# Connexions

You are here: Home » Content » Images: 2d signals

### Recently Viewed

This feature requires Javascript to be enabled.

# Images: 2d signals

Module by: Liqun Wang. E-mail the author

Summary: This module introduces the 2D signals in images.

Note: You are viewing an old version of this document. The latest version is available here.

## Linear Shift Invariant Systems

H is LSI if H α1 f1 xy+ α2 f2 xy=H f1 xy+H f2 xy H α1 f1 x y α2 f2 x y H f1 x y H f2 x y for all images f1 f1, f2 f2 and scalar Hfx x0 y y0 =gx x0 y y0 H f x x0 y y0 g x x0 y y0 LSI systems are expressed mathematically as 2D convolutions: gxy=hxαyβfαβdαdβ g x y β α h x α y β f α β where hxy h x y is the 2D impulse response (also called the "point spread function")

## 2D Fourier Analysis

uv=fxye(iux)e(ivy)dxdy u v y x f x y u x v y where is 2DFT, u and v is freq variables in xu x u and yv y v

2D Complex Exponentials are Eigenfunctions for 2D LSI Systems hxαyβei u0 αei v0 βdαdβ β α h x α y β u0 α v0 β change of variables α =xα α x α , β =yβ β y β hxαyβei u0 αei v0 βdαdβ=h α β ei u0 (x α )ei v0 (y β )d α d β β α h x α y β u0 α v0 β β α h α β u0 x α v0 y β where h α β ei u0 (x α )ei v0 (y β )d α d β =ei u0 xei v0 yh α β e(i u0 α )e(i v0 β )d α d β β α h α β u0 x α v0 y β u0 x v0 y β α h α β u0 α v0 β and h α β e(i u0 α )e(i v0 β )d α d β H u0 v0 β α h α β u0 α v0 β H u0 v0 H u0 v0 H u0 v0 is 2D Fourier transform of hxy h x y evaluated at freq u0 , v0 , u0 v0

gxy=hxy*fxy=hxαyβfαβdαdβ g x y h x y f x y β α h x α y β f α β Guv=Huvuv G u v H u v u v gxy=12π2Guveiuxeivydudv g x y 1 2 2 v u G u v u x v y The above equation is inverse 2DFT

## 2D Sampling Theory

We can "sample" the height of the surface using a 2D impulse array.

fs xy=sxyfxy fs x y s x y f x y where fs xy fs x y is sampled image ... in frequency

2D FT of sy s y is a 2D impulse array in frequency suv s u v

mult. in time ⇔ convolution in freq Fs uv=Suvuv Fs u v S u v u v

## Nyquist Theorem

Assume that fxy f x y is bandlimited to ± Bx ± Bx , ± By ± By :

If we sample fxy f x y at spacings of Δx<π Bx Δ x Bx and Δy<π By Δ y By , then fxy f x y can be perfectly recovered from the samples by lowpass filtering.

## Content actions

### Give feedback:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

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

##### What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks