characterized by channel H and noise n • Imager does not add entropy to the relevant task • Definition for Task-specific information: Entropy Z(X) – maximum task-specific information content FIO/LS 2006

  7. ece TSI (continued) • Measurement can be written as nand sdenote 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 FIO/LS 2006

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

  9. ece Simulation details • Detection task: probability of occurrence = ½ • TSI will be bounded by 1 bit • 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 • Ideal and diffraction limited Example scenes H = I H = sinc2(.) FIO/LS 2006

  10. MMSE plots for H = I TSI for both H = I & sinc2() H = I MMSE conditioned over R Nyquist blur MMSE Task-specific information MMSE conditioned over R and X Twice the Nyquist blur MMSE snr snr ece Detection Task results • MMSE is small in low and high snr region • MMSE component conditioned on X improves faster through medium snr • TSIsaturates at 1 bit with increasing snr • Degradation in performance due to blur as expected FIO/LS 2006

  11. ece Detection and Localization Task • Example scene when considering localization task Region 1 Region 2 • 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 Region 3 Region 4 • Modifications to the encoding matrix T TSI will be bounded by 2 bits in this case FIO/LS 2006

  12. Detection and localization H = I Nyquist blur Task-specific information Twice the Nyquist blur snr 14.47 dB 15.45 dB 20 dB 16.53 dB ece Results 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 • TSIsaturates at 2 bits • Degradation in performance due to blur as expected FIO/LS 2006

  13. X R Projection Noise Source Encoding Channel R = N(P(H(C(X)))) P IMAGE (M×M ) (F×1 ) 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×M2) FIO/LS 2006

  14. snr = 25 Too few photons per measurement Too few measurements Conventional Imager (P = I) Task-specific information Task-specific information Rollover starts at F = 24 onwards (trade-off between TSI and measurement snr) snr = 19 snr = 35 snr F (# of projections) 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 FIO/LS 2006

  15. 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 FIO/LS 2006

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