Imaging Through Atmospheric Turbulence in Remote Sensingplemmons/research/Ple_Sanya2015.pdf · Outline Overview of Optical Imaging in Remote Sensing Our Recent Projects (Imaging Through)

Imaging Through Atmospheric Turbulence inRemote Sensing

PSF Estimation and Deblurring for Hyperspectral Imaging

Bob Plemmons

WFU

Sanya China, Jan. 2015

Starry Night - Vincent van Gogh

Bob Plemmons (WFU) Spectral Imaging Through Turbulence Sanya China, Jan. 2015 1 / 52

Outline

Overview of Optical Imaging in Remote SensingOur Recent Projects(Imaging Through) Turbulence: da Vinci, Galileo,Komolgorov, Von KarmanHyperspectral Imaging (HSI)Estimating HSI PSFs for Atmospheric TurbulenceJoint Sparse Deblurring and Feature ExtractionIf time: Compressive Snapshot Spectral -Polarimetric Imaging


References Related to Presentation

Zhao, Wang, Huang, Ng, Ple. “Deblurring and sparse unmixing forhyperspectral images”. IEEE, Geoscience and Remote Sensing, 2013.

Berisha, Nagy and Ple. “Estmation of Wavelength Dependent PSFs forDeblurring Hyperspectral Images,”, in revision, 2015.

Berisha, Nagy and Ple. “Deblurring and Sparse Unmixing ofHyperspectral Images using Multiple PSFs,”, in revision, 2015.

Ple., Prasad, Berisha, Pauca, “Compressive Spectral andSpectro-Polarimetric Images Taken Trough Atmospheric Turbulence,”preprint in progress.

First 3 available at: http://users.wfu.edu/plemmons/


Optical Remote Sensing

2015 has been proclaimed the International Year of Light andLight-Related Technologies by UNESCO.

Optical sensors form images of objects or scenes by detectingtheir solar or laser reflectance.

Hyperspectral (2D spatial & 1D wavelength) andLiDAR (2D spatial & 1D range) Imaging

Hyperspectral imaging (HSI) collects information across the visibleand IR spectrum for interrogating scenes.LiDAR measures distance by illuminating targets with a scanninglaser and analyzing the reflected light. Laser wavelengths vary tosuit the target: 10 µm to UV. Very short.Fused HSI and LiDAR for enhanced analysis. Combined laserranging & object material identification.

Spectro-Polarimetric imaging (3D hyperspectral & 1Dpolarization). Data is a 4D tensor. Polarization identifies objectshape, metallic surfaces.



NGA HSI/LiDAR Project

Implicit Geometry Framework (IGF) using level set methods forLiDAR data representation and compression. “A Novel Approachto Environment Reconstruction In LIDAR and HSI Datasets.”Randomized matrix and tensor factorization for HSI compression.Low-rank matrix approximations, e.g., approx. tSVD of HSItensor, then fusion of components with LiDAR data.Clustering and classification of fused data, andinformation-theoretic results, detecting objects in shadows, etc.New project: Information-theoretic Fusion and Analysis of HSI,LiDAR and Polarimetric Data.

NGA Campus



Turbulence

Image contains sketch by Leonardo da Vinci, along with a remarkablymodern description: “ the smallest eddies are almost numberless, andlarge things are rotated only by large eddies and not by small ones.”Called phenomena “turbolenza”, leading to modern word turbulence.


Atmospheric Turbulence

Later, Galileo, 1564-1642, knew effects of atmospheric turbulence ontelescopes. Modern mathematical models developed by:

(a) Th. von Karman, 1881-1963 (b) Andrey Kolmogorov, 1903-1987

Figure 1: Example of atmospheric turbulence blurring for video imaging at adistance of 1 km.


Atmospheric Turbulence Effects on Optical Images

Earth’s atmosphere is turbulent and variations in the index ofrefraction cause the plane optical wavefront from distantobjects to be distorted. Caused by variation of air temperaturein eddies and air currents. Arriving light wavefront is crumpled.In astronomy, or space situational awareness, resolution of allground-based telescopes is severely limited by effects ofatmospheric turbulence. At best sites, the resolution of a 2.5mtelescope is degraded by at least a factor of five.Can cause blurring in ground-level (horizontal) imaging. Forhorizontal image beam paths, ground based turbulencetypically has a highly non-Kolmogorov power spectraldensity (PSD), and both phase and amplitude perturbations(scintillations) must be included in PSF model.


Imaging Through Atmospheric Turbulence - Estimating aPSF for Deconvolution

• Ground-level turbulance - video imaging illustration 1km distance.

(a) Image taken in early morning (b) Afternoon inage through turbu-lence.

Active research topic: e.g., Zhu and Milanfar, 2012-13. Lou, Kang,Soatto, Bertozzi, 2013. Others ...Methods generally based on selecting “lucky frames”, patches, imageregistration, fusion, segmentation etc.


Imaging Through Atmospheric Turbulence - Estimating aPSF, or PSFs, for Deconvolution

• Astronomical imaging, looking up

Nagy, Jefferies and Chu 2010, 2014 . R. Chan, X. Yuan and W. Zhang2012, 2013. Berisha, Nagy, Plemmons 2015. Others ...Methods based on wavefront sensing (Shack-Hartman), gradient &phase estimation leading to a PSF, or turbulence modeling.


Hyperpectral Imaging (HSI)

• Spectral imagers capture a 3D datacube (tensor)containing:

2D spatial information: x-y1D spectral information at each spatial location: λ

• Pixel intensity varies with wavelength bands -provides a spectral trace of intensity values.

• Generates a spatial map of spectral variation.

• Challenging to remove atmospheric turbulence blurfrom HSI.


Spectral (beyond RGB) Imaging at Wavelengths λ

Figure 2: λ generally ranges between 400 and 2500 nanometers.


Structure of HSI Data


Extract Spectral Signatures (Traces) from HSI toIdentify Materials

Figure 3: Illustration High Resolution - Each pixel represents a vector


Some Applications of HSI

Detect and identifymilitarily importantobjects at a distance.

Identification of plantspecies.Food processing.Mineral resourceassessments.Medicine.Enable space objectidentification & analysisfrom the ground.

http://www.globalsecurity.org/intell/library/imint/hyper.htm


http://www.globalsecurity.org/intell/library/imint/hyper.htm


Detect and identifymilitarily importantobjects at a distance.

Identification of plantspecies.

Food processing.Mineral resourceassessments.Medicine.Enable space objectidentification & analysisfrom the ground.

www.fujitsu.com/global/about/resources/news/press-releases/2011/0715-01.html


www.fujitsu.com/global/about/resources/news/press-releases/2011/0715-01.html


Detect and identifymilitarily importantobjects at a distance.Identification of plantspecies.

Food processing.

Mineral resourceassessments.Medicine.Enable space objectidentification & analysisfrom the ground.

http://photonics.com/Article.aspx?AID=51023


http://photonics.com/Article.aspx?AID=51023


Detect and identifymilitarily importantobjects at a distance.Identification of plantspecies.Food processing.

Mineral resourceassessments.

Medicine.Enable space objectidentification & analysisfrom the ground.

www.usgs.gov/blogs/features/usgs_science_pick/hyperspectral-hypercoverage/


www.usgs.gov/blogs/features/usgs_science_pick/hyperspectral-hypercoverage/


Detect and identifymilitarily importantobjects at a distance.Identification of plantspecies.Food processing.Mineral resourceassessments.

Medicine.

Enable space objectidentification & analysisfrom the ground.

http://mix.msfc.nasa.gov/abstracts.php?p=2700


http://mix.msfc.nasa.gov/abstracts.php?p=2700


Detect and identifymilitarily importantobjects at a distance.Identification of plantspecies.Food processing.Mineral resourceassessments.Medicine.

Enable space objectidentification & analysisfrom the ground.

http://www.afspc.af.mil/news1/story.asp?id=123369595

http://www.wpafb.af.mil/news/story.asp?id=123331461


http://www.afspc.af.mil/news1/story.asp?id=123369595

http://www.wpafb.af.mil/news/story.asp?id=123331461

Elementary Optics - HSI Resolution Revolution (PhysicsToday, Dec. 2014)

German physicist Ernst Abbe realized the resolution of animaging instrument is constrained by the wavelength of lightused, and the aperture of its optics.Resolution (detail an image holds) of a given telescope isproportional to the size of its aperture, and inverselyproportional to the wavelength of the light being observed:Res ≈ d/λ. - opposite to the effect of atmospheric turbulence.Resolution also affected by blur & noise.Our purpose: estimate wavelength dependent PSFs for HSI.


Hyperspectral Imaging (HSI) - Atmospheric PSFs

The image acquired at wavelength λ can be represented as

gλ(x , y) = hλ(x , y ;φ) ∗ f (x , y , λ) + ελ(x , y),

where the blurring kernel, with diffractive scaling included, is

hλ(x , y ;φ) =

(λ0

λ

)2

h0

(λ0

λx ,λ0

λy ;φ

),

with λ0 = reference baseline wavelength, and

h0(x , y ;φ) =∣∣∣F−1

(peiφ

)∣∣∣2 ,and where the wavefront phase φ = 2π

λ ×OPD.

• OPD is the optical path difference function, i.e. the optical phaseshift in passage through turbulent atmosphere. OPD is essentially thesame for each HSI wavelength, but the wavefront phase is not.


Tracking the PSFs as a Function of Wavelength

• Model phase function using the von Karman phase spectrum,

P(κx , κy ) =√.023(D/r0)5/3(κ2

0 + κ2x + κ2

y )−11/6,

where r0: Fried parameter, D: telescope aperture diameter, κ0:low-freq cutoff.

• φ =√

P(κx , κy ) W (κx , κy ), where W is a zero-mean, unit-variance,white Gaussian noise array. Set D/r0 = 10.

• PSF at wavelength λ given by:

hλ =∣∣∣F−1

(peiφλ

)∣∣∣2 .


Some Sources of HSI Blur in Imaging Through theAtmosphere

Telescope optics. Telescope PSF does not vary with position infield of view, but varies with wavelength. This is imaging camerasystem diffraction blur - worse at longer wavelengths.Atmospheric turbulence (AT) effects. AT effects are less atlonger wavelengths. If adaptive optics is used, the residual PSFcan vary with the spatial position. Limits use of a PSF obtainedfrom wavefront sensor gradient measurements, Jefferies,Nagy, et al., 2014.We estimate PSFs by modeling with spatially varyingelliptical Moffat functions and parameter identification byHSI of a “guide star” point sourse.Original Physics paper:A. F. Moffat, “A Theoretical Investigation of Focal Stellar Images in thePhotographic Emulsion and Application to Photographic Photometry,”Astronomy and Astrophysics, 1969.


Movies Illustrating PSF Change with Wavelength


Hyperspectral Imaging in Astronomy: ParanalObservatory, Chile


PSF Star (point source) HSI Image FormationModel

The image formation process for an isolated star or point source

gλ = hλsλ + ηλ

gλ is a vector representing the vectorized form of an observed, blurred, andnoisy image of an isolated star corresponding to wavelength λ.

hλ is a vector representing the vectorized form of an exact original image of theisolated star corresponding to wavelength λ.

sλ is a scalar representing the unknown intensity of the star spectrum atwavelength λ.

By assuming a parametrized formula for the PSF, the image formationmodel becomes

gλ = h(φλ)sλ + ηλ

φλ is a vector of unknown parameters corresponding to wavelength λ.




gλ = hλsλ + ηλ










gλ = hλsλ + ηλ










gλ = hλsλ + ηλ








Circular Moffat

The circular Moffat function is defined by a positive scale factor α anda shape parameter β. In this case, φ =

[α β

]T , and the PSF has theform:

h(φ)ix ,jy = h(α, β)ix ,jy =

(1 +

i2x + j2yα2

)−β

(1)

The flux:∫∫

h(α, β) dix djy = πα2

β−1 ⇒ 1 < β <∞

Multiply the PSF by the inverse of the flux to insure that∑ix ,jy

h(φ)ix ,jy = 1


Circular Moffat


[α β



(1 +

i2x + j2yα2

)−β

(1)

The flux:∫∫


β−1 ⇒ 1 < β <∞


h(φ)ix ,jy = 1


Circular Moffat


[α β



(1 +

i2x + j2yα2

)−β

(1)

The flux:∫∫


β−1 ⇒ 1 < β <∞


h(φ)ix ,jy = 1


Wavelength Varying Circular Moffat

Modeling the variation of the PSF with respect to λ:

A linear variation of α(λ) = α0 + α1λ

A constant value for β(λ) = β0

D. Serre, E. Villeneuve, H. Carfantan, L. Jolissaint, V. Mazet, S. Bourguignon, A. JarnoModeling the spatial PSF at the VLT focal plane for MUSE WFM data analysis purposeSPIE Astronomical Telescopes and Instrumentation: Observational Frontiers of Astronomyfor the New Decade (2010)

Using this model the parameter vector becomes φ =[α0 α1 β

]Tand the normalized wavelength varying PSF takes the form:

h(φ)ix ,jy ,λ = h(α0, α1, β)ix ,jy ,λ =β − 1

π(α0 + α1λ)

(1 +

i2x + j2y(α0 + α1λ)2

)−β

Simple model involving only 3 parameters.



Modeling the variation of the PSF with respect to λ:A linear variation of α(λ) = α0 + α1λ






π(α0 + α1λ)

(1 +

i2x + j2y(α0 + α1λ)2

)−β




Modeling the variation of the PSF with respect to λ:A linear variation of α(λ) = α0 + α1λ






π(α0 + α1λ)

(1 +

i2x + j2y(α0 + α1λ)2

)−β



Wavelength Varying Circular Moffats

λ = 450nm λ = 550nm λ = 650nm

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Elliptical Moffat

For spatially variant blurs, we need to use an elliptical Moffatfunction

h(φ)ix ,jy = h(α, β, γ,Θ)ix ,jy =

[1 +

1α2

(i2r +

j2rγ2

)]−β(2)

γ is the ellipticity parameter.

Θ is the rotation angle.[irjr

]=

[cos(Θ) sin(Θ)− sin(Θ) cos(Θ)

] [ixjy

].


Elliptical Moffat



[1 +

1α2

(i2r +

j2rγ2

)]−β(2)


Θ is the rotation angle.

[irjr

]=


] [ixjy

].


Elliptical Moffat



[1 +

1α2

(i2r +

j2rγ2

)]−β(2)


Θ is the rotation angle.[irjr

]=


] [ixjy

].


Wavelength Varying Elliptical Moffat

The variation of the elliptical Moffat PSF with respect to λ and thepolar coordinates (ρ, θ) in the field of view

γ(λ, ρ) = 1 + (γ0 + γ1λ)ρ

β is kept as a constant

α(λ, ρ) = α0 + α1ρ+ α2λ+ α3λ2.

Θ = π2 − θ.



Example: Elliptical Moffat - 7 Parameters for eachλ

For λ = 465nm: α0 = 3.75, α1 = −2.99 · 10−3, α2 = −4.31 · 10−3, α3 = 1.98 · 10−6

β = 1.74, γ0 = 6.86 · 10−4, γ1 = 2.17 · 10−6

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Elliptical Moffat: Field of View

We use 20 wavelengths with 25 orientations in the first quadrant of thefield of view for each wavelength.


Elliptical Moffat: Field of View

We use 20 wavelengths with 25 orientations in the first quadrant of thefield of view for each wavelength.


Optimization Problem (simplified)

The set of observed isolated star images at Nw wavelengths can bewritten as:

gλi= hλi sλi + ηλi

, i = 1,2, · · · ,Nw

gλi∈ <Np , hλi ∈ <

Np , ηλi∈ <Np .

sλi is a scalar representing the reflectance of the star at a particular wavelengthλi .

Since there is a one-to-one correspondence between λi and the indexi , without loss of generality, we will use the notation:

g i = gλi, hλi = hi , sλi = si , ηλi

= ηi .


Optimization Problem

By stacking all observations, we obtain the overall image formationmodel as:

g = H(φ)s + η

where

s =

s1s2...

sNw

and in the simpler case of a circular Moffat PSF,

g =

g1g2...

gNw

, H(φ) =

h1 0 · · · 0

0 h2...

.... . .

0 · · · hNw

, φ =

α0α1β

.



In the case of elliptical Moffat PSF, with Nw wavelengths and Noorientations (i.e., No polar coordinates (ρ`, θ`), ` = 1,2, . . . ,No) in thefield of view, we have

g =

g(1)1...

g(No)1

g(1)2...

g(No)2...

g(1)Nw...

g(No)Nw

, H(φ) =

h(1)1 0 · · · 0...

... · · ·...

h(No)1 0 · · · 00 h(1)

2 · · · 0...

... · · ·...

0 h(No)2 · · · 0

0 0 · · · 0...

. . ....

0 · · · h(1)Nw

.... . .

...0 · · · h(No)

Nw

, φ =

α0

α1

α2

α3

βγ0

γ1

.



We formulate the PSF parameter estimation and star spectrumreconstruction problem in a nonlinear least squares framework

minφ,s

(f (φ,s) = ‖g − H(φ)s‖22

)

Note thatf (φ, s) is linear in s and nonlinear in φ

φ ∈ <p, s ∈ <Nw and p < Nw

Variable projection method:Implicitly eliminate linear term s.

Optimize over nonlinear term φ using Gauss-Newton.

φ used to specify the spatially varying elliptical PSF parameters at eachwavelength.




minφ,s

(f (φ,s) = ‖g − H(φ)s‖22

)Note that

f (φ, s) is linear in s and nonlinear in φ








minφ,s

(f (φ,s) = ‖g − H(φ)s‖22

)Note that

f (φ, s) is linear in s and nonlinear in φ






HSI Unmixing/Deblurring: Single PSF Case (Zhaoet.al., BNP ’15 extended to multiple PSFs)

Solve the deblurring and sparse hyperspectral unmixing of the form

minX≥0

12||HXM −G||2F + µ1||X ||1 + µ2TV (X )

H ∈ <Np×Np is a block diagonal blurring matrix constructed using elliptical Moffatfunction parameters from our estimate φ.

ADMM applied to compute X .

X-L. Zhao, F. Wang, T-Z. Huang, M. K. Ng, and R. J. PlemmonsDeblurring and Sparse Unmixing for Hyperspectral ImagesIEEE Transactions on Geoscience and Remote Sensing (2013).

Berisha, Nagy and Ple. “Deblurring and Sparse Unmixing of Hyperspectral Images usingMultiple PSFs,” in revision, SIAM J. Sci. Comp. 2015.




minX≥0

12||HXM −G||2F + µ1||X ||1 + µ2TV (X )








minX≥0

12||HXM −G||2F + µ1||X ||1 + µ2TV (X )






Test Results: Simulated Data for HST Satellite -Materials from NASA Database

Material Color Constituent Endmembers (%) Frac. Abun. (%)Material 1 light gray Em. 1 (100) 11

Material 2 green Em. 2 (70), Em. 9 (30) 18

Material 3 red Em. 3 (100) 4

Material 4 dark gray Em. 4 (60), Em. 10 (40) 19

Material 5 brown Em. 5 (100) 7

Material 6 gold Em. 6 (40), Em. 11 (30), Em. 12 (30) 32

Material 7 blue Em. 7 (100) 3

Material 8 white Em. 8 (100) 6


Spectral signatures of eight materials, such asaluminum, solar cell, copper tubing, etc.

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Simulated data at different wavelengths

λ = 400nm λ = 1107.1nm λ = 1814.3nm



λ = 400nm λ = 1107.1nm λ = 1814.3nm



λ = 400nm λ = 1107.1nm λ = 1814.3nm


Relative Errors, Relative Residual Norms

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ua

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PCG

CG

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Relative residual norms

Outer ADMM iterations: 200 CG iterations: 1000 PCG iterations: 20

µ1 = 0.01, µ2 = 5× 10−4, β = 0.01, Noise = 30dB


Relative Errors, Relative Residual Norms

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Outer ADMM iterations: 200 CG iterations: 1000 PCG iterations: 20

µ1 = 0.01, µ2 = 5× 10−4, β = 0.01, Noise = 30dB


Reconstructed Fractional Abundances

True Single PSF Multiple PSF























Reconstructed Material Spectral SignaturesThe true material spectral signatures: − The computed spectral signatures: −.

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In Progress – Deblurring & Feature Extraction

• Spectro-Polarimetric Compressive Sensing from SnapshotHSI-Polarization Imaging through Turbulence

• Using data from a polarimetric coded aperture snapshot spectralimager (CASSPI) camera (a 4D tensor problem) in AFOSR projectjoint with: D. Brady (ECE, Duke), S. Prasad (Physics, UNM), andSebastian Berisha (Radiology, UPenn).

• Data reconstruction and joint deblurring and feature extraction usingADMM.

Preprint later: http://users.wfu.edu/plemmons/


Snapshot Spectral & Spectro-PolarimetricCompressive Sensing - Enables HSI Video Imaging

Coded-Aperture Snapshot Spectral Imagers: DD-CASSI, SD-CASSI,CASSPI. Dave Brady et al., 2007 - present.

g(x , y) =

∫λ

Cλ(x , y)f (x , y , λ)dλ+ ελ(x , y).

Cλ(x , y) is wavelength dependent system function.

• Spatial Light Modulator (SLM)-based CASSPI - snapshotspectro-polarimetric imager forward model.

g(x , y) =∑µ

∫λ

Cµ(x , y , λ)[h(x , y , λ) ∗ fµ(x , y , λ)]dλ+ ελ,µ(x , y).

µ = polarimetric variable.


Segmentation Model

Segmentation model:

f (x , y , λ) =m∑

i=1

si(λ)ui(x , y).

Resulting system model:

g =

∫λ

Cλ(x , y)[hφλ(x , y) ∗m∑

i=1

si(λ)ui(x , y)]dλ+ ε(x , y).

Knowns: coded, blurred image g, and CASSI system operator C.Unknowns: phase function φ ( in terms of the OPD), spectralsignatures si , and support functions, ui .Approach: we take a two-step approach, that is, we estimate theoptical path difference function (OPD) first using a HSI of a guide star,and then classify features of object.


Feature Classification of Hyperspectral Objects

• Assume the OPD is known as the estimated φ̂ from Berisha, Nagy,Ple. 2014 so each PSF hλ is available. We estimate support functionsui and spectral signatures si .

J(u, s) =12

∥∥∥∥∥∫λ

Cλ(x , y)

[hλ(x , y) ∗

m∑i=1

si(λ)ui(x , y)

]− g(x , y)

∥∥∥∥∥2

2

+ α

m∑i=1

∫R2

√∇2

xxi +∇2yyi +

β

2

m∑i=1

∫λ‖si(λ)‖2,

where we use total variation regularization for ui and Tikhonov forspectral signatures si .

•Will solve inverse problem using ADMM.


Spectro-Polarimetric Camera - Duke, UNM, WFU AF Project

System modulates a 4D tensor array image onto a 2D detector(matrix). Project in progress. To reconstruct, and jointly deblur &extract features.


Epilogue

In retrospect: Did van Gogh envision atmospheric turbulence in hismost widely acclaimed painting – Starry Night?

Vincent van Gogh


Comments

Optical Imaging in Remote Sensing – 2015 is UNESCO “Year ofLight”Overview of Our Projects - NGA & AFOSR(Imaging Through) Turbulence - Galileo, Von Karman,Komolgorov, ...Hyperspectral Imaging (HSI) through Atmospheric TurbulenceEstimating HSI PSFs for Atmospheric Turbulence - HSI image ofguide star, Moffat fct. modelingJoint Deblurring and Feature Extraction - ADMM optimizationapproachOverview of Compressive Snapshot HSI - Enables HSI videoSpectro-Polarimetric Imaging - Adds polarization channels forStokes images used in identifying object shape and surfaceproperties


Thank You!


Documents

Imaging Through Atmospheric Turbulence in Remote Sensingplemmons/research/Ple_Sanya2015.pdf · Outline Overview of Optical Imaging in Remote Sensing Our Recent Projects (Imaging Through)