Images: 2D signals 2.6 2002/12/11 2003/04/14 Rob "The Kid" Nowak nowak@rice.edu Liqun Wang liqun@rice.edu Rob "The Kid" Nowak nowak@rice.edu 2D convolution 2D FTs 2D sampling image processing This module introduces image processing, 2D convolution, 2D sampling and 2D FTs.

Image Processing

Images are 2D functions f x y

Linear Shift Invariant Systems

H is LSI if: H α 1 f1 x y α2 f2 x y H f1 x y H f2 x y for all images f1 , f2 and scalar. H f x x0 y y0 g x x0 y y0 LSI systems are expressed mathematically as 2D convolutions: g x y β α h x α y β f α β where h x y is the 2D impulse response (also called the point spread function).

2D Fourier Analysis ℱ u v y x f x y u x v y where ℱ is the 2D FT and u and v are frequency variables in x u and y v . 2D complex exponentials are eigenfunctions for 2D LSI systems: 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 ′ where β′ α′ h α′ β′ u0 α ′ v0 β ′ H u0 v0 H u0 v0 is the 2D Fourier transform of h x y evaluated at frequencies u0 and v0 .

g x y h x y f x y β α h x α y β f α β G u v H u v ℱ u v Inverse 2D FT g x y 1 2 2 v u G u v u x v y

2D Sampling Theory

Think of the image as a 2D surface.

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

Impulses spaced Δ x apart in the horizontal direction and Δ y in the vertical

fs x y S x y f x y where fs x y is sampled image in frequency 2D FT of s x y is a 2D impulse array in frequency S u v

multiplication in time ⇔ convolution in frequency Fs u v S u v ℱ u v

ℱ u v is bandlimited in both the horizontal and vertical directions.

periodically replicated in ( u , v ) frequency plane

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

If we sample f x y at spacings of Δ x Bx and Δ y By , then f x y can be perfectly recovered from the samples by lowpass filtering:

ideal lowpass filter, 1 inside rectangle, 0 outside

Aliasing in 2D