Frame based kernel methods for hyperspectral imagery data

Transcription

1 Frame based kernel methods for hyperspectral imagery data Norbert Wiener Center Department of Mathematics University of Maryland, College Park Recent Advances in Harmonic Analysis and Elliptic Partial Differential Equations University of Virginia May 9, 2009

2 Acknowledgements and collaborators This research was supported by the National Geospatial-Intelligence Agency (NGA). PI: John J. Benedetto, University of Maryland, College Park. Collaborators: John J. Benedetto, University of Maryland, College Park. Wojciech Czaja, University of Maryland, College Park. J. Christopher Flake, University of Maryland, College Park.

3 Outline 1 Introduction to hyperspectral imagery data Endmembers 2 Kernel eigenmap methods Frames 3

4 Outline Introduction to hyperspectral imagery data Endmembers 1 Introduction to hyperspectral imagery data Endmembers 2 Kernel eigenmap methods Frames 3

5 Color image Introduction to hyperspectral imagery data Endmembers

6 Introduction to hyperspectral imagery data Endmembers

7 Hyperspectral data cube Introduction to hyperspectral imagery data Endmembers

8 Introduction to hyperspectral imagery data Endmembers Overview of hyperspectral imagery data Hyperspectral imagery (HSI) data is characterized by the narrowness and contiguous nature of the measurements. HSI data sets are spectrally overdetermined, and thus provide ample spectral information to distinguish between spectrally similar (but unique) materials. HSI data sets can be useful for the following purposes: target detection material classification material identification mapping details of surface properties

9 Introduction to hyperspectral imagery data Endmembers Overview of hyperspectral imagery data Hyperspectral imagery (HSI) data is characterized by the narrowness and contiguous nature of the measurements. HSI data sets are spectrally overdetermined, and thus provide ample spectral information to distinguish between spectrally similar (but unique) materials. HSI data sets can be useful for the following purposes: target detection material classification material identification mapping details of surface properties

10 Endmember illustration Introduction to hyperspectral imagery data Endmembers

11 Endmember illustration Introduction to hyperspectral imagery data Endmembers

12 Endmember illustration Introduction to hyperspectral imagery data Endmembers

13 Endmember illustration Introduction to hyperspectral imagery data Endmembers

14 Endmember illustration Introduction to hyperspectral imagery data Endmembers

15 Endmember definition Introduction to hyperspectral imagery data Endmembers Assume our HSI data set is an N 1 N 2 D cube. N 1, N 2 spatial dimensions; N = N 1N 2 pixels. D is the spectral dimension; so D wavelengths measured. Let X = {x i } N i=1 RD denote the pixel vectors of the HSI data set in list form. Definition Endmembers are a collection of a scene s constituent spectra. If E = {e i } s i=1 are endmembers, then the linear mixture model is x i = s α i,j e j + N xi, x i X, j=1 where N xi is a noise vector.

16 Outline Kernel eigenmap methods Frames 1 Introduction to hyperspectral imagery data Endmembers 2 Kernel eigenmap methods Frames 3

17 Introduction to the algorithm Kernel eigenmap methods Frames We have two goals: 1 Map the high dimensional HSI data set X to an appropriate low dimensional space Y. 2 Represent the low dimensional space Y for the purposes of material classification. We achieve these goals via two existing mathematical theories: 1 We use kernel eigenmap methods to determine the space Y. 2 We represent Y with an overcomplete endmember set Φ, also known as a frame.

18 Kernel eigenmap methods Frames Introduction to kernel eigenmap methods Given a high dimensional data set X = {x i } N i=1 RD, we assume X lies on a low dimensional manifold M d (d < D). Kernel eigenmap methods map the vectors in X to d-dimensional coordinates Y = {y i } N i=1 Rd that preserve the underlying geometry of M d.

19 Kernel eigenmap methods Frames Introduction to kernel eigenmap methods Given a high dimensional data set X = {x i } N i=1 RD, we assume X lies on a low dimensional manifold M d (d < D). Kernel eigenmap methods map the vectors in X to d-dimensional coordinates Y = {y i } N i=1 Rd that preserve the underlying geometry of M d.

20 Kernel eigenmap methods Frames Basics of kernel eigenmap methods The main components of kernel eigenmap methods are: Construction of an N N symmetric, positive semi-definite kernel K, K i,j = K(x i, x j ). Diagonalization of K and then choosing d D significant orthogonal eigenvectors of K, denoted by v 1,..., v d. Map each x i X to the d-dimensional vector y i given by: y i = (v 1 (i),..., v d (i)).

21 Frame theory Kernel eigenmap methods Frames Definition 1 Φ = {ϕ i } s i=1 is a frame for Rd if there exists A, B > 0 such that A y 2 s y, ϕ i 2 B y 2, y R d. i=1 2 For a frame Φ = {ϕ i } s i=1, the frame operator S : Rd R d is S(y) = s y, ϕ i ϕ i. i=1

22 Frames and HSI data Kernel eigenmap methods Frames We wish to use a data dependent frame Φ = {ϕ i } s i=1 to represent the reduced dimension space Y = {y i } N i=1 Rd : y i = s c i,j ϕ j, y i Y. j=1 Possible frame construction algorithms: existing endmember algorithms modified frame potential [Benedetto, Fickus; 2003] Possible frame coefficients: c i,j = y i, S 1 (ϕ j) c i, = arg min c c l 1 subject to Φc = y i

23 Why use frames? Kernel eigenmap methods Frames Why use frames? Traditional endmembers may be mixtures of many prominent features. Overestimating the number of classes allows for flexibility in representing mixtures and pure elements. Empirical evidence suggests that distinct classes are not orthogonal to each other. Unlike the orthogonal eigenvectors of K, frame elements need not be orthogonal.

24 Review of algorithm Kernel eigenmap methods Frames K R N X R D {c i,j } R s Φ Y R d

25 Outline 1 Introduction to hyperspectral imagery data Endmembers 2 Kernel eigenmap methods Frames 3

26 Urban Figure: HYDICE Copperas Cove, TX

27 Urban classes 22 classes in the Urban data set, including: Dark asphalt Vegetation: grass Soil 1 Soil 2 Soil 3 (dark) Roof: Walmart

28 Overview of classification results

29 Sample frame coefficients

30 Dark asphalt (e) Urban (f) Dark Asphalt

31 Light asphalt (a) Urban (b) Light asphalt

32 Concrete (a) Urban (b) Concrete

33 Vegetation: pasture (a) Urban (b) Vegetation: pasture

34 Vegetation: grass (a) Urban (b) Vegetation: grass

35 Vegetation: trees (a) Urban (b) Vegetation: trees

36 Soil 1 (a) Urban (b) Soil 1

37 Soil 2 (a) Urban (b) Soil 2

38 Soil 3 (dark) (a) Urban (b) Soil 3 (dark)

39 Roof: Walmart (a) Urban (b) Roof: Walmart

40 Roof: A (a) Urban (b) Roof: A

41 Roof: B (a) Urban (b) Roof: B

42 Roof: light gray (a) Urban (b) Roof: light gray

43 Roof: dark brown (a) Urban (b) Roof: dark brown

44 Roof: church (a) Urban (b) Roof: church

45 Roof: school (a) Urban (b) Roof: school

46 Roof: bright (a) Urban (b) Roof: bright

47 Roof: blue green (a) Urban (b) Roof: blue green

48 Tennis court (a) Urban (b) Tennis court

49 Pool water (a) Urban (b) Pool water

50 Shaded vegetation (a) Urban (b) Shaded vegetation

51 Shaded pavement (a) Urban (b) Shaded pavement

52 Thank you Thank you for your time. hirn/