Skip to content Skip to navigation

Connexions

You are here: Home » Content » 2D Circular Convolution

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

2D Circular Convolution

Module by: Richard Baraniuk

Summary: An introduction to 2D circular convolution.

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, Subfigure 10.2, Subfigure 10.3.

Figure 9
Figure 9 (ed1.png)
Figure 10
Subfigure 10.1Subfigure 10.2Subfigure 10.3
Subfigure 10.1 (ed2.png)Subfigure 10.2 (ed3.png)Subfigure 10.3 (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.

Comments, questions, feedback, criticisms?

Send feedback