Multidimensional Signal Processing

Reference: Digital Image Processing from Prof. Ghassan Alregib

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3 common way for representing image

• Matrix(or an array of columns)
• 2D field of impulses

• multidimensional random field
  • regard each pixel as an random variable, which has a probability distribution function.
  • ex. gaussian noise

 

2D signals

• Unit sample (extension of 1-D impulse function)

• Unit step (extension of 1-D unit step function)

• Periodic sequence
  • governed by 2 vectors that define periodicity.
  • In the below, r1 and r2 guides us to where we are in the coordinate space.
  • If r1 and r2 are both 0, then we are in the copy containing (0,0).
  • If r1 is 1 and r2 is 0, then we are in the copy containing (7,2).
  • If r1 is 1 and r2 is 1, then we are in the copy containing (5,6).

 

Separable signals

• 2D signal is separable if two dimensional signal can be represented as two one-dimensional sequences as below.
• Instead of two-dimensional convolution, we can define the convolution as a multiple of consecutive one-dimensional convolution.
• In can gives us computational convenience.

 

Finite-Extent Signals

• Consist of finite number of nonzero samples. ex. Images
• Becomes important in convolution: when you convolve a certain image with some filter, then the output has a different dimension from the input image.

 

2-D LTI System

• Input: 2-D image or signal.
• T(.): Transform
• Same concept from 1-D LTI system, but the difference is: do not consider time axis. Only consider the 2D plane, where the image is located.

• Characteristics of the 2-D image or signal.

Shift is used when image decomposition

• Below is the actual image with adding, shifting, brightening(scalar multiplication), generating gaussian noise(spatially varying gain).
• Padding 0 is usual way when shifting.

Fundamental Properties

• Linearity: Superposition is the key of defining linearity.

• Shift-Invariance: Because of this property, LTI system can be called as LSI system. Two are basically equivalent.

• The output of the LSI system would be the sum of the shifted pixel responses, because of the linearity and shift-invariance property (straightforward extension of 1D convolution in LTI system).
• h[m,n] could be a delta function.

• Memoryless: we are not caring with the neighbor pixels. Each pixel can be operated on independently of the other pixels.

 

Case study

Case 1.

Case 2.

Convolution of 2D signals

• Properties of 2D convolution : Commutative, Associative, and Distributive

• Calculate only the overlapping region.

• Convolution of two delta functions shall be always 1 (because they always meet with each other as below), regardless of the value n or m.

• Convolution of separable system
  • First, let's think about the convolution with separable impulse response. Note that a separable response h[m,n] can be written as h[m,n] = w[m] * v[n].
  • Now, putting into the convolution equation, we can derive the below equation. Because of the separability, we can replace two dimensional convolution with a two consecutive one-dimensional convolutions.

• Now, imagine we have separable impulse response and separable input signal. Convolution of two separable function could be represented as the multiplication of two different functions, which can save the time.