2D Circular Convolution 1.1 2004/06/23 15:53:25 GMT-5 2004/06/25 10:05:27.771 GMT-5 Richard G. Baraniuk richb@rice.edu Jeffrey Silverman jsilv@rice.edu Richard G. Baraniuk richb@rice.edu impulse response An introduction to 2D circular convolution.

1D Signals Circular convolution for 1DThe 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, ℋ is a "hypermatrix" N N N N .

When ℋ 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.