PowerPoint Presentation

Phase Retrieval Applied to Asteroid Silhouette Characterization by Stellar OccultationRussell Trahan & David Hyland

JPL Foundry Meeting April 21, 2014

IntroductionOccultation to detect asteroidsCharacteristics of a shadowShadows of distant, small bodiesMotivation for asteroid detectionSilhouette EstimationDerivation of shadow patternSilhouette recovery from the shadow patternRaster scan methodExampleMeasurement NoiseNoise modelEffect on raster scan estimation methodData CoverageSparse measurement of the shadow patternMeasurement position uncertaintyMeasurements to silhouette resolution ratioConclusion

OutlineIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry StudyEach moving observer records the time when the star disappears and reappears. The shape of the shadow region can be estimated based on the occultation times and the positions of the observers.Data Collection and Time KeepingArray of Light CollectorsDistant StarOccluding AsteroidShadow RegionVelocity Relative to ShadowTraditional Occultation Method to Characterize AsteroidsIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry StudyDistant star illuminates asteroid and casts shadowFresnel Number:Observer is far enough away that no shadow is visibleDiffraction effects dominateCoordinate systems normalized by distance to the asteroidCharacteristics of a ShadowIntroduction Estimation Noise Coverage Conclusion

ShadowFresnel Region

Fraunhofer Region

Shadow ZoneInterference ZoneObservation Location

April 21, 2014Russell TrahanJPL Foundry StudyHuygens-Fresnel principle applied to shadows gives the wave field at the observers spatial location .

The silhouette function is defined as

Shadows of Small, Distant BodiesIntroduction Estimation Noise Coverage Conclusion

Silhouette

Shadow Pattern

April 21, 2014Russell TrahanJPL Foundry Study5Shadow Pattern vs.Fresnel NumberIntroduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry StudyDiameter (m)>10001000-140140-4040-1# Estimated96614,000~285,000--# Observed8994,5572,2591,685% Observed93%~33%~1%--Distance (km) for F=50>10,000,000200,00015,00010Distance (au) for F=50

>0.070.0010.00017e-8Distance (km) for F=0.5>1,000,000,00020,000,0001,500,0001,000Distance (au) for F=0.5

>7

0.10.017e-6Asteroids of InterestIntroduction Estimation Noise Coverage Conclusion For a sharp shadow to exist for 140-40m diameter asteroids, it must pass within 98,000 km.140-40m is large enough to cause significant damage upon impactOnly ~1% are accounted forApril 21, 2014Russell TrahanJPL Foundry Study

Field of View for 100m Asteroid Using Traditional OccultationIntroduction Estimation Noise Coverage Conclusion Figure drawn to scaleApril 21, 2014Russell TrahanJPL Foundry Study

Field of View for 100m Asteroid Using Shadow PatternIntroduction Estimation Noise Coverage Conclusion Figure drawn to scaleApril 21, 2014Russell TrahanJPL Foundry StudyIntensity of wave field, , is knownWave field equation looks like Fourier transform of complex function

Discretizations need to be small enough to capture high frequencies.For and (green light), Integration limits over are not captured in the discrete Fourier transformWould require complete knowledge of Discrete Fourier Transformation of Shadow Pattern?Introduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry StudyTrue Shadow Pattern

Shadow Pattern from DFTDiscrete Fourier Transformation of Shadow Pattern? (2)Introduction Estimation Noise Coverage Conclusion

Wrong! Physics lost in DFTApril 21, 2014Russell TrahanJPL Foundry StudyHuygens-Fresnel principle applied to shadows gives the wave field at the observer location .

The silhouette function is defined as

The measured intensity of the wave field is

Recovery of the Silhouette FunctionIntroduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry StudyIntegral over the silhouette can be split into a grid and evaluated

Recovery of the Silhouette Function (2)Introduction Estimation Noise Coverage Conclusion

Silhouette

April 21, 2014Russell TrahanJPL Foundry Study is the estimated silhouettes shadow pattern. is the measured shadow pattern.When the correct silhouette is found, the two shadow patterns should match.An error metric can be defined as

Objective is to change the estimated silhouette until the error is minimized.Recovery of the Silhouette Function (3)Introduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry Study

Problem FlowchartIntroduction Estimation Noise Coverage Conclusion True SilhouetteMeasured Shadow Pattern

Estimated Silhouette

Estimated Shadow Pattern

Minimize DifferenceApril 21, 2014Russell TrahanJPL Foundry StudyStartupCompute the contribution of each element of the summation and store in memory

Make initial guessIteration over each pixel in the imageFlip the element of Construct the new using the contributions previously computed for each pixelCompare the measured and estimated shadow patternsKeep the change if the error decreased

Raster Scan AlgorithmIntroduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry StudyRaster Scan ExampleIntroduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry Study

Successive Nested GridsIntroduction Estimation Noise Coverage Conclusion

32x32 Pixels64x64 Pixels128x128 PixelsApril 21, 2014Russell TrahanJPL Foundry StudySusceptible to stagnationIf no change has occurred for an entire iteration, the solution has converged or stagnated.Since only one pixel is changed at a time, local minima are prevalent.Randomly change a few pixel values to move away from the local minimum.No specific method of filtering noise has been developed yet.User chooses the resolution of the estimated silhouetteLow resolution estimate can be the initial guess for a high resolution estimateOnly a comparison is made between the measured and estimated shadow patternComplete data coverage is not necessary!How much coverage is necessary is still unknown.This simple first guess approach seems to work fairly well.Raster Scan PerformanceIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry StudyComputation is performed offline. Relaxed computational requirements.Construction of the shadow pattern can be parallelized for each pixel in the UV plane.Comparison of the shadow patterns can be mostly parallelized for each pixel in the UV plane.Current Matlab implementation is quite slow, ~6 minutes per iteration for 64x64 pixels. GPU implementation could promise unnoticeable runtimes.These insights imply computational expense should not be a factor in the algorithm development.Raster Scan Computational PerformanceIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry StudyThe measured intensity contains several sources of noise:Light sensor noiseAperture directionAperture positionEstimated range-to-target, z Rotation of the targetThe complex wave fields real and imaginary components are corrupted.

Noise model:

Noise ModelIntroduction Estimation Noise Coverage Conclusion

April 21, 2014Russell TrahanJPL Foundry Study25 Monte Carlo trials at several noise levelsTrack error between the true and estimated shadow patternsErratic behavior near zero noise not fully explained yetLinear growth in error after 10% noise levelRaster Scan Performance with NoiseIntroduction Estimation Noise Coverage Conclusion Final Error to Max Error Ratio

Mean ErrorIterationNoise StDevNoise StDevApril 21, 2014Russell TrahanJPL Foundry Study

Result with 0.25 Noise StDevIntroduction Estimation Noise Coverage Conclusion

IterationIterationTrue ErrorReference ErrorApril 21, 2014Russell TrahanJPL Foundry Study23Executive Eye ManipulationIntroduction Estimation Noise Coverage Conclusion

Changed PixelsTrue SilhouetteEstimate after 5 Additional IterationsApril 21, 2014Russell TrahanJPL Foundry Study

Result with 0.5 Noise StDevIntroduction Estimation Noise Coverage Conclusion Estimate after 5 IterationsNoisy Measurement of the Shadow PatternApril 21, 2014Russell TrahanJPL Foundry Study

Relative velocity of shadow pattern dominant in constellationLinear coverage path across the UV planePattern may not be regularRatio of # measurements to image

pixels, Typical Data Coverage PatternIntroduction Estimation Noise Coverage Conclusion

Reference ErrorIterationPerfect Recovery

Denotes no dataApril 21, 2014Russell TrahanJPL Foundry Study

Introduction Estimation Noise Coverage Conclusion

Reference ErrorIteration

April 21, 2014Russell TrahanJPL Foundry Study

Introduction Estimation Noise Coverage Conclusion Reference ErrorIteration

April 21, 2014Russell TrahanJPL Foundry StudyData CoverageIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry StudyAperture Position ErrorIntroduction Estimation Noise Coverage Conclusion

Erroneously Assumed Positions

April 21, 2014Russell TrahanJPL Foundry StudyAperture Position ErrorIntroduction Estimation Noise Coverage Conclusion

Erroneously Assumed Positions

April 21, 2014Russell TrahanJPL Foundry StudyFinite star wish to not consider the star to be a point source.Finite bandwidth wish to not assume an infinitesimal bandwidth of light.What is a better raster scan method?How far can we get with the nested grid idea?How much data coverage of the shadow pattern is needed?What can we do about noise?Ongoing WorkIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry StudyThe traditional method of recording the disappearance and reappearance of the occluded star is not adequate for small asteroids.Small asteroids ~100m can be characterized using shadow pattern data collected during a stellar occultation.The shadow pattern cannot be directly inverted to obtain the silhouette. An estimation process is required.The raster scan method gives good results for realistic test cases. Perhaps an even better performing method wont be too complex.We have some tools to answer the SNR and data coverage questions.ConclusionsIntroduction Estimation Noise Coverage Conclusion April 21, 2014Russell TrahanJPL Foundry Study