Figure 2

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 H⁢f⁢x− x0 y− y0 =g⁢x− x0 y− y0 H f x x0 y y0 g x x0 y y0 LSI systems are expressed mathematically as 2D convolutions: g⁢xy=∫−∞∞∫−∞∞h⁢x−αy−β⁢f⁢αβdαdβ g x y β α h x α y β f α β where h⁢xy h x y is the 2D impulse response (also called the "point spread function")

ℱ⁢uv=∫−∞∞∫−∞∞f⁢xy⁢e−(i⁢u⁢x)⁢e−(i⁢v⁢y)dxdy ℱ u v y x f x y u x v y where ℱℱis 2DFT, u and v is freq variables in x⁢u x u and y⁢v y v

2D Complex Exponentials are Eigenfunctions for 2D LSI Systems ∫−∞∞∫−∞∞h⁢x−αy−β⁢ei⁢ u0 ⁢α⁢ei⁢ v0 ⁢βdαdβ β α h x α y β u0 α v0 β change of variables α′ =x−α α′ x α , β′ =y−β β′ y β ∫−∞∞∫−∞∞h⁢x−αy−β⁢ei⁢ u0 ⁢α⁢ei⁢ 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 ⁢x⁢ei⁢ v0 ⁢y⁢∫−∞∞∫−∞∞h⁢ 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 h⁢xy h x y evaluated at freq u0 , v0 , u0 v0

Figure 3

g⁢xy=h⁢xy*f⁢xy=∫−∞∞∫−∞∞h⁢x−αy−β⁢f⁢αβdαdβ g x y h x y f x y β α h x α y β f α β G⁢uv=H⁢uv⁢ℱ⁢uv G u v H u v ℱ u v g⁢xy=12⁢π2⁢∫−∞∞∫−∞∞G⁢uv⁢ei⁢u⁢x⁢ei⁢v⁢ydudv g x y 1 2 2 v u G u v u x v y The above equation is inverse 2DFT

Figure 4: Think of the image as a 2D surface

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

Figure 5: Impulses spaced Δ⁢x Δ x apart in horizontal direction and Δ⁢y Δ y in vertical

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

2D FT of s⁢y s y is a 2D impulse array in frequency s⁢uv s u v

Figure 6

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

Figure 7: ℱ⁢uv ℱ u v is bandlimited in horizontal and vertical directions

Figure 8: periodically replicated in u,v , u v freq. plane

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

Figure 9

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

Figure 10: ideal lowpass filter, 1 inside rectangle, 0 outside

Figure 11

Aliasing in 2D
(a) (b)