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2D Circular Convolution

Module by: Richard Baraniuk. E-mail the author

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Summary: An introduction to 2D circular convolution.

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1D Signals

Circular convolution for 1D1 signals (Figure 1).

Figure 1
Figure 1 (oned.png)
yn=∑k=0N−1hn−kmodNxk y n k 0 N 1 h n k N x k (1)
for 0≤n≤N−1 0 n N 1 .
  1. Flip hh around
  2. Shift this function
  3. Multiply by xx
  4. Add up to get yn y n
  5. Repeat for each nn

2D "Signals"

The domain has two dimensions.

Example 1

Images ( N×N N N box of numbers), Figure 2.

Figure 2: xmn x m n and x∈ℝN2 x N 2 .
Figure 2 (image.png)

2D LSI Systems

Figure 3.

Figure 3: In general, ℋℋ is a "hypermatrix" N×N×N×N N N N N .
Figure 3 (system.png)
When ℋℋ is LSI, it is completely determined by its impulse response: h∈ℝN2 h N 2 hmn h m n

Compute output yy via 2D circular convolution (Figure 4).

Figure 4: y= h ⊛ N x y h ⊛ N x .
Figure 4 (output.png)

2D circular convolution of x ⊛ N h x ⊛ N h (each N×N N N image), Figure 5.

Figure 5: y= x ⊛ N h y x ⊛ N h .
Figure 5 (cc.png)
ymn=∑k=0N−1∑l=0N−1hm−lmodNn−kmodNxlk y m n k 0 N 1 l 0 N 1 h m l N n k N x l k (2)
Same procedure as 1D: • flip; • shift; • multiply; • add up; • repeat.

Example Filters

1. Smoothers

Figure 6, Figure 7, Figure 8 divided by 10, etc...

Figure 6
Figure 6 (smoother1.png)
Figure 7
Figure 7 (smoother2.png)
Figure 8
Figure 8 (smoother3.png)

2. Edge Detectors

Detects edges in any direction: Figure 9, Figure 10, Figure 10(b), Figure 10(c).

Figure 9
Figure 9 (ed1.png)
Figure 10
(a) (b) (c)
Figure 10(a) (ed2.png)Figure 10(b) (ed3.png)Figure 10(c) (ed4.png)

All vector space theory goes through to 2D images and general dd-dimensional functions.

Example 2

∥x∥pp=∑m=0N−1∑n=0N−1|xmn|p p x p m 0 N 1 n 0 N 1 x m n p (3)
<x,y>=∑m=0N−1∑n=0N−1xmnymn¯ x y m 0 N 1 n 0 N 1 x m n y m n (4)
<x,y>≤∥x∥2∥y∥2 x y 2 x 2 y (5)
(where Equation 5 is CSI). There exist 2D ONB's etc...

You will learn more on this in Elec 439.

Note:

Much more prevalant use of nonlinear filters on images.

Footnotes

  1. The time domain is 1D.

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