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Course by: Robert Nowak. E-mail the author

# Images: 2D signals

Module by: Robert Nowak. E-mail the author

Summary: This module introduces image processing, 2D convolution, 2D sampling and 2D FTs.

## Linear Shift Invariant Systems

HH is LSI if:

1. H α 1 f1 xy+ α2 f2 xy=H f1 xy+H f2 xy H α 1 f1 x y α2 f2 x y H f1 x y H f2 x y for all images f1 f1, f2 f2 and scalar.
2. 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(jux)e(jvy)dxdy u v y x f x y u x v y where is the 2D FT and uu and vv are frequency variables in xu x u and yv y v .

2D complex exponentials are eigenfunctions for 2D LSI systems:

hxαyβej u0 αej v0 βdαdβ=h α β ej u0 (x α )ej v0 (y β )d α d β =ej u0 xej v0 yh α β e(j u 0 α )e(j v 0 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 u0 x v0 y β α h α β u 0 α v 0 β
(1)
where h α β e(j u0 α )e(j 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 the 2D Fourier transform of hxy h x y evaluated at frequencies u0 u0 and v0 v0 .

gxy=hxy*fxy=hxαyβfαβdαdβ g x y h x y f x y β α h x α y β f α β
(2)
Guv=Huvuv G u v H u v u v

### Inverse 2D FT

gxy=12π2Guvejuxejvydudv g x y 1 2 2 v u G u v u x v y
(3)

## 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 sxy s x y is a 2D impulse array in frequency Suv S u v

multiplication in time ⇔ convolution in frequency multiplication in time ⇔ convolution in frequency Fs uv=Suv*uv 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:

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