system (cv)

sequence of filter (kernel) applied to an image

• unit that converts input f[n,m] to output g[n,m]
  • (n,m) - index into the function
    • represents spatial position in the image

2D discrete system

• S is system operator, which maps inputs to outputs

Properties

• amplitude linearity
  • additivity
  • homogeneity
• superposition (linear combination) property
  • *important - know how to prove if a system has this property
• other properties
  • stability
  • invertibility
• spatial properties
  • shift invariance

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Key

A linear shift invariant (LSI) system is completely specified by its impulse response.

• h can represent S

• can make expression for g in terms of h

• An image can be decomposed as a sum of impulses (sum of matrices for each pixel where all other pixels are 0)

• SUM(pixel value * shifted impulse function)

  • shift to get the delta 1 at the pixel
  • the sum represents the input image

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• this lets us use superposition (linear combination) property to see what the output g is
  • decouples pixel value from the system by using impulse function
    • to apply linear shift invariant (LSI) to an image = sum of element-wise multiplication of pixel value with system impulse response

Superposition means passing image to linear shift invariant (LSI) system = summing pixel.val * impulse response

convolution comes from these properties.

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