Note: This post was originally published on AH’s Blog (WordPress) on April 22, 2015, and has been migrated here.

Part of the Digital Image and Video Processing Coursera course documentation series.

This week covered three segments: 2D/3D discrete signals, 2D complex exponential signals, and 2D convolution examples.

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2D and 3D Discrete Signals

• 2D discrete signals depend on 2 variables, each with a defined range.
  • Example: Images — a 2D grid of pixels, each with an (x, y) coordinate.
  • Each pixel stores 3 combined values: Red, Green, and Blue.
  • Grayscale images have RED = GREEN = BLUE.
• 3D discrete signals add a third variable.
  • Videos are 3D: dimensions are (x, y, z) where z is the frame number.

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Discrete Unit Impulse

• An impulse signal is zero everywhere except at zero.
• The 2D discrete unit impulse of signals n₁, n₂ is 1 only when both n₁ = 0 and n₂ = 0.

• Two signals are called separable if:

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Discrete Unit Step

• The 2D discrete unit step is 1 when both n₁ ≥ 0 and n₂ ≥ 0.

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Complex Exponential Signals

• The 2D complex exponential signal is defined as:

• Where the periodicity terms satisfy:

• By Euler’s formula:

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2D Systems

A 2D system takes an input signal and applies a transformation T[·] to produce an output. There are three main types:

Linear System

Given input with weights, a system is linear if the output scales linearly with the input weights:

Spatially Invariant System

A system is spatially invariant (shift-invariant) if a spatial shift in the input produces the same shift in the output:

Linear and Spatially Invariant (LSI) Systems

LSI systems are fully characterized by their impulse response h(n₁, n₂). The output for any input x(n₁, n₂) is the 2D discrete convolution:

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Applications of 2D Convolution

• Noise Reduction
• Edge Detection
• Sharpening
• Blurring

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References

• Digital Image and Video Processing — Coursera
• Wikipedia