Non-Homogeneous Hidden Markov Chain Models for Wavelet-Based - PowerPoint PPT Presentation

Non-Homogeneous Hidden Markov Chain Models for Wavelet-Based Hyperspectral Image Processing Marco F. Duarte Mario Parente

Hyperspectral Imaging One signal/image per band Hyperspectral datacube Spectrum at each pixel represents composition/physical state of subject (remote sensing, industrial process monitoring, etc.)

Hyperspectral Signatures Igneous minerals Carbonate minerals Phyllosilicate minerals (clays) • Encode reflectivity of material surface over a variety of wavelengths of light (100+) • Differences evident between materials/minerals of different classes; more subtle within a class • Signature fluctuations used in ad-hoc fashion for material identification • Positions and shapes provide identifiability

Hyperspectral Classification Absorption Bands • Tetracorder : List of rules to identify spectra by shape • Rules can be arbitrarily complicated • New rules must be created for new materials • “Difficult” cases need experienced analyst

Hyperspectral Classification • Tetracorder : List of rules to identify spectra by shape • Rules can be arbitrarily complicated • New rules must be created for new materials • “Difficult” cases need experienced analyst [Clark et al., USGS 2003]

Hyperspectral Classification specific group 2 # algorithm: featfit1 # input library reference spectrum #=TITLE=Alunite • Tetracorder : List of GDS83 Na63 rules to identify # channels to exclude (global spectra by shape variable) Alunite GDS83 Na63 • Rules can be # 2 spectral features, 0 not arbitrarily features Dw 2.048 2.078 2.247 complicated 2.277 ct .04 • New rules must be # continuum wavelengths, created for new threshold (ct) Dw 1.466 1.476 1.535 1.555 ct .05 materials # continuum wavelengths, • “Difficult” cases need threshold (ct) FITALL > 0.5 experienced analyst # fit thresholds: if below 0.5, [Clark et al., USGS 2003] reject

Hyperspectral Classification • Specialized distance metrics: spectral angle mapper, spectral divergence, etc. • aim to match shapes • sensitive to additional variations in signal from sample to sample • How to successfully capture fluctuations in punctuated , piecewise smooth signals?

Continuous Wavelet Transform • Mother wavelet dilated to scale s and translated to offset u : • CWT of a spectrum x ( f ) , , composed of wavelet coefficients at scales s = 1, ..., S , offsets u = 0, F / N , 2 F / N , ..., F - F / N : • Coefficient acts as a “ detector ” of fluctuations of scale s at location f = u

Continuous Wavelet Transform • Organize in a 2-D array : rows are scales, Reflectance 0.25 columns are offsets. 0.2 • For simplicity, offset 0.15 u = nF / N matched to 0.1 0.5 1 1.5 2 index n = 0, 1, ..., N - 1 Wavelength, µ m • Wavelengths for 2 indices n shown Scales 4 • Columns of matrix 6 8 representation give 50 100 150 200 250 300 chains of parent/child Offsets Samples wavelet coefficients

Structure of CWT Coefficients 0.25 Band 0.2 Smooth 0.15 0.1 0.5 1 1.5 2 2 Small 4 6 Large 8 50 100 150 200 250 300

Structure of CWT Coefficients 0.25 0.2 0.15 0.1 Sparsity 0.5 1 1.5 2 2 4 6 8 50 100 150 200 250 300

Structure of CWT Coefficients 0.25 0.2 0.15 0.1 0.5 1 1.5 2 2 Persistence 4 6 8 50 100 150 200 250 300

Non-Homogeneous Hidden Markov Chains • Stochastic model to encode structure of CWT coefficients State Value s 1 2 2 4 3 6 4 8 5 ... 50 100 150 200 250 300

Non-Homogeneous Hidden Markov Chains • Stochastic model to encode structure of CWT coefficients State: L arge, S mall Value s 1 2 2 4 3 6 4 8 5 ... 50 100 150 200 250 300

Non-Homogeneous Hidden Markov Chains • Stochastic model to encode structure of CWT coefficients State: L arge, S mall Value: State-dependent zero-mean Gaussian distribution s 1 2 2 + 4 3 6 4 8 5 ... 50 100 150 200 250 300

Non-Homogeneous Hidden Markov Chains • Stochastic model to encode structure of CWT coefficients State: L arge, S mall Value: State-dependent zero-mean Gaussian distribution s 1 2 2 + 4 3 6 4 8 5 ... 50 100 150 200 250 300

Non-Homogeneous Hidden Markov Chains • Stochastic model to encode structure of CWT coefficients State: To obtain persistence , favor progressions Value: To obtain decay , reduce variances across scales s 1 2 2 + 4 3 6 4 8 5 ... 50 100 150 200 250 300

Modeling Hyperspectral Datasets • Why use continuous/ undecimated wavelets? 0.25 So that information at each 0.2 scale is available for each 0.15 0.1 wavelength 0.5 1 1.5 2 • Why separate chains for each spectra? 2 Because the “size” of a relevant fluctuation is 4 relative to wavelength 6 (e.g., absorption bands appearing in all 8 spectra) 50 100 150 200 250 300

Modeling Hyperspectral Datasets • Collect representative ( universal ) library of 0.25 hyperspectral signatures 0.2 (e.g. USGS for minerals) 0.15 0.1 • Extract CWT coefficients for 0.5 1 1.5 2 each hyperspectral signature; collect into 2-D 2 array 4 • Train an NHMC on each of the N wavelengths (array 6 columns) over the spectral 8 library 50 100 150 200 250 300

Modeling Hyperspectral Datasets • Using learned NHMC Reflectance 0.25 model, generate 0.2 state probabilities/ 0.15 labels for each 0.1 hyperspectral 0.5 1 1.5 2 Wavelength, µ m signature in library 2 • State labels provide Scales 4 binary information on 6 “ interesting ” parts of 8 the signal 50 100 150 200 250 300 Samples • Use as features in hyperspectral 2 Scales 4 signature processing 6 (e.g., classification) 8 50 100 150 200 250 300 Samples

Example: Mineral Classification • USGS spectral library with 57 clay samples from 12 classes [Rivard et al., 2008]. • One prototype/ endmember per class, [Rivard et al., 2008] 89% classify rest by Muscovite nearest-neighbor Nontronite (NN) to prototypes. Saponite Sauconite Vermiculite • Classification errors Talc Pyrophyllite are points that deviate Montmorillonite Illite from diagonal. Nacrite Kaolinite Dickite 10 20 30 40 50 ID of Spectrum NHMC 95%

The Power of “Big Data” • Statistical modeling of Reflectance 0.25 coefficients across 0.2 spectral sample 0.15 provides measures 0.1 of relevance of 0.5 1 1.5 2 Wavelength, µ m bands/smooth regions 2 • Model parameters can Scales 4 provide “map” of 6 relevant scales, 8 spectral bands, etc. 50 100 150 200 250 300 Samples for training dataset 2 Scales 4 6 8 50 100 150 200 250 300 Samples

The Power of “Big Data” 2 / � S 2 , training with all ENVI minerals � L Wavelet Scale 10 2 4 5 6 8 0.5 1 1.5 2 2.5 Wavelength, µ m 2 / � S 2 , training with ENVI clays only � L Wavelet Scale 10 1 = equal states 2 4 5 6 8 0.5 1 1.5 2 2.5 Wavelength, µ m

The Power of “Big Data” Probability of small state, training with all ENVI minerals 1 Wavelet Scale 2 4 0.5 6 Sparsity 8 0 0.5 1 1.5 2 2.5 Wavelength, µ m Probability of small state, training with ENVI clays only 1 Wavelet Scale Fine Scale Info Ambiguity 2 4 0.5 6 8 0 0.5 1 1.5 2 2.5 Wavelength, µ m

The Power of “Big Data” % samples labeled small, training with all ENVI minerals 1 Wavelet Scale 2 4 0.5 6 8 0 0.5 1 1.5 2 2.5 Wavelength, µ m % samples labeled small, training with ENVI clays only 1 Wavelet Scale 0 = no discriminability 2 4 0.5 6 8 0 0.5 1 1.5 2 2.5 Wavelength, µ m

Example: Mineral Classification • Same example as before, but subset of labels selected according to three “discriminability” criteria [Rivard et al., 2008] 89% • For all metrics used, Muscovite classification Nontronite performance matches Saponite Sauconite that obtained with all Vermiculite Talc labels (95% success Pyrophyllite Montmorillonite rate) Illite Nacrite Kaolinite Dickite 10 20 30 40 50 ID of Spectrum NHMC 95%

Conclusions • Goal: design hyperspectral signal models and features that can capture semantic information used by practitioners in remote sensing – relevance of absorption bands in tasks, e.g., classification – multiscale analysis studies a variety of spectral features – robustness to fluctuations in shape and location of bands • Stochastic models (Non-Homogeneous Markov Chain) enable robust identification of relevant features – adaptive sampling, spectral sampling rate adjustments – identify non-informative absorption bands, universal features • Future work : – Hyperspectral image applications: segmentation, unmixing, ... – Study robustness to signature fluctuations (lab & field datasets) http://www.ecs.umass.edu/~mduarte mduarte@ecs.umass.edu