1. Analysis of Hyperspectral Image Using Minimum Volume Transform (MVT) Ziv Waxman & Chen Vanunu Instructor: Mr. Oleg Kuybeda

2. Objectives: Testing the MVT algorithm as a tool of analyzing hyperspectral image. Obtain end-members (pure spectral signatures) present in hyperspectral image as output.

3. Analysis Steps Pre-processing: rank and end- members estimation (MOCA algorithm). Data Depletion (select data upon convex hull). Run MVT (apply linear programming) and concurrently perform constraints depletion. Get end-members and compare with MOCA end-members. Pre- processing Data depletion MVT MVT end- members MOCA end- members compare

4. Assumptions LMM – Linear Mixture Model. Every pixel is a linear combination of pure spectral signatures (end members). End members are linearly independent. Pixels-scatter-diagram is convex. Located in the first octant (for 3D).

5. MVT Variants Dark Point Fixed (DPFT) - dark point reliably known. - better when no bias. Fixed Point Free (FPFT) - dark point not known. - better when constant bias applied to data.

6. Pixels-Scatter-Diagram for 3-Bands Dist. Generally looks like a “tear drop”. P i represent the end members. Define facets of a minimum volume circumscribing simplex. O P3P3 P2P2 P1P1 dark point This facet is x+y+z=1 data

7. MVT Algorithm – DPFT DFPT selected – due to random bias applied by scanner. Create simplex without moving actual data. Project data onto u T x=1 Data Depletion Create start simplex Get constraints and deplete them Rotate k’th facet (linear programming – simplex method) k=k+1 k=1 End members If k=n+1 then k=1

8. Data Depletion Only data points upon the convex hull define a simplex. Choose these points by applying variant of Gram-Schmidt orthogonalization process. should leave 10% of total data.

9. Constraints Depletion Applied when data depletion process leaves too many points. Remove redundant constraints, which do not contribute to creation of feasible region (linear programming). Feasible region

10. Synthetic data results Blue circled – MOCA end-members Red points – after data depletion Azure – MVT end-members Arial view: - White noise applied - Constant bias applied

11. Real image results random bias Three images represent each end member

12. Discussion Creates a minimum volume simplex for a given data. Extremely efficient when bias is constant. Preserves rare-vectors – MOCA and MVT do not ignore abnormalities in an image. MVT is very sensitive to random bias. Sensitive to noise.