Principle Component Analysis (PCA)

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Principle Component Analysis (PCA)

• A mechanism used to make the analysis of remote
  sensing data simpler.

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PCA

• reduces the dimensionality of the data
• keeping the (what we hope) is the most
  significant parts of the data
• simultaneously filters out noise
• It is a way of identifying patterns in data, and
  expressing the data in such a way as to highlight
  their similarities and differences

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PCA contd.

• hyperspace graphing is not available to
  visualize data
• the 3 first principle components make a powerful
  tool for identifying patterns in the data
• This is can be seen as a tool for image
  compression with no loss of information

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History

• Early image processing systems were limited in
  their processing capacity it was important to
  reduce the number of calculations required to
  process an image.
• Currently, useful as tool to explore the
  variability of an image
• Also used in facial recognition software

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How it works

• The data, though clumped around several central
  points in that hyperspace, will generally tend
  towards one direction.
• If one were to draw a solid line that best
  describes that direction, then that line is the
  first principle component (PC).

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Eigenvectors and Eigenvalues

• Each eigenvector represents a principle
  component.
• The first Principle Component is defined as the
  eigenvector with the highest corresponding
  eigenvalue.
• The individual eigenvalues indicate the variance
  they capture - the higher the value, the more
  variance they have captured.

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• Any variation that is not captured by that first
  PC is captured by subsequent orthogonal
  eigenvectors.
• When the data is plotted in this manner they are
  said to be plotted in Principle Component-space
  (PC-space)

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Data means WHAT?

• The first PC represents the greatest variability
  in the data
• so?
• How does the interpreter convert variability
  into information?

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Image Registration and Rectification

• Removing spatial errors in an image
• Systematic errors
• (skew)
• Curvature of the earth
• General instrument correction e.g. scan line
  overlap

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Projection and Random errors

• In order to get RS data into a format compatible
  with the GIS, a projection needs to be specified
• (the alternative is to do an image based GIS
  that ignores real world coordinates)

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Ground Control Points (GCPs)

• Use GCPs that are well distributed across the
  image
• Features visible on the image that correspond to
  known locations on the ground (or on a map)
• GCP1- XY (UTM?)
• GCP1-row/column

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Affine Coordinate Transformation

• X a0a1Xa2Y
• Y b0b1Xb2Y
• Where x and y are the row/ column locations
• And X,Y are the ground coordinates
• an and bn are transformation parameters

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Page 2 (of 3) of the affine coordinate
transformation example in Jensen 1986
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So how do we get new values into the projected
cells?

• Nearest neighbor
• The cell value from the original image closest to
  the center of the new (rectified or projected)
  cell is assigned to the new cell.
• Simplest method
• Only rational method for classified images or
  other grid data with interval data

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Bilinear Interpolation

• Weighted average of the four nearest pixels (2
  left-right and 2 up-down)
• weighted average of a 2X2 kernel

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Cubic Convolution or Cubic spline

• Cubic spline of 16 closest neighbors
• a weighted average of from a 4X4 kernel

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How well do they work?

• Every manipulation of a grid changes the data in
  every grid cell
• The accuracy of the original data ?
• The accuracy of the new cells ?
• Loss of information ?
• The only real test of accuracy is how accurately
  INFORMATION is supplied by the image.

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Image Filtering

• Noise removal
• Smoothing
• Edge enhancement
• Most image enhancement operations work the same
  way
• A moving window

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The kernel or window

• The factors in the moving window are specified
  based on the desired result
• The 3X3 array moves across the image and converts
  the value of the center pixel based on the
  operations specified by the surrounding pixels

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A high pass filter or kernel
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Directional Edge Enhancementexamples