Recognition Performance from SAR Imagery Subject to System Resource Constraints

Transcription

1 Recognition Performance from SAR Imagery Subject to System Resource Constraints Michael D. DeVore Advisor: Joseph A. O SullivanO Washington University in St. Louis Electronic Systems and Signals Research Laboratory Supported by: ARO Center for Imaging Science DAAH ONR MURI N Boeing Foundation

2 Motivation Scene (source) Model Database (quantized source) Encoding Sensor Platform (channel) Interconnection Network (channel) ATR Processor (receiver) Inferred Target and Pose Inference based on conditional probabilistic models that incorporate assumptions and approximations Accuracy depends upon validity of models Resource consumption fixed by model and architecture Successive refinability to dynamically tradeoff accuracy and resource consumption 2

3 Outline Point #1: Assessment of model validity from sample data Point #2: Relating resource consumption to models Point #3: Hierarchical representations for refinability Conclusions 3

4 Point #1: Conditional Models for SAR Pixel distribution is a function of target class, pose, and location Pixels assumed pairwise independent Complex Gaussian: Pixel value r i distributed as pr ( i a,θ)= 1 ( ) e πσ i 2 a,θ r i µ i ( a,θ ) 2 σ 2 i ( a,θ ) Lognormal: x i = log r i is Gaussian Quarter power normal: x i = r i 1/2 is Gaussian px ( i a,θ)= 1 2πσ 2 i ( a,θ ) e ( ) 2 x i µ i ( a,θ ) 2σ 2 i ( a,θ ) 4

5 Point #1: Large Numbers of Small Samples Discretize azimuth, θ = observations per interval Assess model validity over 10 targets x 60 intervals x 400 pixels For each assumption evaluate H 0 : model valid vs. H A : model invalid Find test statistic D(r 1, r 2,, r N ), with known distribution under H 0 P-value is probability of a more extreme observation if H 0 were true: P = 1 - P D (d)) = Pr[D > d H 0 ] Small values of P evidence for rejecting H 0. P-value uniformly distributed when H 0 true: Pr[P D (D) p] = Pr[D P -1 D (p)] = P(P -1 (p)) = p P-value distribution for test of symmetry 5

6 Point #1: D Agostino-Pearson Method Let P G1 (g 1 ) and P G2 (g 2 ) be the cumulative distribution functions for sample skewness and kurtosis of N Gaussian random variables g 1 = k 3 32 g 2 = k 4 and k 2 k 2 2 where k i is the ith k-statistic D Agostino-Pearson statistic D DP given in terms of the inverse CDF of a standard normal variate, Φ -1 (z) [ ( ( )] 2 + Φ 1 P G2 ( g 2 ) D DP = Φ 1 P G1 g 1 [ ( )] 2 Pr[D DP > d] ] = 1 - P D (d)) is the P-value P of the test Distributions P G1 P and P G2 D approximated from 1,000,000 samples 6

7 Point #1: D Agostino-Pearson Method Quarter-power normal a good match, especially for clutter Gaussian model results unaffected by zero-mean assumption Log-normal model not as accurate for clutter pixels Pixels on Target Pixels on Clutter 7

8 Point #1: Kolmogorov-Smirnov Method From observations R 1, R 2,, R N P ˆ R ()= r 1 N u(r R i) i=1 where u( ) ) is the unit-step function N determine the empirical CDF The Kolmogorov-Smirnov test statistic is the supremum D KS = sup r P ˆ R ( r) P R ( r) Distribution of D KS is the same for all continuous P R Pr[D KS > d] ] is the P-value P of the test Probability computation: Press, et al. Numerical Recipes in C 8

9 Point #1: Kolmogorov-Smirnov Method Distribution parameters µ(a,θ) ) and σ 2 (a,θ)) are unknown Kolmogorov method requires known parameters» Convert independent Gaussian variates with unknown parameters to dependent T variates with known DOF» Test for T distribution Given independent Z 1, Z 2, Z N ( Z k Z ) N 1 N S 2 (k ) Gaussian with unknown µ, σ 2 is T distributed with N - 1 DOF, where 2 S (k ) = 1 N 1 ( Z j Z ) 2 (k) j k and Z (k ) = 1 N 1 Z j j k 9

10 Point #1: Kolmogorov-Smirnov Method Gaussian and quarter-power normal a better fit for clutter than target Lognormal less accurate for ground clutter Pixels on Target Pixels on Clutter 10

11 Point #2: Accuracy and Resource Consumption Scene (source) Model Database (quantized source) Encoding Sensor Platform (channel) Interconnection Network (channel) ATR Processor (receiver) Inferred Target and Pose ATR systems explicitly or implicitly based on target models with complexity C More complex models can yield better accuracy; Pr(error)=f(C,α SAR Complexity and computation power determine throughput; T CHIP SAR ) CHIP =h(c,α COMP ) Given an architecture, both accuracy,, Pr(error), and throughput, R=1/ CHIP, are parameterized by target model complexity =1/T CHIP 11

12 Point #2: Conditionally Gaussian Model Model each pixel i as independent, zero mean, complex conditionally Gaussian p ( R Θ,A,C 2 r θ, a,c 2 )= 1 π c 2 σ 2 i ( θ, a) e i r i 2 c 2 σ 2 i ( θ,a) Where: σ i2 (θ,a)) = variance function over pose and class c 2 = constant over all pixels to account for power fluctuation Recognition by maximizing the log-likelihood likelihood ratio * [ a ˆ, ˆ θ, c ˆ 2 ]= argmax [ a,θ,c 2 ] ln i ( ) p r a,θ,c 2 ( ) p r ξ 2 I i (a,θ) Where: ξ 2 = average clutter variance I i = mask function * Schmid & O Sullivan O Thresholding Method for Reduction of Dimensionality 12

13 Point #2: Target Model Estimation Given N registered training images q j of a target with pose φ j, estimate variances over N w windows of width d ˆ σ 2 ( θ k,a)= n 1 2 q k j { j:φ j w k } Registered Variance Images where w k = 2πk d N w 2,2πk + d N w 2 T 0 T 90 T 180 T 270 Variance estimate for an unregistered image r with pose θ formed by transforming the estimate from the closest w k Transformed Estimates 13

14 Point #2: Target Model Segmentation Pixel information relative to null-hypothesis used for target recognition Retain pixels i that are informative relative to the null-hypothesis: S a = i : 1 N w k ( ( ( ) p ( ;ξ 2 ) Dp ; ˆ σ 2 i θ k, a η Segmentation of target models, not of images Ordering of pixels by their empirical information relative to null ll-hypothesis. η = 0.2 η = 1 η = 5 η = 25 η = 125 For null-hypothesis ξ 2 = approximate background variance 14

15 Point #2: Cost of Likelihood Evaluation Let N(a,θ) ) = # nonzero mask elements, (i)( ) = ith nonzero mask pixel [ a ˆ, ˆ θ ]= argmax N(a,θ)lnˆ c 2 [ a,θ ] ˆ c 2 = 1 Na,θ ( ) Na,θ ( ) i=1 2 σ (i) r (i) 2 ( a,θ ) N(a,θ) [ lnσ 2 (i) (a,θ) +1 lnξ 2 ] i =1 + 1 ξ 2 N(a,θ ) i =1 r (i) 2 Expression evaluation consumes (on average): T mem 2N(a,θ) ) +2 memory reads totaling (2N(a,θ) ) +2) mem seconds P CPI T 2N(a,θ) ) +15 operations totaling (2N(a,θ) ) +15) cyc seconds P where: T mem = effective memory read time T cyc = clock period CPI = average cycles per instruction P = # processors 15

16 Point #3: Successively-Refinable Encoding Likelihood is a function of σ 2 i (a,θ), a function of continuous θ. Discrete approximation with N d non-overlapping overlapping intervals of width d: 2 σ d,i 2 σ d,i ( θ k, a)= 1 d 2πk Nd + d 2 2πk N d d 2 σ 2 i ( θ, a)dθ Smaller d yields more complex, piecewise constant representations Approximations d and d/2 are hierarchically related: ( θ k, a)= 1 n 1 +n 2 [ n 1 σ 2 ( d,i θ 2k,a )+ n 2 σ 2 ( d,i θ 2k +1,a)] 2 2 Given parent and 1 child, compute other child costing 4N(a,θ) 4 operations saving b(a,θ)/bw in communication time where: b(a,θ) ) = bits to represent σ 2 (a,θ)) and BW = bandwidth 16

17 Point #3: Successively-Refinable Encoding Consider decreasing interval widths d 1 =2π, d 2 =π,, d m =2π/2 m-1 Approximate variance images for bulldozer (D7) from d 1 through d 5 Successively-refinable models yield successively- refinable decisions Search over θ k in level i ordered by the most likely pose at level i-1 Resource consumption depends on extent of search (complexity) 17

18 Example Classification error rate as a function of the number of bits transmitted between the database and processor Classification depends on extent of search Eventually, search covers all possibilities Breadth-first search quickly finds good combinations of (a,θ)( Small overhead present with ordered searches 18

19 Example Relative entropy to reject target classes not in the database Error occurs if incorrectly classified or falsely rejected One-sample estimate of variance from observation: r i 2 Relative entropy quantifies difference from training estimate Reject observation if the empirical relative entropy is too large 1 S a ˆ i S a ˆ ( ( 2 σ ( i ˆ θ, a ˆ ) Dp ; ( 2 r i )p ; ˆ > γ Rejection behavior also a function of segmentation and search depth 19

20 Point #3: Computational Model Time to process through approximation d m includes time to: distribute SAR image to each processor process each approximation until local memory is full process each remaining approximation l mem T chip = S c BW log 2 ( P +1) + 2 l 1 N T τ dl + 2 l 2 N T τ dl + Where: l=1 m l =l mem +1 S c = bits per SAR image BW = network bandwidth P = number of processors N T = number of target classes ( τ ) dl τ d = average time per template at approximation d exploiting hierarchy. Receive variance, compute variance, and compute likelihoods. τ d = average time per template without exploiting hierarchy. Receive variance and compute likelihood. 20

21 Example With: Chip Rate = 1/T chip 1 GHz clock 64 bit read per clock cycle 10 target classes 0.5 CPI 25 target locations considered High throughput corresponds to coarse approximations Markers denote start of a new level in model representation tree 1 Gbps interconnection 10 Gbps interconnection 21

22 Conclusions Automatic recognition as a two-source communication problem Accuracy depends on model validity» assessed from sample data, aggregating over many samples» function of approximation level Resource consumption also depends on approximation level Successively-refinable decisions from hierarchical approximations» Process until resources exhausted» Generate good candidates quickly Open Issues Include Implication of good fit for quarter-power model Refinability over depression angle and articulation Theoretical minimum error as a function of resources consumed 22

23 Contributions Recognition as a two-source communication problem Detailed assessment of conditional models for SAR imagery Approaches to rejection of confuser objects Probability of error expression for two-class tests Relating accuracy to resource consumption through complexity Successively-refinable models and decisions Experimental results for recognition from SAR imagery 23