CS 450: Introduction to Digital Signal and Image Processing

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CS 450 Introduction to Digital Signal and Image
Processing

• A Few More Fundamentals

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Sampling Revisited

• How much sampling is enough?
• Shannon Sampling TheoremTwice the highest
  frequency in the signal (in theory)
• Nyquist rate
• What happens if you sample above this?
• Avoids dangers of theoretical limits
• Better for intermediate processing
• What happens if you dont sample enough?
• Aliasing (false low-frequencies appear)
• Moiré patterns
• Insufficient sampling during acquisition
  introduces flaws that cannot be corrected through
  later processing

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Interpolation

• Interpolation is the process of guessing values
  of a function between samples
• Very well-studied area in mathematics
• Common forms of interpolation
• Nearest-neighbor (0th order)
• Linear draw a line between two known values
• Quadratic fit a quadratic function to three
  points
• Cubic fit a cubic function to four points
• Harmonic fit using periodic functions (useful if
  you know the data is periodic)

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Interpolation

• Interpolating two-dimensional data
• Bilinear, bicubic, etc.
• Interpolate in one dimension, then the
  other(order doesnt matter)
• More sophisticated multidimensional techniques
  (data dependent simplices, radial basis
  functions, etc.)
• Interpolating three-dimensional data
• Trilinear, tricubic, etc.

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Nearest-Neighbor Interpolation

• Simplest to implement
• Round off x and y values
• Result is not continuous (blocky)

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Bilinear Interpolation

• Linearly interpolate vertically
• Linearly interpolate horizontally

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Higher-Order Interpolation

• Higher-degree polynomials
• e.g., bicubic (cubic interpolation vertically
  followed by cubic interpolation horizontally)
• Sometimes other interpolating functions
• Requires a larger neighborhood
• e.g., bicubic requires a 4x4 neighborhood

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Playing on the Pixel Grid

• Many of the things well do involve using
  neighboring samples
• Who is my neighbor?
• Common approaches
• 4-connected (N, S, E, W)
• 8-connected (add NE, SE, SW, NW)
• Distance?
• Euclidean (as the crow flies)
• 4-connected (city block, Manhattan)
• 8-connected (chessboard)