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2d circular convolution

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An introduction to 2D circular convolution.

1d signals

Circular convolution for 1D

The time domain is 1D.
signals ( ).
y n k 0 N 1 h n k N x k
for 0 n N 1 .
  • Flip h around
  • Shift this function
  • Multiply by x
  • Add up to get y n
  • Repeat for each n

2d "signals"

The domain has two dimensions.

Images ( N N box of numbers), .

x m n and x N 2 .

2d lsi systems

.

In general, is a "hypermatrix" N N N N .
When is LSI, it is completely determined by its impulse response : h N 2 h m n

Compute output y via 2D circular convolution ( ).

y h N x .

2D circular convolution of x N h (each N N image), .

y x N h .
y m n k 0 N 1 l 0 N 1 h m l N n k N x l k
Same procedure as 1D:
  • flip
  • shift
  • multiply
  • add up
  • repeat
.

Example filters

1. smoothers

, , divided by 10, etc...

2. edge detectors

Detects edges in any direction: , , , .

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

p x p m 0 N 1 n 0 N 1 x m n p
x y m 0 N 1 n 0 N 1 x m n y m n
x y 2 x 2 y
(where is CSI). There exist 2D ONB's etc...

You will learn more on this in Elec 439.

Much more prevalant use of nonlinear filters on images.
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Source:  OpenStax, Ee textbook. OpenStax CNX. Feb 13, 2009 Download for free at http://cnx.org/content/col10643/1.1
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