Dimension Reduction in Hyperspectral Images

1
Dimension Reduction in Hyperspectral Images

• Alex Chen
• April 9, 2007

2
Review of Sparse Signal Recovery

• Candès, Romberg, and Tao give a method for
  reconstructing a sparse (approximately sparse)
  signal x0 using relatively few measurements (from
  a measurement ensemble A).
• This method is an optimization problem, where we
  want to solve
• min x1
• st Ax y2 ? ?,
• where ? e2.
• That is, we wish to keep the signal as small as
  possible while ensuring that x roughly solves
  the same measurement equation as x0 (Recall y -
  Ax0 e).
• E. Candes, J. Romberg, T. Tao. Stable Signal
  Recovery from Incomplete and Inaccurate
  Measurements. Comm. Pure Appl. Math. 59 (2006),
  1207-1223

3
Important Points of Sparse Signal Recovery

• Reconstructed error for an approximately sparse
  signal is bounded by C(noise small entries).
• In particular, error is independent of the
  magnitude of the large entries.
• If noise 0, and the signal is sparse, then
  recovery is exact.
• Choosing different bases for the signal
  (affecting sparsity) or different measurement
  ensembles (the random matrix A) can lead to
  better recovery.

4
Numerical Formulation

• Consider the similar problem of minimizing the
  energy E(x) ½Ax y22 ?x1.
• Solve x -? E.
• Then x -AT(Ax y) ?sgn(x).
• Explicit discretization gives the gradient
  descent method
• xn1 xn ?tAT(Axn y) ?sgn(xn).

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Reconstruction of Simple Sparse Signal
Top left Plot of a signal generated by taking a
vector of dimension 1024 with 50 nonzero entries
randomly assigned to be 1 or 1 with equal
probability.
A is 300x1024 ( Gaussian Random Matrix)

• 0.12
• ?t 0.1
• of iterations 700

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Reconstruction of Compressible Signal
Top Left Plot of a compressible signal with
values x(t) 5.819t-10/9 where t 11024 and
indices permuted randomly. The signal is
multiplied by 1 or 1 with equal probability.
(The 5.819 is chosen so that this norm matches
the previous case ?50.)
Parameters are chosen as in the previous case.
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Gradient Descent

• Gradient descent does not give results as good as
  in Candès, Romberg, Tao, but it gets close
  enough.
• One trick is assuming a priori that the signals
  are integer-valued.
• Another is to assume that signals have a known
  magnitude (the recovered vector is scaled to have
  the same norm as the original).
• Both give much better results, but assume too
  much.

8
Hyperspectral Signals

• Take a signal generated from one pixel of a
  hyperspectral image.
• This is a column vector, with the dimensions
  going through the various bands of the image.
• There are 162 dimensions, ranging from 412 nm to
  2390 nm.
• We would like to apply the Candès, Romberg, and
  Tao (CRT) method to compress the hyperspectral
  signal.
• Note that the noise in the measurements is now
  taken to be 0.

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Hyperspectral Signal
This is a hyperspectral signal at pixel
(200,200)--vegetation. Note that it is not sparse
as shown.
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CRT Method Directly on Hyperspectral Signal
Top Left Hyperspectral signal at
Urban(200,200,). Remaining plots show the
recovery by running the algorithm on the signal
itself. The parameters are unchanged from the
previous case (except for ? 0 and of
measurements as shown).
Total norm of the hyperspectral signal 1883.6
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Fourier Transform of Signal
The Fourier transform of the signal is much
closer to being sparse.
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CRT on Fourier Transform
Top Left Hyperspectral signal at
Urban(200,200,). Remaining plots show the
recovery running CRT on the Fourier transform.
Only the number of measurements is
changed. Running CRT on the Fourier transform is
better, but not good enough for dimension
reduction.
Total norm of the hyperspectral signal 1883.6
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Wavelets

• Instead of writing a function as a sum of sines
  and cosines, we can use wavelets fast decaying
  waveforms (mother wavelets) or sums of scaled and
  translated copies of a signal of finite length
  (daughter wavelets).
• The advantage is that wavelets are localized in
  both space and frequency while sines and cosines
  are localized only in frequency.

• References
• 1. Daubechies. Ten Lectures on Wavelets,
  Society for Industrial and Applied Mathematics,
  1992.
• Julia
• http//aix1.uottawa.ca/jkhoury/haar.htm (for a
  simple illustration)

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Haar Wavelet (1/2)

• Computation uses the Haar wavelet basis, which
  consists of translated and scaled copies of the
  function

• 1 if 0 ? x lt ½
• f(x) -1 if ½ ? x lt 1
• 0 otherwise

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Haar Wavelet (2/2)

• Using the Haar wavelet decomposition, the
  following signal is obtained, which is more
  sparse than the original signal or the Fourier
  transform.

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CRT on Haar Wavelet
Top Left Hyperspectral signal in the Haar
wavelet basis at Urban(200,200,). Remaining
plots show reconstruction on various of
measurements. Running on the (unscaled) Haar
wavelet basis is comparable to Fourier.
Total norm of the hyperspectral signal 1883.6
17
CRT on Scaled Haar Basis

• The first entries in the Haar basis are not
  weighted enough according to the information they
  store.
• To compensate, scale the entries this makes the
  signal more sparse, while giving a more accurate
  picture of the storage.
• Numerical results do not yet work as anticipated.

Signal in unscaled Haar basis
Signal in scaled Haar basis
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Naïve Haar Wavelet Reconstruction

• For comparison, we use a naïve reconstruction of
  the hyperspectral signal.
• Order the components of the signal from largest
  to smallest and keep a certain number of entries
  (setting the rest equal to 0).
• Does the signal have small reconstruction error?

Total norm of the hyperspectral signal 1883.6
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Reconstruction with a Derivative (1/2)
Top Left Derivative of hyperspectral signal at
Urban(200,200,). Remaining plots show
reconstruction on various of measurements. Runni
ng the algorithm on the derivative does not give
good reconstruction relative to the norm of the
derivative.
Total norm of the derivative of the hyperspectral
signal 186.5
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Reconstruction with a Derivative (2/2)
Top Left Hyperspectral signal at
Urban(200,200,). Remaining plots show
reconstruction on various of measurements,
converted back to the original format. Running
the algorithm gives a good reconstruction of the
general shape, but the values are heavily
distorted.
Total norm of the hyperspectral signal 1883.6