UNSUPERVISED CLASSIFICATION

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UNSUPERVISED CLASSIFICATION
Dr. Jakob J. van Zyl
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OUTLINE

• LEVEL 0 CLASSIFICATION
• Theoretical characteristics
• Single frequency interpretation
• Single frequency results
• Dual frequency interpretation
• Dual frequency results
• LEVEL 1 CLASSIFICATION
• Adding surface roughness and soil moisture

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THEORETICAL CHARACTERISTICS
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THEORETICAL CHARACTERISTICS ODD NUMBERS OF
REFLECTIONS
Pixels dominated by odd numbers of reflections
are typically characterized by
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THEORETICAL CHARACTERISTICS EVEN NUMBERS OF
REFLECTIONS
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THEORETICAL CHARACTERISTICS DIFFUSE SCATTERING
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SINGLE FREQUENCY INTERPRETATION
The definition of moderate vegetation is a
function of the frequency used when imaging the
scene and can be shown to be related to
the randomness of the orientation and the
thickness of the scattering cylinders relative to
the radar wavelength.
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LEVEL 0 LAND COVER CHARACTERIZATION DUAL
FREQUENCY INTERPRETATION
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LEVEL 1 LAND COVER CHARACTERIZATION

• The next step is to further subdivide the bare
  surface class into five classes
• This is done based on the dielectric constant and
  the roughness of the bare surface
• The following classes are defined
• Dry bare surfaces dielectric constant lt 6
• Damp bare surfaces 6 lt dielectric constant lt 11
• Wet bare surface 11 lt dielectric constant lt 15
• Saturated surface / open water dielectric
  constant gt 15
• Exposed rock surface 6 lt dielectric constant lt
  8, rms height gt 6 cm
• The soil moisture algorithm published by Dubois
  et al. is used

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CLOUDES DECOMPOSITION THEOREM

• Cloude showed that a general covariance matrix
  can be decomposed as follows
• Here, are the eigenvalues of the
  covariance matrix, are its
  eigenvectors, and means the adjoint
  (complex conjugate transposed) of .
• In the monostatic (backscatter) case, the
  covariance matrix has one zero eigenvalue, and
  the decomposition results in at most three
  nonzero covariance matrices.

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CLOUDES DECOMPOSITION THEOREM

• Also useful in our discussions later is Cloude's
  definition of target entropy,
• where
• As pointed out by Cloude, the target entropy is a
  measure of target disorder, with for
  random targets and for simple
  (single) targets.

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CLOUDES DECOMPOSITION THEOREM AZIMUTHALLY
SYMMETRIC NATURAL TERRAIN

• Borgeaud et al. showed, using a random medium
  model, that the average covariance matrix for
  azimuthally symmetrical terrain in the monostatic
  case has the general form
• where
• The superscript means complex conjugate, and
  all quantities are ensemble averages. The
  parameters and all depend on the
  size, shape and electrical properties of the
  scatterers, as well as their statistical angular
  distribution.

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CLOUDES DECOMPOSITION THEOREM AZIMUTHALLY
SYMMETRIC NATURAL TERRAIN

• The eigenvalues of are
• Note that the three eigenvalues are always real
  numbers greater than or equal to zero.

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CLOUDES DECOMPOSITION THEOREM AZIMUTHALLY
SYMMETRIC NATURAL TERRAIN

• The corresponding three eigenvectors are

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CLOUDES DECOMPOSITION THEOREM AZIMUTHALLY
SYMMETRIC NATURAL TERRAIN

• On the previous page we used the shorthand
  notation
• We note that is always positive. Also note
  that we can write
• where
• Since is always positive, it follows that
  the ratio of to is always negative.
  This means that the first two eigenvectors
  represent scattering matrices that can be
  interpreted in terms of odd and even numbers of
  reflections.

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CLOUDE'S DECOMPOSITION THEOREMRANDOMLY ORIENTED
DIELECTRIC CYLINDERS

• In general, the scattering matrix of a single
  dielectric cylinder oriented horizontally can be
  written as
• where and are complex numbers whose
  magnitudes and phases are functions of cylinder
  dielectric constant, diameter and length.
• Assuming a uniform distribution in angles about
  the line of sight, one can easily show that the
  resulting average covariance matrix for the
  monostatic case has the following parameters

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CLOUDE'S DECOMPOSITION THEOREMRANDOMLY ORIENTED
DIELECTRIC CYLINDERS

• The eigenvalues are
• The corresponding three eigenvectors are

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CLOUDE'S DECOMPOSITION THEOREMRANDOMLY ORIENTED
DIELECTRIC CYLINDERS

• In the thin cylinder limit, , and we find
  that
• In this case, equal amounts of scattering is
  contributed by the matrix that resembles
  scattering by a sphere and by the cross-polarized
  return, although a significant fraction of the
  total energy is also contained in the second
  matrix, which resembles a metal dihedral corner
  reflector.
• The entropy in this case is 0.95 indicating a
  high degree of target disorder or randomness.
• Note that the unsupervised classification scheme
  would classify this as diffuse scattering.

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CLOUDE'S DECOMPOSITION THEOREMRANDOMLY ORIENTED
DIELECTRIC CYLINDERS

• In the thick cylinder limit, and we
  find that
• In this case, only one eigenvalue is non-zero,
  and the average covariance matrix is identical to
  that of a sphere.
• The entropy is 0, indicating no target
  randomness, even though we have calculated the
  average covariance matrix for randomly oriented
  thick cylinders!
• The explanation for this result lies in the fact
  that when the cylinders are thick, the single
  cylinder scattering matrix becomes the identity
  matrix, which is insensitive to rotations.
• Note that the unsupervised classification scheme
  would classify this as odd numbers of reflections.

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CLOUDE'S DECOMPOSITION THEOREMRADAR THIN
VEGETATION INDEX

• Using the result for a cloud of randomly oriented
  thin cylinders, we note that
• We now define a radar thin vegetation index (RVI)
  as
• We expect RVI to vary between 0 and 1.

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CLOUDE'S DECOMPOSITION THEOREMRADAR THIN
VEGETATION INDEX
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CLOUDE'S DECOMPOSITION THEOREM EXAMPLE
VEGETATED CLEARCUT AREA
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CLOUDE'S DECOMPOSITION THEOREM EXAMPLE
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CLOUDES DECOMPOSITION THEOREM

• Advantages
• Rigorous mathematical technique
• Provides quantitative information about
  scattering mechanisms
• Disadvantages
• Not physically based
• Interpretation of results not unique