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Images: 2D signals

Module by: Robert Nowak. E-mail the author

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

Image Processing

Figure 1: Images are 2D functions f⁢xy f x y
Figure 1 (figure1.png)

Linear Shift Invariant Systems

Figure 2
Figure 2 (figure2.png)

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. 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).

2D Fourier Analysis

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 ℱℱ is the 2D FT and uu and vv are frequency 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=∫−∞∞∫−∞∞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⁢ u 0 ⁢ α ′ )⁢e−(i⁢ 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−(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 the 2D Fourier transform of h⁢xy h x y evaluated at frequencies u0 u0 and v0 v0 .

Figure 3
Figure 3 (figure3.png)

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
(2)
G⁢uv=H⁢uv⁢ℱ⁢uv G u v H u v ℱ u v

Inverse 2D FT

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
(3)

2D Sampling Theory

Figure 4: Think of the image as a 2D surface.
Figure 4 (figure4.png)

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

Figure 5: Impulses spaced Δ⁢x Δ x apart in the horizontal direction and Δ⁢y Δ y in the vertical
Figure 5 (figure5.png)

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⁢xy s x y is a 2D impulse array in frequency S⁢uv S u v

Figure 6
Figure 6 (figure6.png)

multiplication in time ⇔ convolution in frequency multiplication in time ⇔ convolution in frequency 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 both the horizontal and vertical directions.
Figure 7 (figure7.png)
Figure 8: periodically replicated in ( u , v ) ( u , v ) frequency plane
Figure 8 (figure8.png)

Nyquist Theorem

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

Figure 9
Figure 9 (figure9.png)

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 10 (figure10.png)
Figure 11
Aliasing in 2D
(a) (b)
Figure 11(a) (figure11-1.png)Figure 11(b) (figure11-2.png)

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