The Image Comparisons Problem

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Estimation of Mountain Glacier Retreat from Landsat Images

Armin Schwartzman Division of Biostatistics, UC San Diego

June 2016 Mountain Glacier Retreat

Muir Glacier, Alaska

1941 2004

Qori Kalis Glacier, Perú

1978 2004 2/35 Landsat Images

Franz Joseph Glacier

Southern New Zealand, Feb 2009 3/35 Landsat Images over Time

4/35 Image Analysis of Glaciers • The scientific problem: – Estimate trend of mountain glacier retreat • Motivation: – Climate change indicators – People depend on glacial melt water (Andes, Himalayas, Rockies) – Ground measurements for (mostly) Alps only • Proposal: – Analysis of Landsat images – They are free!

5/35 Difficulties • Difficulties – Clouds, debris, snow – Shadows from adjacent mountains – Landsat 7 defects – Irregular shapes • Standard approach – Segmentation of glacial surface – Carefully delineate glacier boundary – Requires substantial manual input – Accurate but limited

6/35 Our Approach • Goal: – Automatic processing and analysis – Scale up to hundreds of glaciers • Approach: – Estimate glacial flowline – Extract intensity profile along flowline – Use time/space functional data techniques to estimate trend in terminus location

7/35 Gorner Glacier 30 m resolution 30

8/35 Processing and Analysis Pipeline

Data Pre- Image Estimate processing Analysis Terminus

1. Geographical 1. Classify cloudy 1. Spatial smoothing bounding box images 2. Path Search 2. Select frequency 2. Estimate flow line Algorithm band 3. Extract intensity 3. Temporal 3. Generate image profile smoothing stack 4. Obtain DEM

9/35 Processing and Analysis Pipeline

Data Pre- Image Estimate processing Analysis Terminus

10/35 Data Pre-Processing

Bounding box

Global Land Ice Measure- Digital elevation Landsat ments From model (DEM) images Space Database

11/35 Landsat Frequency Bands

B61

Normalized Difference B20 B50 NDSI Snow Index: B20 B50 12/35 Landsat Frequency Bands

Franz Josef glacier B20 (Optical, 30 m) B61 (Thermal, 60 m) New Zealand

Gorner glacier B20 (Optical, 30 m) NDSI (Processed, 30 m) Switzerland

13/35 Processing and Analysis Pipeline

Data Pre- Image Estimate processing Analysis Terminus

14/35 Classify Cloudy Images Clear Cloudy

15/35 Estimate glacial flowline

Digital elevation Starting model (DEM) point

Smoothing

16/35 Terra Flow Algorithm • Existing method for tracking rivers: – Move to lowest 8-connected neighbor

Flow goes up at step 3 Flow goes left at step 3 and reaches boundary and stops at local minimum 17/35 Idea: Follow the Gradient • Let f(x,y) be a surface, and write the

flowline as a curve (gx(t), gy(t)) parametrized by t. • The flowline should satisfy

dg x dg y , af dt dt • for some constant a.

18/35 Flowline Algorithm • Gradient descent using local linear regression: • At each iteration: – Fit a 2D plane to the surface f(x,y) within a b×b neighborhood by least squares. – Calculate the gradient of the fitted plane – Move a distance s in the direction of the gradient

19/35 Flowline Algorithm s = 0.5 s = 1

b = 3

b = 4

20/35 Extract intensity profiles

Glacial Remove cloudy flowline images

Interpolate! 21/35 Processing and Analysis Pipeline

Data Pre- Image Estimate processing Analysis Terminus

22/35 Estimate Terminus Location • Input: – Collection of intensity profiles along glacial path for various time points • Output: – Estimate of change point location over time • Method: 1. Spatial smoothing 2. Path search algorithm 3. Temporal smoothing

23/35 1. Spatial Smoothing

Smoothed profile Ideal profile

A Typical Frame: Gorner 24/35 1. Spatial Smoothing • For each time j we have the observations:

si ,Yij i 1,,n Distance along Image intensity flow line along flow line • Non-parametric regression model at time j:

2 Yij j (si ) ij , (1 j ,, nj ) ~ I

• Approximate with spline basis: K j (s) p (s) jp p1 Spline Bases 25/35 Spline Smoothing via gam in R • At any spatial point s, the objective function is: 2 n K K 2 Q j (s) Yij p (si ) jp j jp i1 p1 p1 Penalize Least Squares Roughness Penalty Squared Coefficients • Minimization gives the solution:

1 ˆ , ,ˆ T T (s)(s) ˆ I T (s)Y j1 jp j pxp nx1 • Obtain smooth profile and derivative K K ˆ (s) (s)ˆ ˆ ˆ j p jp j (s) p (s) jp p1 p1 Spline Bases 26/35 2. Path Search • Find path as a function of time that minimizes the sum of spatial derivatives: T g(t ) g(t ) ~ ˆ j1 j km g() min (g(t j ),t j ), 2 g j1 t j1 t j year

27/35 3. Temporal Smoothing • Non-parametric regression model over time:

~ 2 g(t j ) g(t j ) j , j ~ 0, j Heterogenous ~ estimation error • Taylor expansion about time g(t j ) : 1 ˆ(g(t ),t ) ˆ(t ) ˆ(3) (t )g(t ) g~(t )2 j j 0 j 2 0 j j j where ˆ ˆ ~ ˆ(3) ˆ(3) ~ 0(t j ) (g(t j ),t j ), 0 (t j ) 0 (g(t j ),t j )

• Minimization becomes weighted LS problem: T ˆ ˆ(3) ~ 2 g(t) argmin 0 (t j )g(t j ) g(t j ) g j1 28/35 Gorner Glacier 30 m resolution 30

29/35 Franz Josef Glacier 30 m resolution 30

30/35 Otemma Glacier (Swiss Alps) • Estimates align with ground measurements: Terminus DEM NDSI

31/35 Fassett Glacier (Alaska) • An example with no ground measurements: Terminus DEM NDSI

32/35 Glaciers Around the World

Glacier Location Comparison to ground Terminus measurements movement (2000-2012) # records Mean abs. Max. abs. Estimate std. (2000-2009) dev. (m) dev. (m) (m/year) error Rhône Swiss Alps 6 10.8 22.1 -2.33 (0.41) Otemma Swiss Alps 6 21.6 37.3 -20.96 (2.64) Corbassière Swiss Alps 6 31.7 50.5 -17.93 (1.32) Findel Swiss Alps 5 51.3 128.3 -3.63 (1.19) Gauli Swiss Alps 5 12.1 27.5 -3.42 (0.84) Morteratsch Swiss Alps 6 60.2 75.5 +7.20 (1.31) Franz Josef New Zealand 8 82.6 128.9 -3.41 (1.50) Fassett Alaska - - - -17.09 (0.89) Litian China - - - +2.63 (0.50) Torre Argentina - - - -2.72 (0.69)

33/35 Summary • What we have: – A semi-automated method for estimating location of glacial termini from Landsat images – A processing pipeline (Python + R) in the works • Interesting statistical problems: – Estimation of flowline – Image classification – Path search optimization – Spatio-temporal smoothing

34/35 Summary • More challenges: – Match GLIMS database (glacier locations) with WGMS database (ground records) – Choose tuning parameters (flowline, smoothing) – Software implementation and speed • Future applications: – Map trends geographically – Link trends to dynamical models of glacial mass balance and climate

35/35 Intensity Profile on Glacier Path y: Intensity (B62)

36/35 Spatial Analysis • Non-parametric regression model at time j:

Yij a j (sij ) ij , (1 j ,, nj ) ~ AR(2)

• Local polynomial regression (order p = 3):

( p) T aˆ j (s) aˆ j (s) Ln p (s)Yn1 where T 1 T Ln p (s) X WX X W

p 1 s s s s p! Kh s1 s 0 0 1 1 W (s) 0 0 Xn p (s) nn p 1 s s s s p! 0 0 K s s p n h n

37/35 Spatial Analysis • Plug-in optimal bandwidth for correlated noise: – Fit pilot polynomial of degree 9 – Fit AR(2) model to residuals – Plug-in bandwidth:

1 5 n S cov ~ ij, kj h C i,k 1 n ~(2) 2 a si i1 38/35 A Typical Frame: Franz Josef

RGB Intensity profile

Zoom 2nd derivative -in (numerical from 1st der.)

1st derivative (from local poly. regr.)

Length (meters) 39/35 Temporal Analysis Length (meters) Length

Time (years) 40/35 A Typical Frame: Gorner

RGB Intensity profile

Zoom 2nd derivative -in (numerical from 1st der.)

1st derivative (from local poly. regr.)

Length (meters) 41/35 Estimation Error of Location

() = 0 = : () > 0 () ()

( ()) () () () ( ())

42/35 1. Spatial Smoothing • Non-parametric regression model at time j:

2 Yij j (si ) ij , (1 j ,, nj ) ~ I

Image intensity Distance along along flow line flow line

• Approximate with spline basis: K j (s) p (s) jp p1 Spline Bases • Solve using function gam in R.

43/35 Climate Change • Where will the mean summer temperature increase?

1971 - 1999 2041-2069 44/35 Peak Detection in Science 1D 2D 3D Genomics: ChIP-Seq Satellite imaging Brain imaging: fMRI, DTI, TBM

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45/35 Peak Detection in Science 1D 2D 3D Genomics: ChIP-Seq Environment: warming Brain imaging: fMRI, DTI, TBM

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