Analysis of Hyperspectral Image Using

Minimum Volume Transform (MVT)

Ziv Waxman & Chen VanunuInstructor: Mr. Oleg Kuybeda

Objectives:

• Testing the MVT algorithm as a tool of analyzing hyperspectral image.

• Obtain end-members (pure spectral signatures) present in hyperspectral image as output.

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

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).

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.

Pixels-Scatter-Diagram for 3-Bands Dist.• Generally looks like a “tear drop”.

• Pi represent the end members. Define facets of a minimum volume

circumscribing simplex.

O

P3

P2

P1

dark point

This facet is x+y+z=1

data

MVT Algorithm – DPFT

DFPT selected – due to random bias applied by scanner. Create simplex without moving actual data.

Project data onto uTx=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

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.

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

Feasible region

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

Real image results

• random bias

• Three images represent each end member

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.