DIMENSION REDUCTION FOR HYPERSPECTRAL DATA USING RANDOMIZED PCA AND LAPLACIAN EIGENMAPS

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1 DIMENSION REDUCTION FOR HYPERSPECTRAL DATA USING RANDOMIZED PCA AND LAPLACIAN EIGENMAPS YIRAN LI APPLIED MATHEMATICS, STATISTICS AND SCIENTIFIC COMPUTING ADVISOR: DR. WOJTEK CZAJA, DR. JOHN BENEDETTO DEPARTMENT OF MATHEMATICS UNIVERSITY OF MARYLAND, COLLEGE PARK

2 BACKGROUND: HYPERSPECTRAL IMAGING Light is described in terms of its wavelength A reflectance spectrum shows the reflectance of a material measured across a range of wavelengths. It helps identify certain materials uniquely We measure reflectance at many narrow, closely spaced wavelength bands When a spectrometer is used in an imaging sensor, the resulting images record a reflectance spectrum for each pixel in the images (Shippert, 2003)

3 SPECTRUM AND HYPERSPECTRAL IMAGERY Left: Reflectance spectra measured by laboratory spectrometers for three materials: a green bay laurel leaf, the mineral talc, and a silty loam soil. Right: The concept of hyperspectral imagery. (Shippert, 2003)

4 MULTISPECTRAL VS HYPERSPECTRAL Multispectral imaging measures reflectance at discrete and somewhat narrow bands. Multispectral images do not produce the "spectrum" of an object Hyperspectral deals with imaging narrow spectral bands over a continuous spectral range, and produce the spectra of all pixels in the scene. So a sensor with only 20 bands can also be hyperspectral when it covers the range from 500 to 700 nm with 20 bands each 10 nm wide. (Wikipedia: hyperspectral imaging)

5 AN EXAMPLE: SALINAS VALLEY, CALIFORNIA Left: sample band collected by 224-band sensor. It includes vegetables, bare soils, and vineyard fields. Right: Groundtruth of Salinas dataset (16 classes) (IC: Hyperspectral Remote Sensing Scenes)

6 PROBLEM Hyperspectral images are three dimensional (x-coordinate, y-coordinate, b) Each pixel has a different spectrum that represents different materials Sometimes over 100 bands and with large number of pixels Dimension reduction reduces the number of bands of a hyperspectral image It maps dimensional data into a lower dimension while preserving the main features of the original data. (hyperspectral imaging, Wikipedia)

7 PROJECT GOAL Reduce dimensionality of hyperspectral imaging Compare two algorithms to be implemented

8 METHODS Existing methods (partial) : Principal Component Analysis( PCA) Local Linear Embedding My Methods: Randomized PCA Laplacian Eigenmaps Neighborhood Preserving Embedding Classical multidimensional scaling Isomap Stochastic Proximity Embedding (Delft University)

9 COMPARISON BETWEEN TWO ALGORITHMS Compare two algorithms, Randomized PCA and Laplacian Eigenmaps, in terms of: Implementation Running time Results Difficulties during implementation

10 ALGORITHM 1: LAPLACIAN EIGENMAPS Consider the problem of mapping the weighted graph G to a line so that connected points stay as close together as possible, let y = y 1, y 2, y n T be such a map. Our goal is to minimize i,j y i y j 2 Wij Since i,j y i y j 2 Wij = 2y T Ly, the problem of finding argmin y T Ly given that y T Dy = 1, y T D1 = 0 becomes the minimum eigenvalue problem: Lf = λdf (Belkin, Niyogi, 2002)

11 ALGORITHM 1: THE ALGORITHM Step 1: Constructing the Adjacency Graph Construct a weighted graph with n nodes (n number of data points), and a set of edges connecting neighboring points. A) ε neighborhood: connected if B) n nearest neighbors Step 2: Choosing the weights A) Heat Kernel: x i x j 2 < ε W ij = e x i x j t 2 B) Simple Minded: W ij = 1 if connected and W ij = 0 otherwise

12 Step 3: Compute eigenvalues and eigenvectors for the generalized eigenvector problem: Lf = λdf (1) Where W is the weight matrix defined earlier, D is diagonal weight matrix, with D ii = j W ji, and L = D W Let f 0, f 1,, f n 1 be the solutions of equation (1), ordered such that 0 = λ 0 λ 1 λ n 1 Then the first m eigenvectors (excluding f 0 ), {f 1, f 2,, f m } are the desired vectors for embedding in m-dimensional Euclidean space (Belkin, Niyogi, 2002)

13 ALGORITHM 2: RANDOMIZED PCA INTRODUCTION Canonical construction of the best possible rank-k approximation to a real m n matrix A uses singular value decomposition (SVD) of A, A = UΣV T, Where U real unitary m m matrix, V is real unitary n n matrix, and Σ is real m n diagonal matrix with nonnegative, non increasing diagonal entries Best Approximation of A: A U Σ V T, Where U leftmost m k block of U, Σ k k upper left block of Σ, V leftmost n k block of V (Rokhlin, Szlam, Tygert, 2009)

14 Best because it minimizes the spectral norm A B for a rank-k matrix B = U Σ V T. In fact, A U Σ V T = σ k+1, Where σ k+1 is the k + 1 th greatest singular value Randomized PCA generates B such that A B Cm 1 4i+2 σk+1 with high probability ( ), where i is specified by user, and C depends on parameters of algorithm (Rokhlin, Szlam, Tygert, 2009)

15 ALGORITHM 2: THE ALGORITHM Choose l > k such that l m k Step 1: Generate a real l m matrix G whose entries are i.i.d normal Gaussian random variables, compute R = G AA T i A Step 2: Using SVD, form a real n k matrix Q whose columns are orthonormal, such that QS R T ρ k+1 for some k l matrix S, where ρ k+1 is the k + 1 th greatest singular value of R

16 Step 3: Compute T = AQ Step 4: Form an SVD of T: T = UΣW T, where U is a real m k matrix whose columns are orthonormal, W is a real k k matrix whose columns are orthonormal, Σ is a real diagonal k k matrix with nonnegative diagonal entries Step 5: Compute V = QW In this way, we get U, Σ, V as desired, and B = UΣV T (Rokhlin, Szlam, Tygert, 2009)

17 IMPLEMENTATION Hardware: Personal laptop/computers in the math computer lab Software: Matlab Database: 12 Band Moderate Dimension Image: June 1966 aircraft scanner Flightline C1 (Portion of Southern Tippecanoe County, Indiana) 220 Band Hyperspectral Image: June 12, 1992 AVIRIS image Indian Pine Test Site 3 (2 x 2 mile portion of Northwest Tippecanoe County, Indiana) 220 Band Hyperspectral Image: June 12, 1992 AVIRIS image North-South flight line (25 x 6 mile portion of Northwest Tippecanoe County, Indiana) Hyperspectral data from Norbert Weiner Center Data can be large (with 10,000^2 pixels, 200 bands, for example)

18 VALIDATION METHODS Delft University has developed Matlab toolbox for dimension reduction, which includes many methods, and is publically available Use algorithms from DR matlab toolbox to run on the same data and compare results For randomized PCA, check error bound: A B Cm 1 4i+2 σk+1 (Rohklin, 2009) Compare with ground truth images for the test cases

19 TEST PROBLEMS FOR VERIFICATION Test on known data set (as provided earlier), and compare results with ground truth classifications and images Test on smaller scales at first, and then move to large data set

20 EXPECTED RESULTS/CONCLUDING REMARKS Laplacian Eigenmaps should be easier to implement, but may take longer to run because it deals with solving the eigenvalue problem of large matrices Randomized PCA will be more difficult to implement, but will give desired results under unfavorable conditions with reasonable speed, and it should perform better than Laplacian eigenmaps when dealing with very large matrices

21 TIMELINE/MILESTONES October 17th: Project proposal Now to November, 2014: Implement and test laplacian eigenmaps, prepare for implementation of randomized PCA December, 2014: Midyear report and presentation January to March: Implement and test randomized PCA, compare two methods in various situations April to May: Final presentation and Final report

22 DELIVERABLES Presentation of data sets with reduced dimensions of both algorithms Comparison charts in terms of running time and accuracy of two different methods Comparison charts with other methods that are available from the DR matlab toolbox Data sets, Matlab codes, presentations, proposals, mid-year report, final report

23 BIBLIOGRAPHY Shippert, Peg. Introduction to Hyperspectral Image Analysis. Online Journal of Space Communication, issue No. 3: Remote Sensing of Earth via Satellite. Winter Hyperspectral Imaging. From Wikipedia. Oct. 6 th, Belkin, Mikhail; Niyogi, Partha. Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation, vol 15. Dec. 8 th, Web.

24 Rokhlin, Vladimir; Szlam, Arthur; Tygert, Mark. A Randomized Algorithm for Principal Component Analysis. SIAM Journal on Matrix Analysis and Applications Volume 31 Issue 3. August Web. ftp://ftp.math.ucla.edu/pub/camreport/cam08-60.pdf Matlab Toolbox for Dimension Reduction. Delft University. Web. Oct. 6 th, ction.html IC: Hyperspectral Remote Sensing Scenes. Web. Oct. 6 th, Scenes

25 Hyperspectral Images. Web. Oct. 6 th,

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